Subcategories which still give a Yoneda embedding

Question

If $\mathbf{C}$ is a category, then the Yoneda functor which sends $a$ to $Hom_\mathbf{C}(-,a)$ is a fully faithful embedding of categories $$ \mathbf{C}\rightarrow \mathbf{Func}(\mathbf{C}^{op},\mathbf{Set})$$

Given any subcategory $\mathbf{B}\subseteq \mathbf{C}$, there is a similar functor $$ \mathbf{C}\rightarrow \mathbf{Func}(\mathbf{B}^{op},\mathbf{Set})$$ which restricts $Hom_\mathbf{C}(-,a)$ to arguments in $\mathbf{B}$.

This functor need not be an embedding in general, but there are many examples where it is, and where its an interesting statement that it is. Examples:

• The category of finite sets, inside the category of sets; or more generally, any subcategory of set containing a non-empty set will work.
• The category of sets and maps which factor through a finite set, inside the category of sets (so $\mathbf{B}$ need not be full).
• The category of affine schemes (ie, $\mathbf{Comm}^{op}$), inside the category of schemes.
• The category of open subsets of $\mathbb{R}^n$ and smooth maps between them, inside the category of smooth $n$-dimensional manifolds.
• The category of abelian groups, inside the category of groups.

My question is, is there a name or a nice characterization of this subcategories? I'm writing up some notes for a class this semester, and I want to make a remark to this effect. In the absence of a good name, I was going to call them `Yoneda subcategories'.

Answer 1

Let $B$ be a full subcategory of $C$. The condition that $C \to \mathrm{Set}^{C^{op}} \to \mathrm{Set}^{B^{op}}$ is fully faithful is easily seen to be equivalent to the condition that for every $c \in C$ the set of all morphisms from objects in $B$ to $c$ is a colimit diagram. In other words, $c$ is the colimit of the canonical diagram $(B \downarrow c) \to B \to C$. Therefore one then calls $B$ a dense subcategory of $C$. If you just require that every $c$ is the colimit of some diagram which factors over $B$, then $B$ is called colimit-dense. This is a weaker condition: $\{R\}$ is not dense in $\mathrm{Mod}(R)$, but it is colimit dense. Note that already $\{R \oplus R\}$ is dense in $\mathrm{Mod}(R)$.

You mention that the open subsets of $\mathbb{R}^n$ constitute a dense subcategory of the smooth $n$-manifolds. But actually already $\mathbb{R}^n$ (together with all its endomorphisms) suffices since this is also the local model for the open subsets of $\mathbb{R}^n$. Consequently, we get a fully faithful embedding from the category of smooth $n$-manifolds into the category of right $M$-sets, where $M$ is the monoid of all continuous maps $\mathbb{R}^n \to \mathbb{R}^n$. I don't know if this is of use at all, but I think this is an interesting point of view. It is a sort of algebraic representation from a geometric category of interest, very similar to the functor of points approach in algebraic geometry (where we take affine schemes as a dense subcategory).

You can find more about dense subcategories at the nlab and in "Abstract and Concrete Categories - The Joy of Cats" (Examples 2.11 + Exercise 12.D, online). The definition also plays a central role in the definition of locally presentable categories; see the book by Adamek and Rosicky.

Yoneda's Lemma now just asserts that $C$ is dense in $C$, what else should we expect? There are other formal similarities with the topological notion of the same name: Let $B \subseteq D \subseteq C$ be full subcategories. If $B \subseteq C$ is dense, then $D \subseteq C$ is dense. [If $B \subseteq D$ and $D \subseteq C$ are dense, then $B \subseteq C$ is dense. -- This is not correct, see comment.]