Following are a table of contents and abstracts of articles in Volume 3 of Theory and Applications of Categories. They are accessible on the Web from www.tac.mta.ca/tac/ or by ftp from ftp.tac.mta.ca/pub/tac/html/volumes/1997 Submission of articles to any of the Editors (who are listed after the abstracts) is invited. Please consult the Information for Authors at the Web or ftp sites. For subscription write to tac@mta.ca including a postal address. Robert Rosebrugh, Managing Editor http://www.tac.mta.ca/tac Theory and Applications of Categories tac@mta.ca Department of Mathematics and Computer Science 67 York Street ********NEW STREET ADDRESS******** Sackville, NB E4L 1E6 ********NEW POSTAL CODE*********** Canada +1-506-364-2530 (fax)+1-506-364-2645 ---------------------------------------------------------------------- ISSN 1201-561X THEORY AND APPLICATIONS OF CATEGORIES Volume 3 - 1997 Higher dimensional Peiffer elements in simplicial commutative algebras, Z. Arvasi and T. Porter, 1 Doctrines whose structure forms a fully faithful adjoint string, F. Marmolejo, 23 Note on a theorem of Putnam's, Michael Barr, 45 Lax operad actions and coherence for monoidal n-categories, A_{\infty} rings and modules, Gerald Dunn, 50 Proof theory for full intuitionistic linear logic, bilinear logic, and MIX categories, J.R.B. Cockett and R.A.G. Seely, 85 The reflectiveness of covering morphisms in algebra and geometry, George Janelidze and Max Kelly, 132 Crossed squares and 2-crossed modules of commutative algebras, Zekeriya Arvasi, 160 Monads and interpolads in bicategories, J"urgen Koslowski, 182 On property-like structures, G. M. Kelly and Stephen Lack, 213 Closed model categories for [n,m]-types, J. Ignacio Extremiana Aldana, Luis J. Hernandez Paricio, and M. Teresa Rivas Rodriguez, 251 Multilinearity of Sketches, David B. Benson, 269 ---------------------------------------------------------------------- ABSTRACTS: Higher Dimensional Peiffer Elements in Simplicial Commutative Algebras Z. Arvasi and T. Porter Let E be a simplicial commutative algebra such that E_n is generated by degenerate elements. It is shown that in this case the n^th term of the Moore complex of E is generated by images of certain pairings from lower dimensions. This is then used to give a description of the boundaries in dimension n-1 for n = 2, 3, and 4. ------------------------------------------------------------------------ Doctrines whose structure forms a fully faithful adjoint string F. Marmolejo We pursue the definition of a KZ-doctrine in terms of a fully faithful adjoint string Dd -| m -| dD. We give the definition in any Gray-category. The concept of algebra is given as an adjunction with invertible counit. We show that these doctrines are instances of more general pseudomonads. The algebras for a pseudomonad are defined in more familiar terms and shown to be the same as the ones defined as adjunctions when we start with a KZ-doctrine. ------------------------------------------------------------------------ Note on a theorem of Putnam's Michael Barr In a 1981 book, H. Putnam claimed that in a pure relational language without equality, for any model of a relation that was neither empty nor full, there was another model that satisfies the same first order sentences. Ed Keenan observed that this was false for finite models since equality is a definable predicate in such cases. This note shows that Putnam's claim is true for infinite models, although it requires a more sophisticated proof than the one outlined by Putnam. ------------------------------------------------------------------------ Lax Operad Actions and Coherence for Monoidal n-Categories, A_{\infty} Rings and Modules Gerald Dunn We establish a general coherence theorem for lax operad actions on an n-category which implies that an n-category with such an action is lax equivalent to one with a strict action. This includes familiar coherence results (e.g. for symmetric monoidal categories) and many new ones. In particular, any braided monoidal n-category is lax equivalent to a strict braided monoidal n-category. We also obtain coherence theorems for A_{\infty} and E_{\infty} rings and for lax modules over such rings. Using these results we give an extension of Morita equivalence to A_{\infty} rings and some applications to infinite loop spaces and algebraic K-theory. ------------------------------------------------------------------------ Proof theory for full intuitionistic linear logic, bilinear logic, and MIX categories J.R.B. Cockett and R.A.G. Seely This note applies techniques we have developed to study coherence in monoidal categories with two tensors, corresponding to the tensor-par fragment of linear logic, to several new situations, including Hyland and de Paiva's Full Intuitionistic Linear Logic (FILL), and Lambek's Bilinear Logic (BILL). Note that the latter is a noncommutative logic; we also consider the noncommutative version of FILL. The essential difference between FILL and BILL lies in requiring that a certain tensorial strength be an isomorphism. In any FILL category, it is possible to isolate a full subcategory of objects (the ``nucleus'') for which this transformation is an isomorphism. In addition, we define and study the appropriate categorical structure underlying the MIX rule. For all these structures, we do not restrict consideration to the ``pure'' logic as we allow non-logical axioms. We define the appropriate notion of proof nets for these logics, and use them to describe coherence results for the corresponding categorical structures. ------------------------------------------------------------------------ The reflectiveness of covering morphisms in algebra and geometry G. Janelidze and G. M. Kelly Each full reflective subcategory X of a finitely-complete category C gives rise to a factorization system (E, M) on C, where E consists of the morphisms of C inverted by the reflexion I : C --> X. Under a simplifying assumption which is satisfied in many practical examples, a morphism f : A --> B lies in M precisely when it is the pullback along the unit \etaB : B --> IB of its reflexion If : IA --> IB; whereupon f is said to be a trivial covering of B. Finally, the morphism f : A --> B is said to be a covering of B if, for some effective descent morphism p : E --> B, the pullback p^*f of f along p is a trivial covering of E. This is the absolute notion of covering; there is also a more general relative one, where some class \Theta of morphisms of C is given, and the class Cov(B) of coverings of B is a subclass -- or rather a subcategory -- of the category C \downarrow B \subset C/B whose objects are those f : A --> B with f in \Theta. Many questions in mathematics can be reduced to asking whether Cov(B) is reflective in C \downarrow B; and we give a number of disparate conditions, each sufficient for this to be so. In this way we recapture old results and establish new ones on the reflexion of local homeomorphisms into coverings, on the Galois theory of commutative rings, and on generalized central extensions of universal algebras. ------------------------------------------------------------------------ Crossed squares and 2-crossed modules of commutative algebras Zekeriya Arvasi In this paper, we construct a neat description of the passage from crossed squares of commutative algebras to 2-crossed modules analogous to that given by Conduche in the group case. We also give an analogue, for commutative algebra, of T. Porter's simplicial groups to n-cubes of groups which implies an inverse functor to Conduche's one. ------------------------------------------------------------------------ Monads and interpolads in bicategories Jurgen Koslowski Given a bicategory, 2, with stable local coequalizers, we construct a bicategory of monads Y-mnd by using lax functors from the generic 0-cell, 1-cell and 2-cell, respectively, into Y. Any lax functor into Y factors through Y-mnd and the 1-cells turn out to be the familiar bimodules. The locally ordered bicategory rel and its bicategory of monads both fail to be Cauchy-complete, but have a well-known Cauchy-completion in common. This prompts us to formulate a concept of Cauchy-completeness for bicategories that are not locally ordered and suggests a weakening of the notion of monad. For this purpose, we develop a calculus of general modules between unstructured endo-1-cells. These behave well with respect to composition, but in general fail to have identities. To overcome this problem, we do not need to impose the full structure of a monad on endo-1-cells. We show that associative coequalizing multiplications suffice and call the resulting structures interpolads. Together with structure-preserving i-modules these form a bicategory Y-int that is indeed Cauchy-complete, in our sense, and contains the bicategory of monads as a not necessarily full sub-bicategory. Interpolads over rel are idempotent relations, over the suspension of set they correspond to interpolative semi-groups, and over spn they lead to a notion of ``category without identities'' also known as ``taxonomy''. If Y locally has equalizers, then modules in general, and the bicategories Y-mnd and Y-int in particular, inherit the property of being closed with respect to 1-cell composition. ------------------------------------------------------------------------ On property-like structures G. M. Kelly and Stephen Lack A category may bear many monoidal structures, but (to within a unique isomorphism) only one structure of `category with finite products'. To capture such distinctions, we consider on a 2-category those 2-monads for which algebra structure is essentially unique if it exists, giving a precise mathematical definition of `essentially unique' and investigating its consequences. We call such 2-monads property-like. We further consider the more restricted class of fully property-like 2-monads, consisting of those property-like 2-monads for which all 2-cells between (even lax) algebra morphisms are algebra 2-cells. The consideration of lax morphisms leads us to a new characterization of those monads, studied by Kock and Zoberlein, for which `structure is adjoint to unit', and which we now call lax-idempotent 2-monads: both these and their colax-idempotent duals are fully property-like. We end by showing that (at least for finitary 2-monads) the classes of property-likes, fully property-likes, and lax-idempotents are each coreflective among all 2-monads. ------------------------------------------------------------------------ Closed model categories for [n,m] types J. Ignacio Extremiana Aldana, Luis J. Hernandez Paricio, Maria T. Rivas Rodriguez For m >= n > 0, a map f between pointed spaces is said to be a weak [n,m]-equivalence if f induces isomorphisms of the homotopy groups \pi_k for n <= k <= m~. Associated with this notion we give two different closed model category structures to the category of pointed spaces. Both structures have the same class of weak equivalences but different classes of fibrations and therefore of cofibrations. Using one of these structures, one obtains that the localized category is equivalent to the category of n-reduced CW-complexes with dimension less than or equal to m+1 and m-homotopy classes of cellular pointed maps. Using the other structure we see that the localized category is also equivalent to the homotopy category of (n-1)-connected (m+1)-coconnected CW-complexes. ------------------------------------------------------------------------ Multilinearity of Sketches David B. Benson We give a precise characterization for when the models of the tensor product of sketches are structurally isomorphic to the models of either sketch in the models of the other. For each base category K call the just mentioned property (sketch) K-multilinearity. Say that two sketches are K-compatible with respect to base category K just in case in each K-model, the limits for each limit specification in each sketch commute with the colimits for each colimit specification in the other sketch and all limits and colimits are pointwise. Two sketches are K-multilinear if and only if the two sketches are K-compatible. This property then extends to strong Colimits of sketches. We shall use the technically useful property of limited completeness and completeness of every category of models of sketches. That is, categories of sketch models have all limits commuting with the sketched colimits and and all colimits commuting with the sketched limits. Often used implicitly, the precise statement of this property and its proof appears here. ------------------------------------------------------------------------ Editors of Theory and Applications of Categories John Baez, University of California Riverside baez@math.ucr.edu Michael Barr, McGill University barr@math.mcgill.ca Lawrence Breen, Universite de Paris 13 breen@math.univ-paris13.fr Ronald Brown , University of North Wales r.brown@bangor.ac.uk Jean-Luc Brylinski, Pennsylvania State University jlb@math.psu.edu Aurelio Carboni, Universita della Calabria carboni@unical.it Peter T. Johnstone, University of Cambridge ptj@pmms.cam.ac.uk G. Max Kelly, University of Sydney kelly_m@maths.usyd.edu.au Anders Kock, University of Aarhus kock@mi.aau.dk F. William Lawvere, State University of New York at Buffalo wlawvere@acsu.buffalo.edu Jean-Louis Loday, Universite Louis Pasteur et CNRS, Strasbourg loday@math.u-strasbg.fr Ieke Moerdijk, University of Utrecht moerdijk@math.ruu.nl Susan Niefield , Union College niefiels@union.edu Robert Pare, Dalhousie University pare@mscs.dal.ca Andrew Pitts , University of Cambridge ap@cl.cam.ac.uk Robert Rosebrugh , Mount Allison University rrosebrugh@mta.ca Jiri Rosicky, Masaryk University rosicky@math.muni.cz James Stasheff , University of North Carolina jds@charlie.math.unc.edu Ross Street , Macquarie University street@macadam.mpce.mq.edu.au Walter Tholen , York University tholen@mathstat.yorku.ca Myles Tierney, Rutgers University tierney@math.rutgers.edu Robert F. C. Walters , University of Sydney walters_b@maths.usyd.edu.au R. J. Wood, Dalhousie University rjwood@mscs.dal.ca