nLab free strict cocompletion

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Contents

Context

Category theory

Limits and colimits

Yoneda lemma

Contents

Idea

The small presheaf-construction exhibits the free cocompletion of a locally small category (only) up to equivalence of categories. In other words, it exhibits a universal property that is bicategorical in nature. However, it is also possible to give a description of the free strict cocompletion, i.e. one satisfying a strict 2-categorical universal property, in the sense of CatCat-enriched category theory, which characterises the free cocompletion up to an isomorphism of categories.

Explicitly, the small presheaf construction exhibits a left pseudoadjoint to the forgetful 2-functor from the 2-category of locally small cocomplete categories, whereas a strict free cocompletion exhibits a left strict 2-adjoint (or left CatCat-enriched adjoint).

Abstractly, this follows by the results of Kelly and Lack (2000). However, there is also an explicit description, due to Ehresmann (1981), and expanded upon by Beurier, Pastor & Guitart (2021). The idea is to give a much more “naïve” description of the free cocompletion of a category 𝒞\mathscr{C} whose objects are not presheaves, but simply diagrams into 𝒞\mathscr{C}. While the objects are, in some sense, simpler than presheaves, the morphisms are considerably more complicated: their dual is called called atlases by Ehresmann (1981), and they are called clusters by Ehresmann and Vanbremeersch (1987). The free strict cocompletion is consequently called the category of clusters Clu(𝒞)Clu(\mathscr{C}) by Beurier, Pastor, and Guitart (2021).

The morphisms can be seen to satisfy the limit–colimit formula (Beurier–Pastor–Guitart, Proposition 3.11): given P:𝒫𝒞P \colon \mathscr{P} \to \mathscr{C} and Q:𝒬𝒞Q \colon \mathscr{Q} \to \mathscr{C}, we have Clu(𝒞)(P,Q)lim p𝒫colim q𝒬𝒞(P(p),Q(q))Clu(\mathscr{C})(P, Q) \,\cong\, lim_{p \in \mathscr{P}} colim_{q \in \mathscr{Q}} \mathscr{C}\big(P(p), Q(q)\big).

The category of clusters embeds fully faithfully into the small presheaf category. Its full image is called LClu(𝒞)LClu(\mathscr{C}) by Beurier, Pastor & Guitart (2021).

That the category of clusters forms the free strict cocompletion was stated in Ehresmann (1981) and Ehresmann and Vanbremeersch (1987), but without detailed proofs. A detailed treatment is given by Beurier, Pastor &Guitart (2021).

Ehresmann (1981) also considers the restriction to the free strict cocompletion under a specified class 𝒥\mathscr{J} of diagram shapes: this is given by restricting the objects of the category of clusters.

Universal property

See Beurier–Pastor–Guitart, Theorem 4.4.

Consider a locally small category 𝒞\mathscr{C}, its category of clusters =Clu(𝒞)\mathscr{F}=Clu(\mathscr{C}) (as desribed above), and the canonical inclusion I:𝒞Clu(𝒞)I\colon\mathscr{C}\to Clu(\mathscr{C}) that sends an object of 𝒞\mathscr{C} to a diagram indexed by the terminal category. The triple (𝒞,,I)(\mathscr{C},\mathscr{F},I) has the following additional canonical structure: for every small diagram P:𝒫𝒞P\colon\mathscr{P}\to\mathscr{C} we have a canonical colimit cocone λ P:IPλ(IP)\lambda^P\colon I P\Rightarrow \lambda(I P) in \mathscr{F}. Here λ(IP)\lambda(I P) is an object of \mathscr{F} given by the diagram PP itself, interpreted as an object of \mathscr{F}.

The free strict cocompletion Clu(𝒞)Clu(\mathscr{C}) of a locally small category 𝒞\mathscr{C} satisfies the following universal property: given another such triple (,I,λ)(\mathscr{F}',I',\lambda'), where I:𝒞I'\colon \mathscr{C}\to\mathscr{F}' is a functor landing in a locally small cocomplete category \mathscr{F}' and λ\lambda' is a choice of a colimit cocone for every diagram of the form IPI' P' (P:𝒫𝒞P'\colon \mathscr{P}'\to \mathscr{C}), there is a unique functor J:J\colon\mathscr{F}\to\mathscr{F}' such that JI=IJ I = I' and JJ sends cocones in λ\lambda to the corresponding cocones in λ\lambda'.

In particular, \mathscr{F} is unique up to an isomorphism.

Properties

The assignment 𝒞Clu(𝒞)\mathscr{C}\mapsto Clu(\mathscr{C}) yields a strict 2-functor from locally small categories to locally small categories that implements the free cocompletion construction.

This stands in contrast to the usual construction of small presheaves, which only yields a pseudofunctor.

By restricting the types of diagrams in the construction of Clu(𝒞)Clu(\mathscr{C}), we get strict cocompletion functors for certain types of colimits, e.g., ind-completions.

References

Last revised on January 4, 2024 at 00:26:15. See the history of this page for a list of all contributions to it.