# Is the embedding ${\sf cat}\subset {\sf Cat}$ dense?

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)?

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.
• 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. – Tim Campion Dec 23 '18 at 22:45