direct image (functor)

If $f:X\to Y$ is a continuous map of topological spaces, and if ${\bf Sheaves}(X)$ is the category of sheaves of abelian groups on $X$ (and similarly for ${\bf Sheaves}(Y)$ ), then the direct image functor $f_{*}:{\bf Sheaves}(X)\to{\bf Sheaves}(Y)$ sends a sheaf $\mathcal{F}$ on $X$ to its direct image $f_{*}\mathcal{F}$ on $Y$ . A morphism of sheaves $g:\mathcal{F}\to\mathcal{G}$ obviously gives rise to a morphism of sheaves $f_{*}g:f_{*}\mathcal{F}\to f_{*}\mathcal{G}$ , and this determines a functor.

If $\mathcal{F}$ is a sheaf of abelian groups (or anything else), so is $f_{*}\mathcal{F}$ , so likewise we get direct image functors $f_{*}:{\bf Ab}(X)\to{\bf Ab}(Y)$ , where ${\bf Ab}(X)$ is the category of sheaves of abelian groups on $X$ .

Title	direct image (functor)
Canonical name	DirectImagefunctor
Date of creation	2013-03-22 12:03:13
Last modified on	2013-03-22 12:03:13
Owner	bwebste (988)
Last modified by	bwebste (988)
Numerical id	6
Author	bwebste (988)
Entry type	Definition
Classification	msc 14F05
Related topic	DirectImageSheaf