surjection of localic infinity toposes? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T05:39:55Z http://mathoverflow.net/feeds/question/99616 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/99616/surjection-of-localic-infinity-toposes surjection of localic infinity toposes? Simon Henry 2012-06-14T15:07:30Z 2012-06-14T20:55:59Z <p>Hello!</p> <p>Is there a simple 'topological' condition to detect whenever a morphism of locales $f : X \rightarrow Y$ induces a surjection of infinity-toposes <code>$f : \mathrm{Sh}_{\infty}(X) \rightarrow \mathrm{Sh}_{\infty}(Y)$</code> (i.e. such that $f^*$ is conservative)?</p> <p>It's not enough to assume that f is a surjection of locales: indeed, if we take a topological space $X$ such that <code>$\mathrm{Sh}_{\infty}(X)$</code> is not hypercomplete, and $X^{\mathrm{disc}}$ is its space of points endowed with the discrete topology, then <code>$\mathrm{Sh}_{\infty}(X^{\mathrm{disc}}) \rightarrow \mathrm{Sh}_\infty (X)$</code> can't be a surjection, because the pullback of an $\infty$-connected map in <code>$\mathrm{Sh}_\infty (X)$</code> is a weak equivalence in <code>$\mathrm{Sh}_{\infty}(X^{\mathrm{disc}})$</code>...</p> <p>Thank you!</p>