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).

Definition

If $\Phi$ is a class of colimiting cocones in a category $C$, then we say that a $\Phi$-colimit-preserving functor $j:C\to D$ exhibits a cocomplete category $D$ as a $\Phi$-conservative cocompletion if, for any cocomplete category $E$, composition with $j$ induces an equivalence between the category of $\Phi$-colimit-preserving functors $C\to E$ and the category of small-colimit-preserving functors $D\to E$, and this equivalence is natural in $E$.

Then the conservative cocompletion (with $\Phi$ unspecified) is the $\Phi$-conservative cocompletion for $\Phi$ the class of all colimiting cocones in $C$.

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.

• For a small category $C$, the $\Phi$-conservative cocompletion of $C$ is a reflective subcategory of the presheaf category $[C^op,Set]$, since this is an instance of the orthogonal subcategory problem: $Cont_\Phi(C^op,Set)$ comprises those presheaves that are orthogonal to the canonical natural transformation $colim_i C(-,a_i)\to C(-,\colim_i a_i)$ for every diagram $(a_i)$ in $\Phi$.

• If $C$ is small and symmetric monoidal and $(c\otimes -)$ preserves $\Phi$-colimits, then one can define a symmetric monoidal closed structure on $Cont_\Phi(C^op, Set)$ such that the Yoneda embedding $C\to Cont_\Phi(C^op, Set)$ is strong monoidal. In particular if $C$ is already cartesian closed or symmetric monoidal closed, so too is $Cont_\Phi(C^op, Set)$, since if $(c\otimes -)$ is a left adjoint then it necessarily preserves all colimits. Moreover, the Yoneda embedding preserves this structure. Note that this is not the Day convolution, but a reflection of it into $Cont_\Phi(C^op, Set)$.

• Let $\Psi$ be a sound doctrine of limits. Suppose that $C$ is small and has all $\Psi^{op}$-colimits, then let $\Phi$ be the class of $\Psi^{op}$ colimiting cocones in $C$. The $\Phi$-conservative cocompletion of $C$ is also the free $\Psi$-filtered colimit completion of $C$ (see Theorem 5.5 of ABLR02).

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).

• A small category $C$ with a class of limit cones $\Phi$ is also called a ‘realized limit sketch’. Then the category of models of the sketch is the $\Phi^op$-conservative cocompletion of $C^op$.

• A locally finitely presentable category is the $\Phi$-conservative cocompletion of its full subcategory of compact objects, where $\Phi$ is the class of finite colimit cones. More generally a locally $\kappa$-presentable category is the $\Phi$-conservative cocompletion of its full subcategory of $\kappa$-compact objects, where $\Phi$ is the class of $\kappa$-small colimits.

• Let $L$ be a Lawvere theory, and let $\Phi$ be the class of finite coproducts cones in $L^{op}$ (equivalently, finite copowers). Then the $\Phi$-conservative cocompletion of $L^{op}$ is the category of $L$-algebras. For example, the $\Phi$-conservative cocompletion of FinStoch is the category of abstract convex sets, and the $\Phi$-conservative cocompletion of FinRel is the category of join semilattices. In both convex sets and semilattices, the cartesian product gives a symmetric monoidal structure, since they are commutative algebraic theories, and this gives rise by the Yoneda embedding to the symmetric monoidal closed structure on the categories of $L$-algebras.

• Since categories of algebras are typically not cartesian closed, the previous point illustrates that $Cont_\Phi(C^op,Set)$ is not cartesian closed in general (e.g. monoids, groups, convex sets, rings, vector spaces).

• Let $C$ be a small extensive category. Let $\Phi$ be the class of finite coproduct cones. The category of sheaves for the extensive coverage is the $\Phi$-conservative cocompletion of $C$.

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 the following notes, where cartesian closedness is discussed.

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

The symmetric monoidal case is discussed and proved in Lemma 4.6 of

• Mathys Rennela and Sam Staton, Classical Control, Quantum Circuits and Linear Logic in Enriched Category Theory. LMCS vol 16, 2020. arxiv:1711.05159.

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).

Sound doctrines of limits:

• Jiří Adámek, Francis Borceux, Stephen Lack, Jiri Rosicky, A classification of accessible categories, Journal of Pure and Applied Algebra 175 1–3 (2002) 7-30 [doi:10.1016/S0022-4049(02)00126-3]