Subject: Theory and Applications of Categories: Abstracts, Volume 1

From: categories <[email-masked]>
Date: Fri, 19 Jan 1996 11:22:13 -0400 (AST)

Date: Thu, 18 Jan 1996 23:36:32 -0400 (AST)
From: Bob Rosebrugh <[email-masked]>

The table of contents and abstracts for Volume 1 (1995) of Theory and 
Applications of Categories follows. Access via the WWW is at
http://www.tac.mta.ca/tac/
For subscriptions write to [email-masked]

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                                                        ISSN 1201-561X

             THEORY AND APPLICATIONS OF CATEGORIES 

                    Volume 1, 1995

Oriented singular homology 
Michael Barr                                                            1

Functorial and algebraic properties of Brown's P functor
Luis-Javier Hern\'andez-Paricio                                         10

On finite induced crossed modules, and the homotopy 2-type 
of mapping cones
Ronald Brown and Christopher D. Wensley                                 54

Kan extensions along promonoidal functors 
Brian Day and Ross Street                                               72

Symmetric monoidal categories model all connective spectra
R. W. Thomason                                                          78

Distributive adjoint strings
Robert Rosebrugh and R. J. Wood                                         119

A forbidden-suborder characterization binarily-composable
diagrams in double categories
Robert Dawson                                                           146

Categorical data-specifications
Frank Piessens and Eric Steegmans                                       156

On the size of categories
Peter Freyd and Ross Street                                             174

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Oriented Singular Homology
Michael Barr

We formulate three slightly different notions of oriented singular
chain complexes and show that all three are naturally homotopic to
ordinary singular chain complexes.

Theory and Applications of Categories, Vol 1, No. 1, 1-9.
ftp://ftp.tac.mta.ca/pub/tac/html/volumes/1995/n1/v1n1.{dvi,ps}

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Functorial and algebraic properties of Browns P functor
Luis-Javier Hernandez-Paricio

In 1975 E. M. Brown constructed a functor $\cal P$ which carries 
the tower of fundamental groups of the end of a (nice) space to 
the Brown-Grossman fundamental group. In this work, we study this 
functor and its extensions and analogues defined for pro-sets, 
pro-pointed sets, pro-groups and pro-abelian groups. The new
versions of the $\cal P$ functor are provided with more algebraic 
structure. Examples given in the paper prove that in general the 
$\cal P$ functors  are not faithful, however, one of our main 
results establishes that the restrictions of the corresponding 
$\cal P$ functors to the full subcategories of towers are faithful. 
We also prove that the restrictions of the $\cal P$ functors to 
the corresponding full subcategories of finitely generated towers 
are also full. Consequently, in these cases, the towers of objects 
in the categories of sets, pointed sets, groups and abelian groups, 
can be replaced by adequate algebraic models ($M$-sets, $M$-pointed 
sets, near-modules and modules.) The article also contains the 
construction of left adjoints for the $\cal P$ functors. 

Theory and Applications of Categories, Vol. 1, No. 2, 10-53
ftp://ftp.tac.mta.ca/pub/tac/html/volumes/1995/n2/v1n2.{dvi,ps}     

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On finite induced crossed modules and the homotopy 2-type of mapping cones,
Ronald Brown and Christopher D. Wensley

Results on the finiteness of induced crossed 
modules are proved both algebraically and topologically.  
Using the Van Kampen type theorem for the fundamental crossed module,
applications are given to the 2-types of mapping cones of 
classifying spaces of groups. Calculations of the cohomology 
classes of some finite crossed  modules are given, 
using crossed complex methods. 

Theory and Applications of Categories, Vol. 1, No. 3, 54-71
ftp://ftp.tac.mta.ca/pub/tac/html/volumes/1995/n3/v1n3.{dvi,ps}

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Kan extensions along promonoidal functors
Brian Day and Ross Street

Strong promonoidal functors are defined. Left Kan extension (also 
called "existential quantification") along a strong promonoidal 
functor is shown to be a strong monoidal functor. A construction for the 
free monoidal category on a promonoidal category is provided. A 
Fourier-like transform of presheaves is defined and shown to take 
convolution product to cartesian product. 

Theory and Applications of Categories, Vol. 1, No. 4, 72-78
ftp://ftp.tac.mta.ca/pub/tac/html/volumes/1995/n4/v1n4.{dvi,ps}

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Symmetric monoidal categories model all connective spectra 
R. W. Thomason

The classical infinite loopspace machines in fact
induce an equivalence of categories between a localization
of the category of symmetric monoidal categories and the
stable homotopy category of -1-connective spectra.

Theory and Applications of Categories, Vol. 1, No. 4, 79-118
ftp://ftp.tac.mta.ca/pub/tac/html/volumes/1995/n5/v1n5.{dvi,ps}

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Distributive Adjoint Strings
R. Rosebrugh and R. J. Wood

For an adjoint string V -| W -| X -| Y : B --> C, with Y fully faithful,
it is frequently, but not always, the case that the composite
VY underlies an idempotent monad. When it does, we call the string
distributive. We also study shorter and longer `distributive' adjoint 
strings and how to generate them. These provide a new construction of 
the simplicial 2-category, Delta.

Theory and Applications of Categories, Vol. 1, No. 6, 119-145
ftp://ftp.tac.mta.ca/pub/tac/html/volumes/1995/n6/v1n6.{dvi,ps}

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A forbidden-suborder characterization of binarily-composable 
diagrams in double categories, Robert Dawson

Tilings of rectangles with rectangles, and tileorders (the associated
double order structures) are useful as ``templates'' for composition in
double categories. In this context, it is particularly relevant to ask
which tilings may be joined together, two rectangles at a time, to form
one large rectangle. We characterize such tilings via forbidden suborders,
in a manner analogous to Kuratowski's characterization of planar graphs. 

Theory and Applications of Categories, Vol. 1, No. 7, 146-155
ftp://ftp.tac.mta.ca/pub/tac/html/volumes/1995/n7/v1n7.{dvi,ps}

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Categorical Data-Specifications

F. Piessens and E. Steegmans

We introduce MD-sketches, which are a particular kind of Finite Sum sketches.
Two interesting results about MD-sketches are proved. First, we show that,
given two MD-sketches, it is algorithmically decidable whether their 
model categories are equivalent. Next we show that data-specifications,
as used in database-design and software engineering, can be translated
to MD-sketches. As a corollary, we obtain that equivalence of 
data-specifications is decidable. 

Theory and Applications of Categories, Vol. 1, No. 8, 156-173
ftp://ftp.tac.mta.ca/pub/tac/html/volumes/1995/n8/v1n8.{dvi,ps}

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On the Size of Categories
Peter Freyd and Ross Street

The purpose is to give a simple proof that a category is equivalent 
to a small category if and only if both it and its presheaf category 
are locally small. 

Theory and Applications of Categories, Vol. 1, 1995, No. 9, 174-178
ftp://ftp.tac.mta.ca/pub/tac/html/volumes/1995/n9/v1n9.{dvi,ps}