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)

1. Generator (category theory) #External links 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.
2. 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)