10
$\begingroup$

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!

$\endgroup$
4
  • 1
    $\begingroup$ For the first one, just define the poset category $(\tau,\subseteq)$. Then, continuous functions seem to correspond exactly to contravariant functors between these categories. $\endgroup$
    – WoolierThanThou
    Feb 16, 2020 at 18:46
  • 5
    $\begingroup$ @WoolierThanThou The correspondence isn't "exact". For instance, consider some space $X$ with indiscrete topology $\tau$. Then there are many maps $X\to X$ and they are all continuous, but they induce the same map on the poset $(\tau,\subseteq)$. $\endgroup$
    – Wojowu
    Feb 16, 2020 at 19:07
  • $\begingroup$ There's a way to define a topology on $X$ categorically by looking at the category of subobjects of $X$ and restricting to the open ones, continuity of $f:X\to Y$ is then characterized by the fact that the fibered product in Top of every open subobject of $Y$ along $f$ is an open subobject of $X$, but this not as nice as OP was hoping for I fear $\endgroup$
    – Alessandro Codenotti
    Feb 16, 2020 at 19:14
  • $\begingroup$ The spezialization preorder gives a concrete faithful functor $\mathsf{Top} \to \mathsf{Pre}$. Composing this with the mentioned functor $\mathsf{Pre} \to \mathsf{Cat}$, we get a concrete faithful functor $\mathsf{Top} \to \mathsf{Cat}$. Notice, however, that it isn't full in general. A more useful functor is the fundamental groupoid $\Pi : \mathsf{Top} \to \mathsf{Cat}$. The general question is too general to have a reasonable answer. $\endgroup$
    – Martin Brandenburg
    Feb 23, 2020 at 14:15

0

Reset to default

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.