The Editors of Theory and Applications of Categories are pleased to announce the publication of a special volume dedicated to Joachim Lambek in honour of his 75'th birthday. In addition to the articles abstracted below, the volume includes an Introduction by the guest Editors and a brief biographical essay presented by Michael Barr to the conference held at McGill University on December 5, 1997 in celebration of the same event. The Editors of TAC wish to thank Michael Barr, Philip Scott and Robert Seely who acted as guest editors for this special volume. Abstracts of the articles follow. The journal may be viewed from www.tac.mta.ca/tac/ ------------------------------------------------------------------------- *-Autonomous categories: once more around the track Michael Barr This represents a new and more comprehensive approach to the *-autonomous categories constructed in the monograph [Barr, 1979]. The main tool in the new approach is the Chu construction. The main conclusion is that the category of separated extensional Chu objects for certain kinds of equational categories is equivalent to two usually distinct subcategories of the categories of uniform algebras of those categories. Theory and Applications of Categories, Vol. 6, 1999, No. 1, pp 5-24 http://www.tac.mta.ca/tac/volumes/6/n1/n1.dvi http://www.tac.mta.ca/tac/volumes/6/n1/n1.ps ftp://ftp.tac.mta.ca/pub/tac/html/volumes/6/n1/n1.dvi ftp://ftp.tac.mta.ca/pub/tac/html/volumes/6/n1/n1.ps ------------------------------------------------------------------------- A bicategorical approach to static modules Renato Betti The purpose of this paper is to indicate some bicategorical properties of ring theory. In this interaction, static modules are analyzed. Theory and Applications of Categories, Vol. 6, 1999, No. 2, pp 25-32 http://www.tac.mta.ca/tac/volumes/6/n2/n2.dvi http://www.tac.mta.ca/tac/volumes/6/n2/n2.ps ftp://ftp.tac.mta.ca/pub/tac/html/volumes/6/n2/n2.dvi ftp://ftp.tac.mta.ca/pub/tac/html/volumes/6/n2/n2.ps ------------------------------------------------------------------------- The categorical theory of self-similarity Peter Hines We demonstrate how the identity $N\otimes N \cong N$ in a monoidal category allows us to construct a functor from the full subcategory generated by $N$ and $\otimes$ to the endomorphism monoid of the object $N$. This provides a categorical foundation for one-object analogues of the symmetric monoidal categories used by J.-Y. Girard in his Geometry of Interaction series of papers, and explicitly described in terms of inverse semigroup theory in [6,11]. This functor also allows the construction of one-object analogues of other categorical structures. We give the example of one-object analogues of the categorical trace, and compact closedness. Finally, we demonstrate how the categorical theory of self-similarity can be related to the algebraic theory (as presented in [11]), and Girard's dynamical algebra, by considering one-object analogues of projections and inclusions. Theory and Applications of Categories, Vol. 6, 1999, No. 3, pp 33-46 http://www.tac.mta.ca/tac/volumes/6/n3/n3.dvi http://www.tac.mta.ca/tac/volumes/6/n3/n3.ps ftp://ftp.tac.mta.ca/pub/tac/html/volumes/6/n3/n3.dvi ftp://ftp.tac.mta.ca/pub/tac/html/volumes/6/n3/n3.ps ------------------------------------------------------------------------- A Note on Rewriting Theory for Uniqueness of Iteration M. Okada and P. J. Scott Uniqueness for higher type term constructors in lambda calculi (e.g. surjective pairing for product types, or uniqueness of iterators on the natural numbers) is easily expressed using universally quantified conditional equations. We use a technique of Lambek [18] involving Mal'cev operators to equationally express uniqueness of iteration (more generally, higher-order primitive recursion) in a simply typed lambda calculus, essentially Godel's T [29,13]. We prove the following facts about typed lambda calculus with uniqueness for primitive recursors: (i) It is undecidable, (ii) Church-Rosser fails, although ground Church-Rosser holds, (iii) strong normalization (termination) is still valid. This entails the undecidability of the coherence problem for cartesian closed categories with strong natural numbers objects, as well as providing a natural example of the following computational paradigm: a non-CR, ground CR, undecidable, terminating rewriting system. Theory and Applications of Categories, Vol. 6, 1999, No. 4, pp 47-64 http://www.tac.mta.ca/tac/volumes/6/n3/n3.dvi http://www.tac.mta.ca/tac/volumes/6/n3/n3.ps ftp://ftp.tac.mta.ca/pub/tac/html/volumes/6/n3/n3.dvi ftp://ftp.tac.mta.ca/pub/tac/html/volumes/6/n3/n3.ps ------------------------------------------------------------------------- Contravariant Functors on Finite Sets and Stirling Numbers Robert Pare Contravariant Functors on Finite Sets and Stirling Numbers We characterize the numerical functions which arise as the cardinalities of contravariant functors on finite sets, as those which have a series expansion in terms of Stirling functions. We give a procedure for calculating the coefficients in such series and a concrete test for determining whether a function is of this type. A number of examples are considered. Theory and Applications of Categories, Vol. 6, 1999, No. 5, pp 65-76 http://www.tac.mta.ca/tac/volumes/6/n5/n5.dvi http://www.tac.mta.ca/tac/volumes/6/n5/n5.ps ftp://ftp.tac.mta.ca/pub/tac/html/volumes/6/n5/n5.dvi ftp://ftp.tac.mta.ca/pub/tac/html/volumes/6/n5/n5.ps ------------------------------------------------------------------------- Comparing coequalizer and exact completions M. C. Pedicchio and J. Rosicky We characterize when the coequalizer and the exact completion of a category $\cal C$ with finite sums and weak finite limits coincide. Theory and Applications of Categories, Vol. 6, 1999, No. 6, pp 77-82 http://www.tac.mta.ca/tac/volumes/6/n6/n6.dvi http://www.tac.mta.ca/tac/volumes/6/n6/n6.ps ftp://ftp.tac.mta.ca/pub/tac/html/volumes/6/n6/n6.dvi ftp://ftp.tac.mta.ca/pub/tac/html/volumes/6/n6/n6.ps ------------------------------------------------------------------------- Enriched Lawvere theories John Power We define the notion of enriched Lawvere theory, for enrichment over a monoidal biclosed category $V$ that is locally finitely presentable as a closed category. We prove that the category of enriched Lawvere theories is equivalent to the category of finitary monads on $V$. Moreover, the $V$-category of models of a Lawvere $V$-theory is equivalent to the $V$-category of algebras for the corresponding $V$-monad. This all extends routinely to local presentability with respect to any regular cardinal. We finally consider the special case where $V$ is $Cat$, and explain how the correspondence extends to pseudo maps of algebras. Theory and Applications of Categories, Vol. 6, 1999, No. 7, pp 83-93 http://www.tac.mta.ca/tac/volumes/6/n7/n7.dvi http://www.tac.mta.ca/tac/volumes/6/n7/n7.ps ftp://ftp.tac.mta.ca/pub/tac/html/volumes/6/n7/n7.dvi ftp://ftp.tac.mta.ca/pub/tac/html/volumes/6/n7/n7.ps ------------------------------------------------------------------------- Epimorphic regular contexts Robert Raphael A von Neumann regular extension of a semiprime ring naturally defines a epimorphic extension in the category of rings. These are studied, and four natural examples are considered, two in commutative ring theory, and two in rings of continuous functions. Theory and Applications of Categories, Vol. 6, 1999, No. 8, pp 94-104 http://www.tac.mta.ca/tac/volumes/6/n8/n8.dvi http://www.tac.mta.ca/tac/volumes/6/n8/n8.ps ftp://ftp.tac.mta.ca/pub/tac/html/volumes/6/n8/n8.dvi ftp://ftp.tac.mta.ca/pub/tac/html/volumes/6/n8/n8.ps ------------------------------------------------------------------------- Natural deduction and coherence for non-symmetric linearly distributive categories Robert R. Schneck In this paper certain proof-theoretic techniques of [BCST] are applied to non-symmetric linearly distributive categories, corresponding to non-commutative negation-free multiplicative linear logic (mLL). First, the correctness criterion for the two-sided proof nets developed in [BCST] is adjusted to work in the non-commutative setting. Second, these proof nets are used to represent morphisms in a (non-symmetric) linearly distributive category; a notion of proof-net equivalence is developed which permits a considerable sharpening of the previous coherence results concerning these categories, including a decision procedure for the equality of maps when there is a certain restriction on the units. In particular a decision procedure is obtained for the equivalence of proofs in non-commutative negation-free mLL without non-logical axioms. Theory and Applications of Categories, Vol. 6, 1999, No. 9, pp 105-146 http://www.tac.mta.ca/tac/volumes/6/n9/n9.dvi http://www.tac.mta.ca/tac/volumes/6/n9/n9.ps ftp://ftp.tac.mta.ca/pub/tac/html/volumes/6/n9/n9.dvi ftp://ftp.tac.mta.ca/pub/tac/html/volumes/6/n9/n9.ps