Concurrent computation can be given an abstract mathematical treatment very similar to that provided for sequential computation by domain theory and denotational semantics of Scott and Strachey. A simple domain theory for concurrency is presented. Based on a categorical model of linear logic and associated comonads, it highlights the role of linearity in concurrent computation. Two choices of c...

In this paper we characterize those accessible V-categories that have limits of a specified class. We do by introducing the notion companion C for class weights Ψ, as collection special types colimit diagrams are compatible with Ψ. then Ψ-limits accessibly embedded and C-virtually reflective in presheaf V-category, C-models sketches. This allows us to recover standard theorems locally presentab...

In the model presented in [1, 4] a type is interpreted by a nominal set A equipped with two “face” operations: if u : A and i is a symbol we can form u(i = 0) : A(i = 0) and u(i = 1) : A(i = 1) elements independent of i. The unit interval is represented by the nominal set I, whose elements are 0, 1 and the symbols. Alternatively, we have a presentation as a presheaf category over the category w...

We prove that the class of numerable open covers topological spaces is smallest contains with pairwise disjoint elements and two elements, closed under composition coarsening covers. apply this result to establish an analogue Brown–Gersten property for spaces: a simplicial presheaf on site satisfies homotopy descent all if only it elements. also strengthening these results manifolds, ensuring c...

We give a generalized version of the Freyd conjecture and a way to think about a possible proof. The essential point is to describe an elementary formal reduction of the question that holds in any triangulated category. There are no new results, but at least one known example drops out very easily. In algebraic topology, the generating hypothesis, or Freyd conjecture, is a long-standing conject...

For a monad S on a categoryK whose Kleisli category is a quantaloid, we introduce the notion of modularity, in such a way that morphisms in the Kleisli category may be regarded as V(bi)modules (= profunctors, distributors), for some quantale V . The assignment S // V is shown to belong to a global adjunction which, in the opposite direction, associates with every (commutative, unital) quantale ...

(1) (restriction) if U ⊂ V ⊂ W , then ρWU = ρV U ◦ ρWV (2) (gluing) if {Vi} is an open covering of U , and we have si ∈ F(Vi) such that for all i, j, ρVi,Vi∩Vj (si) = ρVj ,Vi∩Vj (sj), then there exists a unique s ∈ F(U) such that ρU,Vi(s) = si for all i. If one omits the gluing condition from the above definition, one has a presheaf of abelian groups. Elements of F(U) are called sections of F o...

In this paper we introduce the sheaf of stratified Whitney jets of Gevrey order on the subanalytic site relative to a real analytic manifold X. Then we define stratified ultradistributions of Beurling and Roumieu type on X. In the end, by means of stratified ultradistributions, we define tempered-stratified ultradistributions and we prove two results. First, if X is a real surface, the tempered...

The concept of generalised species of structures between small categories and, correspondingly, that of generalised analytic functor between presheaf categories are introduced. An operation of substitution for generalised species, which is the counterpart to the composition of generalised analytic functors, is also put forward. These definitions encompass most notions of combinatorial species c...

For a small quantaloid Q, a Q-closure space is a small category enriched in Q equipped with a closure operator on its presheaf category. We investigate Q-closure spaces systematically with specific attention paid to their morphisms and, as preordered fuzzy sets are a special kind of quantaloid-enriched categories, in particular fuzzy closure spaces on fuzzy sets are introduced as an example. By...