When does the left-adjoint to a geometric morphism preserve epis? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T09:11:54Z http://mathoverflow.net/feeds/question/19862 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/19862/when-does-the-left-adjoint-to-a-geometric-morphism-preserve-epis When does the left-adjoint to a geometric morphism preserve epis? David Carchedi 2010-03-30T18:36:35Z 2010-04-01T08:20:47Z <p>Suppose I have a functor $f:(C,J)\to(D,K)$ between Grothendieck sites. Is there a condition on $f$ such that $f_!$ (the left adjoint to $f^*$) sends "$J$-epimorphisms", to $K$-epimorphisms, where by $J$-epimorphism I mean:</p> <p>$h:X\to Y$ such that for all $C$, and all $y \in Y(C)$, there exists a cover $(g_i:C_i\to C)$ in $J$ and $y_i \in X(C_i)$ such that for all $i$, $Y(g_i)(y)=h(y_i)$.</p> <p>EDIT: If X and Y are sheaves, then the notion of "J-epimorphism" coinincides with the categorical epis. As mentioned by David Brown, ANY left adjoint will preserves epis.</p> <p>In fact, in the situation in which I was interested, I actually have such a (appropriate analogue of a) J-epimorphism between a sheaf and a stack, so, since f_! is a left adjoint, it will preserve this.</p> http://mathoverflow.net/questions/19862/when-does-the-left-adjoint-to-a-geometric-morphism-preserve-epis/19869#19869 Answer by David Zureick-Brown for When does the left-adjoint to a geometric morphism preserve epis? David Zureick-Brown 2010-03-30T19:14:52Z 2010-03-30T19:14:52Z <p>A left adjoint functor always takes epimorphisms to epimorphisms. This is easy to see using the <a href="http://en.wikipedia.org/wiki/Epimorphism" rel="nofollow">usual</a> definition of epimorphism and checking that this is equivalent to your definition for sheaves on a site.</p>