Although in an abstract category the morphisms are not explicitly defined, in a concrete example (model theory?), morphisms are (always/usually?) mappings that preserve some properties.
In the category of topological spaces, the morphisms are continuous mappings. One property that is preserved under a continuous mapping is for example convergence of a filter/net/sequence, as Arturo Magidin noted in a comment
In order to view topological maps (i.e., continuous maps) as "preserving a structure", you really need to think of them in terms of preserving the notion of "closeness", not the notion of "open sets" (by the "inverse" of the mapping). It just so happens that the right way to say "f sends points that are close-to-one-another to points that are close-to-one-another" is via inverse images when you consider open sets. To define it in terms of direct images, you consider instead the filter of neighborhoods of a point. – Arturo Magidin
In the category of measurable spaces with measurable mappings being the morphisms, I was wondering what properties are preserved by a measurable mapping? Quickly browsing through Wikipedia and some other sources doesn't provide me the answer.
I used to think that a measurable mapping preserving structures is in the sense that the inverse of the mapping maps a measurable subset to a measurable subset. But as Arturo said, preserving structure should be done by the mapping in the forward direction not its inverse in the backward direction.
Thanks and regards!