$S$-local objects of presheaves are reflective and characterize local presentability

Question

Let $PSh(\mathcal{C})$ be the category of presheaves on a small category $\mathcal{C}$. Let also $S$ be any fixed set of morphisms in $PSh(\mathcal{C})$. I say that an object $F\in PSh(\mathcal{C})$ is $S-$local if, for all $g\in S$, the map $PSh(\mathcal{C})(g,F)$ is an isomorphism of sets. Let now $PSh(\mathcal{C})_{S}$ denote the full subcategory of $PSh(\mathcal{C})$ given by $S-$local objects.

1. I am told that $PSh(\mathcal{C})_{S}$ is a reflective subcategory of $PSh(\mathcal{C})$, but I can not find a left adjoint to the inclusion functor. Somehow, such a left adjoint should be something like the localization of $PSh(\mathcal{C})$ at $S$, I guess, but I can not see how.
2. nlab says that, up to equivalence, these subcategories of $S-$local objects are all and only the locally presentable category, but it seems to provide no references for this result and I did not manage to find it mentioned explicitely somewhere else. Is there any kind of reference for this statement? How could one prove it?

Any help with any of the above questions would be greately appreciated. (In particular, for the second problem above, even a simple list of intermediate statements that lead to the desired result would suffice).

Answer 1

All references below refer to the following

Source: Adamek, Rosicky: Locally Presentable and Accessible Categories

The import theorem for you is the following, appearing as Cor 2.45 (see Rem 0.7 for the connection to adjoints):

Theorem: An accessible functor $F:{\mathscr C}\to{\mathscr D}$ is a right adjoint if and only if it commutes with limits.

For completeness, here are the definitions:

• Def. 2.1: Given a regular cardinal $\lambda$, a category ${\mathscr C}$ is called $\lambda$-accessible if it has $\lambda$-directed colimits and there is a set of $\lambda$-presentable objects generating ${\mathscr C}$ under $\lambda$-directed colimits. It is called accessible if it is $\lambda$-accessible for some $\lambda$.
• Def. 2.16: Given a regular cardinal $\lambda$, a functor $F:{\mathscr C}\to{\mathscr D}$ is called $\lambda$-accessible if ${\mathscr C}$ and ${\mathscr D}$ are $\lambda$-accessible and $F$ preserves $\lambda$-directed colimits. It is called accessible if it is $\lambda$-accessible for some $\lambda$.

Now, the presheaf category over a small category is always accessible (Thm 2.39)

Theorem: For any accessible category ${\mathscr K}$ and any small category ${\mathscr A}$ the category ${\mathscr K}^{\mathscr A}$ is accessible.

For ${\mathscr K}=\textbf{Set}$ one can even say that $\textbf{Set}^{\mathscr A}$ is locally finitely presented (Def 1.9): Every representable presheaf $\text{Hom}_{\mathscr A}(X,-)$ is finitely presented in $\textbf{Set}^{\mathscr A}$, and as usual any presheaf is a colimit of representable ones. This is Example 1.12.

Now, given any set of objects $S$ is a locally presentable (:= accessible + cocomplete) category ${\mathscr K}$, you can consider the full subcategory ${\mathscr K}_S$ of $S$-local objects as you defined in your question. As ${\mathscr K}_S$ is closed under limits in ${\mathscr K}$, we see that it for proving that it is reflective in ${\mathscr K}$ it suffices to show that ${\mathscr K}_S\hookrightarrow {\mathscr K}$ is accessible.

In a locally presentable category, any object is $\lambda$-presentable for some regular cardinal $\lambda$ (Prop 1.16), so we may pick a $\lambda$ such that all objects of $S$ are $\lambda$-presentable. Then it is clear that ${\mathscr K}_S$ is closed under $\lambda$-directed colimits in ${\mathscr K}$, so ${\mathscr K}_S\hookrightarrow{\mathscr K}$ is an accessible embedding (Def 2.35). Hence, what remains to show is that ${\mathscr K}_S$ is itself accessible, which (Cor 2.36) is the same as asking that it is closed under $\mu$-pure subobjects for some $\mu$. The definition of $\mu$-pure is a bit technical (Def 2.27), but I encourage you to look at it and check that under our assumptions (and using that $\mu$-pure morphisms are always monomorphisms in accessible categories, Prop 2.29), the $S$-local objects are indeed closed under $\mu := \lambda$-pure subobjects, and you get the desired result.