Using inverse image for morphism definitions

Question

We all know that if $(X_1,\tau_1)$ and $(X_2,\tau_2)$ are topological spaces, then a continuous function $f:X_1\to X_2$ is defined to satisfy the requirement that the inverse image of any open subset of $X_2$ is an open subset of $X_1$. That is, $f^{-1}(A) \in \tau_1$ for $A \in \tau_2$. Similarly, if $(X_1,\mathcal{A}_1)$ and $(X_2,\mathcal{A}_2)$ are measure spaces, then a measurable function $f:X \to Y$ is defined to satisfy the requirement that the inverse image of any measurable subset of $X_2$ is a measurable subset of $X_1$. That is, $f^{-1}(A) \in \mathcal{A}_1$ for $A \in \mathcal{A}_2$.

While I understand why morphisms in these specific categories are defined this way, in the abstract I assumed that we choose inverse image because inverse image is well-behaved under unions and intersections. That is, lets say $X$ and $Y$ are sets, and $T_X \subseteq \mathcal{P}(X)$ and $T_Y \subseteq \mathcal{P}(Y)$ are some prescribed subset families we use to abstract a category (like open sets for topology or $\sigma$-algebra for measures). If $f:X \to Y$ is a function of sets, then $f(A \cap B) \subseteq f(A) \cap f(B)$, but with inverse image $f^{-1}(C \cap D)=f^{-1}(C) \cap f^{-1}(D)$. So while we have two candidates $f:\mathcal{P}(X) \to \mathcal{P}(Y)$ and $f^{-1}:\mathcal{P}(Y) \to \mathcal{P}(X)$ we could use to define the "right" notion of morphisms, the latter is better behaved than the former, since $T_X$ and $T_Y$ are closed under unions and intersections.

But what exactly is going here? For instance, defining continuous functions to be open maps fails in capturing our intuition of continuity, sure, but abstractly, what is wrong with it?

I am asking because I encountered another category: simplicial complexes. A simplicial complex too can be described abstractly as a set $V$ (of vertices), and a certain collection $S$ of finite subsets of $V$ (the simplices). However, we don't require $S$ to be closed under unions now. What we do prescribe is that $S$ contains all singletons and is downward closed under subsets. In this category, a morphism of simplicial complexes $(V_1,S_1)$ to $(V_2,S_2)$ is a map $f:V_1 \to V_2$ such that for any $A \in S_1$, $f(A) \in S_2$.

So here too we define the category in terms of a collection of subsets satisfying certain conditions, but for defining morphisms, we don't use inverse image.

My question is:

If we want to define morphisms on a some category that is defined by prescribing subset collections, when do we use image and when do we use preimage? What is the "right" or "natural" notion and how should we abstractly understand this?

Answer 1

It does not directly answer your general question (which has no general answer probably anyway), but the confusion or awkwardness about the definition of continuous maps just disappears when you choose a different, equivalent definition of topological spaces. In other words, you can work with a category which is isomorphic to $\mathbf{Top}$. For example, you can work with Kuratowski spaces: A map $f : X \to Y$ between Kuratowski spaces is continuous by definition if the following holds: If $x \in X$ touches $A \subseteq X$, then $f(x) \in Y$ touches $f(A) \subseteq Y$. So as you can see, no inverse image, and the definition of continuity is much more geometric! (See also Vectornaut's answer here.) You can also work with net convergence spaces: A map $f : X \to Y$ between such spaces is continuous by definition if the following holds: If $x_{\alpha} \to x$ in $X$, then $f(x_{\alpha}) \to f(x)$ in $Y$. With this, you can even see topological spaces as "multialgebraic structures", see Edgar, The class of topological spaces is equationally definable (here), and homomorphisms of multialgebraic structures are defined just as for algebraic structures.

In any case, the "right definition" of the morphisms between topological spaces should not be a consequence of some formal properties of set-theoretic operations. Topological spaces should model geometric spaces, and continuous maps should model maps which are "graphically continuous" (which can be formalized pretty well with Kuratowski spaces above).

Edit. For the general question, something which comes to my mind is that algebraic structures are built up from their elements using algebraic operations, and simplicial complexes are built up from their simplices. This might explain that we want to consider maps which preserve these building blocks. But topological spaces are not built from open sets. In fact, open sets are just auxiliary gadgets (again, see here). This might be an explanation why we do not consider maps which preserve open sets. However, when we see topological spaces as being "built from" from their points and the touch relation / convergence relation as above, it makes sense to consider maps of points which preserve this relation. Similarly, measurable sets can be considered to be auxiliary gadgets to define integration.