The importance of accessible categories has been widely recognized; they can be described as those freely generated in some precise sense by a small set objects and, because that, satisfy many good properties. More specifically finitely characterized as: (a) free cocompletions under filtered colimits, and (b) flat presheaves on category. equivalence between is what makes the theory so general f...

Homotopy theorists have traditionally studied topological spaces and used methods which rely heavily on computation, construction, and induction. Proofs often go by constructing some horrendously complicated object (usually via a tower of increasingly complicated objects) and then proving inductively that we can understand what’s going on at each step and that in the limit these steps do what i...

It is shown that the “constant sheaves” functor ∇ : Sets → Eff does not preserve 1-filtered colimits, and that as a consequence of this, the full subcategory of Eff on the countable projective objects is not dense. © 2005 Elsevier B.V. All rights reserved.

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 ...

Categorical structures and their pseudomaps rarely form locally presentable 2-categories in the sense of Cat-enriched category theory. However, we show that if categorical structure question is sufficiently weak (such as monoidal, but not strict categories) then 2-category accessible. Furthermore, explore flexible limits such possess interaction with filtered colimits.

In this paper, we prove that the polygraphic homology of a small category, defined in terms resolutions category ?Cat strict ?- categories, is naturally isomorphic to its nerve, thereby extending result Lafont and Métayer. Along way, investigate homotopy colimits with respect Folk model structure deduce theorem which formally resembles Quillen's Theorem A.

We show that a derivator is stable if and only homotopy finite limits colimits commute, limit functors have right adjoints, colimit left adjoints. These characterizations generalize to an abstract notion of "stability relative class functors", which includes in particular pointedness, semiadditivity, ordinary stability. To prove them, we develop the theory derivators enriched over monoidal weig...