From cat-dist Mon Nov 2 14:45:49 1998 Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id MAA01190 for categories-list; Mon, 2 Nov 1998 12:57:17 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Date: Mon, 2 Nov 1998 16:46:48 +1100 (EST) Message-Id: <199811020546.QAA25507@algae.socs.uts.EDU.AU> To: categories@mta.ca Subject: categories: Re: FISh performance From: Barry Jay Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: Dear Ralph, thank you for taking an interest in the FISh performance results for quicksort. You raise two basic questions. Is the FISh program faster? If so, why? Is quicksort in FISh faster than qsort (on arrays of doubles)? ============================================================== > (1) Your C code compares using two floating point operations >(subtraction, compare), while the fish program uses just one (no >subtraction). This makes a small difference to the overall performance of the algorithm. I re-ran the test for qsort with the following comparison function int comp(const void *i, const void *j) { return (*(double*)i > *(double*)j ) ; } that has only one operation. The C results are better by about 10% but FISh is still twice as fast. C benchmark1 C benchmark2 FISh benchmark 3.59 3.35 1.65 >(3) The C library qsort takes a three-valued comparison function, >while your fish code uses a two-valued comparison. I'm not sure what point you're making. The designers of qsort had a free choice. I suspect that the three-valued comparison is used to exploit equality of values, to obtain greater efficiency. In any event, it is unlikely to make much difference in the light of (1) above. Conclusion: The FISh program *is* faster. Why is quicksort in FISh faster? ================================= >(4) As you note, your fish code gets optimised by specialising to the >datatype and comparison function used. Yes. This is a given. The question is to pinpoint the source of the speedup. > ... This is the main >performance issue, and has nothing to do with boxing/unboxing issues. Point taken. The data supplied to quicksort is not boxed, but this is because (the length of the array and) the size of the array entries are given as explicit parameters. In this sense qsort is *not* polymorphic in its choice of data type. If it were not provided by the C programmer then boxing would add another layer of indirection to the program. Such information about the size of entries etc. is inferred by the FISh compiler. >... Whether [FISh is faster] will depend on >things like instruction scheduling in the compiler & CPU, & register >allocation in the caller & in any case the difference will be tiny. Your conclusion does not match the performance figures above. The FISh program avoids making a function call at all. That is, instead of passing the addresses to a function, a primitive comparison is made. The key point is that this can only be done because information like the length of the array and the size of the entries (in bytes) can be inferred by the FISh compiler, instead of being supplied by the user. Yours, Barry Jay ************************************************************************* | Associate Professor C.Barry Jay, | | Reader in Computing Sciences Phone: (61 2) 9514 1814 | | Head, Algorithms and Languages Group, Fax: (61 2) 9514 1807 | | University of Technology, Sydney, e-mail: cbj@socs.uts.edu.au | | P.O. Box 123 Broadway, 2007, http://www-staff.socs.uts.edu.au/~cbj | | Australia. FISh homepage ... ~cbj/FISh | ************************************************************************* From cat-dist Mon Nov 2 17:54:05 1998 Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id QAA12026 for categories-list; Mon, 2 Nov 1998 16:58:42 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Subject: categories: CT for CS? From: rjwood@mscs.dal.ca To: categories@mta.ca Date: Mon, 2 Nov 1998 16:55:10 -0400 (AST) X-Mailer: ELM [version 2.4 PL24alpha3] MIME-Version: 1.0 Content-Type: text/plain; charset=US-ASCII Content-Transfer-Encoding: 7bit Message-Id: <19981102205515Z23968-361+3@mscs.dal.ca> Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: The new Faculty of Computer Science at Dalhousie is asking Mathematics for new classes and the chairman of Mathematics is willing to suggest a Category Theory class. Is there now a `standard CT for CS class' in the same way that there is a `standard first class in measure theory' and so on? If so, can you advise on 1) text(s), including material to be explicitly covered in a 39 hour class; 2) background required, both for a CS major and a Mathematics major; 3) universities which actually do this --- particularly those with pretensions and realities similar to Dalhousie's? Please send responses directly to me rather than the categories list. I will summarize them for the list. Thanks RJ From cat-dist Tue Nov 3 21:18:47 1998 Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id UAA18254 for categories-list; Tue, 3 Nov 1998 20:18:28 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Message-ID: <363F38EF.3EFC@dcs.ed.ac.uk> Date: Tue, 03 Nov 1998 17:10:07 +0000 From: Samson Abramsky Organization: Dept. of Computer Science, University of Edinburgh X-Mailer: Mozilla 3.04 (X11; U; SunOS 5.6 sun4m) MIME-Version: 1.0 To: categories Subject: categories: Post-doctoral fellowship Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: Dear Colleagues, A 1-year postdoctoral fellowship is available on the UK EPSRC-funded grant ``Foundational Structures in Computer Science'', to work on topics related to game semantics. Further details of this position can be found at www.personnel.ed.ac.uk/vacs/external (Click vac3ext.htm, and then ``8 Postdoctoral Research fellowships''. The relevant posts are the first two of these.). I would be very grateful if you could bring this position to the attention of anyone in your group who might be interested. Anyone wishing to discuss the project informally is welcome to contact me by email. Best wishes, Samson Abramsky From cat-dist Tue Nov 3 21:22:34 1998 Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id UAA17314 for categories-list; Tue, 3 Nov 1998 20:16:58 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Message-Id: <199811031118.HAA12375@mailserv.mta.ca> From: boerger Organization: FernUniversitaet - Gesamthochschule To: categories@mta.ca Date: Tue, 3 Nov 1998 12:19:11 +0100 MIME-Version: 1.0 Content-type: text/plain; charset=US-ASCII Content-transfer-encoding: 7BIT Subject: Re: categories: Reference? In-reply-to: X-mailer: Pegasus Mail for Win32 (v2.54DE) Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: The result that finite products coincide with finite coproducts in categories enriched commutative monoids can be found in Herrlich`s and Strecker`s book under 40.8 (p.308 in the 2nd edition). The converse is given there under 4.12 (p.310). They use the term "semi-additive category", which I am also used to. Though I agree with Bill Lawvere that prefixes like "semi" should be omitted if possible, I am not convinced by his suggestion "linear categeory" because for me subtraction seems essential fo linear algebra. Maybe somebody invents a better term. Greetings Reinhard From cat-dist Tue Nov 3 21:25:17 1998 Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id UAA18050 for categories-list; Tue, 3 Nov 1998 20:17:49 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Message-ID: <363E8966.F3F23F3D@maths.usyd.edu.au> Date: Tue, 03 Nov 1998 15:41:11 +1100 From: Robbie Gates X-Mailer: Mozilla 4.05 [en] (X11; I; OSF1 V4.0 alpha) MIME-Version: 1.0 To: categories@mta.ca Subject: categories: Re: FISh performance References: <199811020546.QAA25507@algae.socs.uts.EDU.AU> Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: Warning: there's no category theory in this letter - it's purely a reponse to the discussion of execution speed of FISh quicksort. Barry Jay wrote: > Why is quicksort in FISh faster? > ================================= I would like to propose another reason the FISh quicksort appears to be faster than the library qsort - they aren't the same algorithm. In particular, the standard library qsort and the FISh sort are using different methods to pick the pivot. Choice of pivot in quicksort is important to ensure the worst case doesn't degenerate to O(n^2). The choice of pivot in the FISh code is always the first element, which does give very poor performance in the case the array is already sorted. I'm not sure what the qsort in the libc i have here does. However, typical tricks include selecting the median of three elements at the bottom, middle and top of the array, choosing a random element, choosing the median of three random elements, and so on. Importantly, each of these choices requires some work, and thus negatively impacts on the average speed of the sort in order to alleviate the worst case speed. Caveat: I'm not a professional benchmarker/tester, and the tests below are probably very flawed. I have tried to give details on everything that seems relevant, but your mileage may vary. This data is more given to suggest an alternative, plausible explanation for the observed speed difference in the FISh and C benchmarks, than to claim i have identified all the issues in the timing results found for FISh and C quicksort. To test this hypothesis, i coded up quicksort in C++ using the same pivot picking strategy as the FISh code. I then modified my C++ code, and the C benchmark code and C translated FISh code (CTFC) from the FISh website (by CTFC i mean the contents of test01.c), by altering the random number generator to allow specification of two command line arguments giving the multiplier and step for the seeding algorithm. i.e rather than seed = seed * 25173 + 17431 i used seed = seed * mult + 17431 for varying mult. This allowed me to test the extent to which the algorithm was sensitive to the data being sorted. What follows are timing values on a DGUX AlphaStation 255 (233 MHz) with mult = 1,2,3, ... ,10 (on an array of 10000 values). For each program, i give the total (user + system) time reported by the builtin time function of bash version 2.02.0(1)-release (alpha-dec-osf4.0) (averaged over 10 trials in each case). The time columns are libqsort (= the standard library qsort), cppqsort (= my C++ qsort), fishqsort (= CTFC). They were compiled with gcc, g++, gcc in each case, and -O optimization. NOTE: With these mult values, the data supplied to the qsort is deliberately nonrandom to try and detect the effect proposed in the second paragraph. mult libqsort cppqsort fishqsort 1 .0350 3.462 3.410 2 .0348 4.039 4.039 3 .1003 .0334 .0391 4 .0350 4.085 4.052 5 .1130 .0332 .0393 6 .0356 4.042 4.039 7 .1025 .0341 .0382 8 .0357 4.061 4.054 9 .1088 .0336 .0382 10 .0355 4.053 4.041 Observe that the libqsort has much better average performance (and that its profile is different to the other two - one could probably determine something about the choice of pivots with enough tests ;-). Also, the CTFC at best beats the library by a factor a bit under 3, whereas the library beats the CTFC by a factor of over 110. More importantly, the C++ and fish code run in very similar time, neither consistently beating the other. Also, their profiles are almost identical. I also ran one bigger test - in the case of a 90000 element array consisting of 1 to 90,000 in order, the library took around .3 seconds, whereas the CTFC overflowed the stack and crashed just under 200 seconds into the test (the C++ code took 315 seconds to overflow the stack and crash - i'm not sure what's causing the difference here, but i suspect it has to do with the fact my C++ code used paramaters rather than a static struct). Based on this, I'm proposing that the standard library quicksort is trading off speed for the average cases to bring the worst cases down to something acceptable. This is presumably a serious issue for a library quicksort - its better to run on average somewhat slower in order to get consistency, as this makes applications easier to test for market acceptance of speed, rather than the application apparently hanging when the user manages to sort the worst case database with about 1 million entries in it (when your testers happily sorted their million entry db's with no trouble). one other point - the compile time polymorphism is not specific to FISh (as Ralph Loader mentioned). You write: > The FISh program avoids making a function call at all. That is, > instead of passing the addresses to a function, a primitive comparison > is made. The key point is that this can only be done because > information like the length of the array and the size of the entries > (in bytes) can be inferred by the FISh compiler, instead of being > supplied by the user. C++ templates also give the ability to supply a compile time comparision function to sort routines, the code will likewise be specialised on the given comparison and on the size of the entries. One could write sort templates that take the length as well, however since the size of stuff to be sorted typically varies depending on the current application state (i.e. in a manner unknown at compile time) this tends not to be enough of a win to be worth doing. - robbie -- *--------------------------------> + <---------------------------------* |_| robbie gates | pgp key available |_| V category theorist V //cat.maths.usyd.edu.au/~robbie V *--------------------------------> + <---------------------------------* From cat-dist Fri Nov 6 19:18:24 1998 Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id SAA29437 for categories-list; Fri, 6 Nov 1998 18:04:12 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Date: Fri, 6 Nov 1998 14:28:03 -0500 (EST) From: Andre Scedrov Message-Id: <199811061928.OAA02735@saul.cis.upenn.edu> To: categories@mta.ca Subject: categories: MFPS deadline extension Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: Dear Colleagues, We have had a number of requests to extend the MFPS deadline for submissions for next spring's meeting. The deadline was November 3, but because of these requests, we have decided to extend the deadline until next Tuesday, November 10. Detailed information about submissions is available on the MFPS 15 home page, http://www.math.tulane.edu/mfps15.html. Best regards, The Organizers and Program Committee Chairmen From cat-dist Mon Nov 9 16:31:20 1998 Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id PAA00232 for categories-list; Mon, 9 Nov 1998 15:09:35 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Date: Mon, 9 Nov 1998 11:06:13 +0100 (MET) From: Philippe Gaucher Message-Id: <199811091006.LAA04175@irma.u-strasbg.fr> To: categories@mta.ca Subject: categories: computation in CPS Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-MD5: QqGGoKoDrptNPwmdXmVFYw== Content-Transfer-Encoding: 8bit X-MIME-Autoconverted: from quoted-printable to 8bit by mailserv.mta.ca id GAA18911 Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: Bonjour, Here is a question about composable pasting schemes (CPS). In the omega-category In generated by the n-cube, is it possible to find a kind of "general" formula for the (n-1)-source (target) of the n-morphism corresponding to the interior of In ? I can do mechanical computation in low dimension but I am not able for the moment to imagine a formula for any dimension. In high dimension , computations become very long. For example, using notations of Crans/Johnson/Street etc..., in I2, we have (R(x) means the CPS generated by x, sometimes also denoted by (x)) : s_1(00)=R(-0,0+) (almost the definition in a CPS) and t_0(-0)=s_0(0+)=-+ => s_1(00)=R(-0) o_0 R(0+) (1) because the union is the composition in the framework of CPS. And t_1(00)=R(+0,0-)=R(0-) o_o R(+0) (2). Obvious with a picture. In I3, we have : s_2(000)=R(-00,0+0,00-)=R(-00,0++,-0-,0+0,00-,++0) t_0(-00)=s_0(0++) => R(-00,0++) = R(-00) o_0 R(0++) t_0(-0-)=s_0(0+0) => R(-0-,0+0) = R(-0-) o_0 R(0+0) t_0(00-)=s_0(++0) => R(00-,++0) = R(00-) o_0 R(++0) and t_1(R(-00) o_0 R(0++)) = t_1(-00) o_0 R(0++) (axiom of omegaCat) = (-0-) o_0 (-+0) o_0 (0++) with (2) s_1(R(-0-) o_0 R(0+0)) = R(-0-) o_0 s_1(0+0) (axiom of omegaCat) = (-0-) o_0 (-+0) o_0 (0++) with (1) => R(-00,0++,-0-,0+0) = R(-00,0++) o_1 R(-0-,0+0) and in the same way, we preove that t_1(R(-0-,0+0))=s_1(R(00-,++0)) => s_2(000) =((-00) o_0 (0++)) o_1 ((-0-) o_0 (0+0)) o_1 ((00-) o_0 (++0)) For t_2(000), read the above formula from the right to the left and replace - by +. Almost obvious with a picture. For I4 now : I have found a formula for s_3(0000)... A little bit long and not interesting. For I5 : Too long. More generally, the question is : for a CPS, is there a way to compute the source and target of a R({x}) using only compositions of elements like R({y}) ? Thanks in advance for your help. pg. From cat-dist Mon Nov 9 16:32:31 1998 Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id PAA32643 for categories-list; Mon, 9 Nov 1998 15:15:18 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Message-ID: <36472B4E.7E74@dcs.ed.ac.uk> Date: Mon, 09 Nov 1998 17:50:06 +0000 From: Samson Abramsky Organization: Dept. of Computer Science, University of Edinburgh X-Mailer: Mozilla 3.04 (X11; U; SunOS 5.6 sun4m) MIME-Version: 1.0 To: categories Subject: categories: Post-doctoral fellowship Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: I recently posted a message about the availability of some post-doc positions on this list. I gave a reference to the University personnel department web pages for further details. Unfortunately, as soon as I had done so the personnel department expunged this information. I therefore repeat the posting, including some further details of the positions. Best wishes, Samson Abramsky ---------------- Two 2-year postdoctoral fellowships are available on the UK EPSRC-funded grant ``Foundational Structures in Computer Science'', to work on topics related to game semantics. I would be very grateful if you could bring these positions to the attention of anyone in your group who might be interested. Anyone wishing to discuss the project informally is welcome to contact me by email. Further details of the positions: 1. To investigate the applications of game semantics to programming languages and the computational interpretation of Classical Logic. The applicant should have a strong background in logic and semantics, and preferably some familiarity with recent developments in Game Semantics. Duration: 12 months. Ref: 896767 2. To investigate applications of game semantics to concurrent systems. The applicant should have a good background in concurrency and semantics. Both these posts are associated with the project ``Foundational Structures in Computer Science''. The Principal Investigator is Professor Samson Abramsky. Duration: 12 months. Ref: 896768 Salary for these positions will be in the range 15735 to 20107 (UK pounds). Further particulars including details of the application procedure should be obtained from: The Personnel Office, The University of Edinburgh. 1 Roxburgh Street, Edinburgh EH8 9TB Tel: 0131-650-2511 (24 hour answering service) quoting the appropriate reference number closing date: 5 November 1998 Late applications will b considered. From cat-dist Tue Nov 10 11:02:58 1998 Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id IAA11876 for categories-list; Tue, 10 Nov 1998 08:55:21 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Date: Mon, 9 Nov 1998 16:31:58 -0500 (EST) From: Sjoerd CRANS Message-Id: <199811092131.QAA18657@scylla.math.mcgill.ca> To: categories@mta.ca Subject: categories: Re: computation in CPS Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: Philippe Gaucher asked: > for a CPS, is there a way to compute > the source and target of a R({x}) using only compositions of elements > like R({y}) ? Yes and no. Yes in the sense that because the source and the target are pasting schemes themselves, Johnson's pasting theorem gives that 1. they are compositions of R({y})'s and 2. *any* way you do this gives the same result. No in two senses: although Johnson's proof actually gives an algorithm, I don't think this algorithm has ever been implemented (in AXIOM for example); and secondly, there is (as far as I know) no *general* expression which works for cubes of all dimensions. Sjoerd Crans From cat-dist Tue Nov 10 21:49:50 1998 Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id UAA11252 for categories-list; Tue, 10 Nov 1998 20:36:24 -0400 (AST) Message-Id: <199811110036.UAA11252@mailserv.mta.ca> X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Date: Tue, 10 Nov 1998 18:53:15 +0100 (MET) From: Perez Garcia Lucia Subject: categories: Gödel and category theory To: categories@mta.ca Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: I am interested in the foundations of mathematics -more concretely, in the claim that category theory can serve as a superior substitute for set theory in the foundational landscape. In this context, I would like to point out a footnote which appears in 'What is Cantor's Continuum Problem?', written by Kurt G?del in 1947, revised and expanded in 1964, and finally published in Benacerraf P. and Putnam H. (eds.) 1983: Philosophy of Mathematics. Selected Readings, Cambridge University Press, pp. 470-485. It reads as follows: It must be admitted that the spirit of the modern abstract disciplines of mathematics, in particular of the theory of categories, transcends this concept of set*, as becomes apparent, e.g., by the self-applicability of categories (see MacLane, 1961**). It does not seem however, that anything is lost from the mathematical content of the theory if categories of different levels are distinguished. If there exist mathematically interesting proofs that would not go through under this interpretation, then the paradoxes of set theory would become a serious problem for mathematics. *(the concept of set G?del was referring to is the iterative one). **(MacLane, S. 1961. "Locally Small Categories and the Foundations of Set Theory". In Infinitistic Methods, Proceedings of the Symposium on Foundations of Mathematics (Warsaw, 1959). London and N.Y., Pergamon Press). I need some help to grasp the following questions: - In what sense the self-applicability of categories transcends the concept of set?. (It is obvious that categories transcend the concept of well- founded set but, what's the matter with non-well-founded sets?. - In what sense do you think G?del proposed distinguishing different levels of categories?. Would it be possible that G?del was thinking of something like type theory?. - Do you agree with G?del's intuition that nothing would be lost with such a distinction?. - Finally, in the last lines of the note G?del seems to suggest a research programme for category theory as an alternative foundation of mathematics. To what extent has it been carried out?. Thanks for your help. Regards, Luc?a P?rez Dpt.L?gica y Filosof?a de la Ciencia University of Valencia -Spain- -- *********************************************** lperez *********************************************** From cat-dist Tue Nov 10 21:56:08 1998 Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id UAA10911 for categories-list; Tue, 10 Nov 1998 20:40:50 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Message-ID: <36479E64.97C61D0C@iswest.com> Date: Mon, 09 Nov 1998 21:01:09 -0500 From: Zhaohua Luo X-Mailer: Mozilla 4.5 [en] (Win98; U) X-Accept-Language: en MIME-Version: 1.0 To: categories@mta.ca Subject: categories: Axioms of Algebraic Geometry Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: Axioms of Algebraic Geometry I Zhaohua Luo (11/7/98) (a draft) The axioms of algebraic geometry given below consist of three (well known) algebraic axioms (A1) - (A3) and three geometric axioms (G1) - (G3), based on Diers's axioms of Zariski categories. The complete html version of this note (with links) is available at http://www.azd.com/axioms.html. Comments and suggestions are welcome. Consider a functor U: A --> Set from a category A to the category Set of sets. Algebraic Axioms: (Axiom A1) U has a left adjoint. (Axiom A2) Any bijective morphism in A is an isomorphism. (Axiom A3) Any pair of parallel morphisms in A has a surjective coequalizer. Recall that a functor satisfying the axioms (A1) - (A3) is an algebraic functor, and the pair (A, U) is an algebraic category (or algebraic construct, or quasivariety). An algebraic functor U is finitary if it preserves direct colimits. Any algebraic functor is faithful. In the following we shall regard A as a concrete category over Set via an algebraic functor U, and identify an object X with its underlying set U(X). A difference of an object is an ordered pair (a, b) of elements of A, denoted formally by a - b. (a) A difference a - b is a zero if a = b. (b) A difference a - b is a unit if for any morphism f: A --> B, f(a) = f(b) implies that B is terminal. (c) A difference a - b is nilpotent if for any morphism f: A --> B, f(a) - f(b) is a unit implies that B is terminal. (d) An object is reduced if it has no non-zero nilpotent difference; (A, U) is reduced if any object is reduced. (e) A morphism f: A --> B is flat if the pushout functor C/A --> C/B along it preserves monomorphisms. (f) A difference a - b is invertible (or disjunctable) if there is a flat epimorphism i: A --> A(a, b) such that i(a) - i(b) is a unit, and any morphism j: A --> B factors through i if j(a) - j(b) is a unit. Suppose U x V is the product of two objects U and V with the projections u: U x V --> U and v: U x V --> V. The product U x V is co-universal if for any morphism f: U x V --> Z, let Z --> ZU and Z --> ZV be the pushouts of u and v along f, then the induced morphism Z --> ZU x ZV is an isomorphism. Geometric Axioms: (Axiom G1) Any object has a unit difference. (Axiom G2) The product of any two objects isco-universal. (Axiom G3) Any difference of an object is invertible. We call any functor U: A --> Set satisfying the above six axioms an algebraic-geometric functor. An algebraic-geometric category is a pair (A, U) consisting of a category A and an algebraic-geometric functor U on A. Remark. (cf. [Luo, Categorical Geometry]) (a) An algebraic-geometric category is the opposite of an analytic geometry. (b) A finitary algebraic-geometric category is the opposite of a coherent analytic geometry. (c) Any finitary algebraic-geometric category satisfies the first five of the six axioms of Zariski categories defined in Diers's book Categories of Commutative Algebras, Oxford University Press, 1992. The sixth axiom simply means that the category is strict, i.e. the Grothendieck topology defined by open subsets is subcanonical. Example. The following categories are algebraic-geometric categories: (a) The category of frames (non-finitary, reduced, non-strict). (b) The category of distributive lattices (finitary, reduced, non-strict). (c) The category of Boolean algebras (finitary, reduced, strict). (d) The category of commutative rings with identity (finitary, non-reduced, strict). (e) The category of reduced commutative rings (finitary, reduced, strict). (f) The category of commutative regular rings (finitary, reduced, strict). From cat-dist Wed Nov 11 13:14:14 1998 Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id LAA04069 for categories-list; Wed, 11 Nov 1998 11:15:30 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Date: Wed, 11 Nov 1998 10:03:10 -0500 (EST) From: Michael Barr To: Perez Garcia Lucia cc: categories@mta.ca Subject: categories: Re: Gödel and category theory In-Reply-To: <199811110036.UAA11252@mailserv.mta.ca> Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: I should really let people with more interest in foundations field this question, but fools rush in ... On Tue, 10 Nov 1998, Perez Garcia Lucia wrote: > > I am interested in the foundations of mathematics -more concretely, > in the claim that category theory can serve as a superior substitute > for set theory in the foundational landscape. In this context, I would > like to point out a footnote which appears in 'What is Cantor's > Continuum Problem?', written by Kurt G?del in 1947, revised and expanded > in 1964, and finally published in Benacerraf P. and Putnam H. (eds.) 1983: > Philosophy of Mathematics. Selected Readings, Cambridge University Press, > pp. 470-485. It reads as follows: > > It must be admitted that the spirit of the modern abstract disciplines > of mathematics, in particular of the theory of categories, transcends > this concept of set*, as becomes apparent, e.g., by the self-applicability > of categories (see MacLane, 1961**). It does not seem however, that > anything is lost from the mathematical content of the theory if categories > of different levels are distinguished. If there exist mathematically > interesting proofs that would not go through under this interpretation, > then the paradoxes of set theory would become a serious problem for > mathematics. > *(the concept of set G?del was referring to is the iterative > one). > **(MacLane, S. 1961. "Locally Small Categories and the > Foundations of Set Theory". In Infinitistic Methods, > Proceedings of the Symposium on Foundations of Mathematics > (Warsaw, 1959). London and N.Y., Pergamon Press). > > I need some help to grasp the following questions: > > - In what sense the self-applicability of categories transcends the concept > of set?. (It is obvious that categories transcend the concept of well- > founded set but, what's the matter with non-well-founded sets?. > I will pass on this one. As far as I know, using category theory as foundations gives an equally powerful, but not more powerful, foundation. But I would say the same about non-well-founded set theory. Basically, it is a matter of convenience and, perhaps, coherence. > - In what sense do you think G?del proposed distinguishing different levels > of categories?. Would it be possible that G?del was thinking of something > like type theory?. > I would assume that is what he meant. > - Do you agree with G?del's intuition that nothing would be lost with such > a distinction?. > In a word: no. > - Finally, in the last lines of the note G?del seems to suggest a research > programme for category theory as an alternative foundation of mathematics. > To what extent has it been carried out?. > Quite a bit; it is called elementary topos theory. I want to add something here. I have taught a course in set theory (twice, actually). I didn't much enjoy it, so perhaps I am prejudiced. But I have a specific complaint. In all the fields of mathematics that I have worked with (all the axiomatic fields, I should say), structures are defined and then functions that preserve those structures. The structure of a set is that of an epsilon tree, but this structure is ignored when it comes to defining functions. A good thing too, since the only functions that preserve that structure are inclusions of subsets. And the only endomorphism of a set is the identity. Believing that theory should follow practice, I am unhappy with the standard foundations that build this elaborate structure only to ignore it. When categories are used as foundations, then the undefined terms are object, arrow, domain, codomain, identity, and the relation (partial function) of composition. They are all used regularly in category theory; they are not just there to give a formal foundation. And functors are exactly what preserve these things. > > Thanks for your help. > > Regards, > > Luc?a P?rez > Dpt.L?gica y Filosof?a de la Ciencia > University of Valencia -Spain- > > -- > *********************************************** > lperez > *********************************************** > > From cat-dist Wed Nov 11 15:29:11 1998 Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id OAA32079 for categories-list; Wed, 11 Nov 1998 14:08:27 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f From: ct99@mat.uc.pt Message-Id: <3.0.5.32.19981111170241.007a6340@mat.uc.pt> X-Sender: ct99@mat.uc.pt X-Mailer: QUALCOMM Windows Eudora Light Version 3.0.5 (32) Date: Wed, 11 Nov 1998 17:02:41 +0000 To: categories@mta.ca Subject: categories: CT99 - first announcement Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: Please circulate to all interested colleagues. =================================================================== FIRST ANNOUNCEMENT =================================================================== CT99 International Category Theory Meeting Department of Mathematics University of Coimbra Coimbra, Portugal July 19-24, 1999 The Department of Mathematics of the University of Coimbra is organizing an International Meeting on Category Theory to be held at the University of Coimbra, from 19th to 24th of July, 1999. The program will consist of 50mn plenary lectures delivered by invited speakers and 25mn contributed talks. Scientific Committee: -------------------- Jiri Adamek (Technische Universitat Braunschweig, Germany) Bernhard Banaschewski (McMaster University, Canada) Peter T. Johnstone (University of Cambridge, UK) Andre Joyal (Universite du Quebec a Montreal, Canada) F. William Lawvere (SUNY at Buffalo, USA) Dana Scott (Carnegie Mellon University, USA) Ross Street (Macquarie University, Australia) Walter Tholen (York University, Canada) Organizing Committee: -------------------- Manuela Sobral (Universidade de Coimbra) Maria Manuel Clementino (Universidade de Coimbra) Jorge Picado (Universidade de Coimbra) Lurdes Sousa (Instituto Superior Politecnico de Viseu) Goncalo Gutierres (Universidade de Coimbra) Additional Information: ---------------------- Important information, as well as the registration form, will be sent to you on January 1999. Updated information can be found at the conference home page: http://www.mat.uc.pt/~ct99/ Pre-Registration: ---------------- If you plan to attend the conference please pre-register at the web page, or fill the pre-registration form below and send it by email (mailto:ct99@mat.uc.pt), by fax (+351-39-832568) or by mail (to CT99, Departamento de Matematica, Universidade de Coimbra, Apartado 3008, 3000 Coimbra, Portugal). ------------------------------------------------------------------- CATEGORY THEORY 99 Coimbra, Portugal, July 19-24, 1999 PRELIMINARY REGISTRATION FORM (Please send until 31 December, 1998) I plan to attend CT99. I plan to give a talk: YES ___ NO ___ Family name (please print): .................................. First name: .................................................. Affiliation: ................................................. .............................................................. .............................................................. Mailing address (if different): .............................. .............................................................. .............................................................. Email: ....................... Fax: .......................... From cat-dist Wed Nov 11 15:47:03 1998 Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id OAA30265 for categories-list; Wed, 11 Nov 1998 14:22:28 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Date: Wed, 11 Nov 1998 17:16:37 GMT Message-Id: <199811111716.RAA26437@merlin.mat.uc.pt> X-Sender: mmc@mat.uc.pt X-Mailer: Windows Eudora Light Version 1.5.2 Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" To: categories@mta.ca From: "M. M. Clementino" Subject: categories: SCTA - first announcement Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: =================================================================== FIRST ANNOUNCEMENT =================================================================== School on Category Theory and Applications Department of Mathematics University of Coimbra Coimbra, Portugal July 13-17, 1999 The Department of Mathematics of the University of Coimbra is also organizing a School on Category Theory and Applications from 13th to 17th of July, 1999. This school will consist of the following 7 hours intensive courses, at a postgraduate level: n-Categories by John Baez (University of California, USA) Algebraic Theories by M. Cristina Pedicchio (University of Trieste, Italy) Chu Spaces: duality as a common foundation for computation and mathematics by Vaughan Pratt (Stanford University, USA). The second announcement will appear on January 1999. Updated information can be found at the school home page http://www.mat.uc.pt/~scta/ Pre-registration: ---------------- If you plan to attend the conference please register at the web page, or fill the pre-registration form below and send it by email (to scta@mat.uc.pt), by fax (+351-39-832568) or by mail (to SCTA, Departamento de Matematica, Universidade de Coimbra, Apartado 3008, 3000 Coimbra, Portugal). ---------------------------------------------------------------------------- SCHOOL ON CATEGORY THEORY AND APPLICATIONS Coimbra, Portugal, July 13-17, 1999 PRELIMINARY REGISTRATION FORM (Please send until 31 December, 1998) I plan to attend SCTA. Family name (please print): .................................. First name: .................................................. Affiliation: ................................................. .............................................................. .............................................................. Mailing address (if different): .............................. .............................................................. .............................................................. Email: ....................... Fax: .......................... From cat-dist Wed Nov 11 16:07:34 1998 Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id OAA02911 for categories-list; Wed, 11 Nov 1998 14:32:48 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f From: Koslowski Message-Id: <199811111549.QAA03784@lisa.iti.cs.tu-bs.de> Subject: categories: Norddeutsches Kategorienseminar To: categories@mta.ca (categories list) Date: Wed, 11 Nov 1998 16:49:22 +0100 (MET) X-Mailer: ELM [version 2.4ME+ PL28 (25)] MIME-Version: 1.0 Content-Type: text/plain; charset=US-ASCII Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: Dear Colleagues, The following announcement is directed primarily, but not exclusively, at German readers of this list. We simply don't know how many categorically interested people are hiding in (northern) Germany :-) (An English version comes after the German version.) Thank you for your attention. -- J"urgen Koslowski =================================================================== NORDDEUTSCHES KATEGORIENSEMINAR (NDKS) am 20./21.\ Februar 1999 in Braunschweig Vor der Jahrtausendwende und in Anlehnung an das wohlbekannte "Nordwestdeutsche Kategorienseminar" der 70'er Jahre m\"ochten wir an Kategorien interessierten Mathematikern und Informatikern im norddeutschen Raum (und dar"uber hinaus) ein Forum bieten, ihre Ergebnisse zu pr\"asentieren und sich mit anderen auszutauschen. Insbesondere soll dem kategoriellen Nachwuchs die M\"oglichkeit geboten werden, im informellen Rahmen Gleichgesinnte kennenzulernen und evtl.\ erste Tagungserfahrungen zu sammeln. Insofern wird die Tagungssprache \"uberwiegend Deutsch sein. Aber nat\"urlich sind auch ausl\"andische KollegInnen herzlich eingeladen, falls sie sich gerade im norddeutschen Raum aufhalten. Wir verstehen diese Veranstaltung nicht als Konkurrenz zur erfolgreichen europ"aischen PSSL-Reihe, sondern eher als regionale Vorbereitung. Beim Zeitplan wollen wir uns an das bew\"ahrte Schema halten mit Vortr\"agen am Samstag und am Sonntag Vormittag, sowie An- und Abreise am Freitag bzw.\ Sonntag. Sie k\"onnen nat\"urlich auch l\"anger bleiben! Um die Unterbringung der Teilnehmer in Braunschweig k"ummern wir uns. Bitte bringen Sie diese Ank\"undigung auch eventuell interessierten KollegInnen zur Kenntnis. Wir freuen uns schon auf Ihren Besuch in Braunschweig! Jiri Adamek J"urgen Koslowski ------------------------------------------------------------------------------ NORDDEUTSCHES KATEGORIENSEMINAR (NDKS) February 20/21, 1999, in Braunschweig Before the third millennium and in the tradition of the well-known "Nordwestdeutsches Kategorienseminar" of the 1970's we would like invite categorically interested mathematicians and computer scientists from Northern Germany (and beyond) to present their results and to exchange ideas. In particular, we hope that students interested in category theory will have a chance to meet like-minded people in an informal atmosphere and gain experience at meetings. Hence we expect most talks to be in German. But, of course, foreign colleagues who happen to be in the area are cordially invited to attend as well. This meeting is not intended as competition for the highly successful European PSSL series. We would like it to serve as regional preparation instead. We want to keep the familiar timetable of talks on Saturday and Sunday morning, which leaves Friday and Saturday for arrival and departure, respectively. Of course, you can stay longer, if you wish! We will try to find accommodations for the participants. Please notify your colleagues who might be interested. We are looking forward to your visit in Braunschweig! Jiri Adamek J"urgen Koslowski ------------------------------------------------------------------------------- Bitte an J"urgen Koslowski schicken, m\"oglichst bis zum 10. Januar: Please return to J"urgen Koslowski , preferably before January 10: Postanschrift / postal address: Institut f"ur Theoretische Informatik TU Braunschweig Postfach 3329 D-38023 Braunschweig Germany Ich m\"ochte am NDKS am 20./21.\ Februar 1999 in Braunschweig teilnehmen / I intend to come to the NDKS on February 20/21, 1999, in Braunschweig: * Ich m\"ochte vortragen zum Thema / I intend to give a talk entitled ................................................................... * ( ) 25 Minuten/minutes ( ) 40 Minuten/minutes * Bitte reservieren Sie ein Zimmer f\"ur / Please reserve accommodation for Donnerstag/Freitag/Samstag/Sonntag / Thursday/Friday/Saturday/Sunday * ( ) Einzelzimmer / single room ( ) Doppelzimmer / double room * ( ) Ich bin VegetarierIn / I am a vegetarian Name / name : Adresse / address: Email : Telefon / phone : -- J"urgen Koslowski % If I don't see you no more in this world ITI % I meet you in the next world TU Braunschweig % and don't be late! koslowj@iti.cs.tu-bs.de % Jimi Hendrix (Voodoo Child) From cat-dist Wed Nov 11 20:16:47 1998 Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id TAA08282 for categories-list; Wed, 11 Nov 1998 19:08:00 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Message-Id: <199811112120.IAA12919@macadam.mpce.mq.edu.au> X-Sender: street@macadam.mpce.mq.edu.au Mime-Version: 1.0 Content-Type: text/plain; charset="iso-8859-1" Date: Thu, 12 Nov 1998 08:21:09 +1000 To: categories@mta.ca From: street@mpce.mq.edu.au (Ross Street) Subject: categories: Re: computation in CPS Content-Transfer-Encoding: 8bit X-MIME-Autoconverted: from quoted-printable to 8bit by mailserv.mta.ca id RAA29277 Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: Dear Categories: I sent this yesterday but I notice I had mixed up the address. I'll try again. --Ross >In the omega-category In generated by the n-cube, is it possible to >find a kind of "general" formula for the (n-1)-source (target) of >the n-morphism corresponding to the interior of In ? I can do >mechanical computation in low dimension but I am not able for the >moment to imagine a formula for any dimension. In high dimension , >computations become very long. There are two algorithms: my excision of extremals and the Aitchison-Pascal triangle. The former starts with the top dimension cell and works down while the former builds up the cocycle identities recursively from dimension 0. Excision of extremals was done for simplexes on page 325 of [1] resulting in the formulas on page 330 and 331. The cubes case was done around the same time but not published. At the end of the 1980s, Ross Moore implemented this algorithm on a Mac; but things do get big quickly after the cases given in [1]. For general parity complexes, see page 330 of [2] where, even after [3], two expository mistakes remain: in the first line of the Algorithm, "largest" should be "smallest"; on the fourth line the plus signs should be unions; and on the fifth line the element w should be chosen to be not in "mu"(u)_(n+1). Aitchison presented his algorithm for simplexes and cubes at the 1987 category conference in Louvain-la-Neuve, Belgium. I explain it somewhat on page 68 of [5]; also see page 559 of [4]. The simplex case can be obtained from the cube case which has more symmetry. 1. The algebra of oriented simplexes, J. Pure Appl. Algebra 49 (1987) 283-335 2. Parity complexes, Cahiers topologie et géométrie différentielle catégoriques 32 (1991) 315-343 3. Parity complexes: corrigenda, Cahiers topologie et géométrie différentielle catégoriques 35 (1994) 359-361 4. Categorical structures, Handbook of Algebra, Volume 1 (editor M. Hazewinkel; Elsevier Science, Amsterdam 1996; ISBN 0-444-82212-7) 529-577 5. Higher categories, strings, cubes and simplex equations, Applied Categorical Structures 3 (1995) 29-77 & 303 Don't be surprised that the formula was hard to imagine. John Roberts said that no amount of staring revealed the pattern. Yet, the Aitchison-Pascal triangle is the solution! I seem to remember Iain Aitchison got it first for the cubes and then, by geometry, transferred to simplexes (which is where Roberts was looking). <== Ross From cat-dist Wed Nov 11 20:28:32 1998 Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id TAA06714 for categories-list; Wed, 11 Nov 1998 19:08:47 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Message-Id: <199811112141.QAA06255@po.cwru.edu> Date: Wed, 11 Nov 1998 16:41:48 -0500 (EST) X-Sender: cxm7@pop.cwru.edu X-Mailer: Windows Eudora Version 1.4.4 Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" To: categories@mta.ca From: cxm7@po.cwru.edu (Colin McLarty) Subject: categories: Re: Gödel and category theory Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: Perez Garcia Lucia wrote, among other things: >- In what sense the self-applicability of categories transcends the concept > of set?. (It is obvious that categories transcend the concept of well- > founded set but, what's the matter with non-well-founded sets?. Well-founding is an irrelevant detail. Take any non-wellfounded set theory which includes the axiom of choice, such as Aczel's AFA. Then every set is isomorphic to an ordinal, that is to a well-founded set. Since categorical methods are all isomorphism invariant, any categorical structure available in this set theory is also available in well-founded sets. I have discussed this in an article "Anti-foundation and self-reference" Journal of Philosophical Logic 22 (1993) 19-28. There is no real chance that abandoning the axiom of choice will help either--say by adopting AFA without Axiom of Choice. Rather, the apparent issue is existence of a universal set--a set of all sets, so that you make a category of all categories. If you want to use membership based set theory this will require non-wellfounding, but again the details of membership and wellfounding are irrelevant. Anyway, the problem here is that functions are hard to work with in set theory with a universal set. I have shown that in any such set theory meeting a few weak conditions there is a category of all categories, and it is not cartesian closed. The result is clear from the more particular case "Failure of cartesian closedness in NF" Journal of Symbolic Logic57 (1992) 555-56. Working with such a poor 'category of all categories' is much more difficult than just doing without. I think a more promising approach is to use Benabou's theory of fibrations and definability as in Benabou J. (1985). "Fibered categories and the foundations of naive category theory". Journal of Symbolic Logic 50, 10-37. I have discussed this briefly in "Category theory: Applications to the foundations of mathematics" Routledge Encyclopedia of Philosophy (1998); and in "Axiomatizing a category of categories" Journal of Symbolic Logic56 (1991) 1243-60. I see no good arguments that there SHOULD be a genuine "category of all categories" in any strong sense. But it seems an interesting question. >- In what sense do you think G?del proposed distinguishing different levels > of categories?. Would it be possible that G?del was thinking of something > like type theory?. More likely he was thinking of Eilenberg and Mac Lane's use of Goedel-Bernay's set theory as a foundation in "The general theory of natural equivalences", so there are set categories and class categories. To study Goedel's claim here, you should look at any of Mac Lane's papers on foundations that Goedel might have seen by this time. Maybe the foundational parts of "The general theory of natural equivalences" are all he could have seen, I don't know. Then it would be good to know what people around Princeton were saying about category theory at this time--and that might be very hard to find out. Colin From cat-dist Thu Nov 12 11:00:59 1998 Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id IAA27774 for categories-list; Thu, 12 Nov 1998 08:42:50 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Message-Id: <3.0.3.32.19981112003347.006956a0@imiucca.csi.unimi.it> X-Sender: acarboni@imiucca.csi.unimi.it X-Mailer: QUALCOMM Windows Eudora Light Version 3.0.3 (32) Date: Thu, 12 Nov 1998 00:33:47 +0100 To: categories@mta.ca From: aurelio carboni Subject: categories: Godel Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: AS for the Perez calling for references, I found quite surprising that nobody quoted Lawvere's work on the subject. Carboni. From cat-dist Thu Nov 12 12:56:04 1998 Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id LAA09704 for categories-list; Thu, 12 Nov 1998 11:10:30 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Message-Id: Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" Date: Thu, 12 Nov 1998 15:26:28 +0100 To: categories@mta.ca From: grandis@dima.unige.it (Marco Grandis) Subject: categories: Preprint available Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: The following preprint: M. Grandis, "An intrinsic homotopy theory for simplicial complexes with applications to image processing" is available at: ftp://pitagora.dima.unige.it/WWW/FTP/GRANDIS/ as: Lnk.Nov98.ps *** Abstract. A simplicial complex is a set equipped with a down-closed family of distinguished finite subsets; this structure is mostly viewed as codifying a triangulated space. Here, this structure is used directly to describe "spaces" of interest in various applications, where the associated triangulated space would be misleading. An intrinsic homotopy theory, not based on topological realisation, is introduced. The applications considered here are aimed at metric spaces and digital topology; concretely, at image processing and computer graphics. A metric space X has a structure t_e(X) of simplicial complex at each "resolution" e > 0; the resulting n-homotopy group \pi_n(t_e(X)) detects those singularities which can be captured by an n-dimensional grid, with edges bound by e; this works equally well for continuous or discrete regions of euclidean spaces. *** Comments would be appreciated. In particular, I am uneasy about a question of terminology. In my opinion, the term "simplicial complex", quite appropriate when the structure is viewed as codifying a triangulated space, is unfit when such objects are treated as "spaces" in themselves (somewhat close to bornological spaces, which have similar axioms on objects and maps). In other words, "simplicial complex" should not refer to the category itself, say C, but to its usual embedding in Top, the simplicial realisation. The two aspects may clash, e.g. with respect to initial or final structures: the coarse C-object on three points (final structure, all parts are distinguished) is realised as a euclidean triangle; a C-subobject is sufficient to produce a topological subspace (a regular subobject in Top), but a C-subspace (a regular subobject in C) is a stronger notion. Moreover, from a more concrete point of view, the simplicial realisation is quite inappropriate in most of the applications considered in this work. The opposition "C-object / simplicial complex" is in part similar to "sequence / series": the second term refers to a more specific view & use of the same data; the clashing of the opposition is particularly evident in the notions of convergence, for a sequence or a series. That's why I am calling a C-object a "combinatorial space". (The term "combinatorial complex" has also been used for simplicial complex; and I wanted a term of the form "attribute + space", to use freely of topological terms like discrete, coarse, subspace...) But of course it is embarassing to propose a new term for a classical notion. Marco Grandis From cat-dist Thu Nov 12 13:31:50 1998 Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id LAA21977 for categories-list; Thu, 12 Nov 1998 11:56:16 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Message-ID: <364AAD06.739C69F4@server.ru> Date: Thu, 12 Nov 1998 12:40:22 +0300 From: Danilov Nikita Organization: Server Ltd. X-Mailer: Mozilla 4.06 [en] (WinNT; I) MIME-Version: 1.0 To: Categories Mailing List Subject: categories: Yoneda lemma Content-Type: text/plain; charset=koi8-r Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: Hello all, I got a proof that if Hom-functor h of category C factors through complete category M up to natural isomorphism (that is $h \cong RP : C* \times C --> Set$, where P is faithful forgetful functor P : M --> Set and R : C* \times C --> M) and P preserves limits, then 1. for every category C' Hom-functor Nat of Funct(C',C) factors through M up to isomorphism: Nat = Nat' P, and 2. if Hom-functor of M itself factors through M and is right adjointed to multiplication functor (i.e., M is cartesian closed), P preserves exponential adjoints and for every object x in C there is e_x : 1 --> R(x,x), such that e_xP selects 1_x in Hom(x,x), then for every functor F : C --> M, compatible with R in obvious sense, these exists natural (on x) isomorphism Nat'(R^x,F) \cong xF as objects in M, that is Yoneda lemma holds. Is this known/trivial/known-to-be-false? Nikita. From cat-dist Mon Nov 16 18:04:23 1998 Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id QAA22989 for categories-list; Mon, 16 Nov 1998 16:23:30 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f X-Received: from callisto.acsu.buffalo.edu (qmailr@callisto.acsu.buffalo.edu [128.205.7.122]) by mailserv.mta.ca (8.8.8/8.8.8) with SMTP id QAA19707 for ; Mon, 16 Nov 1998 16:08:23 -0400 (AST) X-Received: (qmail 7653 invoked by uid 39883); 16 Nov 1998 20:07:52 -0000 Date: Mon, 16 Nov 1998 15:07:52 -0500 (EST) From: F W Lawvere To: categories Reply-To: wlawvere@ACSU.Buffalo.EDU Subject: categories: re: Sets Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: Conceptualizing and axiomatizing Mike Barr's experience with teaching membership-based set theory is shared by many mathematicians, and quite a few share his conclusions. One conclusion is that clarification is needed on even more basic questions than just the large/small issue (which concerned Goedel, Mac Lane, and Perez), in order to arrive at conceptions and axiomatizations compatible with the practice of mathematics. For example, I was aiming at such a clarification in pp. 118-128 of my 1976 paper in honor of Professor Eilenberg's 60th birthday, where I advocated some rational connection between conceptualizing and axiomatizing. The complete lack of such a connection in a recent article in the journal "Mathematical Structures in Computer Science" could have been avoided by the editors, if not by the authors. In a section labeled "Basic Set Theory" (p.510) they quote from my above paper a description of the notion of abstract set: 1. ...each element of X has no structure whatsoever. 2. X itself has no internal structure except for equality and inequality of pairs of elements.... immediately followed by their absurd conclusion: "axiomatically this corresponds to taking the membership relation epsilon as the only primitive notion of set theory and to postulating .." some axioms typical to Zermelo-style membership-based theory! Of course those axioms are NOT compatible with the conception quoted: they violate (1) because according to the Zermelo primitives and axioms, an element usually has elements, which would be structure; and they violate (2) since according to those primitives and axioms, a pair of elements of X may stand itself in the membership relation, which would be an internal structure other than equality. The authors neglected to quote the third clause which (as in the example that Mike mentions) their axioms also violate. The notion of abstract set (Kardinalen in Cantor's sense) is basic among the many other notions of cohesive and/or variable sets (Mengen) to the extent that we can model the Mengen via diagrams of maps between abstract sets. Abstract sets may be "abstracted from" less abstract sets, as Cantor did, or used, as most modern mathematics in practice does, as all-purpose memory cells or parameterizers or nodes in such diagrams. In addition to the papers by Colin McLarty mentioned in his message of November 11, 1998, the following papers should help to clarify this notion and its role in mathematics. J. Isbell Adequate sub-categories Illinois J. Math. vol. 4, pp 541-552, 1960 F. W. Lawvere An elementary theory of the category of sets Proc. Nat. Acad. Sc. USA, vol. 52, 1964, pp 1506-1511 F. W. Lawvere Variable quantities and variable structures in topoi, (see especially pp 118-128) in Algebra, Topology, and Category Theory, ed. Heller & Tierney, Academic Press, 1976 F. W. Lawvere Cohesive toposes and Cantor's lauter Einsen (concerning Cantor's neglected Kardinalen) Philosophia Matematica, vol. 2, 1994, pp 5 - 15 W. Mitchell Boolean topoi and the theory of sets (the membership-free content of Goedels constructible sets still needs to be clarified further) Journal of Pure and Applied Algebra, vol. 2, 1972, pp 261-274 ******************************************************************** F. William Lawvere Mathematics Dept. SUNY wlawvere@acsu.buffalo.edu 106 Diefendorf Hall 716-829-2144 ext. 117 Buffalo, N.Y. 14214, USA ********************************************************************* From cat-dist Mon Nov 16 20:43:05 1998 Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id TAA18523 for categories-list; Mon, 16 Nov 1998 19:45:46 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Date: Mon, 16 Nov 1998 17:08:09 -0500 (EST) From: James Stasheff To: categories@mta.ca Subject: categories: query Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: The language of higher category theory in more than analogous to homotopy theory, especially in the cellular version. What is the appropriate reference for a non-categorical reader? Have the A_\infty categories of Smirnov or Fukaya been treated in the categorical literature?? thanks .oooO Jim Stasheff jds@math.unc.edu (UNC) Math-UNC (919)-962-9607 \ ( Chapel Hill NC FAX:(919)-962-2568 \*) 27599-3250 http://www.math.unc.edu/Faculty/jds From cat-dist Tue Nov 17 11:44:55 1998 Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id JAA30432 for categories-list; Tue, 17 Nov 1998 09:40:08 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Message-Id: Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" Date: Tue, 17 Nov 1998 09:26:51 +0100 To: categories@mta.ca From: grandis@dima.unige.it (Marco Grandis) Subject: categories: Re: preprint available (on simplicial complexes) Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: Reply to James Stasheff > is it also available on your web page without ftp? No, from my home page: http://pitagora.dima.unige.it/webdima/STAFF/GRANDIS/ you would just have access to all papers available by ftp, at the address I gave: ftp://pitagora.dima.unige.it/WWW/FTP/GRANDIS/ *** > will you be posting it to the math archive at lanl? yes, in a while and possibly after revision. *** With best wishes Marco Grandis From cat-dist Wed Nov 18 12:51:08 1998 Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id KAA23834 for categories-list; Wed, 18 Nov 1998 10:53:47 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f From: john baez Message-Id: <199811180412.UAA15669@charity.ucr.edu> Subject: categories: Re: query To: categories@mta.ca Date: Tue, 17 Nov 1998 20:12:45 -0800 (PST) X-Mailer: ELM [version 2.4 PL24 PGP3 *ALPHA*] MIME-Version: 1.0 Content-Type: text/plain; charset=US-ASCII Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: Jim Stasheff writes: > The language of higher category theory in more than analogous > to homotopy theory, especially in the cellular version. What > is the appropriate reference for a non-categorical reader? > Have the A_\infty categories of Smirnov or Fukaya been treated > in the categorical literature?? Unfortunately the truly appropriate reference has not yet been written, because the equivalence between weak infinity-groupoids and homotopy types has not yet worked out in full detail, at least not in the cellular version. The *dream* of translating all of homotopy theory into higher category theory is outlined in: John Baez and James Dolan, Categorification, to appear in Proceedings Workshop on Higher Category Theory and Mathematical Physics at Northwestern University, Evanston, Illinois, March 1997, eds. Ezra Getzler and Mikhail Kapranov, preprint available as math.QA/9802029. This also has lots of references to different places where various bits of the dream have been realized. Basically the dream consists of working out the following correspondence: HIGHER CATEGORY THEORY HOMOTOPY THEORY omega-groupoids homotopy types n-groupoids homotopy n-types k-tuply groupal omega-groupoids homotopy types of k-fold loop spaces k-tuply groupal n-groupoids homotopy n-types of k-fold loop spaces k-tuply monoidal omega-groupoids homotopy types of E_k spaces k-tuply monoidal n-groupoids homotopy n-types of E_k spaces stable omega-groupoids homotopy types of infinite loop spaces stable n-groupoids homotopy n-types of infinite loop spaces Z-groupoids homotopy types of spectra How do A_infinity categories fit in? As far as I can tell they should correspond to omega-categories where all j-morphisms are invertible for j > 1. From cat-dist Wed Nov 18 12:58:44 1998 Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id LAA22834 for categories-list; Wed, 18 Nov 1998 11:01:58 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f X-Received: from uts.univ.trieste.it (pedicchi@uts.univ.trieste.it [140.105.48.16]) by mailserv.mta.ca (8.8.8/8.8.8) with ESMTP id GAA00219 for ; Wed, 18 Nov 1998 06:00:49 -0400 (AST) X-Received: from localhost (pedicchi@localhost) by uts.univ.trieste.it (8.8.8/8.8.8) with SMTP id LAA10270 for ; Wed, 18 Nov 1998 11:00:37 +0100 (MET) Date: Wed, 18 Nov 1998 11:00:36 +0100 (MET) To: categories From: "Pedicchio M. Cristina" Subject: categories: PSSL in Trieste In-Reply-To: Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: PSSL TRIESTE 28-29 November 1998 Saturday: 9.30 - 10.10 F.W. Lawvere: TBA 10.15 - 10.45 A. Kock: Strength of Amazing Right Adjoints (with G. Reyes) 10.50- 11.20 G. Janelidze: Homological Algebra in additive categories via Descent Theory break 11.50 - 12.20 M. Grandis: An intrinsic homotopy theory for simplicial complexes 12.25 - 12.55 E. Vitale: Factorization systems for symmetric cat-groups lunch 15.00 - 15.30 J. Adamek: M-completeness is seldom monadic over graphs (joint with G.M. Kelly) 15.35 -16.05 J. Rosicky: Generalized varieties 16.10 - 16.40 I. Le Cruerer: Descent of internal structures break 17.10 - 17.40 D. Bourn: Normal monomorphisms and Mal'cev objects in protomodular categories 17.45 - 18.15 M. Gran: Internal groupoids in Maltsev categories 18.20 - 18.50 H. Kleisli: A model of linear logic revisited. Sunday: 9.00 - 9.30 J. Goguen: Sheaf Theory and cocurrency in Computer Science 9.35 - 10.05 R.F.C. Walters: On the algebra of feedback and systems with boundary (joint work with P Katis and N Sabadini) 10.10 -10.40 J. Koslowski: *-Linear Bicategories (joint with Robert Seely and Robin Cockett.) break 11.10 - 11.40 F. Linton: Banach Spaces *are* monadic - just not quite over Sets 11.45 - 12.15 W. Tholen: Total cocompleteness of the formal product completion (joint with J.Adamek and L.Sousa) 12.20 - 12.50 M. Clementino: A categorical setting for local compactness. M.Cristina Pedicchio - Dept.Mathematics - Univ.Trieste 34100 Trieste - Italy 39 40 6763256/3727 - fax 39 40 6763256 - pedicchi@univ.trieste.it From cat-dist Wed Nov 18 13:03:19 1998 Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id LAA23129 for categories-list; Wed, 18 Nov 1998 11:23:13 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Message-Id: Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" Date: Wed, 18 Nov 1998 13:27:09 +0100 To: categories@mta.ca From: kock Subject: categories: preprint available (Kock and Reyes) Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: The following preprint A. Kock and G.E. Reyes: Fractional Exponent Functors and Categories of Differential Equations is available via the Home Page http://www.mi.aau.dk/~kock/ or directly by ftp (200 KB) ftp://ftp.imf.au.dk/pub/kock/ODE5.ps (also available in .dvi format, 100 KB). Abstract: This paper grew out of a question/suggestion of Lawvere: to use the "amazing right adjoints" (= fractional exponents) of Synthetic Differential Geometry, to get information on the category of second order differential equations. As a byproduct of our investigations, we derive some information about the strength (enrichment) of fractional exponent functors in general. From cat-dist Wed Nov 18 15:36:38 1998 Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id OAA20916 for categories-list; Wed, 18 Nov 1998 14:07:57 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f MIME-Version: 1.0 Content-Type: text/plain; charset=iso-8859-1 Date: Wed, 18 Nov 1998 16:09:09 +0000 (GMT) To: categories@mta.ca Subject: categories: preprint available X-Mailer: VM 6.43 under 20.4 "Emerald" XEmacs Lucid Message-ID: <13906.60796.497804.182184@mhuilinn.dcs.ed.ac.uk> From: "Martin Escardo" Content-Transfer-Encoding: 8bit X-MIME-Autoconverted: from quoted-printable to 8bit by mailserv.mta.ca id MAA30554 Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: The following preprint is available at http://www.dcs.ed.ac.uk/home/mhe/pub/papers/patch-CSLC.ps.gz & http://www.dcs.ed.ac.uk/home/mhe/papers.html & ftp://ftp.dcs.ed.ac.uk/pub/mhe/patch-CSLC.ps.gz On the compact-regular coreflection of a compact stably locally compact locale. ABSTRACT: The Scott continuous nuclei form a subframe of the frame of all nuclei. We refer to this subframe as the patch frame. We show that the patch construction exhibits (i) the category of Stone locales and continuous maps as a coreflective subcategory of the category of coherent locales and coherent maps, (ii) the category of compact regular locales and continuous maps as a coreflective subcategory of the category of compact stably locally compact locales and perfect maps, and (iii) the category of regular locally compact locales and continuous maps as a coreflective subcategory of the category of stably locally compact locales. We relate our patch construction to Banaschewski and Brümmer's construction of the dual equivalence of the category of compact stably locally compact locales and perfect maps with the category of compact regular biframes and biframe homomorphisms. Comments are welcome. ----------------------------------------------------------------- Martin H. Escardo, LFCS, Computer Science, Edinburgh University King's Buildings, Mayfield Road, Edinburgh EH9 3JZ, Scotland office: 2606 (JMCB) fax: +44 131 667 7209 phone: +44 131 650 5135 mailto:mhe@dcs.ed.ac.uk http://www.dcs.ed.ac.uk/home/mhe ----------------------------------------------------------------- From cat-dist Wed Nov 18 22:23:15 1998 Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id VAA27550 for categories-list; Wed, 18 Nov 1998 21:28:48 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Message-Id: <199811190014.LAA28074@macadam.mpce.mq.edu.au> X-Sender: street@macadam.mpce.mq.edu.au Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" Date: Thu, 19 Nov 1998 11:15:46 +1000 To: categories@mta.ca From: street@mpce.mq.edu.au (Ross Street) Subject: categories: Re: query Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: >The language of higher category theory in more than analogous >to homotopy theory, especially in the cellular version. What >is the appropriate reference for a non-categorical reader? >Have the A_\infty categories of Smirnov or Fukaya been treated >in the categorical literature?? Fortunately, since the A_\infty categories (described sketchily in the preprint I have of Fukaya) use chain complexes rather than topological spaces (so that I believe a 1-object A_\infty category is an algebra for the A_\infty non-permutative operad on chain complexes - an A_\infty DG-algebra), we do not need to realise the whole homotopy types dream (as described by John Baez) to give a categorical description of A_\infty categories. Several years ago I remember discussing this by email with Jim Stasheff. Dominic Verity was here at the time. I cannot remember all the details but the two basic ingredients were some free strict n-categories made from the set (whose elements are to be the objects of the A_\infty category) in much the way the orientals are constructed, and the construction (below) mentioned in my Oberwolfach Descent Theory notes of September 1995. There is a connection between the Gray tensor product and ordinary chain complexes. Each chain complex R gives rise to a (strict) omega-category J(R) whose 0-cells are 0-cycles a in R, whose 1-cells b : a --> a' are elements b in R_1 with d(b) = a'- a, whose 2-cells c : b --> b' are elements c in R_2 with d(c) = b'- b, and so on. All compositions are addition. This gives a functor J : DG --> omega-Cat from the category DG of chain complexes and chain maps. In fact, J : DG --> omega-Cat is a monoidal functor where DG has the usual tensor product of chain complexes and omega-Cat has the Gray tensor product. By applying J on homs, we obtain a (2-) functor J_* : DG-Cat --> V_2-Cat, where V_2 is omega-Cat with the Gray-like tensor product (extending the natural tensor product of oriented cubes as described in Sjoerd Crans thesis). In particular, since DG is closed, it is a DG-category and we can apply J_* to it. The V_2-category J*(DG) has chain complexes as 0-cells and chain maps as 1-cells; the 2-cells are chain homotopies and the higher cells are higher analogues of chain homotopies. Best regards, Ross From cat-dist Thu Nov 19 13:59:01 1998 Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id MAA15732 for categories-list; Thu, 19 Nov 1998 12:23:05 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f X-Sender: sjv@pop.doc.ic.ac.uk Message-Id: Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" Date: Thu, 19 Nov 1998 11:53:12 +0000 To: categories@mta.ca From: s.vickers@doc.ic.ac.uk (Steven Vickers) Subject: categories: Has this topos ever been found useful? Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: After seeing Eric Goubault's work on directed homotopy, where he defines a notion of "locally partially ordered space", I found myself considering the following topos. Does anyone know if it has been used, for instance in algebraic geometry or algebraic topology? What I am trying to capture by the definition is the process of taking the lower semicontinuous reals in [0,1] (classically, [0,1] with its Scott topology, so that the numerical order becomes the specification order) and identifying 0 and 1. Then the specialization from 0 to 1 becomes a non-trivial endomorphism e of 0, so we are forced to consider the modified space as a topos, not a locale. In addition, an extra point will spring into existence, namely the (filtered) colimit of 0 --> 0 --> 0 --> ... e e e The topos is given by a site as follows. Let Q+ be the additive monoid of non-negative rationals and Q/Z the rationals modulo the integers. Q+ acts on Q/Z by addition. Let C be the corresponding category of elements (object = element [q] of Q/Z, morphisms ([q],f): [q] -> [q+f] for f in Q+) and generate a Grothendieck topology on C^op by letting any object [q] of C be cocovered by all the morphisms ([q],f) for f > 0. Let E be the topos corresponding to this site on C^op. Its points are flat presheaves F on C with the condition that if x is in F([q]) then x = ([q],f)y for some f > 0, y in [q+f]. The points arising from the wrapped-round [0,1] are like half-infinite helices (F([q]) = N for all q), and the new point is like an infinite helix (F([q]) = Z for all q). Steve Vickers. From cat-dist Thu Nov 19 14:05:04 1998 Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id MAA15445 for categories-list; Thu, 19 Nov 1998 12:22:16 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Message-Id: Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" Date: Thu, 19 Nov 1998 10:31:09 +0100 To: categories@mta.ca From: grandis@dima.unige.it (Marco Grandis) Subject: categories: Re: query Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: I have not seen Fukaya's paper. If the problem is to formalise higher homotopies for, say, topological spaces or chain complexes, together with their operations, there is a simple solution based on the path endofunctor (or, dually, the cylinder endofunctor; or their adjunction) and its powers. The path endofunctor P is constructed so that a homotopy a: f -> g: X -> Y amounts to a map a: X -> PY; the maps f, g are recovered by means of the two faces d-, d+: PY -> Y. (For Top, PY is obviously the space Y^[0, 1] of paths in Y, with the compact-open topology; for chain complexes (PY)_n = Y_n + Y_(n+1) + Y_n, with suitable differential.) P comes equipped with various natural transformations (faces, degeneracy, connections, symmetries, concatenation...), satisfying "algebraic" coherence axioms (a sort of "cubical comonad" with additional structure). These transformations represent the basic structure of lower order homotopies. But now you have, practically for free: - n-tuple homotopies, represented by the power endofunctor P^n, - n-homotopies, represented by a subfunctor P_n of P^n, - their operations, via the usual algebra of natural transformations, - the deduced coherence relations of the latter. Eg, a double homotopy, represented by a map X -> P^2(Y), has four faces given by the four natural transformations d-P, d+P, Pd-, Pd+: P^2 -> P; if k is the concatenation of homotopies, kP and Pk are the vertical and horizontal concatenation of double homotopies, and so on; if k is associative, as is the case for chain complexes but not for spaces, so are all higher concatenations. A double homotopy is said to be a 2-homotopy when its "vertical" faces (say) are trivial, i.e. factor through the degeneracy 1 -> P. (For Top, P^2(Y) is the space of maps from the standard square [0, 1]^2 to Y; P_2(Y) is the subspace of those maps which are constant on the vertical faces of the square.) This way of deducing higher homotopies and their operations from the lower ones can be found in: M. Grandis, Categorically algebraic foundations for homotopical algebra, Appl. Categ. Structures 5 (1997), 363-413. M. Grandis, On the homotopy structure of strongly homotopy associative algebras, J. Pure Appl. Algebra, 134 / 1 (1999 ?), 15-81. Regards, Marco Grandis Dipartimento di Matematica Universita' di Genova via Dodecaneso 35 16146 GENOVA, Italy e-mail: grandis@dima.unige.it http://pitagora.dima.unige.it/webdima/STAFF/GRANDIS/ ftp://pitagora.dima.unige.it/WWW/FTP/GRANDIS/ From cat-dist Thu Nov 19 14:09:44 1998 Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id MAA11116 for categories-list; Thu, 19 Nov 1998 12:21:21 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f X-Received: from zeus.mpce.mq.edu.au (zeus.mpce.mq.edu.au [137.111.219.12]) by mailserv.mta.ca (8.8.8/8.8.8) with ESMTP id GAA09013 for ; Thu, 19 Nov 1998 06:34:47 -0400 (AST) X-Received: (from joyal@localhost) by zeus.mpce.mq.edu.au (8.9.1/8.9.1) id VAA05255; Thu, 19 Nov 1998 21:34:42 +1100 From: Andre Joyal Message-Id: <199811191034.VAA05255@zeus.mpce.mq.edu.au> Subject: categories: category-homotopy To: categories@mta.ca Date: Thu, 19 Nov 1998 21:34:42 +1100 (EST) X-Mailer: ELM [version 2.4 PL23] Content-Type: text Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: This is in reply to a message of James Stasheff and of other participants. I agree that the language of category is more than analogous to homotopy theory, and think that the connection is deep. The work on A-infinity spaces is a striking early illustration of the connection between homotopy coherence and categorical coherence. Homotopy theory is the main guideline for the emergence of higher category theory. For example, the first correct concept of 3-category was discovered as a direct outcome of modelling homotopy 3-types with Gray groupoids. Before this example, all existing 3-categories were equivalent to strict ones. In my opinion, after a first stage of developpement, higher category theory should induce new progress in homotopy theory. This is the real test. In particular, the new theory should help understanding homotopy groups of spheres. I am including an abstract of my talk on a related subject. Regards, Andre Joyal > University of Sydney Algebra Seminar > Friday 20th November, 12-1pm, Carslaw 273 > > Title: The homology of symmetric and braided monoidal categories > Abstract: > We wish to comment on some aspects of the connection between > category theory, topology and homological algebra. The > connection is at the root of higher K-theory and it is guiding > much of the actual research on higher dimensional categories. > We shall concentrate on the relation between monoidal categories > and iterated loop spaces. To each category C we can associate a > space BC called the (Milgram) classfying space of C. The > homology of C is defined to be the homology of BC. The space BC > is a monoid when C has a tensor product, and it has the > structure of an E-infinity space (resp. of an E-2 space) if the > tensor product is symmetric (resp. braided). We shall briefly > discuss the work of P. May and F. Cohen on the homology of E-n > spaces. It shows that the homology of a symmetric (resp. of a > braided) monoidal category is a graded-commutative algebra > admitting Dyer-Lashof operations (resp. is a poisson algebra). > These structures play a crucial role in determining the homology > of the symmetric groups and of the braid groups. The poisson > algebra structure also appears in the recent work of Lehrer and > Segal on the rational homology of classical regular semisimple > varieties. > > ------------------------------ > > Best wishes, > Andrew Mathas > From cat-dist Thu Nov 19 15:41:26 1998 Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id MAA20135 for categories-list; Thu, 19 Nov 1998 12:55:27 -0400 (AST) X-Received: from triples.math.mcgill.ca (rags@Triples.Math.McGill.CA [132.206.150.30]) by mailserv.mta.ca (8.8.8/8.8.8) with ESMTP id LAA02732 for ; Thu, 19 Nov 1998 11:06:24 -0400 (AST) X-Received: from localhost (rags@localhost) by triples.math.mcgill.ca (8.8.5/8.6.10) with SMTP id KAA25936 for ; Thu, 19 Nov 1998 10:08:32 -0500 X-Authentication-Warning: triples.math.mcgill.ca: rags owned process doing -bs Date: Thu, 19 Nov 1998 10:08:32 -0500 (EST) From: "R.A.G. Seely" To: categories@mta.ca Subject: categories: Michael Barr named Peter Redpath Professor (fwd) Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: ================================= It gives me great pleasure to announce to the department that Professor Michael Barr has now been appointed as the Peter Redpath Professor of Mathematics. I am sure that you will join me in congratulating him on this richly deserved honour. This chair was previously held by Professor Edward Rosenthall, Professor Jim Lambek (who still holds the title of Peter Redpath Emeritus Professor) and by Professor Carl Herz. Georg Schmidt, Chair Department of Mathematics and Statistics McGill University ================================= From cat-dist Fri Nov 20 13:23:36 1998 Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id LAA00792 for categories-list; Fri, 20 Nov 1998 11:19:59 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Message-ID: <3655805B.38C188D2@bangor.ac.uk> Date: Fri, 20 Nov 1998 14:44:43 +0000 From: "T.Porter" X-Mailer: Mozilla 4.06 [en] (Win95; I) MIME-Version: 1.0 To: "categories@mta.ca" Subject: categories: Jim's query Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: This is not really an adequate reply to Jim's query. The reason is that as I understand it, he is asking for a source that will explore the homotopy theory of the `globular' models of weak infty categories. My approach to the area is in some sense the dual. Starting with the various models for homotopy types or bits of them, try to see what is mirrored in the weak infty category theory by the homotopy structure. This is in some sense the converse to his query but may none-the-less be relevant. My intuition was, and still is, that the Kan condition on simplicial sets gives a composition/pasting up to coherent homotopy. Moreover the `filler' gives the justification for the composite. (Compare the filler structure of the nerve of a category with that in an arbitrary Kan complex.) Accepting that as a starting point, and the idea that the `category' of weak infinity categories should be a weak infinity category, Jean-Marc Cordier and I looked at `locally Kan' simplicially enriched categories (e.g. Trans AMS 349(1997)1-54). With that viewpoint, it becomes clear that the structure of an A_\infty category is needed to make things really `coherent', but that many of the constructions of `ordinary' category theory have A_\infty or homotopy coherent analogues in this setting, which thus serves as a test-bed' for the development of the more general theory. In part this relates to Michael Batanin's paper in the Cahiers where explicit consideration of A_\infty structure is given. That theory looks at the `global' structure to some extent, but simplicially enriched groupoids model all homotopy types, so a corollary of the simplicial to globular type of transition should be that one should be able to construct weak \infty categroies DIRECTLY from the algebra of a simplicially enriched groupoid. The obvious place to look for this is in the Moore complex which carries a hypercrossed complex structure in the sense of Pilar Carrasco and Antonio Cegarra. (This is related to the n-hypergroupoid structures of Jack Duskin.) Exploring the \infty category structure, potentially in their definition, is the aim of another line of research and in low dimensions, this has been attacked by Ali Mutlu and myself, (see very recent articles in TAC or Bangor's preprint list on the web). Getting nearer to Jim's query, any bridge between homotopy theory and higher dimensional category theory should I feel aim to be approachable by algebraic topologists and therefore should start with a recognisable model for homotopy types. Another approach that must be mentioned is that of Tamsamani and Simpson using multisimplicial objects. Presumably this also links in with the cat^n-groupoid approach pioneered some 14 years ago by Loday. This only handles n-types but can be extended to a model that has higher information but in those dimensions above n, the Whitehead products are trivial. (Has anyone looked at the Whitehead and Samelson products from a globular or weak \infty category viewpoint?) The question of simplicial rather than cubical theory is a difficult one. Marco Grandis made a good case for the cubical formulation the other day, and the use of Kan filler conditions in a cubical setting has been for a long time a speciality of Heiner Kamps (see the recent book by him and me!). That theory does not directly address the question of A_\infty category structures, but it does raise a question that deserves some attention. Suppose you want an analogue of a given theorem from homotopy theory but in another non-topological context and you can get a weak composition structure in your setting (typically given by fillers for boxes in a cubical `enrichment'). What fillers/composites do you need for your particular theorem (e.g.Dold's theorem on homotopy equivalence between cofibrations, or long Dold-Puppe type sequences)? This last question also raises that of the motivation for the generalisations. I am convinced these are useful, even important, but perhaps some debate on directions to explore and goals to seek out might help in providing a fuller answer to Jim's question. Tim. ************************************************************ Timothy Porter |tel direct: +44 1248 382492 School of Mathematics |mathematics office: 382475 University of Wales, Bangor | fax: 383663 Dean St. |World Wide Web Bangor |homepage http://www.bangor.ac.uk/~mas013/ Gwynedd LL57 1UT |Mathematics and Knots : exhibition United Kingdom |http://www.bangor.ac.uk/ma/CPM/ From cat-dist Fri Nov 20 13:24:51 1998 Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id LAA13273 for categories-list; Fri, 20 Nov 1998 11:21:34 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f X-Received: from callisto.acsu.buffalo.edu (qmailr@callisto.acsu.buffalo.edu [128.205.7.122]) by mailserv.mta.ca (8.8.8/8.8.8) with SMTP id XAA05617 for ; Thu, 19 Nov 1998 23:49:19 -0400 (AST) X-Received: (qmail 8435 invoked by uid 39883); 20 Nov 1998 03:49:16 -0000 Date: Thu, 19 Nov 1998 22:49:16 -0500 (EST) From: F W Lawvere To: categories Reply-To: wlawvere@ACSU.Buffalo.EDU Subject: categories: announcement of pre-prints Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: There are two items now available for downloading (PDF) from my homepage http://www.acsu.buffalo.edu/~wlawvere/ They are: Volterra's functionals and the covariant cohesion of space Abstract: Volterra's principle of passage from finiteness to infinity is far less limited than a linearized construal of it might suggest; I outline in Section III a nonlinear version of the principle with the help of category theory. As necessary background I review in Section II some of the mathematical developments of the period 1887-1913 in order to clarify some more recent advances and controversies which I discuss in Section I. Some relevant historical and current literature is discussed in relation to the categorical analysis: Volterra and Hadamard on the notion of element, Fichera's critique of the relation between functional analysis and continuum physics, and the recent Michor & Kriegl book published by the AMS. Outline of Synthetic Differential Geometry Abstract: These rough notes were distributed to the geometry seminar at Buffalo in February 1998, sketching the background of categorical dynamics in anticipation of the April 1999 AMS Meeting in which there will be a special session on related matters. In particular, some of the results stated in my September 1997 AMS talk in Montreal "Toposes of laws of motion" are outlined, especially the relation of second order differential equations to a.t.o.m's (amazingly tiny object models). An additional appendix has been added (November 1998) to these rough notes concerning recent advances on these questions by Kock and Reyes. Toposes of laws of motion will be added to the web page as soon as the transcript of the original video is completed. ********************************************************************** F. William Lawvere Mathematics Dept. SUNY wlawvere@acsu.buffalo.edu 106 Diefendorf Hall 716-829-2144 ext. 117 Buffalo, N.Y. 14214, USA ********************************************************************** From cat-dist Fri Nov 20 13:36:28 1998 Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id LAA13498 for categories-list; Fri, 20 Nov 1998 11:15:38 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Date: Thu, 19 Nov 1998 19:06:47 +0000 (GMT) From: Ronnie Brown X-Sender: mas010@publix To: categories@mta.ca Subject: categories: higher order category theory and homotopy theory Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: There is a more down to earth and historically rooted approach. A concern of the early topologists (Dehn, Hopf, Alexandroff,...) was to find a higher dimensional version of the fundamental group, since the were irritated by the facts that (i) \pi_1 gave good geometric information (e.g. for complex variable theory) which often involved the non commutativity, (ii) H_1 was \pi_1 made abelian, for connected spaces, (iii) H_n existed for all n \ge 0 (and was abelian). There was also the intuition of big things made out of composing little bits (simplices in the Poincare formulation). The `solution' was to use formal sums (homology), and this gave abelian results. Cech's 1932 submission on Higher homotopy groups to the Zurich ICM was withdrawn at the request of Alexandroff and Hopf once they had proved these groups abelian for n \ge 2. The whole aim was to get immediate geometric results and computations. My suggestion in 1967 was to use `higher homotopy groupoids' since these did not have to be abelian. It was only in 1974 that Philip Higgins and I found a definition of a homotopy double groupoid of a pair; with this and the work with Chris Spencer we could prove a 2-D Van Kampen theorem. This is still giving actual computations of homotopy 2-types and invariants not previously known. For an exposition see my notes Groupoids and crossed objects in algebraic topology from the 1997 School in Algebraic Topology at Grenoble http://www.bangor.ac.uk/~mas010/brownpr.html which also go on briefly to the higher dimensional case (for strict higher groupoids) parts of which were announced in 1977 (with Philip Higgins, Compte Rendue Acad. Sci. Paris S\'er. A. 285 (1977) 997-999, 286 (1978) 91-93. Also, with the use of n-fold groupoids one can recover all n-types (Loday) a result Grothendieck described to me in 1986 (in Montpellier) as `absolutely beautiful'. Again, the intricate structure of these can give new computations of homotopy invariants and homotopy types, while studying pushouts of these (strict again!) one can find new algebraic constructions, such as a non abelian tensor product of groups (which act on each other). We have to develop new methods to compute even with these strict objects, so there is still a lot to be done in getting explicit information on specific examples using the non strict ones. One overall setting is to have functors \Pi ----------> (topological data) <--------- (Algebraic data) IB | | U | | B | | V V ======== (spaces)====== where IB (\mathhbb{B}) and B are classifying data or space functors, U is forget, and \Pi is a functor which should satisfy Van Kampen Theorem (preserve certain colimits), and also \Pi IB is equivalent to 1. At the level of 3-types we have such a set up with topological data = squares of (pointed) spaces, algebraic data = crossed squares (a kind of triple groupoid). We do not (so far) have such a setup when (algebraic data) is say the candidate proposed by Andre (which are equivalent to 2-crossed modules). Graham Ellis has shown how to compute 3-types and homotopy classes of maps using the above set up (Math. Z., 461 (1993) 93-110. ). The existence of the functor \Pi (satisfying VKT, so one can apply free algebraic objects) helps a lot. So I find it interesting that even strict 3-fold groupoids have an intricate structure (they model 3-types!). The relations and uses of all these models needs lots of work. The tensor product of crossed complexes (corresponding to a Gray tensor product of strict \infinity groupoids) yields interesting models for loop spaces, by considering monoids for this tensor: (see for example Baues and Tonks On the twisted cobar construction. Mathematical Proceedings of the Cambridge Philosophical Society 121 (1997) pp.229-245 ). But the nice thing about crossed complexes is one can write down explicitly the Eilenberg-Zilber theorem (Tonks' thesis), which I have not seen for infinity groupoids! Ronnie Brown From cat-dist Sun Nov 22 01:01:47 1998 Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id AAA15425 for categories-list; Sun, 22 Nov 1998 00:02:59 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Date: Fri, 20 Nov 1998 17:40:38 -0500 (EST) From: James Stasheff To: "categories@mta.ca" Subject: categories: Re: Jim's query In-Reply-To: <3655805B.38C188D2@bangor.ac.uk> Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: Amazing what deep interpretations have been given to my shallow question though i appreciate the answers. Al I was after was some place for a novice reader to see how category theory and homtopy theory interacted in the concept of A_\infty-category cf. Smirnov or Fukaya .oooO Jim Stasheff jds@math.unc.edu (UNC) Math-UNC (919)-962-9607 \ ( Chapel Hill NC FAX:(919)-962-2568 \*) 27599-3250 http://www.math.unc.edu/Faculty/jds From cat-dist Sun Nov 22 01:01:57 1998 Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id AAA19995 for categories-list; Sun, 22 Nov 1998 00:01:52 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Date: Fri, 20 Nov 1998 15:53:55 -0500 (EST) From: Michael Barr To: Categories list Subject: categories: staragain.dvi Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: A paper titled *-Autonomous categories: once more around the track has just been posted: ftp.math.mcgill.ca/pub/barr/staragain.dvi Basically, it redoes nearly all of the original Lecture Notes volume in about one fifth the space, using the Chu construction and proving a very general theorem. The only example from the original monograph that is not convered by this theorem is the category of Banach balls, the subject of a recent paper by Kleisli and me. That is also thee under the name balls.dvi (if I have remembered it correctly). This paper is to be submitted to tac for the Lambekfestschrift. From cat-dist Sun Nov 22 01:04:05 1998 Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id XAA16062 for categories-list; Sat, 21 Nov 1998 23:58:38 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Date: Fri, 20 Nov 1998 18:58:35 GMT Message-Id: Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" X-Mailer: Eudora F1.5.4 To: categories@mta.ca From: carlos@picard.ups-tlse.fr (Carlos Simpson) Subject: categories: Joyal's message + query Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: A. Joyal writes: >In particular, the new theory should help understanding >homotopy groups of spheres. I think it is safe to say that the ``n-category crowd'' (myself included) would love to make some progress on this. The main thing I am wondering about is: what exactly does one want to know about the homotopy groups of spheres? for example, is there a concrete question which needs answering? Up until not too long ago, I was under the impression that the question was how to calculate them; but it turns out that E. Brown in 1956 gave a perfectly good algorithm for calculating homotopy groups; another was subsequently given by Kan and refined by Curtis: one just has to calculate the ``nonabelian homology groups'' of a simplicial complex of free groups, and by Curtis one can divide out by a higher commutator subgroup so this actually only concerns a simplicial complex of nilpotent groups. (I give another algorithm in q-alg/9710011 using n-categorical type ideas, but this doesn't necessarily seem particularly useful and in any case I have more recently learned that it is basically the same thing as the ``Milgram model'' see the Handbook, see also an article by Baues for the double loop space etc.etc....) One might complain that these are ``algorithms'' rather than ``formulae'' but I have never quite understood the distinction. One generally needs a computer algorithm to evaluate one of Zagier's formulas! Maybe there is a subtle type of distinction about the type of machine the algorithm is supposed to run on? Or a quantitative question about the asymptotics of the computing time? Another possible version of the question (related to the computing time question in the previous sentence) might be: does one know how to bound the size of the $\pi _i (S^k)$? I can imagine applying n-categorical ideas to give a bound here, but it seems to be that there must be known bounds using spectral sequences or other. If there are known bounds, does someone know what the best one is? Other than that, does Prof. Joyal or anyone else have an idea of the type of question about the $\pi _i(S^k)$ which could/should be attacked by n-categories? -------------------- On an introduction to $n$-categories: it would be great if J. Baez could bundle together his ``This week's finds in math. phys.'' which concern $n$-categories. This would make a really nice introduction specially for friends and family. For the more mathematically minded, a recommended ``bapteme de feu'' would be Grothendieck's ``Pursuing stacks''. Another good place to look is Benabou's LNM 47. ---- On A_{\infty}-categories and \infty-categories: the response of R. Street looks quite adequate to the question (and in fact is a nice illustration of what the ``Gray tensor product'' really means, a point I had never clearly understood up to now...). Here are a few more remarks, specially related to J. Baez's comment. The notion of Segal category (which, again I recently found out, first appeared in the paper of Dwyer-Kan-Smith...) corresponds to the notion (that J. Baez mentionned) of \infty-category in which the $i$-morphisms are invertible for $i>1$ (I call this condition ``$1$-groupic''). A first remark is that Segal categories are equivalent to strict simplicially enriched categories, as was already shown by Dwyer-Kan-Smith. (in particular, you can read ``simplicially enriched category'' wherever I write ``1-groupic $\infty$-category'' below...). This suggests that A_{\infty}-categories are also probably equivalent to strict i.e. DG categories, but it isn't a proof (see below for why not) and it would be interesting to know if someone has an answer to that question. Segal categories or $1$-groupic $\infty$-categories, are not actually totally equivalent to A_{\infty} categories, because they don't encode the ``spectrum'' structure on Hom sets. In fact the notion of $A_{\infty}$-category in the DG world probably has a topological analogue where the Hom's are indeed spectra (cf Vogt et al on E_{\infty} ring spaces and the like???). Here is a proposition for an $\infty$-categorical notion which should correspond to this placing of a spectra structure on the Hom spaces. Suppose $C$ is a 1-groupic $\infty$-category. We say that $C$ is {\em spectric} if it satisfies the following conditions: ---C has an initial and final object and these coincide (call them *); ---if $X\in C$ then the fiber product $\Omega X := * \times _{X} *$ exists (as a homotopy-limit) in $C$; (also that $\Omega$ becomes a functor...?) ---the functor $\Omega$ is an equivalence of categories. These conditions are inspired by M. Hovey's definition of ``stable model category'' in ``Model categories''. (In particular, if M is a stable model category then the simplicial category, Dwyer-Kan localization of M inverting wk equivs L(M), will be a spectric $\infty$-category.) It follows from this definition that the $Hom$ spaces in C (i.e. the $\infty$-groupoids which are the Hom's of the $\infty$-category C) have structures of spectra: Hom(x,y)=\Omega Hom(x, \Omega ^{-1}y) = \Omega Hom(\Omega x, y) etc. Thus, C will correspond to the version of A_{\infty} category mentionned above where the Homs are spectra. On the other hand, given an A_{\infty}-category B, formally adjoin the shifts of objects, by setting for example Hom(X, Y[n]):= \Omega^nHom(X,Y) (for integers n) etc. It looks likely that this will give a spectric $\infty$-category C, and that these two constructions will be (as much as possible) a correspondence between A_{\infty}-categories and spectric $\infty$-categories. .... Sorry to clog up the e-waves so... Carlos Simpson From cat-dist Sun Nov 22 01:05:22 1998 Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id AAA13274 for categories-list; Sun, 22 Nov 1998 00:04:40 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Message-ID: <3655F5F5.4C15@mpce.mq.edu.au> Date: Sat, 21 Nov 1998 10:06:29 +1100 From: Michael Batanin Organization: Maquarie University, Sydney X-Mailer: Mozilla 3.01Gold (Win95; I) MIME-Version: 1.0 To: categories@mta.ca Subject: categories: Re: query References: <199811190226.NAA02248@macadam.mpce.mq.edu.au> Content-Type: text/plain; charset=koi8-r Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: This is a summary of my correspondence with J.Stasheff. James Stasheff wrote: > As monoids can be described as categories with one object, > one can consider \Aoo structures on categories with a notion of > homotopy, e.g. topological categories or differential graded categories. > To be more precise, the set of objects and the set of morphisms > carry a notion of homotopy. As usual, one deals with composable > morphisms and then weakens the axiom of associativity up to homotopy > in the strong sense in order to > define \Aoo - categories. This was first done by Smirnov by 1987 > \cite{smirnov:baku} to handle functorial homology operations and > their dependence on choices (cf. indeterminacy). More recently, Fukaya > \cite{fukaya:1} reinvented \Aoo -categories with remarkable applications to > Morse theory and Floer homology. > Inspired by this work, Nest and Tsygan have proposed an \Aoo -category > with automorphisms of an associative algebra as objects and for > the space of morphisms, a twisted version of the Hochschild complex > of the corresponding endomorphism algebras. Michael Batanin: One can generalize "ordinary" category theory in the different ways. One can consider internal category theory, enriched category theory. We can also consider a category as a special sort of simplicial set. All this points of view have their own A_{\infty}-analogues. I realize, that the approach of Smirnov, Fukaya and others is a generalization of "internal" category theory. In my paper "Monoidal globular categories as a natural environment ..."(Adv.Math. 136, 39-103 (1998)) I also consider a Cat-internal version of A_{\infty}-\omega-category (so it involves a weak form of interchange law)that I call monoidal globular category. A surprising coherence theorem sais that a general monoidal globular category is equivalent to a strict one (the internal category structure on objects aloows to strictify interchange low). In another my paper "Homotopy coherent category theory and A_{\infty}-structures in monoidal categories" (JPAA 123(1998),67-103) I defined an enriched version of A_{\infty}-category. So we have a honnest set of objects but morphisms are objects of a monoidal simplicial categories with a Quillen model structure. I also can define what A_{\infty} functor is and prove an appropriate coherence and homotopy invariance theorems. Another nice theorem sais that A_{\infty}-categories and their A_{-infty}-functors form an A_{infty}-category in a natural way. The simpliocial point of view on A_{\infty}-categories goes back to Boardman and Vogt book. The corresponding notion is a simplicial set satisfying some weak Kan conditions. This approach was extensively used by T.Porter and J.-M.Cordier (see T.Porter's answer on Jim's query). James Stasheff: > Since in a category we are concerned with $n$-tuples of morphisms > only when they are composable, it is appropriate to similarly relax > the composition operations for in defining an operad. the result is > known as a partial operad and appears in two different contexts: > in the mathematical physics of vertex operator algebras (VOAs) > \cite{yizhi} and Mivhael Batanin: In my work "Globular monoidal categories ..." I introduced the n-dimensional operads over trees. A 1-dimensional operad in this sense is not exatly the same as usual non-symmetric operad as every operation may have source and target and we can multiply just composable chains of operations. A one object version of this may be identify with a usual nonsymmetric operad. (In my paper I use \omega-operads). I wonder if a partial operad is the same as my 1-operad? Michael. From cat-dist Mon Nov 23 13:31:06 1998 Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id LAA13734 for categories-list; Mon, 23 Nov 1998 11:13:20 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Date: Fri, 20 Nov 1998 21:07:05 +0100 From: magne@ii.uib.no (Magne Haveraaen local) Message-Id: <199811202007.VAA09953@hemlock.uib.no> To: categories@mta.ca Subject: categories: Professorship in Programming Theory, University of Bergen X-Sun-Charset: US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: Department of Informatics, University of Bergen announces a position Professorship in Computer Science (Programming Theory) The successful applicant will be expected to join the research group in Foundations and Theory of Programming, whose current interests focus on: 1. formal design and development of software systems 2. basic theory of software systems, in particular, the algebraic and logical foundations 3. theory, design and implementation of specification and programming languages and tools. If no applicants are found qualified for a full professorship, the applications will be evaluated for a position of associate professor. An additional associate professorship is expected to be announced early 1999. Currently, the members of the group are: Magne Haveraaen - magne@ii.uib.no - http://www.ii.uib.no/~magne Khalid Mughal - khalid@ii.uib.no - http://www.ii.uib.no/~khalid Michal Walicki - michal@ii.uib.no - http://www.ii.uib.no/~michal Valentinas Kriauciukas - valis@ii.uib.no - http://www.ii.uib.no/~valis Any one of them may be contacted for more information concerning the research activity and the position. Details concerning the application procedure can be found at: http://www.ii.uib.no/gen/profpututl.html General information about the group, the department, the university and Bergen may be found at http://www.ii.uib.no/pt http://www.ii.uib.no/ http://www.uib.no/ http://www.uib.no/guide Notice that the deadline is unreasonably tight - 15.december 1998. We hope to extend this somewhat, but we are not sure this is possible. From cat-dist Mon Nov 23 13:42:35 1998 Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id LAA16527 for categories-list; Mon, 23 Nov 1998 11:13:45 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Date: Fri, 20 Nov 1998 21:06:59 +0100 (MET) Message-Id: <199811202006.VAA27751@brics.dk> From: Uffe Henrik Engberg To: categories@mta.ca Subject: categories: BRICS Int. PhD School: Call for Admission and Grant Applications Mime-Version: 1.0 (generated by tm-edit 7.106) Content-Type: text/plain; charset=US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: [Please accept our apologies if you receive this more than once] B R I C S International PhD School in Computer Science University of Aarhus Denmark Call for Admission and Grant Applications This is a call for admission and grant applications from students to BRICS International PhD School in Computer Science at University of Aarhus, Denmark. The call is aimed at students starting August 1999, with application deadline January 15th, 1999. BRICS International PhD School is an integrated part of the BRICS (Basic Research in Computer Science) Research Centre, and both are funded by the Danish National Research Foundation. The school admits 10-15 students (Danish and foreign) annually, and it provides a substantial number of student grants. The core areas of the PhD School are: Semantics of Computation; Logic in Computer Science; Computational Complexity; Design and Analysis of Algorithms; Programming Languages; Distributed Computing; Verification; and Data Security and Cryptology. The PhD school will provide its students with a solid background in the theoretical foundation of computer science, and centred around BRICS activities. From this foundation, the students may either continue in one of the core areas or venture into areas of a more applied or experimental nature as possible areas of thesis specialisation. The PhD School wishes to recruit PhD students of the highest international standards. It provides an excellent research environment and scientific training facilities, and aims at making its PhD graduates attractive for a wide spectrum of employers - in private and public research and development institutions, both in Denmark and abroad. So, if you are a student with at least four years of full-time study by the summer of 1999, highly motivated and well prepared for a PhD study in computer science within a truly international environment, please send us an application following the instructions below. For more details, please visit http://www.brics.dk, or contact us by e-mail at phdschool@brics.dk. Also, we would appreciate your passing on this information to interested students and colleagues at your university or research institute. BRICS The Research Centre BRICS (Basic Research In Computer Science) was founded in 1994 by the Danish National Research Foundation at the Universities of Aarhus and Aalborg. BRICS is a centre of basic research in Algorithmics, Logic and Semantics, and has a scientific staff of 30 (permanent and long term visitors), and around 100 short term visitors annually. Admission Prerequisites and Study Structure Admission is based on knowledge corresponding to four years of full-time studies, including basic courses in programming and programming languages, computer systems, algorithms and data structures, computability and mathematics (this list will be interpreted in a flexible way). The time allocated for the PhD studies is a further four years, where the first two years include some mandatory course work and introductory research, concluded by a qualifying examination, whereas the last two years are dedicated to the writing of the thesis, finishing with a defence. All students admitted to the school will enter this study structure. We may, however, take into account merits from previous study. Students are normally admitted for the semester start September 1st. --- but admission may take place throughout the year. PhD Student Grants The school offers student grants of different types: 1. tuition waiver and full studentship 2. tuition waiver and partial studentship 3. tuition waiver Only a limited amount of type 1. grants are available. How to apply 1. Fill in the application form on http://www.brics.dk/PhDSchool/Application.html Alternatively, send an e-mail to phdschool@brics.dk with subject "PhD Application" including: - your full name, personal address and phone number, college / university, URL of home page (if applicable), and e-mail address, - in case you apply for grants from the PhD School, the type of your application (see above - for details see http://www.brics.dk). 2. Send by ordinary mail to the address below - a covering letter, including the information from your application form/e-mail, - a short curriculum vitae, - complete official transcripts from colleges or universities, documenting minimally four years of full time study, - names of three people whom we may ask for letters of recommendation, - an indication of your motivation for a PhD study, and particular research area of initial interest (max two pages). Applications for admission August 1st, 1999 should be sent as soon as possible and before January 15th, 1999. Decisions will be announced in March, 1999. Postal Address BRICS International PhD School Department of Computer Science University of Aarhus Ny Munkegade, Bldg. 540 DK-8000 Aarhus C Denmark Phone: +45 8942 3264 Fax: +45 8942 3255 From cat-dist Mon Nov 23 14:19:22 1998 Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id MAA17932 for categories-list; Mon, 23 Nov 1998 12:01:44 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Date: Mon, 23 Nov 1998 16:45:58 +0100 From: Philippe Gaucher Message-Id: <199811231545.AA14182@irmast1.u-strasbg.fr> To: categories@mta.ca Subject: categories: about dimension of morphisms Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Content-Md5: Ew+TX2rFg2ed4CvwDWs2Ig== Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: Bonjour, I think that I have found a small pathological behaviour with dimension of morphisms in omega-categories. This is the following one. Let us first recall that min{m/s_m(x)=x} = min{n/t_n(x)=x} =: dim(x). Because x = s_m(x) = t_m s_m(x) (because m=s_0(u) (<> means different). Then u o_0 v is of dimension 1. Proof : Suppose that u o_0 v of dimension 0. Then s_0(u) = s_0(u o_0 v) = u o_0 v and t_0(v) = t_0(u o_0 v) = u o_0 v So s_0(u) = t_0(v) : false. Here is now the pathological behaviour : If s_0(u) = t_0(v), here is an example of category with u and v of dimension 1 and u o_0 v of dimension 0 : We take A={alpha,beta,u,v} (four elements) ------------------------------------- alpha beta u v ------------------------------------- s_0 alpha beta alpha beta ------------------------------------- t_0 alpha beta beta alpha ------------------------------------- and for o_0 : ------------------------------------- alpha beta u v ------------------------------------- alpha alpha xxxx u xxxx ------------------------------------- beta xxxx beta xxxx v ------------------------------------- u xxxx u xxxx alpha ------------------------------------- v v xxxx beta xxxx ------------------------------------- (at the intersection of row x and column y, read x o_0 y) All axioms of categories seem to be verified (?). The only composition is a "word" of finite length "... u (beta) v (alpha) u (beta) v..." In (A,s_0,t_0,o_0), u and v are of dimension 1 and u o_0 v of dimension 0 because u o_0 v = alpha. Where am I wrong ? (observe that this behaviour cannot appear with for example a category coming from a composable pasting scheme). pg. From cat-dist Tue Nov 24 12:12:36 1998 Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id KAA21250 for categories-list; Tue, 24 Nov 1998 10:13:17 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f From: john baez Message-Id: <199811240408.UAA19352@charity.ucr.edu> Subject: categories: HDA4: 2-Tangles To: categories@mta.ca (categories) Date: Mon, 23 Nov 1998 20:08:31 -0800 (PST) X-Mailer: ELM [version 2.4 PL24 PGP3 *ALPHA*] MIME-Version: 1.0 Content-Type: text/plain; charset=US-ASCII Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: The following preprint is now available at the places listed below. Comments and corrections are welcome! -------------------------------------------------------------------- Higher-Dimensional Algebra IV: 2-Tangles John C. Baez, Laurel Langford Just as knots and links can be algebraically described as certain morphisms in the category of tangles in 3 dimensions, compact surfaces smoothly embedded in R^4 can be described as certain 2-morphisms in the 2-category of `2-tangles in 4 dimensions'. Using the work of Carter, Rieger and Saito, we prove that this 2-category is the `free semistrict braided monoidal 2-category with duals on one unframed self-dual object'. By this universal property, any unframed self-dual object in a braided monoidal 2-category with duals determines an invariant of 2-tangles in 4 dimensions. --------------------------------------------------------------------- This paper is math.QA/981139 on the mathematics preprint server, so you can get it at: http://xxx.lanl.gov/abs/math.QA/9811139 It's also available as a Postscript file at my website: http://math.ucr.edu/home/baez/hda4.ps If you have any trouble, let me know and I can send you a copy. From cat-dist Tue Nov 24 12:20:15 1998 Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id KAA19160 for categories-list; Tue, 24 Nov 1998 10:15:08 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Date: Tue, 24 Nov 1998 14:35:47 +0100 From: Matthieu Amiguet Subject: categories: Continuity To: categories@mta.ca Message-id: <365AB632.F03C741@info.unine.ch> MIME-version: 1.0 X-Mailer: Mozilla 4.07 (Macintosh; I; 68K) Content-type: text/plain; x-mac-creator=4D4F5353; x-mac-type=54455854; charset=iso-8859-1 Content-transfer-encoding: 8BIT Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: Dear categoricians, I'm wondering about the possibility of speaking of trajectories in a category. That is, given a category C, in what sense - if any - can we consider a continuous function R->C (where R are the real numbers). What extra structure is needed on C to consider it as a topological space? what relation with the category structure? I guess there's a link with topos theory, but I can't make it really clear... Subsidiary question: in what case is there a notion of differentiability for the function f? Thank you for direct answers or pointers to (quite easyly understandable) literature. Regards, ---------------------------------- Matthieu Amiguet, doctorant Institut d'Informatique et d'Intelligence Artificielle Université de Neuchatel rue Emile Argand 11 CH - 2000 Neuchatel tel: +41 32 718 27 36 matthieu.amiguet@info.unine.ch ---------------------------------- From cat-dist Tue Nov 24 16:55:12 1998 Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id PAA02592 for categories-list; Tue, 24 Nov 1998 15:03:02 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f X-Sender: sjv@pop.doc.ic.ac.uk Message-Id: Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" Date: Tue, 24 Nov 1998 17:42:53 +0000 To: categories@mta.ca From: s.vickers@doc.ic.ac.uk (Steven Vickers) Subject: categories: Re: Continuity Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: >Dear categoricians, > >I'm wondering about the possibility of speaking of trajectories in a >category. That is, given a category C, in what sense - if any - can we >consider a continuous function R->C (where R are the real numbers). >What extra structure is needed on C to consider it as a topological >space? what relation with the category structure? >I guess there's a link with topos theory, but I can't make it really >clear... >Subsidiary question: in what case is there a notion of differentiability >for the function f? >Thank you for direct answers or pointers to (quite easyly >understandable) literature. >Regards, >---------------------------------- >Matthieu Amiguet, doctorant The key is indeed via toposes, specifically Grothendieck toposes. As has always been recognized, these are generalized topological spaces and the geometric morphisms are generalized continuous maps. Instead of a category C whose objects are XXX's, you replace it by a topos whose points are XXX's (i.e. the classifying topos for the theory of XXX's). For instance, the category of set would be replaced by the classifying topos for the theory of sets (i.e. what's described in Johnstone's book and elsewhere as the "object classifier"). I'll write S[set] for this classifying topos. This works provided the objects and morphisms of C are the models and homomorphisms of a geometric theory. Turning to topological spaces X, if X is sober then its points are also the models of a geometric theory (to do proper justice to this you need to take the localic approach to topology) and so there is a corresponding classifying topos SX which is actually the category of sheaves over X. The "continuous maps" between toposes are just the geometric morphisms. They transform points of the source topos to points of the target. Specific examples: 1. For two toposes SX and SY obtained from topological spaces, the geometric morphisms from SX to SY really do correspond to the continuous maps from X to Y. 2. Geometric morphisms from SX to S[set] are the sheaves on X, and these can indeed be thought of as continuous maps from X to the category of sets (each sheaf maps points of X to their stalks). By no means can every category C be considered the category of models of a geometric theory. One necessary condition (not sufficient) is for C to have all filtered colimits. For a small category C, one can make a corresponding topos S^C (the functor category from C to sets) that is an analogue of the ind completion (which freely adjoins filtered colimits). Its points are the flat presheaves over C and include (by the Yoneda embedding) a full and faithful image of C. A particular link with category structure lies in specialization. Any sober topological space has the specialization order on its points, and a topos has analogous specialization morphisms between points - they are just homomorphisms between models of the geometric theory. For a topos SX obtained from a space, this "specialization category" structure on the points is a poset, the original specialization order. Any geometric morphism is, when considered as a points transformer, functorial with respect to the specialization morphisms - this generalizes the topological fact that a continuous map is monotone with respect to the specialization order. Though the "generalized space" ideas are not at all new, they are often hidden expositionally by the idea of topos as generalized universe of sets. If you look at my paper "Topical Categories of Domains" (to appear in Maths Structures in Computer Science but at present available through my Web home page) you will find a careful attempt to discuss toposes directly as generalized spaces. The home page is http://theory.doc.ic.ac.uk:80/people/Vickers/ There is also a short paper "Strongly Algebraic = SFP (Topically)" that includes a summary in these purely spatial terms of established results about points of presheaf toposes. Regarding differentiability, I too would be interested to hear of answers or pointers. I guess it would most conveniently be discussed using Caratheodory's approach. If X is a topological ring, or a ring object in the category of toposes, then f: X -> X is C^1 iff there is some continuous phi: XxX -> X such that for all x, y. f(y) - f(x) = (y-x).phi(x,y) (The use of quantified "elements" of X can easily be translated into the equality of two geometric morphisms from XxX to X.) The derivative of f is f'(x) = phi(x,x). Steve Vickers. From cat-dist Tue Nov 24 21:33:11 1998 Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id UAA15638 for categories-list; Tue, 24 Nov 1998 20:24:50 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f From: john baez Message-Id: <199811242328.PAA02098@charity.ucr.edu> Subject: categories: n-categorical miscellany To: categories@mta.ca (categories) Date: Tue, 24 Nov 1998 15:28:09 -0800 (PST) X-Mailer: ELM [version 2.4 PL24 PGP3 *ALPHA*] MIME-Version: 1.0 Content-Type: text/plain; charset=US-ASCII Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: 1) Carlos Simpson writes: >A. Joyal writes: >>In particular, the new theory should help understanding >>homotopy groups of spheres. >I think it is safe to say that the ``n-category crowd'' (myself included) >would love to make some progress on this. The main thing I am wondering >about is: what exactly does one want to know about the homotopy groups of >spheres? for example, is there a concrete question which needs answering? There are a lot of complicated and interesting patterns in the stable homotopy groups of spheres. A lot of these patterns are currently studied using a generalized cohomology theory called complex cobordism theory, together with an intricate mess of spectral sequences: Complex cobordism and stable homotopy groups of spheres / Douglas C. Ravenel. Orlando: Academic Press, 1986. Nilpotence and periodicity in stable homotopy theory / by Douglas C. Ravenel. Princeton, N.J. : Princeton University Press, 1992. It's all far too technical for me to understand, so I don't have much hope of suddenly answering some question that homotopy theorists are stuck on! However, I think that the n-category crowd can give more conceptual explanations of some facts that appear as miracles in the current approach. For example, there's an important relation between complex cobordism theory and formal group laws: the complex cobordism of a point is "the universal formal group law". There should be an elegant conceptual proof of this, but as far as I know, there are just elegant *heuristics* for why it should be true, followed by a grungy computation. 2) Carlos Simpson writes: >On an introduction to n-categories: it would be great if J. Baez could >bundle together his ``This week's finds in math. phys.'' which concern >n-categories. This would make a really nice introduction specially for >friends and family. Especially during the holiday season! I should try to bundle them together more nicely at some point, but right now you can start at: http://math.ucr.edu/home/baez/week73.html , skip the bit about left-right asymmetry in amino acids, read the stuff on n-categories, and then keep clicking on the thing that says: "To continue reading the `Tale of n-Categories', click here." It fizzles out around week100, though I plan to resume it with an issue discussing Carlos Simpson's paper on the stabilization hypothesis and Mark Hovey's book on model categories. 3) Some people have noted that http://xxx.lanl.gov/abs/math.QA/9811139 doesn't give them the paper "HDA4 - 2-Tangles". Sorry! - the delay before the preprint server makes papers publicly available is longer than it used to be. It should be working by tomorrow. If you still have trouble after that (it's a big file and may bust your printer), feel free to email me and ask me to send you a copy. From cat-dist Thu Nov 26 17:34:43 1998 Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id QAA07473 for categories-list; Thu, 26 Nov 1998 16:14:21 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Subject: categories: commutative monoids To: categories@mta.ca Date: Thu, 26 Nov 1998 19:56:18 +0000 (GMT) X-Mailer: ELM [version 2.4 PL25] MIME-Version: 1.0 Content-Type: text/plain; charset=US-ASCII Content-Transfer-Encoding: 7bit Message-Id: From: Tom Leinster Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: A commutative monoid is a strict monoidal category with one object, so in this sense one can talk about a category enriched in a commutative monoid. Has anyone looked very hard at such things? Thanks, Tom