A very basic definition in category theory is the definition of morphism between objects.
If the category is a construct, i.e., a category $\mathcal C$ equipped with a faithful functor $U\colon \mathcal C\to Set$, how can the morphisms be seen as structure-preserving?
In other words, what exactly does "structure-preserving" mean? If I have two topological spaces, why are continuous functions "structure-preserving" functions?
Has it something to do with the "final topology" or "initial topology" of a continuous function?
Best Answer
The definition of a category does not talk about any structure of the objects. All you need is a class of objects, a class of morphisms and some rule to compose morphisms.
It just happens that taking topological spaces as objects, continuous maps as morphisms and the ordinary composition of maps as the composition of morphisms turns out to be a category, namely $\mathbf{Top}$. In a category, two objects are isomorphic if there are mutually inverse morphisms between them. In the context of topological spaces and continuous maps this turns out to describe homeomorphic spaces, thus "isomorphism" translates to "homeomorphism" in this case. So any statement that is true in category theory "up to isomorphism" will be true in this category "up to homeomorphism".
You could as well form a category by again taking the topological spaces as objects but instead of continuous maps take all maps as morphisms. Isomorphisms in this category will just be bijective maps, so $S^1$ and $\mathbb R$ will be isomorphic objects in this category, but not in the category of topological spaces and continuous maps. This category is really just the category of sets, namely $\mathbf{Set}$, since the morphisms don't capture any topological features of the spaces.
For most structures in mathematics, we have some idea of two of a kind being "isomorphic", for topological spaces that is homeomorphic spaces, for groups it's isomorphic groups, for vector spaces it's isomorphic vector spaces. To capture these "structural properties" in a category you need to pick the class of morphism so your definition of "isomorphism" becomes "$A$ and $B$ are isomorphic if there are mutually inverse morphisms between them". For topological spaces you need to choose continuous maps to capture homeomorphism of spaces as isomorphism of objects.
Another example is the homotopy category of topological spaces $\mathbf{hTop}$: If we want two topological spaces to be "isomorphic" in our category whenever they are homotopic as topological spaces, we need a different class of morphisms. This time what we need is equivalence classes of homotopic continuous maps, with a well defined composition of these equivalence classes. Then we have "$A$ and $B$ are isomorphic objects" if and only if $A$ and $B$ are homotopic spaces, so this category captures the homotopy class of spaces.