Question

Suppose C is a category with small colimits, and G $\in$ ob(C) is a strong generator. (This means that $f: X \to Y$ is an isomorphism iff $f_{*}: Hom(G,X) \to Hom(G,Y)$ is a bijection.)

How does one prove that every object X is a colimit of G (ie, a quotient of coproducts of G)?

I think X $\simeq$ colim_{G --> X} G. I can see there is a natural map e: colim_{G --> X} G $\to$ X, and that the induced map $e_{*}: Hom(G,colim_{G --> X} G) \to Hom(G, X)$ is surjective, but I cannot see that it map is injective.

Answer 1

I think there are actually three possible things that you might be asking, but the answer to all of them is no. Suppose that G is a strong generator in a cocomplete category C. Then you can ask:

1. Is every object X of C the colimit of G over the canonical diagram of shape $(G\downarrow X)$? (If so, then G is called dense in C.)

2. Is every object some colimit of a diagram all of whose vertices are G? (If so, then G is called colimit-dense in C.)

3. Is C the smallest subcategory of itself containing G and closed under colimits? (If so, then G is a colimit-generator of C.)

The category of compact Hausdorff spaces is a counterexample to the first two. It is monadic over Set (the monad is the ultrafilter monad, aka Stone-Cech compactification of the discrete topology), and hence cocomplete, and the one-point space is a strong generator. But any colimit of a diagram consisting entirely of 1-point spaces must be in the image of the free functor from Set (the one-point space being its own Stone-Cech compactification), and hence (since that functor is a left adjoint and preserves colimits) must be the free object on some set. However, not every compact Hausdorff space is the Stone-Cech compactification of a discrete set.

As Todd pointed out in the comments, though, CptHaus is the colimit-closure of the one-point space, since every object is a coequalizer of maps between free ones (because the category is monadic over Set); thus it isn't a counterexample to the third question. Counterexamples to the third question are actually much harder to come by, and in fact if you assume additionally that C has finite limits and is "extremally well-copowered", then it is true that any strong generator is a colimit-generator. I think there's a proof of this somewhere in Kelly's book "Basic concepts of enriched category theory," and a brief version can be found here. However, without these assumptions, one can cook up ugly and contrived counterexamples, such as example 4.3 in the paper "Total categories and solid functors" by Borger and Tholen.

Answer 2

See cor. 4.4 in "Strong regular and dense generators" Tholen , Borger:

http://www.numdam.org/numdam-bin/fitem?id=CTGDC_1991__32_3_257_0