Homomorphism

A Homomorphism (not to be confused with homeomorphism) is a structure preserving map.

For example, given vector spaces [ilmath]V\text{ and }W[/ilmath] then [math]\text{Hom}(V,W)[/math] is the vector space of all linear maps of the form [math]f:V\rightarrow W[/math], as linear maps will preserve the vector space structure.

Group homomorphism

Given two groups [ilmath](A,\times)[/ilmath] and [ilmath](B,+)[/ilmath] a map [ilmath]f:A\rightarrow B[/ilmath] is a homomorphism if:

• [math]\forall a,b\in A[f(ab)=f(a)+b(b)][/math] - we need not use different operations (we could use multiplicative for both) but I wanted to emphasise the different groups.

Topological homomorphism

Isn't a thing! I've seen 1 book ever (and nothing online) call a continuous map a homomorphism, Homeomorphism is a big thing in topology though. If something in topology (eg [math]f_*:\pi_1(X)\rightarrow\pi_2(X)[/math]) it's not talking topologically (as in this case) it's a group (in this case the Fundamental group and just happens to be under the umbrella of Topology