Descent of Morphisms of Sheaves

2012-05-09T21:12:12Z

While reading Brylinski I am trying to understand the descent of morphisms of sheaves.

In trying to form a new definition of a presheaf $A$ over a space $X$, we associate to each surjective local homeomorphism $f:Y \to X$ a set, denoted $A(Y\xrightarrow{f}X)$. The "restriction" condition of a presheaf amounts to: given a surjective local homeomorphism $g:Z \to Y$ we have a pullback map $g^{-1}:A(Y\xrightarrow{f}X) \to A(Z \xrightarrow{fg}X)$. The transitivity property for these "restriction" (pullback) maps is that given any diagram $$W \xrightarrow{h} Z \xrightarrow{g} Y \xrightarrow{f} X$$ having $(gh)^{-1} = h^{-1} \circ g^{-1}$ as pullbacks $A(Y\xrightarrow{f}X) \to A(W \xrightarrow{fgh} Z)$. $\ \$

If $A$ is already a presheaf, in the good 'ol fashioned sense, then we can define our assignment $A(Y\xrightarrow{f}X)$ to be the global sections of $Y$ given by the inverse image of $A$ on $X$, i.e. $\Gamma(Y, f^{-1}A)$

I have 2 questions:

Is it true that if $A$ is already a sheaf in the good 'ol fashioned sense, then the above property (transitivity of the "restriction") is satisfied? My proof feels trivial, hence my worry. Also, I am uneasy since Brylisnki doesn't state this fact but instead says it "should" be true.
He later comments that as functors from the category of sheaves on $Y$ to the category of sheaves on $W$ , $h^{-1}\circ g^{-1}$ and $(gh)^{-1}$ are NOT equal; but there is a natural transformation. Why are these two functors not equal? It seems like they send the same sheaves to the same places, unless of course I am making identifications of categories that I don't realize?

Answer by Moshe for Descent of Morphisms of Sheaves

2012-05-10T13:46:02Z

Yes, the property is satisfied (note that $\Gamma(Y,f^{-1}A)=A(f(Y))$, since $f$ is open). I didn't read the book, but I imagine that the point is not to give a strange definition of a sheaf on a topological space, but rather to motivate the generalisation to situations where you know what maps you wish to consider to be local homeomorphisms, but they are not actually local homeomorphisms for any (classical) topology. The classical example is the etale (Grothendieck) topology on schemes. The book "Sheaves in geometry and logic" has a good exposition of these ideas.

Descent of Morphisms of Sheaves - MathOverflow

Descent of Morphisms of Sheaves

Answer by Moshe for Descent of Morphisms of Sheaves