Fiberwise skeleton vs. category of isomorphism types

Question

What is the relationship (if any) between the process of taking a skeleton of a category, taking a fibered skeleton of a fibered category, and taking isomorphism classes of a category as objects of a new category? Which process do we use to move from the category of monomorphisms to the category of subobjects? Any references would be greatly appreciated.

This came up for me in the construction of the subobject fibration; we begin with a category $\mathcal{B}$ having pullbacks and observe that the codomain functor $cod:\mathcal{B}^\rightarrow\to\mathcal{B}$ is a fibration, then restrict successively to the subcategory of monos ${\sf Mono}(\mathcal{B})\subseteq\mathcal{B}^\rightarrow$ then the category of subobjects ${\sf Sub}(\mathcal{B})$. I'm currently trying to understand how the process of moving from ${\sf Mono}(\mathcal{B})$ to ${\sf Sub}(\mathcal{B})$ preserves the fact that the codomain functor is a fibration, and asked a question over at MSE to try and better understand the process.

It was my understanding that ${\sf Sub}(\mathcal{B})$ was a skeleton of ${\sf Mono}(\mathcal{B})$, however it wasn't clear how to extend the codomain fibration to a fibration on the skeleton (see link for details) if it wasn't already a specific skeleton containing the correct pullback projections, which seemed clunky.

Zhen Lin pointed out that we actually need to take the fibered skeleton of ${\sf Mono}(\mathcal{B})$ to get ${\sf Sub}(\mathcal{B})$ meaning we should only take skeletons of the fibers $cod({\sf Mono}(\mathcal{B}))_X$ and then reconstruct a new overcategory using these skeletons, and the details of this process seem simple enough to work out although a reference would be nice.

Upon reviewing the book I'm working through (Categorical Logic and Type Theory by Jacobs) I realized that the author doesn't seem to mention this process, and further seems to be taking a different route. Specifically, he says:

I tried to find a reference on the process of actually taking the isomorphism classes themselves of a category $\mathscr{C}$ to be the objects of a new category and came across this paper, however it also proves that the resulting category $[\mathscr{C}]$ is not equivalent to $\mathscr C$ in general so this process is different than taking skeletons.

Specifically, if we let $A$ be the class of objects of $\mathscr C$ and $B$ be the class of isomorphism classes of $A$, then let $\mathscr A$ and $\mathscr B$ be the discrete categories on $A$ and $B$ respectively, the author constructs $[\mathscr{C}]$ as the pushout in ${\bf cat}$ of the diagram below

and gives an explicit (somewhat cumbersome) description of the arrows. It's unclear to me which of the above processes is the correct one to use in this situation, although using the regular skeleton doesn't seem to work. Any assistance is appreciated.

Answer 1

As you say, for a general category it is not possible to form an equivalent category from its isomorphism classes. However, if the category is thin (i.e. a preorder) then this is possible, and gives an isomorphic result to taking a skeleton.

More generally, it is possible if the category contains no nonidentity automorphisms. If $C$ is a category with $C/\mathord\cong$ its set of isomorphism classes, then for $R,S\in C/\mathord\cong$ to define a hom-set we have to choose some $x\in R$ and $y\in S$ and take the homset $C(x,y)$. In order to define composition, this choice must be invariant under different choices of $x$ and $y$ up to unique isomorphism, which requires that any two objects in the same isomorphism class must be uniquely isomorphic. This is the same as every automorphism being the identity.

This construction can be done fiberwise to a fibration, as can taking skeleta, and the results are again isomorphic. I believe the only thing to be aware of is that fiberwise skeleta won't in general preserve splitness of a fibration (though this is not an issue for thin categories, of course).

Under this identification, Jacobs is describing the same process that Zhen did. By "a subobject" he means a specific object (not an isomorphism class) together with an isomorphism class of monomorphisms into it, which object of the fiberwise skeleton.