Enriching some categories

Start with a plain old category C. You have some objects A and B and Hom(A,B) to describe all the arrows from A to B.

What structure does Hom(A,B) have?

Not much. Ok, now I'm stuck. I can't say any more.

If the Hom(A,B) were just sets, it would be locally small. Not good enough.

If I somehow knew they were abelian groups, then I would be in pre-additive categories, and I would start to have some hope about doing all the abelian category chases.

What are the structures I could possibly ask Hom(A,B) to have? I need to have a well defined map Hom(B,C) x Hom(A,B) to Hom(A,C) so that tells me that it has to be monoidal in order for the left hand side to make sense.

And there should be the identity in Hom(A,A).  That means there should be a morphism from the monoidal unit to Hom(A,A) in this monoidal category.

AND coherence relations time:

Now another example, what if I enrich over chain complexes of complex vector spaces. Each hom is now differential graded. So this is a dg-category.

I can relax conditions a little bit more and give you an example of an A_\infty category. The specific one I have in mind is the Fukaya category of some symplectic manifold.

Picture time

Draw more holomorphic polygons to give the higher maps that mess up the dg-category structure.