# Intuition behind the notion of full subcategory?

Maybe this is too trivial but category theory is not my expertise.

Well, let $\mathcal{A}$ be a subcategory of $\mathcal{B}$ and let $\imath: \mathcal{A}\longrightarrow \mathcal{B}$ be the "inclusion" functor. We say $\mathcal{A}$ is a full subcategory of $\mathcal{B}$ if $\imath$ is full.

Can anyone give me some intuiton behind the concept of full subcategory in contrast with that of a subcategory? What should I have in mind in order to identify full subcategories in practice?

I'd also appreciate examples of full subcategories.

Thanks.

## 1 Answer

let $\mathcal{A}$ be a subcategory of $\mathcal{B}$. This says $\mathcal{A}$ contains a subset of the objects of $\mathcal{B}$ and a subset of the morphisms (arrows, henceforth) of $\mathcal{B}$ and meets the criteria to still be a category.

One way to do that is to discard all the arrows that aren't demanded by the category axioms, leaving only the identity arrows. This seems a rather paltry thing.

Another way to do that is to keep all the arrows of $\mathcal{B}$ that happen to have domain and codomain in $\mathcal{A}$. This would be the "fullest" that a sub-thing of $\mathcal{B}$ could be. Such a thing is called a full subcategory.

A nice thing about full subcategories is that we only need to specify which objects are in the subcategory -- all arrows that could be in the subcategory are in the subcategory.

A functor is full if the map of arrows is surjective. For the inclusion functor, which is injective, to be full, the map from arrows to arrows must be a bijection. That is, in the subcategory context, the inclusion functor takes all the objects to themselves and every possible (compatible?) arrow to itself as well.

In my limited experience, full subcategories are typical. One has to restrict away certain arrows to get a less than full subcategory. For instance, the set of $3$-manifolds with homeomorphisms as arrows has as a subcategory the set of $3$-manifolds with isotopies as arrows (... and if we have been talking implicitly about smooth manifolds, the first has as a subcategory the set of $3$-manifolds with diffeomorphisms as arrows and the second a subcategory with isotopic diffeomorphisms as arrows. And now we're ankle-deep in mapping class groups.) Your particular application domain may not have the same pattern of categorification as mine...