internal version of a flat functor? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T03:42:20Z http://mathoverflow.net/feeds/question/36156 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/36156/internal-version-of-a-flat-functor internal version of a flat functor? David Roberts 2010-08-20T02:11:05Z 2010-08-20T12:00:44Z <p>I'm working out of <em>Sheaves in geometry and logic</em>, for reference.</p> <p>There is a characterisation of flat functors $A:C \to Set$ as those such that the Grothendieck construction $\int_C A$ is a filtering category. There are more general versions of this result, in which $Set$ is replaced by a more general topos. One should also be able to characterise those discrete opfibrations that arise from flat functors (up to iso/equiv?). How about if we replace $C$ by an internal category, in a topos $E$ say? Then functors out of $C$ are replaced by discrete opfibrations over $C$ in $E$. </p> <p>My question is this:</p> <blockquote> <p>What sort of thing should be considered as the analogue of a flat functor in the internal setting? </p> </blockquote> http://mathoverflow.net/questions/36156/internal-version-of-a-flat-functor/36182#36182 Answer by Finn Lawler for internal version of a flat functor? Finn Lawler 2010-08-20T12:00:44Z 2010-08-20T12:00:44Z <p>A flat internal presheaf/discrete fibration $F \to C$ is simply one whose total category F is filtered in the internal sense (see <em>Topos Theory</em> 2.51 (filteredness), 4.31 (flatness); <em>Elephant</em> B.2.6.2, B.3.2.3). The definition just reexpresses filteredness by requiring that certain maps (e.g. $F_0 \to 1$) are regular epis (presumably you can replace these with covers for your favourite Grothendieck topology).</p> <p>I <em>think</em> that if C = BG is a group then these are exactly the G-torsors in E. I'm not sure whether this extends to flat = locally representable for C a general internal category, though. </p>