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A quick question.

Let $j : {\sf cat} \to {\sf Cat}$ be the inclusion of small categories in locally small categories; is $j$ a dense functor?

If it is not (as I expect, since it's hard for a generic large category $X$ to be determined by the canonical colimit over $(j\downarrow X))$), is it true that there is a natural isomorphism $$ \int^{A\in \sf cat}{\sf Cat}(A,PA')\times A\cong PA' $$ (of course, the coend/Kan extension on the left might not exist due to size issues; but I guess it's clear what I have in mind)?

If it's not clear what I have in mind, here's the full story: provided $\text{Lan}_jj$ exists, it is the density comonad of $j$: it is a comonad, thus it has a counit $\sigma : \text{Lan}_jj\Rightarrow 1$ which is invertible iff $j$ is dense.

Even though $\sigma$ is not invertible, is the whiskering $\sigma * P$ an isomorphism (where $P$ sends a small category to its presheaf category)?

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Yes it is. In fact, the full subcategory of $Cat$ on the walking commutative triangle is dense. So $cat$, being a full supercategory of a dense category, is dense.

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  • $\begingroup$ That's a nice explanation. Thank you. $\endgroup$ – Fosco Loregian Dec 22 '18 at 20:33
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    $\begingroup$ The full subcategory on the arrow category {0 < 1} is not dense in Cat, since the forgetful functor from Cat to the category of reflexive graphs is not full. Instead, take the full subcategory on the category {0 < 1 < 2}, which is dense in Cat. $\endgroup$ – Alexander Campbell Dec 23 '18 at 5:49
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    $\begingroup$ @AlexanderCampbell oh wow, that’s embarrassing. Thanks! $\endgroup$ – Tim Campion Dec 23 '18 at 6:21
  • $\begingroup$ Wait a second you both, I have a doubt now. A similar argument would be that Delta is dense, but I've always thought that it was dense in small categories, not in locally small ones (the nerve is fully faithful on simplicial sets, not simplicial classes) $\endgroup$ – Fosco Loregian Dec 23 '18 at 10:34
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    $\begingroup$ A more canonical way to think about it is that $Cat$ is a full subcategory of the category $CAT=Ind_\kappa(cat)$ of large categories where $\kappa$ is the size of the universe. And a category is always dense in its $Ind_\kappa$ completion. $\endgroup$ – Tim Campion Dec 23 '18 at 22:45

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