When does the left-adjoint to a geometric morphism preserve epis?

2010-03-30T18:36:35Z

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:

$h:X\to Y$ 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 $i$, $Y(g_i)(y)=h(y_i)$.

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.

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.

Answer by David Zureick-Brown for When does the left-adjoint to a geometric morphism preserve epis?

2010-03-30T19:14:52Z

A left adjoint functor always takes epimorphisms to epimorphisms. This is easy to see using the usual definition of epimorphism and checking that this is equivalent to your definition for sheaves on a site.

When does the left-adjoint to a geometric morphism preserve epis? - MathOverflow

When does the left-adjoint to a geometric morphism preserve epis?

Answer by David Zureick-Brown for When does the left-adjoint to a geometric morphism preserve epis?