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nLab > Latest Changes: free strict cocompletion

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    • CommentRowNumber1.
    • CommentAuthorvarkor
    • CommentTimeDec 11th 2023

    It seemed useful to give an explicit discussion of this construction on a separate page to free cocompletion. At the moment, there is not a lot of content, so it could have instead been a subsection on free cocompletion, but there is much more that can be said, at which point I think having a separate page will be useful.

    v1, current

    • CommentRowNumber2.
    • CommentAuthorvarkor
    • CommentTimeDec 11th 2023

    I’m not sure whether it is more appropriate to use the terminology “free strict cocompletion” or “strict free cocompletion”. Since we talk of strict monoidal categories, talking also of strict cocomplete categories seems reasonably, so for now I have called this page “free strict cocompletion”.

    v1, current

    • CommentRowNumber3.
    • CommentAuthorDmitri Pavlov
    • CommentTimeDec 11th 2023
    • (edited Dec 11th 2023)

    This seems like another construction of the free cocompletion. Perhaps it would make sense to explain how the other constructions do not satisfy the 2-categorical universal property, if that is the point of this page?

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeDec 11th 2023

    I have turned “2-categorical” into “strict 2-categorical”, for clarity.

    (Many entries here say “2-category” for the weak definition.)

    diff, v3, current

    • CommentRowNumber5.
    • CommentAuthorvarkor
    • CommentTimeDec 13th 2023
    • (edited Dec 13th 2023)

    Tweaked the wording in introduction to make clearer the distinction between the free (non-strict) cocompletion and the free strict cocompletion. It would be nice to have a simple explicit counterexample to the strictness of the category of presheaves (though that it is not strict may be deduced from the results of Beurier, Pastor, and Guitart).

    diff, v4, current

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeDec 13th 2023

    Is it really about bicategorical vs. strict 2-categorical, not rather about 2-categorical vs 1-categorical universality?

    • CommentRowNumber7.
    • CommentAuthorvarkor
    • CommentTimeDec 13th 2023

    Yes, the category of presheaves and the category of clusters both have two-dimensional universal properties (though the latter does not seem to be explicitly spelled out in BPG21), but the category of presheaves only has the universal property up to equivalence, whereas the category of clusters has the universal property up to isomorphism.

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeDec 13th 2023
    • (edited Dec 13th 2023)

    I hear you, but it remains confusing that the difference between “equivalence” and “isomorphism” is that between 2-categorical (strict or weak!) and 1-categorical notions, not between weak or strict 2-categorical notions, no?

    In particular, equivalence of categories is a notion in the strict 2-category of categories, so that it is hard to see what you mean when you insist this to be a bicategorical notion in contrast.

    Maybe you could write out explicitly the universal property that this is about?

    • CommentRowNumber9.
    • CommentAuthorvarkor
    • CommentTimeDec 13th 2023

    Let me be more explicit about what I mean here, and then we can work out how the entry should be refined.

    My understanding of the terminology that is used in the literature (though maybe this does not quite align with the nLab conventions, which tend to be a little different) is that strict 2-categorical universal properties are defined in terms of (strict) 2-adjunctions, i.e. Cat-enriched adjunctions. Objects satisfying such universal properties are therefore determined up to (Cat-enriched) isomorphism. On the other hand, bicategorical universal properties are defined in terms of pseudoadjunctions/biadjunctions, where one has pseudonatural equivalences of the form B(Lx,y)≃A(x,Ry)B(Lx, y) \simeq A(x, Ry). Objects satisfying such universal properties are therefore determined up to equivalence. A strict 2-categorical universal property is a 1-categorical universal property in addition to a universal property involving the 2-cells.

    We can talk about both strict 2-categorical universal properties and bicategorical universal properties in a strict 2-category (whereas only the bicategorical universal properties make sense for bicategories). However, it is often the bicategorical universal properties that are more appropriate, e.g. the appropriate kinds of two-dimensional limits, monoidal structure, or closed structure. This is the case for the presheaf construction: though it is a universal property of an object in a strict 2-category, its universal property is one appropriate to a bicategory. This is why I call it a bicategorical universal property on the page. As far as I am aware, this terminology “bicategorical universal property”, even in the setting of strict 2-categories, is fairly standard. However, perhaps it would be clearer to simply write “weak 2-categorical universal property” versus “strict 2-categorical universal property”, or something like this?

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeDec 13th 2023

    Thanks, now I see what you mean! Yes, mentioning in the entry something like “universal properties in the CatCat-enriched context” would help.

    • CommentRowNumber11.
    • CommentAuthorvarkor
    • CommentTimeDec 13th 2023

    Attempted to clarify the meaning of “strict” in this context.

    diff, v5, current

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeDec 13th 2023

    Thanks for this!

    • CommentRowNumber13.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJan 4th 2024

    Added:

    Universal property

    See Beurier–Pastor–Guitart, Theorem 4.4.

    Consider a locally small category 𝒞\mathscr{C}, its category of clusters ℱ=Clu(𝒞)\mathscr{F}=Clu(\mathscr{C}) (as desribed above), and the canonical inclusion I:𝒞→Clu(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,ℱ,I)(\mathscr{C},\mathscr{F},I) has the following additional canonical structure: for every small diagram P:𝒫→𝒞P\colon\mathscr{P}\to\mathscr{C} we have a canonical colimit cocone λ P:IP⇒λ(IP)\lambda^P\colon I P\Rightarrow \lambda(I P) in ℱ\mathscr{F}. Here λ(IP)\lambda(I P) is an object of ℱ\mathscr{F} given by the diagram PP itself, interpreted as an object of ℱ\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 (ℱ′,I′,λ′)(\mathscr{F}',I',\lambda'), where I′:𝒞→ℱ′I'\colon \mathscr{C}\to\mathscr{F}' is a functor landing in a locally small cocomplete category ℱ′\mathscr{F}' and λ′\lambda' is a choice of a colimit cocone for every diagram of the form I′P′I' P' (P′:𝒫′→𝒞P'\colon \mathscr{P}'\to \mathscr{C}), there is a unique functor J:ℱ→ℱ′J\colon\mathscr{F}\to\mathscr{F}' such that JI=I′J I = I' and JJ sends cocones in λ\lambda to the corresponding cocones in λ′\lambda'.

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

    Properties

    The assignment 𝒞↦Clu(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.

    diff, v6, current

1 to 13 of 13

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