For a locally presentable abelian category B with projective generator, we construct the derived and contraderived model structures on of complexes, proving in particular existence enough homotopy complexes objects. We also show that D(B) is generated, as triangulated coproducts, by generator B. Grothendieck A, injective coderived complexes. Assuming Vopěnka's principle, prove D(A) products, co...

Markov categories, having tensors with copying and discarding, provide a setting for categorical probability. This paper uses finite colimits what we call uniform states in such categories to define (fixed size) multiset functor, basic operations sums zips of multisets, graded monad structure. Multisets can be used represent both urns filled coloured balls also draws multiple from urns. The mai...

If A is a small category and E a Grothendieck topos, the Kan extension LanF of a flat functor F : A → E along any functor A → D preserves whatever finite limits may exist in D; this is a well known fundamental result in topos theory. We shall present a metamathematical argument to derive out of this some other left exactness results, for Kan extensions with values in a (possibly large) site E. ...

Abstract For a category $\mathcal {E}$ with finite limits and well-behaved countable coproducts, we construct model structure, called the effective on of simplicial objects in , generalising Kan–Quillen structure sets. We then prove that is left right proper satisfies descent sense Rezk. As consequence, obtain associated $\infty $ -category has limits, colimits satisfying descent, locally Carte...

This work discusses three aspects of net theory: compositionality of nets, analysis of nets and high-level nets. Net theory has often been criticised for the diiculty of giving a compositional semantics to a net. In this work we discuss this problem form a category theoretic point of view. In category theory compositionality is represented by colimits. We show how a high-level net can be mapped...

We study diagrams associated with a finite simplicial complex K, in various algebraic and topological categories. We relate their colimits to familiar structures in algebra, combinatorics, geometry and topology. These include: right-angled Artin and Coxeter groups (and their complex analogues, which we call circulation groups); Stanley-Reisner algebras and coalgebras; Davis and Januszkiewicz's ...

such that C and D are compact objects in S (an object X in S is compact if the representable functor Hom(X,−) preserves arbitrary coproducts). The concept of a coherent functor has been introduced explicitly for abelian categories by Auslander [1], but it is also implicit in the work of Freyd on stable homotopy [9]. In this paper we characterize coherent functors in a number of ways and use the...

This thesis provides a logical and mathematical foundation for object-oriented specification languages with a further modularisation unit between the system and object classes. The unit is denoted object-oriented module, or module for short, and initially described in an informal way. Modules offer a better approach to reusability and provide better structuring of large, complex and distributed...