direct image (functor)

If f:XY is a continuous map of topological spacesMathworldPlanetmath, and if 𝐒𝐡𝐞𝐚𝐯𝐞𝐬(X) is the category of sheaves of abelian groupsMathworldPlanetmath on X (and similarly for 𝐒𝐡𝐞𝐚𝐯𝐞𝐬(Y)), then the direct image functor f*:𝐒𝐡𝐞𝐚𝐯𝐞𝐬(X)𝐒𝐡𝐞𝐚𝐯𝐞𝐬(Y) sends a sheaf on X to its direct imagePlanetmathPlanetmath f* on Y. A morphism of sheaves g:𝒢 obviously gives rise to a morphism of sheaves f*g:f*f*𝒢, and this determines a functorMathworldPlanetmath.

If is a sheaf of abelian groups (or anything else), so is f*, so likewise we get direct image functors f*:𝐀𝐛(X)𝐀𝐛(Y), where 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