Lex colimits

Abstract

Many kinds of categorical structure require the existence of finite limits, of colimits of some specified type, and of "exactness" conditions between the finite limits and the specified colimits. Some examples are the notions of regular, or Barr-exact, or lextensive, or coherent, or adhesive category. We introduce a general notion of exactness, of which each of the structures listed above, and others besides, are particular instances. The notion can be understood as a form of cocompleteness "in the lex world" -- more precisely, in the 2-category of finitely complete categories and finite-limit preserving functors.Comment: 38 pages; v2: final journal version, various minor changes and new Section 5.9 dealing with filtered colimit

Fundamental group functors in descent-exact homological categories

“…It turns out that in a Barr-exact category C , every regular epimorphism f : E → B is effective for descent, which means that the pullback functor p * : (C ↓ B) → (C ↓ E) is monadic. This can be viewed as a form of exactness condition on a category, that we call here descent-exactness (see [24] for a general notion of exactness). Thus, the kind of categories we consider are pointed + protoadditive + regular homological + descent-exact where, in presence of the other axioms, the protomodularity [6] condition can be equivalently expressed by saying that the split short five lemma holds.…”

Two-dimensional regularity and exactness

“…We obtain this canonical choice from the theory of lex colimits developed by the second author and Lack in [12]. This is a framework for dealing with V-categorical structures involving limits, colimits, and exactness between the two; one of the key insights is that, for a given class of colimits, the appropriate exactness conditions to impose are just those which hold between finite limits and the given colimits in the base V-category V; more generally, in any "V-topos" (lex-reflective subcategory of a presheaf V-category).…”

“…However, the exactness conditions required do not simply amount to stability under pullback of codescent morphisms; one must also impose the extra condition that, if A → B is a codescent morphism, then so also is the diagonal map A → A × B A. This condition, forced by the general theory of [12], has not been noted previously and is moreover, substantive: for example, the category Set, seen as a locally discrete 2-category, satisfies all the other prerequisites for regularity in this sense, but not this final condition. Finally, the regularity notion associated with the factorisation system (iii) appears to be new, although an abelian version of it is considered in [14].…”

“…The theory of kernel-quotient systems was first investigated by Street in unpublished work [24], and developed further in a preprint of Betti and Schumacher [3]; a published account of some of their work may be found in [10]. As indicated above, the new element we bring is the use of the ideas of [12] to justify the exactness conditions appearing in the notions of F-regularity and F-exactness.…”

“…In fact, we define three notions of regularity and exactness, each based on one of the three canonical ways of factorising a functor in Cat: as (surjective on objects, injective on objects and fully faithful), as (bijective on objects, fully faithful), and as (bijective on objects and full, faithful). The correctness of our notions is justified using the theory of lex colimits [12] introduced by Lack and the second author. Along the way, we develop an abstract theory of regularity and exactness relative to a kernel-quotient factorisation, extending earlier work of Street and others [24,3].…”