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

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.

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(Note that you can use latex as you would normally. It makes subscripty expressions easier to read.) – François G. Dorais Mar 30 '10 at 18:52
Certainly $f^*$ does not always have a left adjoint (e.g. $f^*$ might not even be exact). If the functor f induces a morphism of sites (so that in particular $f^*$ is exact), I believe $f^*$ still may not have a left adjoint. – David Zureick-Brown Mar 30 '10 at 19:00
f^* certainly DOES have a left adjoint SINCE the geometric morphism arises from a functor from C to D. – David Carchedi Mar 30 '10 at 19:10
(the left adjoint is given by the left-kan extension of $C \mapsto Hom( blank, f(U))$) – David Carchedi Mar 30 '10 at 19:11
@David C: The trick to get good answers is to ask good questions. Well motivated questions usually get excellent answers. As is, your question is hard to understand and motivate. It's only after reading your comment to David B's answer that I understood what was going on. That comment would fit in very well within the question and make it much easier to understand. (A few notational fixes wouldn't hurt either.) – François G. Dorais Mar 31 '10 at 12:46

Maybe this works in general: it seems like your definition of `J-epimorphism' is equivalent to the sheafification of the morphism being an epi-morphism in the category of sheaves. Then your question follows from the case of sheaves (since, in any case, you have to sheafify $f_!$ to get a left adjoint to $f^*$). – David Zureick-Brown Mar 30 '10 at 20:06
You don't need to think about sheaves at all to see that left adjoints preserve epis. An arrow $f$ is epic iff a certain diagram is a colimit, and left adjoints preserve colimits. The diagram in question is the commuting square whose top and left morphisms are $f$ and bottom and right morphisms are identities (arrows pointing down and to the right): $f$ is epic iff this square is a pushout. – Tim Campion Feb 10 '14 at 23:56