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In mathematics, specifically in algebraic topology and algebraic geometry, an inverse image functor is a contravariant construction of sheaves; here “contravariant” in the sense given a map :, the inverse image functor is a functor from the category of sheaves on Y to the category of sheaves on X.
Similarly, the inverse image (or preimage) of a given subset of the codomain is the set of all elements of that map to a member of . The image of the function f {\displaystyle f} is the set of all output values it may produce, that is, the image of X {\displaystyle X} .
direct image with compact support f! : Sh(X) → Sh(Y) exceptional inverse image Rf! : D(Sh(Y)) → D(Sh(X)). The exclamation mark is often pronounced "shriek" (slang for exclamation mark), and the maps called "f shriek" or "f lower shriek" and "f upper shriek"—see also shriek map. The exceptional inverse image is in general defined on the ...
If f(x)=y, then g(y)=x. The function g must equal the inverse of f on the image of f, but may take any values for elements of Y not in the image. A function f with nonempty domain is injective if and only if it has a left inverse. [21] An elementary proof runs as follows: If g is the left inverse of f, and f(x) = f(y), then g(f(x)) = g(f(y ...
A co-fibred-category is an -category such that direct image exists for each morphism in and that the composition of direct images is a direct image. A co-cleavage and a co-splitting are defined similarly, corresponding to direct image functors instead of inverse image functors.
Let f: X → Y be a continuous map of topological spaces or a morphism of schemes. Then the exceptional inverse image is a functor Rf!: D(Y) → D(X) where D(–) denotes the derived category of sheaves of abelian groups or modules over a fixed ring. It is defined to be the right adjoint of the total derived functor Rf! of the direct image with ...
On the other hand, the inverse image or preimage under f of an element y of the codomain Y is the set of all elements of the domain X whose images under f equal y. [6] In symbols, the preimage of y is denoted by f − 1 ( y ) {\displaystyle f^{-1}(y)} and is given by the equation
The set is closed in and its image under is closed in because is a closed map. Hence the set V k = Y ∖ f ( X ∖ ∪ a ∈ γ k U a ) {\displaystyle V_{k}=Y\setminus f\left(X\setminus \cup _{a\in \gamma _{k}}U_{a}\right)} is open in Y . {\displaystyle Y.}