Whats the relation between preorders and (small) thin categories?

Question

In introductory category theory books, it is often given the example that given a preordered set, we can obtain a thin category from it in the obvious way. Sometimes they go as far as to say that preorders and (small) thin categories are the same thing. This is perhaps forgivable because at that point the notion of equivalence of categories isn't defined yet and/or they mean (small) strict thin categories instead. But it still makes me wonder to what relation is there between the usual category of preorders and the 2-category of (pre)orders. The latter is equivalent to the 2-category of partial orders so it seems to have less information. Is there some natural way of obtaining the usual 1-category of preorders from the 2-category of partial orders or from the 1-category of partial orders without breaking the principle of equivalence? Should we really be thinking of thin categories as partial orders as opposed to preorders?

For extra motivation, some other wrong example that is often given in introductory category books is that they say groups are the same thing as groupoids with exactly one object. However we can recover the usual 1-category of groups from the 2-category of connected groupoids: we do this by considering pointed connected groupoids. I was hoping/wondering if something similar can be done to recover preorders from thin categories.

Answer 1

As I said in the comments, we must break the principle of equivalence in order to get the 1-category of preordered sets back from the 2-category of thin categories. But this can be done in a black-box fashion, and exactly the same technique works to reconstruct the 1-category of categories from the 2-category of categories, so let me describe it in that generality.

A strict category consists of the following data:

• A category $C$ (i.e. an object in the abstract 2-category $\textbf{Cat}$).
• A set $O$ (i.e. a discrete object in $\textbf{Cat}$ – I omit the definition).
• A functor $i: O \to C$ that is essentially surjective on objects.

Given strict categories $(C, O, i)$ and $(D, P, j)$, a morphism $(C, O, i) \to (D, P, j)$ consists of the following data:

• A functor $f : C \to D$.
• A functor $f' : O \to P$.
• An isomorphism $j \circ f' \cong f \circ i$.

Thus, we obtain the 2-category of strict categories as a full sub-2-category of $[\mathbf{2}, \textbf{Cat}]_\textrm{ps}$. It is straightforward to check that this 2-category is equivalent to a 1-category, i.e. that every 2-morphism is invertible and there is at most one 2-morphism between any parallel pair of 1-morphisms.

Conceptually, this is saying that the objects in the 1-category $\textbf{Cat}$ are not categories per se but rather categories equipped with a set-indexed collection of objects. This extra structure enables us to distinguish between isomorphic objects and also causes the 2-morphisms to become trivial, as desired.