Date: Wed, 24 Apr 1996 09:39:27 -0300 (ADT) Subject: Symmetric Monoidal Closed Categories Date: Wed, 24 Apr 1996 09:20:57 +0100 From: Chris Francis Townsend I am just about to finish the writing up of my thesis and have one remaining difficulty. I needed to use the following result about a symmetric monoidal closed categorie C, `If C has coequalizers then so does the category CMon(C) of commutative monoids over C' I feel sure that this result is well known by anybody who has worked with symmetric monoidal closed categories and so have been frustrated by the fact that can't find a reference for it. Does anybody know where I can find such a reference for this result (or of some generalization of it)? Thanks Chris Townsend Date: Wed, 24 Apr 1996 15:20:28 -0300 (ADT) Subject: Re: Symmetric Monoidal Closed Categories Date: Wed, 24 Apr 96 17:17 BST From: Dr. P.T. Johnstone In my "Topos Theory" book, you will *very nearly* find a proof that, if C is symmetric monoidal closed, then the forgetful functor CMon(C) --> C creates coequalizers of reflexive pairs. (See Lemma 0.17 and Exercise 0.1; in the latter, the cartesian products have to be replaced by tensor products, but that is a triviality.) Given that all finite colimits can be constructed from coproducts and reflexive coequalizers (and that CMon(C) always has finite coproducts, for a different reason), that is all you need. Peter Johnstone Date: Tue, 8 Sep 1998 13:58:02 -0400 (EDT) From: Michael Barr Subject: categories: Chew on this Has anybody seen the following symmetric closed monoidal category? Let #A# be a category tripleable over Set. Let #V# be the comma category (F,#V#), F being the free functor. So an object (S,s,A) is an arrow s:FS --> A and a map (S,s,A) to (T,t,B) is a pair f: S --> T and g: A --> B making the obvious square commute. The closed structure (S,s,A) --o (T,t,B) is a certain arrow of the form (Hom((S,s,A),(T,t,B),?,B^S). The monoidal structure is fairly ugly, but it exists. Of course, an object (S,s,A) can also be thought of as an S-tuple of elements of A, by adjointness. Michael Date: Mon, 14 Sep 1998 01:55:11 +0100 (BST) From: Paul Taylor Subject: categories: Barr's symmetric monoidal closed comma category Mike Barr asked whether anyone had been aware that the comma category (F,A) or equivalently (S,U) is symmetric monoidal closed, where A ^ | F| -| |U | V S is a monoidal (tripleable) adjunction over S=Set. As he said, an object (S,s,A) is an arrow s:FS--> A (or, by adjointness, an S-tuple of elements of A) and a map (S,s,A) to (T,t,B) is a pair f: S --> T and g: A --> B making the obvious square commute. He said that the closed structure (S,s,A) --o (T,t,B) is a certain arrow of the form (Hom((S,s,A),(T,t,B),?,B^S). I have certainly seen comma categories like (S,U); they are referred to as gluing, sconing or the Freyd cover. (BTW Peter says that "scone" is a corruption of Sierpinski cone, so its correct pronunciation is presumably "shco:ne".) This construction provides an almost magical proof of strong normalisation, consistency and similar results for various fragments of symbolic or categorical logic. The algebraic theory that I have in mind here for Mike's monadic adjunction is that which describes the fragment of logic in question (the Australians would prefer us to talk about a 2-monad here). An early example of such a proof in the unification of these two traditions was given by Yves Lafont in an appendix to his thesis. He proved that the embedding of a category in the free CCC that it generates is full and faithful. In the course of this he described the exponential in (S,U), which is what Mike's formula gives, though I didn't notice (and apparently nor did Yves) that the A-object which we have is the internal version of Mike's Hom-set. Yves Lafont's result appears as part of a generic account of the gluing construction in Section 7.7 of my forthcoming book. Paul Subject: categories: Re: Barr's symmetric monoidal closed comma category Date: Tue, 15 Sep 1998 13:42:59 +0900 From: HASEGAWA Masahito Related to Mike and Paul's messages to categories, I had the following observation in my recent draft paper (http://www.kurims.kyoto-u.ac.jp/~hassei/papers/basic.ps.gz "Logical predicates for intuitionistic linear logic", Lemma A.1): Suppose that C and D are symmetric monoidal closed categories and that G:C->D is a symmetric monoidal functor. Moreover suppose that D has pullbacks. Then the comma category (D,G) can be given a symmetric monoidal closed structure, so that the obvious projection (D,G)->C is strict symmetric monoidal closed. I used this (together with the free symmetric monoidal cocompletion) for deriving "logical predicates (logical relations)" for intuitionistic linear type theories, thus in a similar way as Lafont's use of glueing for typed lambda calculi and ccc. In this paper I also have another lemma for glueing symmetric monoidal adjunctions, for interpreting the modality !. I have been wondering if this is a sort of folklore, but never found a reference. Best Regards, Masahito. Masahito Hasegawa Research Institute for Mathematical Sciences Kyoto University Kyoto 606-8502 Japan MAIL: hassei@kurims.kyoto-u.ac.jp URL: http://www.kurims.kyoto-u.ac.jp/~hassei Date: Sat, 19 Sep 1998 11:10:26 -0400 (EDT) From: F W Lawvere Subject: categories: Symmetric monoidal closed comma categories This concerns the discussion by Mike Barr, Paul Taylor, and Masahito Hasegawa regarding symmetric monoidal closed comma categories: The construction (later called 'comma') in the category of categories was introduced in 1963 primarily for foundational simplification (though it was clear that certain particular cases, such as slice, were already in direct use). Besides the 2-categorical equational description of adjointness, one needs the description in terms of a bijection between arrows, but that does not require the complicated assumption that there exists a category of sets in which two given categories can be enriched. Namely, an adjunction between two given categories can be described by giving a third 'adjunction category', related by appropriate functors to them, which is isomorphic to two differently-constructed 'comma' categories. It seems that there are many cases in which this third category is of interest in itself, whether or not one of the two given categories is or is not monadic or comonadic over the other. Emilio Faro's notes from my Fall 1990 Buffalo course Categories of Space and of Quantity, mention essentially the result cited by Masahito Hasegawa. If an adjunction involves monoidal functors, then the adjunction category tends to be a monoidal closed category. This remark was essentially intended to supply semantically-based examples of closed categories which have one aspect which is linear (in the straightforward sense that coproducts equal products) and an opposite aspect which is cartesian (in the sense that the tensor is the categorical product). Of course, the most immediate subclass of examples, based on the data of a rig in a cartesian closed category, involve monadic adjunctions. On the other hand, several published papers on related matters axiomatically assume comonadic adjunctions. However, the simple algebraic stance, as Masahito Hasegawa points out, is that both aspects, as well as the relation between them, are all regarded as equally given. As part of the logic (= natural structure) of the resulting situation there will be unary (= modal?) operators reflected on each aspect by composing. A further step is to investigate to what extent the data can be approximated via data which is reconstructed on the basis of only one aspect or the other using this additional reflected structure. Both that step a la M. Stone, as well as the simple algebraic stance in the spirit of Chu and G. Mackey, are of course involved in the full study of any related pair of aspects (e.g. algebra and geometry). A problem from topology, where related considerations may help, concerns the operation of collapsing a connected subspace to a point (the effect of this operation on relative homology is part of the content of algebraic topology). In extending this operation to apply to not-necessarily-connected subspaces (and more generally, from inclusion maps to arbitrary maps), collapsing all these to a point would be an unnecessarily discontinuous functor. Rather, within the category whose objects are continuous maps, consider the subcategory wherein the domains of these structural maps are discrete (or zero-dimensional, if that is different in the model of continuity being considered). That subcategory is reflective (with the help of pushout) in case the model admits a left adjoint connected-components functor. In the case of a subspace, the reflector collapses each of its components to a distinct point in the new ambient space, and the lifted unit of the adjunction is epimorphic if the original one (to the connected components pi zero) is, even where the subspace is empty. I am wondering: under what conditions are these categories and functors cartesian monoidal closed? Indeed these things are probably folklore, but listed below are some references containing partial indications. Bill Functorial Semantics of Algebraic Theories Thesis Columbia University (1963) The Category of Categories as a Foundation for Mathematics, Proceedings of La Jolla Conference, Springer-Verlag (1966) 1 - 20 Categories of Space and of Quantity, Buffalo Course Notes by Emilio Faro (1990) ******************************************************************************* F. William Lawvere Mathematics Dept. SUNY wlawvere@acsu.buffalo.edu 106 Diefendorf Hall 716-829-2144 ext. 117 Buffalo, N.Y. 14214, USA ******************************************************************************* Date: Fri, 25 Sep 1998 15:55:51 +0300 (EET DST) From: Mamuka Jibladze Subject: categories: Re: Chew on this One more comment on that fascinating ugly monoidal structure. Many years ago D. Pataraia as a student was asked to realise tensor product of vector spaces (V and W over k) as a colimit. He then came up with a diagram (sorry for still more ugly notation) k_{v,w} / \ / \ |_ _| V_w W_v That is, vertices of the diagram consist of U(W) copies of V, U(V) copies of W, and U(V)xU(W) copies of k (U is the forgetful functor to sets). And the maps... well, you guess. The reason this is relevant is that in the Barr's monoidal category, the product of (S->U(A)) and (T->U(B)) is (SxT->U(C)) where C is the colimit, in the category of algebras, of F(1)_{s,t} / \ / \ |_ _| A_t B_s It does not look so ugly after all, does it? :), Mamuka