Abstract. We introduce the category of set-theoretic representations of a matched pair of groupoids. This is a monoidal category endowed with a monoidal functor fl to the category of quivers over the common base of the groupoids in the matched pair. We study monoidal functors between two such categories of representations which preserve the functor fl. We show that the centralizer of such a mon...

Just as knots and links can be algebraically described as certain morphisms in the category of tangles in 3 dimensions, compact surfaces smoothly embedded in R4 can be described as certain 2-morphisms in the 2-category of ‘2-tangles in 4 dimensions’. Using the work of Carter, Rieger and Saito, we prove that this 2-category is the ‘free semistrict braided monoidal 2-category with duals on one un...

It is well-known that every monoidal category is equivalent to a strict one. We show that for categories of sets with additional structure (which we define quite formally below) it is not even necessary to change the category: The same category has a different (but isomorphic) tensor product, with which it is a strict monoidal category. The result applies to ordinary (bi)modules, where it shows...

Just as knots and links can be algebraically described as certain morphisms in the category of tangles in 3 dimensions, compact surfaces smoothly embedded in R4 can be described as certain 2-morphisms in the 2-category of ‘2-tangles in 4 dimensions’. Using the work of Carter, Rieger and Saito, we prove that this 2-category is the ‘free semistrict braided monoidal 2-category with duals on one un...

The notion of proof-net category defined in this paper is closely related to graphs implicit in proof nets for the multiplicative fragment without constant propositions of linear logic. Analogous graphs occur in Kelly’s and Mac Lane’s coherence theorem for symmetric monoidal closed categories. A coherence theorem with respect to these graphs is proved for proof-net categories. Such a coherence ...

We introduce the notion of strong concatenable process for Petri nets as the least refinement of non-sequential (concatenable) processes which can be expressed abstractly by means of a functor Q[ ] from the category of Petri nets to an appropriate category of symmetric strict monoidal categories with free non-commutative monoids of objects, in the precise sense that, for each net N , the strong...

The Reshetikhin-Turaev invariant, Turaev's TQFT, and many related constructions rely on the encoding of certain ribbon tangles (n-string links, or n-ribbon handles) as n-forms on the coend of a ribbon category. We introduce the monoidal category of Hopf diagrams, and describe a universal encoding of ribbon string links as Hopf diagrams. This universal encoding is an injective monoidal functor a...