Ingredients
Incarnations
Properties
Universal aspects
Classification
Induced theorems
…
In higher category theory
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 -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 -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 whose objects are not presheaves, but simply diagrams into . 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 by Beurier, Pastor, and Guitart (2021).
The morphisms can be seen to satisfy the limit–colimit formula (Beurier–Pastor–Guitart, Proposition 3.11): given and , we have .
The category of clusters embeds fully faithfully into the small presheaf category. Its full image is called 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 of diagram shapes: this is given by restricting the objects of the category of clusters.
See Beurier–Pastor–Guitart, Theorem 4.4.
Consider a locally small category , its category of clusters (as desribed above), and the canonical inclusion that sends an object of to a diagram indexed by the terminal category. The triple has the following additional canonical structure: for every small diagram we have a canonical colimit cocone in . Here is an object of given by the diagram itself, interpreted as an object of .
The free strict cocompletion of a locally small category satisfies the following universal property: given another such triple , where is a functor landing in a locally small cocomplete category and is a choice of a colimit cocone for every diagram of the form (), there is a unique functor such that and sends cocones in to the corresponding cocones in .
In particular, is unique up to an isomorphism.
The assignment 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 , we get strict cocompletion functors for certain types of colimits, e.g., ind-completions.
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]
Last revised on January 4, 2024 at 00:26:15. See the history of this page for a list of all contributions to it.