# Examples of homomorphisms outside algebra and topology?

I recently started looking at category theory and I see that homomorphisms are a pretty important concept. So far I have seen two "flavours" of homomorphisms in mathematics:

1. homomorphisms between algebraic structures (groups, rings, fields, vector spaces, etc.) which preserve the algebraic operations, whatever they may be
2. continuous functions between topological spaces

I get the sense that there are probably more examples, but I just can't think of any. Moreover it seems odd to me that there are so many examples of homomorphisms in algebra, and yet we have continuous functions from topology just sitting alone all by itself.

Could anyone point out other examples of homomorphisms?

I am primarily looking for examples that have to do with so-called "concrete categories" which non-category theorists would be familiar with prior to studying category theory. I'm aware that functors are a sort of homomorphism between categories, but I'm hoping to find more concrete examples than that.

• An easy example: monotone functions between posets are the "homomorphisms" in the category of all posets. – ZeroXLR Sep 21 '15 at 14:52
• A more "exotic" example. In the category of proofs, the objects are logical formulae and the "homomorphisms" are logical deductions between formulae! – ZeroXLR Sep 21 '15 at 15:01
• You can find plenty of examples like these in Awodey's Category Theory which will also teach you the subject from a pretty elementary level. The author provides the textbook for free I think. Check google... – ZeroXLR Sep 21 '15 at 15:14
• We can connect the two types of morphisms you describe by consider continuous linear operators between topological vector spaces (like Banach or Hilbert spaces). Now the continuous functions from topology don't sit alone by themselves any more ;-) – Ben Sep 21 '15 at 15:15
• Back in your school days and geometry lessons, did you notice that all problems that asked for constructing triangles were really about isomorphism classes of triangles? – Hagen von Eitzen Sep 21 '15 at 15:46