# Inverse Image as the left adjoint to pushforward

Assume $X$ and $Y$ are topological spaces, $f : X \to Y$ is a continuous map. Let ${\bf Sh}(X)$, ${\bf Sh}(Y)$ be the category of sheaves on $X$ and $Y$ respectively. Modulo existence issues we can define the inverse image functor $f^{-1} : {\bf Sh}(Y) \to {\bf Sh}(X)$ to be the left adjoint to the push forward functor $f_{*} : {\bf Sh}(X) \to {\bf Sh}(Y)$ which is easily described.

My question is this: Using this definition of the inverse image functor, how can I show (without explicitly constructing the functor) that it respects stalks? i.e is there a completely categorical reason why the left adjoint to the push forward functor respects stalks?

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edit: I Changed the title to better reflect the question! –  DBr Apr 13 '11 at 22:53

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A functor which is a left adjoint preserves colimits (see for instance Mac Lane, "Categories for the working mathematician", chapter V, section 5 "Adjoints on Limits"); particularly, stalks.

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See, I have thought about this. I understand that inverse image preserves colimits of sheaves. How does this imply that inverse image preserves colimit of sections?? –  DBr Apr 14 '11 at 0:06
You're right: it's not the same thing. –  a.r. Apr 14 '11 at 0:50