Turning a category into a poset

Question

Let $C$ be some category, for example $Set$. Is there a way to "mod out isomorphisms" in a way that the resulting category is a poset, i.e. a thin category and $hom(x,y) \cong hom (y,x)$ implies $x=y$? For $Sets$ this category would just consist of cardinals and $hom(x,y)$ has one element iff $x \leq y$. I am not speaking of the skeleton here.. under which circumstance is this possible.. does this procedure have a name?

Answer 1

One can always define a preorder on the class of objects of a category. You define $A \leq B$ if there exists a morphism from $A$ to $B$. After that you can quotient by the equivalence relation defined by $A \sim B$ if $A\leq B$ and $B \leq A$ to obtain a partially ordered class.

However I doubt that is what you have in mind (since if do this to the category of sets you end up with a two element poset isomorphic to $\{0,1\}$ with $0\leq 0$, $0\leq 1$ and $1\leq 1$). Perhaps what you have in mind is the following. Given a category $\mathbb C$ take the sub-category $\tilde {\mathbb C}$ of $\mathbb C$ with objects the same, but with morphisms monomorphisms. Now preform the operation described in the previous paragraph. The Cantor–Bernstein theorem means that for $\mathbb{C}$ the category of sets, we have $A \sim B$ if and only if $A$ is isomorphic to $C$.