Talk:Subcategory

If a full subcategory contains all identity morphisms, and for every pair of id. morphisms it contains, it contains all morphisms between the corresponding objects, then what is the difference between a full subcategory and the original category? Is it so, that we take a "subset" of the objects, and a subcategory is required to contain only all identitiy morphisms of the objects in that subset? Mikolt 13:26, 9 Jul 2004 (UTC)

Yes, that's my understanding. A subcategory contains all identity morphisms between a subset of objects. For example, Ab is a full subcategory of Grp but Ring isn't (there are group homomorphisms between rings which aren't ring homomorphisms). The page seems worded badly, but this isn't my area of expertise. – Fropuff 15:46, 2004 Jul 9 (UTC)

It actually wasn't right, really; and was also too compressed. I've tried another kind of definition. Charles Matthews 16:43, 9 Jul 2004 (UTC)

Definition

Why is this definition so complicated? How about:

Let C be a category. A subcategory S of C is a category with

$\mathrm{Ob}(\mathcal{S})\subseteq\mathrm{Ob}(\mathcal{C}) \ \text{and} \ \mathrm{Mor}(\mathcal{S})\subseteq\mathrm{Mor}(\mathcal{C})$.

That is, a category whose objects resp. morphisms form a subclass of the objects resp. morphisms of the original category. Spaetzle (talk) 14:45, 7 June 2011 (UTC)

You can do something like this, it is quite sensible, but you do need to require that the inclusions induce a functor S → C. Otherwise, according to your definition, the dual of the category of sets (Set^op) is a "subcategory" of the category of sets Set, which is not true. Similarly, according to your definition, every countable group is isomorphic to a subgroup of $(\mathbb N,+)$. ComputScientist (talk) 21:21, 7 June 2011 (UTC)

I agree about functoriality. But I don't agree about Set^op. Set does have morphisms and they do go in the opposite direction in Set^op. So Mor(Set^op) is not a subclass of Mor(Set). Spaetzle (talk) 13:27, 9 June 2011 (UTC)

Hi -- It depends what you mean by $\mathrm{Mor}(\mathcal{C})$ -- does that somehow contain the information about what the source and target of each morphism is? How does that work? Usually there is extra data in the definition of a category, saying what the domain/codomain of each morphism is. You need to check that this data is the same in S and C. I'd say that's part of what the functoriality does. ComputScientist (talk) 14:38, 9 June 2011 (UTC)