Talk:Generator (category theory)

Question about the definition
Must a generator have a morphism to every other object? Or does it only need a morphism to X when X has two different morphisms to some Y? E.g., consider the category with two objects and only the identity morphisms: are both objects generators? If one forms a category from a partially ordered set, by making a single morphism from x to y whenever x ≤ y, is every object a generator, or is only the unique minimal object (if it exists) a generator? The definition as stated in the article is the more inclusive one (e.g. both objects in the arrowless category are generators and all objects in a partial order are generators) but that makes me uncomfortable. A more restrictive statement of the definition would be that G is a generator if, whenever f and g are both morphisms from X to Y, there exists a morphism h from G to X, such that f=g iff hf=hg. But I don't know enough category theory to be sure how the term is actually used. —David Eppstein 06:12, 18 February 2007 (UTC)


 * There seem to be different definitions. The one in the article is, e.g. in the book of Schubert on categories. There it is also mentioned, that Grothendieck has a different definition, which I don't have at hand right now. I'll look it up. Jakob.scholbach 17:49, 23 February 2007 (UTC)


 * Generator (category theory) links to https://ncatlab.org/nlab/show/generator, which links to https://ncatlab.org/nlab/show/separator with a different definition from that given here; it does not require that a generator have a morphism to every other object. I find the definition here to be more useful.
 * What's wrong with the definition already in the article? It amounts to the same thing and seems clearer. Shmuel (Seymour J.) Metz Username:Chatul (talk) 22:21, 27 December 2018 (UTC)