Let $X$ be some given object of a concrete category $\mathcal{C}$. It is possible to define another category $\mathcal{C}/X$ . An object of $\mathcal{C}/X$ is a morphism of $\mathcal{C}$ with codomain $X$. (Namely $A_0\to X$ for $A_0 \in \mathcal{C}$ is such an object of this category). Now one constructs $\mathcal{P}(X)\subseteq \mathcal{C}/X$ (another category) in which one restricts the morphisms of $\mathcal{C}/X$ to monomorphisms. If I'm not mistaken this is called category of parts of $X$.

My question is this: is $\mathcal{P}(X)$ the same as the topology $\mathcal{O}_X$ one chooses to construct a topological space? Topology is a choice of collection of open subsets of some set $X$, $\mathcal{P}(X)$ seems to me to fit perfectly in here.

Continuing on the same lines how can one categorically define structure preserving maps for topological spaces?


No; $\mathcal{P}(X)$ (usually called $\operatorname{Sub}(X)$) is just (equivalent to) the poset of all subobjects of $X$, ordered by inclusion.

When $\mathcal{C} = \mathbf{Set}$, $\operatorname{Sub}(X)$ is merely (equivalent to) the power set of $X$. Even when $\mathcal{C} = \mathbf{Top}$, $\operatorname{Sub}(X)$ doesn't look anything like $\mathcal{O}_X$.

I'm not sure how closely the below touches what you're looking for....

Top has an open set classifier; if you let $S$ be the Sierpinski space and $u : 1 \to S$ be the arrow whose image is the open point of $S$, then the open subsets $U \subseteq X$ are given by characteristic functions $\chi_U : X \to S$.

That is, open subobjects are in one-to-one correspondence with pullback squares of the form

$$ \begin{matrix} U &\to& 1 \\ \downarrow & & {\ \ \ \!}\downarrow{u} \\ X &\xrightarrow{\chi_U}& S \end{matrix} $$

In terms of the category Top, the usual formulation of a topological space is equivalent to specifying the two sets $\hom(1, X)$ and $\hom(X, S)$. Then, a structure preserving map $X \to Y$ (that is, an arrow of Top) induces the corresponding functions $\hom(1,X) \to \hom(1,Y)$ and $\hom(Y,S) \to \hom(X,S)$.

Incidentally, one can formulate a similar notion of "space" called a "locale" by getting rid of the points.

One takes the algebraic properties of the collection of open sets to axiomatize something called a frame, and to define morphisms of frames. The category of locales is then defined to be the opposite category to the category of frames.

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