At the risk of annoying some of the categorists I feel urged to pose this beginner-ish question:

If one talks about a specific category such as *the**category of sets with functions* or *the**category of groups with group-homomorphisms* or *the**category of topological spaces with* *homeomorphisms**continous maps* (let's restrict to these), what should I have in mind, how should I think of it?

a sheer structure of point-like objects and arrows which is merely isomorphic to a class of set-theoretic objects with set-theoretically definable morphisms between them (e.g. functions as sets) or

the class of set-theoretic objects

**itself**(plus morphisms) orwhat else?

In case of (1) shouldn't for example the category of sets been termed "the (abstract) category which is isomorphic to the (concrete) class (not *category*!) of all sets with functions" (as we would talk about "the unlabelled graph X which is isomorphic to the labelled graph Y")? And only because this is inconvenient, we talk of "the category of sets"?

[Added:] It's common talk to say "**Set** is the category whose objects **are** all sets...". This sounds like taking position (2).

~~
Side-question: There is the notion of "the category of models of a theory with elementary maps". Is the category of groups with group-homomorphisms the same as the category of models of group theory with elementary maps? If not so: why?~~ (Made a separate question out of this.)

Setas "the category of all small sets" (p. 12) and I read this "whose objectsarethe small sets". Can you point me to a location where ML faces my question, or is it just implicitly answered (like the "inner structure" of an object is just implicitly determined by its hom-sets)? – Hans Stricker Jan 22 '10 at 13:49