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I would like to define a category as $\bf Cat \downarrow Set$ (which would be the slice category of $\bf Cat$ over the object $\bf Set$. However, since $\bf Set$ is not an object of $\bf Cat$, I cannot do that. However, using the slice definition still gives us some category where the objects are $\bf Set$-valued functors and arrows are functors in $\bf Cat$ along with some 2-isomorphism.

So I don't think this is the result of some slice but it seems so close ! Moreover I could also extend this definition to all the categories, but again, since their is no category of all categories, I don't think it is the result of any slicing.

My question is : is there a standard notation for these kind of construction ? Am I completely lost with those size issues?

Thanks a lot :)

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What you're describing is the (2-)comma category $(\mathbf{Cat} \hookrightarrow \mathbf{CAT}) \downarrow (\mathbf{Set} : \mathbf{1} \to \mathbf{CAT})$, where $\mathbf{Cat}$ is the (2-)category of small categories, $\mathbf{1}$ is the terminal category and $\mathbf{CAT}$ is the (2-)category of locally small categories. The objects of this comma category are small categories with functors into $\mathbf{Set}$ and morphisms are functors between small categories commuting with the functors into $\mathbf{Set}$. Comma categories are a generalisation of (co)slice categories that often allow you to describe structure which is "slice-like", but isn't quite.

Alternatively, if you're happy with the objects being locally small categories, you can simply take the (2-)slice category $\mathbf{CAT}/\mathbf{Set}$.

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  • $\begingroup$ Thanks for your answer, that's exactly what I was looking for. Do you have any references on $\bf CAT$ ? I was not able to find something about it. I looked into enriched categories to see that it is equivalent to $\bf Cat$ enriched with $\bf Set$. $\endgroup$
    – 141592653
    Commented Jun 18, 2020 at 18:19
  • $\begingroup$ I'm not sure what might be considered a canonical reference, but see for instance Section 1.7 of M-Completeness Is Seldom Monadic Over Graphs, which describes foundational issues regarding treating $\mathbf{CAT}$ as a category versus a 2-category. $\endgroup$
    – varkor
    Commented Jun 18, 2020 at 20:07
  • $\begingroup$ @141592653 for category related things it is always nice to check ncatlab.org/nlab/show/2-category - references are on the bottom of the page. $\endgroup$
    – Gerrit Begher
    Commented Jun 19, 2020 at 8:15
  • $\begingroup$ Thank you very much, you were very helpful $\endgroup$
    – 141592653
    Commented Jun 19, 2020 at 10:04
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Here is how you get rid of the "size" issues in constructions like this:

An object of the slice (2-)category that you want is a functor $F:{\mathbf C}\to{\mathbf{Set}}$, which is otherwise known as a presheaf on ${\mathbf C}^{\mathsf{op}}$.

The Grothendieck construction transforms this a functor $P:{\mathbf E}\to{\mathbf C}^{\mathsf{op}}$ that is a discrete fibration.

It's an exercise to work out what the 1- and 2-cells of this 2-category look like in terms of discrete fibrations.

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