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More precisely, is every concrete category C isomorphic to a category C' of small categories such that the morphisms between two elements of C are precisely the functors between their images in C'?

At some point I started adopting this point of view as a philosophy without ever bothering to actually verify it.

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Isomorphic or equivalent? And equivalent or (weak) equivalent-as-2-categories? –  Theo Johnson-Freyd Oct 19 '09 at 6:41
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No. There is no full subcategory of Cat equivalent to • ⇉ •, because for any two objects C and D of Cat, either both Hom(C, D) and Hom(D, C) are nonempty, or one of C and D is the empty category and then one of Hom(C, D) and Hom(D, C) is empty and the other is the category •.

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Darn. You probably understand why I asked this, though: this seems to be true of the standard concrete categories students are first introduced to, such as Grp, Rng, Top, ... are these categories characterized by a stronger property than concreteness? –  Qiaochu Yuan Oct 19 '09 at 7:08
    
How do you do it for Rng? –  Reid Barton Oct 19 '09 at 7:21
    
Hmm. I meant Ring, but I may have to take that back. It looks like the standard category-theoretic description of a ring is as a monoid object in Ab, so we need to require that the functors preserve the monoid structure. –  Qiaochu Yuan Oct 19 '09 at 7:40
    
There's a subtlety for Grp also: the full sub-2-category of Cat on the one-object groupoids is not 2-equivalent to the category of groups, because the former has nontrivial natural transformations. The right way to fix this is to talk about pointed one-object groupoids, pointed functors and pointed natural transformations--again extra structure that must be preserved. –  Reid Barton Oct 19 '09 at 22:41
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One way to think of categories algebraically is as a "monoid-oid". So, do for monoids what you do for groups to make groupoids. A (small) category is a set, with an associative, unital, partial multiplication, i.e. composition. So, anything that is an example of this, (monoids, groups, abelian groups, groupoids, etc) is certainly representable as a category.

But there are other examples of things which we aren't used to thinking of as partial monoids. Pre-orders for instance can be thought of as a set of arrows, where the identities are the set elements in a the usual sense and multiplication "witnesses" transitivity. This seems to be kindof a "Lawvere-esque" way of thinking about the category (cf. elementary theory of abstract categories, ETAC).

Warning: this could be a digression... A better way to recover lots of concrete things in a categorical way is to look at functors that preserve some structure (e.g. products or monoidal structure) as the "things" and natural transformations as the "thing homomorphisms". Lawvere theories give a good example of this, and provide a general way to talk about groups, rings, etc. Incidentally, they do pretty much the same job as monads (and there are some nice equivalences). PROPs are another good example. These are both specific cases of a type of 2-monad called a Doctrine. For a prolonged, but interesting discussion, see here.

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