Grothendieck Topos

A Grothendieck topos $\mathcal{E}$ is a category that admits a Geometric Embedding to a Presheaf Category.

Explicitly, $\mathcal{E}$ is a Grothendieck Topos if there is some category $\mathcal{C}$, a Fully Faithful Functor $\iota:\mathcal{E}\to\hat{\mathcal{C}}$, and a Left Exact Left Adjoint $L\dashv\iota$.

As Categories of Sheaves

Every Grothendieck topos is equivalently a category of Sheaves over a small Site.

In Infinity Topoi

This section comes from Left-Exact Localizations of $\infty$-Topoi I: Higher Sheaves

Note that this is a particular feature of 1-Topos theory, and does not extend to Infinity Topoi. It remains true that every infinity topos is a Left Exact Localization of a category of infinity presheaves for a small Infinity Category $\mathcal{C}$, but these localizations can no longer be completely described via a Grothendieck Topology.

The crux of the issue is that every left-exect localization is not generated by inverting Embeddings. Rather, every left-exact localization $\hat{\mathcal{C}}\to\mathcal{E}$ is a composite of a Topological Localization $\hat{\mathcal{C}}\to\mathcal{E^{\prime}}$ and a Cotopological Localization $\mathcal{E^{\prime}}\to\mathcal{E}$. The topological localization can still be described entirely as inverting a class of Embeddings, but the cotopological class is a localization that inverts no embeddings. The classic theory of Grothendieck Topologies applies to the topological embeddings, but not the cotopological ones.