Is there a canonical way to go back from $\hat{\mathbb{C}}$ (presheaves) to $\mathbb{C}$

Question

Is there a “canonical” way to go back from $\hat{\mathbb{C}}$ (category of presheaves) to $\mathbb{C}$. Here we define $\hat{\mathbb{C}}$ to be the category of functors from $\mathbb{C}^{op} \rightarrow \mathrm{SET}$. I just started category theory, and it seems that the Yoneda embedding is sort of the canonical way to go from $\mathbb{C}$ to $\hat{\mathbb{C}}$. So I was wondering if going back is just as “easy”.

Cheers!

Answer 1

You can recover a category $\mathcal{C}$ from its category of presheaves, up to Cauchy completion. In particular that means that if $\mathcal{C}$ is Cauchy complete, then you can recover it (up to equivalence) from its category of presheaves. A lot of information about this can be found on the nLab page about Cauchy complete categories.

Let me just state the major relevant points here. The numbering in this answer refers to the (current) numbering on the nLab page.

Definition. A category $\mathcal{C}$ is called Cauchy complete if every idempotent splits. That is, for every $e: C \to C$ such that $ee = e$, there are $r: C \to D$ and $i: D \to C$ such that $ri = Id_D$ and $ir = e$.

Note that if a category has equalizers, then it is Cauchy complete (take $i$ to be the equalizer of $e$ and $Id_C$, and $r$ the universal arrow corresponding to $e$).

Then Proposition 2.3 states that if $\mathcal{C}$ is Cauchy complete, then it can be recovered from $\mathbf{Set}^{\mathcal{C}^\text{op}}$ as the full subcategory of "tiny objects" or "small projective objects".

Definition. In a cocomplete category $\mathcal{E}$ we say that an object $E$ is tiny or small projective if the hom-functor $\operatorname{Hom}(E, -): \mathcal{E} \to \mathbf{Set}$ preserves all small colimits.

There are equivalent characterisations possible. For example, we could also recover $\mathcal{C}$ as the indecomposable projectives.

If $\mathcal{C}$ is not Cauchy complete, then taking the tiny objects will give us the Cauchy completion $\bar{\mathcal{C}}$. We cannot hope to do better, because there is an equivalence of categories $$ \mathbf{Set}^{\mathcal{C}^\text{op}} \simeq \mathbf{Set}^{\bar{\mathcal{C}}^\text{op}}. $$

Answer 2

For another sense of "go back", you might ask for a functor $\hat C\to C$ somehow nicely related to the Yoneda embedding. While this seems a bit unlikely since $\hat C$ is much "bigger" than $C$, it is actually extremely common for locally small $C$ that the Yoneda embedding has a left adjoint $L:\hat C\to C$. Such categories are called total.