nLab large cocompleteness

Idea

A locally small category cannot admit all large discrete colimits unless it is a preorder (by an argument of Freyd see complete small category). However, many nice locally small categories admit some large colimits.

This page is intended to act as a reference for classes of large colimits that are commonly encountered, and the class of categories that admit them.

On this page, categories will be assumed locally small unless stated otherwise.

Colimits versus limits

While this page focuses on categories with large colimit properties, these often imply strong completeness properties. For instance, compact categories (below) are complete, but not always cocomplete.

Cointersections of epimorphims

Many locally small categories admit small colimits and E-colimits for $E$ a class of epimorphisms, i.e. cointersections of morphisms in $E$.

Evidently, a small-cocomplete category that is co-well-powered will admit cointersections of all epimorphisms, though not all relevant examples are co-well-powered.

Strong epimorphisms

When $E$ is the class of strong epimorphisms, such a category is called well-cocomplete.

Regular epimorphisms

The free cocompletion of a locally small category $A$ under small colimits and cointersections of regular epimorphisms is the full subcategory of the presheaf category $[A^{op}, Set]$ on the weakly multirepresentable presheaves (also called petty presheaves). See Remark 4.39 of Lack & Tendas 2024 (such categories are called well-cocomplete in this paper, though conflicts with the earlier terminology).

Theorem 1 of Kelly & Koubek 1981 states that every functor $F \colon K \to A$, where $A$ has such colimits, admits a colimit if $F$ has a weakly terminal set.

Adjointness

Total category

A total category is a category whose Yoneda embedding admits a left adjoint. Every total category is compact in the sense below.

Compact categories

A category $A$ is compact in the sense of Isbell (1968) if every (large-)cocontinuous functor from $A$ has a right adjoint.

Beware that this is un-related to the notion of compact closed category.

Every compact category has small limits and intersections of monomorphisms, but not necessarily small colimits. A counterexample is mentioned in §3.15 of Börger et al..

A category $A$ is called strongly compact in Brandenburg if every small-[cocontinuous functor]] from $A$ has a right adjoint. Every strongly compact category is consequently compact.

All large colimits

Despite its ill behaviour, it is possible to describe the cocompletion of a locally small category $A$ under large colimits (assuming the law of excluded middle): it is given by the functor category $[A^{op}, 2]$ where $2$ is the interval category. In particular, this gives rise to a lax-idempotent pseudomonad whose unit components are (unusually) not representably fully faithful. (One might argue this disqualifies it from being a “cocompletion” in the traditional sense.) See the discussion following Example 26 of Walker.

Related pages

• total category

References

• John Isbell. Small subcategories and completeness. Mathematical systems theory 2 1 (1968) 27-50 [doi:10.1007/BF01691344]

• Max Kelly and V. Koubek. The large limits that all good categories admit. Journal of Pure and Applied Algebra 22.3 (1981): 253-263.

• R. Börger, Walter Tholen, M. B. Wischnewsky & H. Wolff. Compact and hypercomplete categories. Journal of Pure and Applied Algebra 21.2 (1981): 129-144.

• Max Kelly, A survey of totality for enriched and ordinary categories, Cahiers de Topologie et Géométrie Différentielle Catégoriques, 27 no. 2 (1986), p. 109-132, numdam

• Charles Walker?. “Distributive laws via admissibility.” Applied Categorical Structures 27.6 (2019): 567-617.

• Stephen Lack and Giacomo Tendas. Virtual concepts in the theory of accessible categories. Journal of Pure and Applied Algebra 227.2 (2023): 107196.

• Martin Brandenburg. Large limit sketches and topological space objects. arXiv preprint 2106.11115 (2021).