nLab conservative cocompletion

Contents

Context

Category theory

Limits and colimits

Yoneda lemma

Contents

Idea

Given a category CC, we may construct the free cocompletion of CC, freely adding some class of colimits. Often, however, CC will already have some colimits, which we wish to preserve. A conservative cocompletion of a category CC is a cocompletion that preserves the colimits in CC.

For a small category CC with Φ\Phi-colimits, there is a simple description of the Φ\Phi-conservative cocompletion (for a class Φ\Phi of colimits). It is the the full subcategory [C ,Set] Φ[C^\circ, Set]_\Phi of the presheaf category on CC spanned by the functors sending Φ\Phi-colimits in CC to limits in the presheaf category.

For a large category, this description does not suffice in general, nor does it suffices to consider categories of small presheaves: in fact, there are locally small categories that do not admit locally small conservative cocompletions (see AV02) (however, they do admit conservative cocompletions that are large and not locally small).

Properties

  • For a small category CC, the conservative cocompletion Cont(C op,Set)Cont(C^op, Set) is complete and cocomplete, and the embedding CCont(C op,Set)C \to Cont(C^op, Set) creates limits and colimits. Consequently, every small category may be continuously and cocontinuously fully embedded into a complete and cocomplete locally small category. (Note that Cont(C op,Set)Cont(C^op, Set) will rarely be closed unless CC is (counterexample).) However, not every locally small category admits such an embedding: see Example III.1 of Trnková 1966.

References

  • Joachim Lambek, Completions of categories: Seminar lectures given 1966 in Zürich, Vol. 24. Springer.

  • Věra Trnková, Limits in categories and limit-preserving functors, Commentationes Mathematicae Universitatis Carolinae 7.1 (1966): 1-73.

  • J. F Kennison, On limit-preserving functors, Illinois Journal of Mathematics 12.4 (1968): 616-619.

  • Max Kelly, Basic Concepts of Enriched Category Theory, Cambridge University Press, Lecture Notes in Mathematics 64 (1982)

  • Jiřı́ Adámek and Jiřı́ Velebil?. A remark on conservative cocompletions of categories. Journal of Pure and Applied Algebra 168.1 (2002): 107-124.

See also Theorem 11.5 of:

  • Marcelo Fiore. Enrichment and representation theorems for categories of domains and continuous functions. University of Edinburgh.

See section 6.4 (and Theorem 6.4.3 in particular) of:

Last revised on April 25, 2024 at 12:32:43. See the history of this page for a list of all contributions to it.