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).

Definition

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

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

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.

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

  • If CC is small and symmetric monoidal and (c)(c\otimes -) preserves Φ\Phi-colimits, then one can define a symmetric monoidal closed structure on Cont Φ(C op,Set)Cont_\Phi(C^op, Set) such that the Yoneda embedding CCont Φ(C op,Set)C\to Cont_\Phi(C^op, Set) is strong monoidal. In particular if CC is already cartesian closed or symmetric monoidal closed, so too is Cont Φ(C op,Set)Cont_\Phi(C^op, Set), since if (c)(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 Φ(C op,Set)Cont_\Phi(C^op, Set).

  • Let Ψ\Psi be a sound doctrine of limits. Suppose that CC is small and has all Ψ op\Psi^{op}-colimits, then let Φ\Phi be the class of Ψ op\Psi^{op} colimiting cocones in CC. The Φ\Phi-conservative cocompletion of CC is also the free Ψ\Psi-filtered colimit completion of CC (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 CC with a class of limit cones Φ\Phi is also called a ‘realized limit sketch’. Then the category of models of the sketch is the Φ op\Phi^op-conservative cocompletion of C opC^op.

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

  • Let LL be a Lawvere theory, and let Φ\Phi be the class of finite coproducts cones in L opL^{op} (equivalently, finite copowers). Then the Φ\Phi-conservative cocompletion of L opL^{op} is the category of LL-algebras. For example, the Φ\Phi-conservative cocompletion of FinStoch is the category of abstract convex sets, and the Φ\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 LL-algebras.

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

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

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

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

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:

Last revised on July 23, 2025 at 08:45:11. See the history of this page for a list of all contributions to it.