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While thinking about Yoneda embedding, I came up with following two questions (I should apologies, if those are too vague):

  1. Does the Yoneda embedding $y :\mathcal{C}\to\mathbf{Set}^{\mathcal{C}^{\mathbf{op}}}$ has a left adjoint? If there is a such adjoint, how does it work?
    Since $y$ preserves limits, according to adjoin functor theorem, it can be a right adjoint. But so far, I haven't see anything like this.
  2. The Yoneda embedding is never an equivalence between categories (so a proper embedding). Now suppose we iterate this process $$y_0=\mathcal{C}\to y_1=\mathbf{Set}^{\mathcal{C}^{\mathbf{op}}}\to y_2=\mathbf{Set}^{y_1^{\mathbf{op}}}\to y_3=\mathbf{Set}^{y_2^{\mathbf{op}}}\to\cdots.$$ This is a diagram in 2-category $\mathbf{CAT}$ indexed by the ordinal $\omega.$ Does this diagram has (a sort of) colimit? If so, how does it looks like?
Bumblebee
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  • These should probably be separate questions. – jgon Apr 15 '20 at 01:30
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    That said, the answer to 2 should be yes, it should just be the "union" of all the categories, with morphisms between objects being taken in the larger of the two categories. I'm not sure what else you might want to know about this category. – jgon Apr 15 '20 at 01:32
  • @jgon: Thank you, I think, I got it. Would you like write this as an answer. – Bumblebee Apr 15 '20 at 01:36
  • Sorry I don't really have the time to write a full answer, though you're more than welcome to write up an answer yourself if the comment helped you figure it out. Q1 looks pretty interesting though, so I hope someone will come along and give an answer. – jgon Apr 15 '20 at 01:39
  • I think the answer to 1 is yes, assuming $\mathcal{C}$ is cocomplete(?), with the left adjoint being something like $P\mapsto \int^{c\in\mathcal{C}} P(c)\cdot c$ by a co/end calculus manipulation, and $\cdot$ being the tensoring of $\mathcal{C}$ by $\mathbf{Set}$, but I'm not at all sure of this. – jgon Apr 15 '20 at 01:45
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    I must be missing something: see here for info on when a left adjoint exists. – jgon Apr 15 '20 at 01:48
  • Actually, I think my left adjoint is valid assuming $\mathcal{C}$ is small: see this answer to see that cocompleteness is all that's necessary for a small category to be total, i.e. admit a left adjoint to Yoneda. In general, the problem is that my coend is large. – jgon Apr 15 '20 at 01:58
  • Well that was an interesting rabbit hole. Thank you for this question! :) Hopefully my links will be helpful to you. – jgon Apr 15 '20 at 01:59
  • @jgon: Thank you for your helpful comments. Yes, it is. – Bumblebee Apr 15 '20 at 02:00
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    @jgon but note that small categories can only be cocomplete if they’re posets! – Kevin Carlson Apr 15 '20 at 02:43
  • @KevinCarlson Yeah, I realized that the formula for the left adjoint I gave wasn't going to be terribly helpful once I'd realized my mistake. But we can probably adapt it to the case where we have a cocomplete category with a small dense subcategory. – jgon Apr 15 '20 at 06:33

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Regarding 1, a few points, some just gathering remarks from the comments: such categories-when locally small, at least-are called total. Unless they’re posets, they can’t be small; they must be cocomplete but in fact more is true, since not every presheaf on a large category is a small colimit of representables and cocompleteness refers only to small colimits. Being total is intuitively the requirement that $C$ should admit every large colimit that it plausibly could. Essentially all naturally occurring cocomplete large categories are in fact total, although the dual notion of cototality is less common; such categories satisfy the perfect form of the co-adjoint functor theorem, namely that any cocontinuous functor out admits a right adjoint. One explanation for categories like Grothendieck toposes and Grothendieck abelian categories which satisfy both versions of the perfect adjoint functor theorem is that they are lucky enough to be cototal, which is intuitively a weak form of having a cogenerator.

Kevin Carlson
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