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 tac@mta.ca ---------------------------------------------------------------------- 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 -------------------------------------------------------------------- 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} --------------------------------------------------------------------- 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} ----------------------------------------------------------------------- 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} ------------------------------------------------------------------------ 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} -------------------------------------------------------------------------- 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} -------------------------------------------------------------------------- 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} -------------------------------------------------------------------------- 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} --------------------------------------------------------------------------- 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} --------------------------------------------------------------------------- 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}