Cohesive ∞-Toposes - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T10:59:51Z http://mathoverflow.net/feeds/question/42911 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/42911/cohesive-toposes Cohesive ∞-Toposes Urs Schreiber 2010-10-20T16:47:17Z 2010-10-20T19:34:53Z <p>Say an $\infty$-topos $\mathbf{H}$ is <strong>cohesive</strong> if its global section geometric morphism $\Gamma : \mathbf{H} \to \infty \mathrm{Grpd}$ admits a further left adjoint $\Pi$ and a further right adjoint $\mathrm{Codisc}$: </p> <p>$$(\Pi \dashv \mathrm{Disc} \dashv \Gamma \dashv \mathrm{Codisc}) : \mathbf{H} \to \infty \mathrm{Grpd}$$ </p> <p>with $\mathrm{Disc}$ and $\mathrm{Codisc}$ full and faithful and such that $\Pi$ moreover preserves finite products.</p> <p>Here</p> <ol> <li><p>$\mathrm{Codisc}$ induces an $\infty$-quasitopos $\mathrm{Conc}(\mathbf{H}) \hookrightarrow \mathbf{H}$ of <em>concrete</em> objects, those that look like $\infty$-groupoids <em>equipped with extra cohesive structure</em> : for instance with topology, or with smooth structure.</p></li> <li><p>$\Pi$ sends an object $X$ to its geometric path $\infty$-groupoid, which co-classifies locally constant $\infty$--stacks on $X$;</p></li> </ol> <p>More details are at <a href="http://ncatlab.org/nlab/show/cohesive+(infinity,1)-topos" rel="nofollow">http://ncatlab.org/nlab/show/cohesive+(infinity,1)-topos</a> , where also some (classes of) examples are discussed.</p> <p><strong>Question.</strong> What other (classes of) examples can you find? </p> <p>Specifically, which "derived" $\infty$-toposes (over $\infty$-sites of certain duals of algebras over $\infty$-algebraic theories) are cohesive?</p>