The structure theorem of Joyal, Street and Verity says that every traced monoidal category C arises as a monoidal full subcategory of the tortile monoidal category IntC. In this paper we focus on a simple observation that a traced monoidal category C is closed if and only if the canonical inclusion from C into IntC has a right adjoint. Thus, every traced monoidal closed category arises as a mon...

Motivated by a model for syntactic control of interference, we introduce a general categorical concept of bireeectivity. Bireeective subcategories of a category A are subcategories with left and right adjoint equal, subject to a coherence condition. We characterize them in terms of split-idempotent natural transformations on id A. In the special case that A is a presheaf category, we characteri...

We consider a nitely branching transition system as a coalgebra for an endofunctor on the category Set of small sets. A map in that category is a functional bisimulation. So, we study the structure of the category of nitely branching transition systems and functional bisimulations by proving general results about the category H-Coalg of H-coalgebras for an endofunctor H on Set. We give conditio...

This paper introduces axioms for an abstract trace on a monoidal category. This trace can be interpreted in various contexts where it could alternatively be called contraction, feedback, Markov trace or braid closure. Each full submonoidal category of a tortile (or ribbon) monoidal category admits a canonical trace. We prove the structure theorem that every traced monoidal category arises in th...

The most familiar example of higher, or vertically iterated enrichment is that in the definition of strict n-category. We begin with strict n-categories based on a general symmetric monoidal category V. Motivation is offered through a comparison of the classical and extended versions of topological quantum field theory. A sequence of categorical types that filter the category of monoidal catego...

In this paper, it is shown that the category of stratified $L$-generalized convergence spaces is monoidal closed if the underlying truth-value table $L$ is a complete residuated lattice. In particular, if the underlying truth-value table $L$ is a complete Heyting Algebra, the Cartesian closedness of the category is recaptured by our result.

It is shown that for every monoidal bi-closed category C left and right (semi)dualization by means of the unit object not only defines a pair of adjoint functors, but that these functors are monoidal as functors from C, the dual monoidal category of C into the transposed monoidal category C. We, thus, generalize the case of a symmetric monoidal category, where this kind of dualization is a spec...

We investigate the abelian category which is the target of intersection homology. Recall that, given a stratified space X, we get intersection homology groups IHnX depending on the choice of an n-perversity p. The n-perversities form a lattice, and we can think of IHnX as a functor from this lattice to abelian groups, or more generally R-modules. Such perverse R-modules form a closed symmetric ...

An action ∗ : V × A−→ A of a monoidal category V on a category A corresponds to a strong monoidal functor F : V−→ [A,A] into the monoidal category of endofunctors of A. In many practical cases, the ordinary functor f : V−→ [A,A] underlying the monoidal F has a right adjoint g; and when this is so, F itself has a right adjoint G as a monoidal functor—so that, passing to the categories of monoids...

Operads were originally defined as V-operads, that is, enriched in a symmetric or braided monoidal category V. The symmetry or braiding in V is required in order to describe the associativity axiom the operads must obey, as well as the associativity that must be a property of the action of an operad on any of its algebras. A sequence of categorical types that filter the category of monoidal cat...