nLab conservative cocompletion

Idea

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

For a small category $C$ with $\Phi$-colimits, there is a simple description of the $\Phi$-conservative cocompletion (for a class $\Phi$ of colimits). It is the the full subcategory $[C^\circ, Set]_\Phi$ of the presheaf category on $C$ spanned by the functors sending $\Phi$-colimits in $C$ 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 $C$, the conservative cocompletion $Cont(C^op, Set)$ is complete and cocomplete, and the embedding $C \to Cont(C^op, Set)$ 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 $Cont(C^op, Set)$ will rarely be closed unless $C$ is (counterexample).) However, not every locally small category admits such an embedding: see Example III.1 of Trnková 1966.

Examples

• The category of finite semilattices is the conservative cocompletion of the category of finite sets and relations under finite colimits (and also its free cocompletion under reflexive coequalisers) (AMMU14).
• The category Pos of partial orders is the free conservative cocompletion of the category of total orders (Tataru24).

Related concepts

• free cocompletion

References

• Joachim Lambek, Completions of categories: Seminar lectures given 1966 in Zürich, Lecture Notes in Mathematics 24 (1966), Springer. doi, ISBN: 978-3-540-03607-4 (softcover), 978-3-540-34840-5 (electronic).

• 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). TAC reprint.

• 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:

• Michael Makkai, Robert Paré, Accessible categories: The foundations of categorical model theory, Contemporary Mathematics 104. American Mathematical Society, Rhode Island (1989) (ISBN:978-0-8218-7692-3)

For examples:

• Adámek, J., Myers, R. S., Urbat, H., & Milius, S. “On continuous nondeterminism and state minimality.” Electronic Notes in Theoretical Computer Science 308 (2014): 3-23.

• Mimram, Samuel, and Cinzia Di Giusto. “A categorical theory of patches.” Electronic notes in theoretical computer science 298 (2013): 283-307.

• Calin Tataru, “Partial orders are the free conservative cocompletion of total orders.” arXiv:2404.12924 (2024).