# 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.

• Abelian groups inside groups. Commented Dec 26, 2017 at 21:31

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.
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.
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...