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

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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.)

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I just changed "homeomorphism" to "continous map". – Hans Stricker Jan 22 2010 at 12:08
Hans, it sounds to me like what you should do is curl up with a nice book like Mac Lane which will presumably explain many of these issues and more. – Qiaochu Yuan Jan 22 2010 at 13:26
Even Mac Lane defines Set as "the category of all small sets" (p. 12) and I read this "whose objects are the 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 2010 at 13:49
And then I stumbled over this (p. 10): "A category (as distinguished from a metacategory) will mean any interpretation of the category axioms within set theory." So Mac Lane is siding with position (2)? What then about your answer below? – Hans Stricker Jan 22 2010 at 13:52
My understanding is that what Mac Lane calls a metacategory is what most people are happy to call a category, for example the nLab (ncatlab.org/nlab/show/category).  But I'll wait until a real category theorist weighs in on the issue. – Qiaochu Yuan Jan 22 2010 at 13:57
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## 6 Answers

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.

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If I think of $\mathbb{Z}$'s standard model (pairs of naturals in the von Neumann standard model) I would firstly think of the group $\mathbb{Z}/3\mathbb{Z}$ as a set consisting of three equivalence classes on $\mathbb{Z}$ and after that I would abstract from this model and think of the group abstractly (like you do in 1). It never would come into my mind, to think of this group as in 2. – Hans Stricker Jan 23 2010 at 14:54

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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

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I've never heard of those "extra structures". How do they come in? – Hans Stricker Jan 22 2010 at 12:44

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."

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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.

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Would you agree then, that "the category of sets" is a convenient shortcut for "the (abstract) category that is isomorphic to the class of all sets with all functions between them". Or doesn't this make sense? – Hans Stricker Jan 22 2010 at 12:01
@Hans: To my knowledge, every definition of a category uses some of the theory of sets to develop the basic results in category theory. As I said before, and as you can read the discussion on the nLab, ETCC is not very good at all. I am not aware of a single useful result from ETCC that doesn't also appear as a consequene of ETCS. – Harry Gindi Jan 22 2010 at 12:07

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@Qiaochu, we specifically want an adjoint pair of functors rather than just a particular forgetful functor. – Harry Gindi Jan 22 2010 at 12:10

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Why should we have to think about everything the same way? We don't even think of all groups the same way! See mathoverflow.net/questions/2551/… – Charles Siegel Jan 22 2010 at 13:45

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@Hans: trying to think about everything the same way is not "trying to be unreasonably rigorous", as there is no rigorousness in it at all, but being a bit silly :) – Mariano Suárez-Alvarez Jan 22 2010 at 16:06
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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.

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