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Consider the following well-known statement:

Let $C$ be a small category, $E$ a cocomplete category, and $F\colon C\to E$ a functor. Then there is a (up to isomorphism) unique cocontinuous functor $F'\colon \mathbf{Set}^{C^\mathrm{op}}\to E$ such that $F'\circ y \cong F$, where $y\colon C\to \mathbf{Set}^{C^\mathrm{op}}$ is the Yoneda embedding.

Question: Is there a way to phrase this as an adjunction (between bicategories, of course)?

Idea: Consider the bicategory $\mathbf{SmallCat}$ of small categories and functors and the bicategory $\mathbf{CocompCat}$ of cocomplete categories and cocontinuous functors. Then the above statement almost states that $C\mapsto \mathbf{Set}^{C^\mathrm{op}}$ is a left adjoint to the forgetful functor $\mathbf{CocompCat}\to \mathbf{SmallCat}$, except that this forgetful functor isn't well-defined: the underlying category of a cocomplete category isn't small.

How can one fix this size issue?

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    $\begingroup$ It's a relative pseudoadjunction. $\endgroup$
    – varkor
    Commented Nov 13, 2021 at 11:39
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    $\begingroup$ This is part of an adjunction between (possibly large) categories and (possibly large) categories with small colimits (and small colimit preserving functors between them). You may restrict to locally small categories to keep things simple. The left adjoint assigns to each locally small category $C$ the category of small presheaves (i.e. those which are small colimits of representable ones). If $C$ is small, all presheaves on $C$ are small, which is why we get the usual universal property of presheaves on a small category as a particular case. $\endgroup$
    – D.-C. Cisinski
    Commented Nov 13, 2021 at 11:44
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    $\begingroup$ @IvanDiLiberti If it's a duplicate of a duplicate of a duplicate of a duplicate it would be more appropriate and polite to just link the original thread. $\endgroup$
    – user984603
    Commented Nov 13, 2021 at 11:54
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    $\begingroup$ @varkor Not at all. I wouldn't have recognized that as an answer to my question since I don't know about profunctors, Kleisli bicategories, pseudomonads, pseudoalgebras, Prof, ... As I said, Cisinski's comment answers my question. :-) $\endgroup$
    – user984603
    Commented Nov 13, 2021 at 12:34
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    $\begingroup$ @user984603: since you were asking questions about (pseudo)adjunctions between bicategories, I had expected you were familiar with 2-categorical terminology. Even if not, if you just ignore the prefix "pseudo" in the introduction, most of the words are standard category theory terms. But if you're satisfied with Cisinksi's comment, then it doesn't matter :) (It follows from Example 3.9 of that paper.) $\endgroup$
    – varkor
    Commented Nov 13, 2021 at 13:16

1 Answer 1

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As Denis-Charles says in the comments, the best way to handle this is to replace the presheaf category $\mathbf{Set}^{C^{\mathrm{op}}}$ by the full subcategory $\hat{C}$ of small presheaves. By definition, a presheaf is small if it satisfies any of these equivalent conditions:

  • it is a small colimit of representables;

  • it is the left Kan extension of its restriction to some small full subcategory of $C$;

  • it is the left Kan extension of some presheaf on some small category along some functor into $C$.

Every presheaf on a small category is small. But, for instance, a presheaf on a large discrete category is small iff its support is small; hence the terminal presheaf is not small.

The functor $C \mapsto \hat{C}$ is left adjoint to the forgetful functor $$ (\text{cocomplete locally small categories}) \to (\text{locally small categories}), $$ in a suitable 2-categorical sense.

A standard reference for this is:

Brian J. Day and Stephen Lack. Limits of small functors. Journal of Pure and Applied Algebra 210 (2007), 651–663.

But it goes back further than 2007. The introduction to Day and Lack's paper recounts some of the history.

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