Between abstract and concrete: What's the right way to think of specific categories?

Question

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?

1. 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

2. the class of set-theoretic objects itself (plus morphisms) or

3. what 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.)

Answer 1

There is nothing really particular to categories about this question. You may as well ask:

If one is talking about a specific group, say the group $\mathbb{Z}/3\mathbb{Z}$, should one think of it as:

1. a set of three "atoms" labeled a, b, c, together with a multiplication law (aa = a, ab=b, ...) and a zero element (a), or
2. a set {a, b, c} of three particular *sets*, say a = {}, b = {{}}, c = $\aleph_4$, together with an addition law...?

I'm sure everyone has their own personal preference. For me, (1) corresponds more closely to my intuition, but as long as you understand the relevance of the notion of evil concepts, there's nothing you can't do with (2) as well.

Answer 2

There's a "really close correspondence" between quivers and categories, where quivers are directed graphs that can have multiple arrows from one vertex to another one and also loop arrows, which are arrows from a vertex to itself. Isomorphisms become undirected edges. This is a really good and precise way to think about it, because this viewpoint generalizes very nicely to some models of higher catgory theory, specifically A. Joyal's theory of quasicategories. The whole beauty of category theory is that all of the information about an object is contained within its arrows, and that the underlying thing that the category represents is not actually important. That is, we have all of the information about the category by: a.) Knowing the structure of the graph of the category. b.) knowing the structure of the hom-sets (which don't always have to be sets), and c.) in extra structure that lives over the graph (like a grothendieck topology or a model structure (this is unrelated to the models you were talking about. It has to do with abstract homotopy theory). The only place that it's nice to have sets is for defining the hom-sets in an unenriched setting. Without some notion of a set, it's hard to get important theorems like yoneda's lemma. Lawvere famously came up with two categorical foundational theories, ETCC and ETCS. At the moment, ETCC is pretty much useless. It contains ETCS as a subaxiomatization, but all of the structure axiomatized in ETCC can be constructed from ETCS (depending on if you take the topos of sets to be boolean or not, and some other unimportant technicalities).

ETCS = Elementary theory of the category of sets

ETCC = Elementary theory of the category of categories

Answer 3

Regarding the vivid discussion in the comments after the question (and hopefully, also of some interest for the question itself): I think that a "metacategory" is a definition by axioms, using only first order language, while "interpretation" means: an interpretation as in logic (say, as in p. 29 of Ebbinghaus-Flum-Thomas).

So such an interpretation (a category) is a set, or for convenience, several sets: A set of "objects," a set of "arrows" two function (that is, two more sets) "dom, cod" from the set of arrows to the set of objects, a function "1" from the objects to the arrows, a function "$\circ$" on the pairs of composable arrows, etc., that satisfy the first order axioms of a metacategory.

In summary, I agree with the comment of Qiaochu Yuan: set theory is involved, but not because the objects should somehow be "sets with structure."

Answer 4

Position 2 is only tenable because the categories you describe automatically come with forgetful functors to $\text{Set}$. But in order to think about more general categories (say, homotopy categories) you can't and shouldn't think this way. One way to resolve this situation is to define "concrete category" to mean a category together with a particular forgetful functor to $\text{Set}$, since a particular abstract category may be concrete in more than one way and the functor encodes extra information. In other words, I guess I'm siding with Position 1.

Edit: With regard to your edit, as Harry says, there is some set theory necessary to set up category theory, so it all depends on your approach. But I would say that defining a category to be "the category of these kinds of sets with these kinds of functions between them" is no different from defining a group via one of its faithful actions or representations or defining a manifold via one of its embeddings into $\mathbb{R}^n$. While we pick a particular instantiation to describe what we're talking about, we then talk about the abstract thing.

Answer 5

Some (abstract) categories allow their objects and arrows to be "interpreted by themselves", without recurring to sets from set theory, especially poset categories via Dedekind completions.

Answer 6

Is this a helpful analogy?

Any categorical theory (whose models are all isomorphic) describes a structure uniquely up to isomorphism. In this case there is no need to start with a set-model and forget about it after it has done its work.

On the other hand: Only non-categorical theories give rise to full-blown concrete categories like that of groups with homomorphisms, topological spaces with continuous maps and so on.