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  2. Inverse image functor - Wikipedia

    en.wikipedia.org/wiki/Inverse_image_functor

    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.

  3. Image (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Image_(mathematics)

    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} .

  4. Image functors for sheaves - Wikipedia

    en.wikipedia.org/wiki/Image_functors_for_sheaves

    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 ...

  5. Inverse function - Wikipedia

    en.wikipedia.org/wiki/Inverse_function

    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 ...

  6. Fibred category - Wikipedia

    en.wikipedia.org/wiki/Fibred_category

    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.

  7. Exceptional inverse image functor - Wikipedia

    en.wikipedia.org/wiki/Exceptional_inverse_image...

    Let f: XY 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 ...

  8. Function (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Function_(mathematics)

    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

  9. Proper map - Wikipedia

    en.wikipedia.org/wiki/Proper_map

    The set is closed in and its image under is closed in because is a closed map. Hence the set V k = Yf ( 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.}