ct.category theory - categorical description of elements in a direct limit

Sunday, 27 May 2007

ct.category theory - categorical description of elements in a direct limit

As with that previous question, I don't understand precisely what the rules of the game are when you say "without using an explicit construction". But maybe I can say something useful.

First I'll answer the first part of the first question. Then I'll use it to explain a little of why I don't think there's going to be a precise way to formulate the "no explicit construction" rule.

We have a sequence $A_1 to A_2 to cdots$ in the category of sets, and its colimit $A$ , and we wish to show that every element of $A$ comes from an element of some $A_n$ . And we wish to do it without using the explicit formula for colimits. Being a category theorist, I'll write 1 rather than $*$ for a one-element set.

I'll use the fact that the category of sets is a well-pointed topos, E say. (Those hypotheses can probably be weakened.) Take an element $a$ of $A$ , that is, a map $a: 1 to A$ . Since colimits are stable under pullback in any topos, pulling the colimit cocone back along $a$ gives a sequential colimit
$mathrm{colim}(X_1 to X_2 to cdots) = 1$
in the category E, together with a map $f_n: X_n to A_n$ for each $n$ , making the evident square commute. (Draw a diagram!) Since any colimit of initial objects is initial, and in a well-pointed topos $1$ is not initial, at least one $X_n$ is not initial. It's a fact that in a well-pointed topos, every object is either initial or admits an element (map from $1$ ). So, at least one $X_n$ admits an element, $x$ say. Then $f_n x$ is an element of $A_n$ , and a quick diagram chase shows that it maps to $a in A$ .

So, I've answered the first question without apparently using any explicit constructions involving sets --- in the sense that I just assumed certain axioms on the category of sets and did the proof categorically. But the thing is, you can always do that. "Explicit constructions" can always be translated into categorical arguments (and though it's not apparent from the proof above, that's a totally mechanical process).

If your point of view on sets is that they are what ZFC says they are, then here's an equivalent categorical formulation: sets and functions form a well-pointed topos with natural numbers object and choice, satisfying a first-order axiom scheme of replacement. ZFC and this entirely categorical axiomatization are in a precise sense equivalent. Anything you can do in one context, you can do in the other.

On coproducts: it can be shown that in any topos, coproducts are disjoint and the coprojections are jointly epic. I think that's in Mac Lane and Moerdijk's book Sheaves in Geometry and Logic. In a well-pointed topos, epic = surjective and monic = injective (where by sur/injective I'm referring to the elementwise notion implicit in the question). Hence your statements on coproducts hold in any well-pointed topos.

On the question about algebraic structures:

Colimits (="direct limits") of sequences are an example of filtered colimits --- see Categories for the Working Mathematician Chapter IX, for instance. The forgetful functors Group $to$ Set, Ring $to$ Set, etc, all preserve filtered colimits. The jargon for this is: "the free group/ring/... monad is finitary". As the terminology hints, preservation of filtered colimits corresponds to the fact that the theory of groups, rings, etc. only involves finitary operations.

For example, the theory of groups has an operation with 2 arguments (multiplication), an operation with 1 argument (inverse), and an operation with 0 arguments (identity). The numbers 2, 1, and 0 are all finite, so the theory of groups is finitary, so the forgetful functor Group $to$ Set preserves filtered colimits.

(A non-example would be the theory of ordered sets in which every countable subset had a supremum, and maps that preserved those suprema. One of the operations in the theory is "take the supremum of a countably infinite subset", so this theory isn't finitary. Correspondingly, the forgetful functor from these ordered sets to Set won't preserve filtered colimits.)

AnswerDesk

Sunday, 27 May 2007