André Joyal

Une théorie combinatoire des séries formelles

Abstract This paper presents a combinatorial theory of formal power series. The combinatorial interpretation of formal power series is based on the concept of species of structures. A categorical approach is used to formulate it. A new proof of Cayleys formula for the number of labelled trees is given as well as a new combinatorial proof (due to G. Labelle) of Lagranges inversion formula. Polyas enumeration theory of isomorphism classes of structures is entirely renewed. Recursive methods for computing cycle index polynomials are described. A combinatorial version of the implicit function theorem is stated and proved. The paper ends with general considerations on the use of coalgebras in combinatorics.

The geometry of tensor calculus. I

This paper defines and proves the correctness of the appropriate string diagrams for various kinds of monoidal categories with duals. Mathematics Subject Classifications (1991). 18D10, 52B11, 53A45 , 57M25, 68Q10, 82B23.

Traced monoidal categories

Traced monoidal categories are introduced, a structure theorem is proved for them, and an example is provided where the structure theorem has application.

Bisimulation from Open Maps

An abstract definition of bisimulation is presented. It makes possible a uniform definition of bisimulation across a range of different models for parallel computation presented as categories. As examples, transition systems, synchronisation trees, transition systems with independence (an abstraction from Petri nets), and labelled event structures are considered. On transition systems the abstract definition readily specialises to Milners strong bisimulation. On event structures it explains and leads to a strengthening of the history-preserving bisimulation of Rabinovitch and Traktenbrot and van Glabeek and Goltz. A tie-up with open maps in a (pre)topos, as they appear in the work of Joyal and Moerdijk, brings to light a new model, presheaves on categories of pomsets, into which the usual category of labelled event structures embeds fully and faithfully. As an indication of its promise, this new presheaf model has “refinement” operators. The general approach yields a logic, generalising Hennessy?Milner logic, which is characteristic for the generalised notion of bisimulation.

Quasi-categories and Kan complexes

A quasi-category X is a simplicial set satisfying the restricted Kan conditions of Boardman and Vogt. It has an associated homotopy category hoX. We show that X is a Kan complex iff hoX is a groupoid. The result plays an important role in the theory of quasi-categories (in preparation). Here we make an application to the theory of initial objects in quasi-categories. We briefly discuss the notions of limits and colimits in quasi-categories.

Tortile Yang-Baxter operators in tensor categories

Abstract The free tortile tensor category T ∼ on one generating object is shown here, by purely algebraic techniques, to also be the free tensor category containing an object equipped with a tortile Yang-Baxter operator. Shum has a geometric characterization of T ∼ as the category whose arrows are tangled ribbons. This alternative universal property can be used to construct representations of T ∼ .

Bisimulation and open maps

An abstract definition of bisimulation is presented. It allows a uniform definition of bisimulation across a range of different models for parallel computation presented as categories. As examples, transition systems, synchronization trees, transition systems with independence (an abstraction from Petri nets), and labeled event structures are considered. On transition systems, the abstract definition readily specialises to Milners (1989) strong bisimulation. On event structures, it explains and leads to a revision of the history-preserving bisimulation of Rabinovitch and Traktenbrot (1988), and Goltz and van Glabeek (1989). A tie-up with open maps in a (pre)topos brings to light a promising new model, presheaves on categories of pomsets, into which the usual category of labeled event structures embeds fully and faithfully. As an indication of its promise, this new presheaf model has refinement operators, though further work is required to justify their appropriateness and understand their relation to previous attempts.<<ETX>>

A completeness theorem for open maps

This paper provides a partial solution to the completeness problem for Joyal’s axiomatization of open and etale maps, under the additional assumption that a collection axiom (related to the set-theoretical axiom with the same name) holds. In many categories of geometric objects, there are natural classes of open maps, of proper maps and of etale maps. Some of the forma1 properties of these classes in the category of schemes were isolated by Grothendieck [S]. In the late 70’s, an attempt was made by the first author to give a full axiomatization of classes of open (and of etale) maps in topoi: along with a list of axioms, he suggested some concrete candidates for “universal” classes of open (etale) maps in path-topoi such as T2 and Y’ which could be used to test the completeness of the axioms. In 1988 E. Dubuc proved a completeness theorem for these axioms for etale maps relative to the Sierpinski topos Y2, under some special assumptions on the site; see [2]. In this paper we prove a completeness theorem for any class R of open maps (in any topos b) which satisfies an additional axiom called the Collection Axiom. More precisely, for any topos Y-, the class of morphisms in Y2 which are quasi-pullback squares (in T) satisfies the axioms for open maps as well as this Collection Axiom. Given d and R, we construct a suitable topos LT and a geometric morphism cp : F2 + 8, with the property that an arrowfin I belongs to the class R iff its inverse image q*(f) is a quasi-pullback. We also prove a completeness theorem of the following form: for a topos d and a class R of open maps satisfying the Collection Axiom, there exists a suitable geometric realization functor from d into a category of locales, with the property that a mapfin d belongs to R iff its geometric realization is an open (in the usual sense) map of locales. Analogous results hold for pretopoi and exact categories. Furthermore, we show that these completeness theorems yield similar theorems for classes of etale maps, thereby improving Dubuc’s result.

Feynman Graphs, and Nerve Theorem for Compact Symmetric Multicategories (Extended Abstract)

We describe a category of Feynman graphs and show how it relates to compact symmetric multicategories (coloured modular operads) just as linear orders relate to categories and rooted trees relate to multicategories. More specifically we obtain the following nerve theorem: compact symmetric multicategories can be characterised as presheaves on the category of Feynman graphs subject to a Segal condition. This text is a write-up of the second-named authors QPL6 talk; a more detailed account of this material will appear elsewhere [Andre Joyal and Joachim Kock. Manuscript in preparation].

Coherence completions of categories

Abstract This is the first of a series of papers on coherence completions of categories. Here we show that there is a close connection between Girards coherence spaces and free bicomplete categories. We introduce a new construction for creating models of linear logic, the coherence completion of a category. By presenting coherence completions as categories enriched over the category of pointed sets and the category of coherence spaces, the free structures on coherence completions are obtained in a very natural way. We show that if C is monoidal closed or ★-autonomous then so is its coherence completion. We also prove that if C is a model of linear logic then so is its coherence completion. A key idea of the paper which is introduced into linear logic is the notion of softness. We hope that this idea could be of use in solving the full completeness for larger fragments of linear logic.