The Sigma D localization | Higher Topos Theory notes

Construction 7.32.

Consider the functor

D^!\colon \Shv_{\tau_D^{\ad}}(C^{\ad})\catop\longrightarrow\Cat.

Let $\Sigma_D$ be the largest congruence contained in the inverse image of the isomorphisms under this functor. More explicitly,

\Sigma_D=\{f\mid \text{every base change of every $\Delta^nf$ is a $D^!$-isomorphism}\}.

Every morphism of $\Sigma_D$ is $\infty$ -connected: the condition for $$n=0$$ implies that every base change is a $$D$$ -cover and hence an effective epimorphism, and the same applies to every diagonal.

Assume that $\Sigma_D$ has small generation, so that it defines an accessible left exact localization. We define $\Shv_{\tau_D}^+(C)$ by the pullback square

where the bottom functor is the fully faithful right adjoint of the localization. The top functor is likewise the fully faithful right adjoint of a left exact localization. The left vertical functor admits a left adjoint, and the pair

\left(\Shv_{\tau_D}^+(C),\Shv_{\tau_D^{\ad}}(C^{\ad})[\Sigma_D^{-1}]\right)

is a fractured topos.