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
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
the equivalence classes of left exact reflective subcategories $\mathcal{E} \hookrightarrow PSh(\mathcal{C})$ of the category of presheaves
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).
(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.
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.).
(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).
Francis Borceux, vol. 3, section 3.5 of Handbook of Categorical Algebra, Cambridge University Press (1994)
Shane Kelly, What is the relationship between Grothendieck pretopologies and Grothendieck topologies?, 2022 (web) .
Last revised on February 9, 2024 at 03:18:44. See the history of this page for a list of all contributions to it.