nLab free strict cocompletion

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 $Cat$-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 $Cat$-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 $\mathscr{C}$ whose objects are not presheaves, but simply diagrams into $\mathscr{C}$. 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 $Clu(\mathscr{C})$ by Beurier, Pastor, and Guitart (2021).

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

The category of clusters embeds fully faithfully into the small presheaf category. Its full image is called $LClu(\mathscr{C})$ 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 $\mathscr{J}$ 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 $\mathscr{C}$, its category of clusters $\mathscr{F}=Clu(\mathscr{C})$ (as desribed above), and the canonical inclusion $I\colon\mathscr{C}\to Clu(\mathscr{C})$ that sends an object of $\mathscr{C}$ to a diagram indexed by the terminal category. The triple $(\mathscr{C},\mathscr{F},I)$ has the following additional canonical structure: for every small diagram $P\colon\mathscr{P}\to\mathscr{C}$ we have a canonical colimit cocone $\lambda^P\colon I P\Rightarrow \lambda(I P)$ in $\mathscr{F}$. Here $\lambda(I P)$ is an object of $\mathscr{F}$ given by the diagram $P$ itself, interpreted as an object of $\mathscr{F}$.

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

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

Properties

The assignment $\mathscr{C}\mapsto Clu(\mathscr{C})$ 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 $Clu(\mathscr{C})$, we get strict cocompletion functors for certain types of colimits, e.g., ind-completions.

References

• Andrée Ehresmann, Pro-objects and atlases, Comments 199.1…3, p. 368. in Charles Ehresmann, Œuvres complètes et commentées. Vol. IV-1. Esquisses et complétions,

  Amiens 1981, supplément no. 1 au volume XXII (1981) des Cahiers de topologie et géométrie différentielle (pdf)

• Andrée Ehresmann and J-P. Vanbremeersch, Hierarchical Evolutive Systems: A mathematical model for complex systems, Bulletin of Mathematical Biology 49.1 (1987): 13-50.

• G. M. Kelly, Stephen Lack, On the monadicity of categories with chosen colimits, Theory and Applications of Categories 7 7 (2000) 148-170 [TAC]

• Erwan Beurier, Dominique Pastor, René Guitart Presentations of clusters and strict free-cocompletions, Theory and Applications of Categories 36 17 (2021) 492-513 [tac:36-17]