Cotopological Localization

Let $\mathcal{E}$ be an Infinity Topos, and $\iota:\mathcal{F}\hookrightarrow\mathcal{E}$ be an Accessible Left Exact Localization of $\mathcal{E}$. We say that $\mathcal{F}$ is a cotopological localization of $\mathcal{E}$ if the left adjoint $L$ of $\iota$ satisfies one of the following equivalent conditions:

• For every Embedding $u$ in $\mathcal{E}$, if $L(u)$ is an Equivalence in $\mathcal{F}$, then $u$ is an equivalence in $\mathcal{E}$. In other words, $L$ is a Conservative Infinity Functor when restricted to embeddings.
• For every morphism $u$ in $\mathcal{E}$, if $L(u)$ is an Equivalence in $\mathcal{F}$, then $u$ is Infinity Connected.

This second condition explains why cotopological localizations do not appear in the study of n-topoi; the theory of infinity connected morphisms there is trivial!

References

• Lurie, Jacob. “Higher Topos Theory,” July 31, 2008. https://doi.org/10.48550/arXiv.math/0608040. (6.5.2.17)