nLab sheaf toposes are equivalently the left exact reflective subcategories of presheaf toposes

Statement

Proposition

Let $(\mathcal{C}, \tau)$ be a small site. Then the full subcategory inclusion $i \colon Sh(\mathcal{C},\tau) \hookrightarrow PSh(\mathcal{C})$ of its category of sheaves (Grothendieck topos) into its category of presheaves is a reflective subcategory inclusion

$$Sh(\mathcal{C},\tau) \underoverset {\underset{i}{\hookrightarrow}} {\overset{L}{\longleftarrow}} {\bot} PSh(\mathcal{C})$$

such that the reflector $L \colon PSh(\mathcal{C}) \to Sh(\mathcal{C})$ preserves finite limits (the reflector being sheafification).

Moreover, up to equivalence, every Grothendieck topos arises this way: Given a small category $\mathcal{C}$ there is a bijection between

1. the equivalence classes of left exact reflective subcategories $\mathcal{E} \hookrightarrow PSh(\mathcal{C})$ of the category of presheaves

2. Grothendieck topologies$\tau$ on $\mathcal{C}$,

which is such that $\mathcal{E} \simeq Sh(\mathcal{C}, \tau)$.

(e.g. Borceux 94, prop. 3.5.4, cor. 3.5.5, Johnstone, C.2.1.11, or Kelly).

Proposition

(accessible embedding is implied)

In the situation of prop. it follows that the inclusion $i \colon Sh(\mathcal{C},\tau) \hookrightarrow PSh(\mathcal{C})$ is an accessible functor, hence an accessible reflective subcategory inclusion.

Proof

Every Grothendieck topos like $Sh(\mathcal{C}, \tau)$ and $PSh(\mathcal{C})$ is a locally presentable category (this prop., Borceux 94, vol 3, prop. 3.4.16). Therefore with prop. the statement follows by the adjoint functor theorem for locally presentable categories (this prop.).

Remark

(generalization to (∞,1)-toposes)

For (∞,1)-toposes the accessibility of the reflection is no longer implied in general, contrary to prop. above, but needs to be required. It is however still implied for topological localizations (Lurie, Cor. 6.2.1.6).

References

• Francis Borceux, vol. 3, section 3.5 of Handbook of Categorical Algebra, Cambridge University Press (1994)

• Peter Johnstone, Sketches of an Elephant

• Jacob Lurie, Higher Topos Theory

• Shane Kelly, What is the relationship between Grothendieck pretopologies and Grothendieck topologies?, 2022 (web) .