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)

Definition incomplete?

The definition of subcategory seems to be missing a statement that the law of composition must be the same as in the parent category. If we have a parent category of just 3 objects A,B,C and arrows f:A->B, g:B->C, h,h':A->C and the same objects and arrows in the child category, then I can define gf=h in the parent category and gf=h' in the child category. All requirements are fulfilled, but this probably would not be called a subcategory. Leen Droogendijk (talk) 08:37, 22 December 2012 (UTC)

The definition was correct. I have tried to clarify the article. ComputScientist (talk) 20:10, 28 December 2012 (UTC)

Why is "full subcategory" missing from the section Types of categories?

The section Types of categories mentions "strictly full" and "lluf" categories, but makes no mention of the extremely common term "full subcategory".

Even if there is some good reason for this, it ought to be addressed within that section.

I hope someone knowledgeable on this subject can fix this blatant gap in the article. 2601:200:C000:1A0:81B6:62D0:48E8:FCF8 (talk) 18:30, 15 April 2021 (UTC)