What objects can be turned into a category?

Question

A poset $(P,\le)$ can be turned into a category in a standard way. A group $(G,\cdot)$ can also be turned into a category in a standard way. Can we topological space $(X,\tau)$ into a category in a meaningful/useful way? In general, what pairs $(X,\eta)$ consisting of a set $X$ and a collection of subsets $\eta\subseteq\mathcal{P}(X)$ (with some properties) can be turned into a category?

I'm assuming that "meaningful way" means something like

1. you can recover the object from the category and
2. functors between these categories correspond to the appropiate morphisms between the objects

Thanks!

Answer 1

One way of understanding the process of "turning something into a category" is to admit a functor $X \to \mathbf{Cat}$, where $X$ is some category of the things you are interested in. To establish your criteria for "meaningful" you could then ask that such a functor is

1. essentially injective,
2. fully faithful.

(In the literature, these are sometimes called full embeddings).

These requirements pretty much line up with being able to

1. recover the original object from the category, by (essential) injectivity,
2. for functors between the categories obtained this way to bijectively correspond to morphisms in $X$.

If you want $X$ to be a category of "sets equipped with extra structure", one way of making that precise is to ask for $X$ to be a concrete category: that is, for there to be some faithful forgetful functor $X \rightarrow \mathbf{Set}$.

For your first example, observe that $\mathbf{Pos} \hookrightarrow \mathbf{Cat}$, which is a full embedding, and moreover we have a faithful functor $\mathbf{Pos} \to \mathbf{Set}$ as every poset is a set, and monotone maps between them injectively correspond to functions.

For your second example, we have $\mathbf{Grp} \hookrightarrow \mathbf{Grpd} \hookrightarrow \mathbf{Cat}$, where the first functor is a full embedding of a group as a one-object of groupoid, and the second is the functor takes a groupoid and gives a category by forgetting that all of its morphisms are invertible. $\mathbf{Grp}$ is a concrete category also.

For topological spaces, the answer is a bit more complicated, but roughly (handwaving a bit here) if you restrict your answer to certain "nice" topological spaces you can recover a categorical notion quite nicely as locales, which are the opposite category of frames, which are also a concrete category. There are more details here - it's quite a deep topic, and one of the main motivations for the study of toposes.