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) or
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.)