In what categories does the "classical" notion of function make sense?

Question

I'm new to category theory, and I often struggle to choose the right level of abstraction when working with categories. I also found that many textbooks are rather inconsistent in their conventions with regards to the terminology (eg. they often interchangeably use terms like epimorphism and surjection, etc). So I wondered what's a minimal set of requirements on a category so that it makes sense to say that morphism are functions? How about Abelian categories?

Answer 1

The notion you are looking for is probably that of a concrete category. A concrete category is a category that is embedded in the category of sets; thus its objects are associated with actual sets, and its morphisms are associated with actual functions.

Answer 2

Beware that even in conrete categories the terms epimorphism and surjection are not interchangeable.

For example, consider

• the category Mon where objects are monoids and morphisms their algebraic homomorphisms
• and the forgetful functor making it concrete being the obvious $U: \mathrm{Mon} \to \mathrm{Set}$.

Now the embedding morphism $m: (\mathbb{Z}, \cdot, 1) \to (\mathbb{Q}, \cdot, 1)$ is a monomorphism and epimorphism as can be seen by some small, but tedious calculations. However, the associated actual function given by $U(m): \mathbb{Z} \to \mathbb{Q}$ is obviously not an epimorphism in Set. Notably, in Set the terms epimorphism and surjection really coincide — at least with AoC.