Preserving structures

Category theory abstracts the notion of the preservation of structure by means of morphisms. Is there a description of what it means to preserve structure of different types of mathematical structures (topological spaces, modules, rings, etc.)?

I interpret the question as follows: What does it mean for a map to preserve structure (and therefore to be called a morphism)?

Very roughly, there are two sorts of structures: Algebraic structure and topological structure.

A) Algebraic structure on a set is given by a set of operations of certain arities such that certain equations hold. A map preserves this algebraic structure if it commutes with the operations in the obvious sense. Examples are group homomorphisms between groups, ring homomorphisms between rings, and isometries between metric spaces (viewing the metric as a binary real-valued operation).

B) A topological structure is given by a set of "admissible" subsets satisfying certain properties (often closure properties). Examples are topological spaces, measurable spaces, partial orders (consider the down-sets) and filters. A map preserves this structure iff the preimage of an admissible subset is admissible. In the above examples we get continuous maps, measurable maps, increasing maps and filter maps.

We can also mix algebraic and topological structures. There are lots of interesting and important examples, but let me only mention three of them:

1) Topological groups and Lie groups. A morphism of topological (Lie) groups is a continuous (smooth) group homomorphism.

2) Schemes, or more generally ringed spaces. A morphism of ringed spaces $(X,\mathcal{O}_X) \to (Y,\mathcal{O}_Y)$ is a continuous map $f : X \to Y$ (topological part) together with a homomorphism of sheaves of rings $f^\# : \mathcal{O}_Y \to f_* \mathcal{O}_X$ (algebraic part).

3) C*-algebras, more generally Banach algebras. A morphism of C*-algebras is a continuous algebra homomorphism which also preserves the involution. Actually it turns out that we get continuity for free, and the norm is always $\leq 1$.

The most general notions of "structure" and "homomorphism" are developed and studied within model theory.

A map between structured sets preserves structure when it preserves true statements about the elements of the sets (not about the sets themselves). A decision about what kind of structure to preserve is a decision about what kind of true statements you care about. For example, group homomorphisms preserve true statements involving the group operations, e.g. $g^3 = h^2$ (for $g, h$ elements of some group), ring homomorphisms preserve true statements involving the ring operations, e.g. $a^2 + b^3 = c^4$ (for $a, b, c$ elements of some ring), and so forth.

The case of topological spaces is more interesting. Here the true statements being preserved are essentially statements about limits. More formally, using the Kuratowski closure axioms you can think of a continuous function as a function that preserves true statements of the form "point $x$ is contained in the closure of subset $S$."

Typically, structures on a set $A$ are relations, that is subsets $R_A$ of some $A^n$. In that case $f\colon A\to B$ preserves the structure if $(a_1,\ldots,a_n)\in R_A$ implies $(f(a_1),\ldots,f(a_n))\in R_B$. An example of this are groups where we can consider the group law as ternary relation $\{(a,b,c)\in G^3\mid a\circ b=c\}$. Of course this also works with infinitary relations.

Topological spaces are a bit different, as here the structure on $A$ is a subset $T_A$ of $\mathcal P(A)$, which introduces some contravariance. As you know, $f\colon A\to B$ is continuous if the inverse images of open $B$-sets are open $A$-sets. That is, $f\colon A\to B$ preserves the stucture if the induced map $f^{-1}\colon \mathcal P(B)\to\mathcal P(A)$ behaves as above, i.e. $U\in T_B$ implies $f^{-1}(U)\in T_A$.

Of course, all these can be combined and even more complex structures can be considered.