* The Lambek Festschrift: Theory and Applications of Categories
@ 1999-12-01 15:45 Bob Rosebrugh
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From: Bob Rosebrugh @ 1999-12-01 15:45 UTC (permalink / raw)
To: Categories
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/
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*-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
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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
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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
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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
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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
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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
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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
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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
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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
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1999-12-01 15:45 The Lambek Festschrift: Theory and Applications of Categories Bob Rosebrugh
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