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Say an $\infty$-topos $\mathbf{H}$ is cohesive if its global section geometric morphism $\Gamma : \mathbf{H} \to \infty \mathrm{Grpd}$ admits a further left adjoint $\Pi$ and a further right adjoint $\mathrm{Codisc}$:

$$(\Pi \dashv \mathrm{Disc} \dashv \Gamma \dashv \mathrm{Codisc}) : \mathbf{H} \to \infty \mathrm{Grpd}$$

with $\mathrm{Disc}$ and $\mathrm{Codisc}$ full and faithful and such that $\Pi$ moreover preserves finite products.

Here

  1. $\mathrm{Codisc}$ induces an $\infty$-quasitopos $\mathrm{Conc}(\mathbf{H}) \hookrightarrow \mathbf{H}$ of concrete objects, those that look like $\infty$-groupoids equipped with extra cohesive structure : for instance with topology, or with smooth structure.

  2. $\Pi$ sends an object $X$ to its geometric path $\infty$-groupoid, which co-classifies locally constant $\infty$--stacks on $X$;

More details are at http://ncatlab.org/nlab/show/cohesive+(infinity,1)-topos , where also some (classes of) examples are discussed.

Question. What other (classes of) examples can you find?

Specifically, which "derived" $\infty$-toposes (over $\infty$-sites of certain duals of algebras over $\infty$-algebraic theories) are cohesive?

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