specifically in algebraic topology and algebraic geometry, an inverse image functor is a contravariant construction of sheaves; here “contravariant”...

In mathematics, the direct image functor is a construction in sheaf theory that generalizes the global sections functor to the relative case. It is of...

algebraic geometry, the exceptional inverse image functor is the fourth and most sophisticated in a series of image functors for sheaves. It is needed to express...

direct image functor, taking sheaves and their morphisms on the domain to sheaves and morphisms on the codomain, and an inverse image functor operating...

We conclude that left adjoint to the inverse image functor f ∗ {\displaystyle f^{*}} is given by the direct image. Here is a characterization of this result...

pullback functor taking bundles on Y to bundles on X. Fibred categories formalise the system consisting of these categories and inverse image functors. Similar...

In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic...

quantifier can be understood as the right adjoint of a functor between power sets, the inverse image functor of a function between sets; likewise, the existential...

that colim is a left inverse of Δ. Note that both the limit and the colimit functors are covariant functors. One can use Hom functors to relate limits and...

adjoint functor f ! {\displaystyle f^{!}} , called twisted or exceptional inverse image functor, to a higher direct image with compact support functor R f...

bundle f ∗ E {\displaystyle f^{*}E} over M . {\displaystyle M.} Inverse image functor – functor between categories of Abelian-group-valued sheaves induced...

quantifier can be understood as the right adjoint of a functor between power sets, the inverse image functor of a function between sets; likewise, the existential...

functors between triangulated categories. In particular, Neeman used it to simplify and generalize the construction of the exceptional inverse image functor...

topological space. The functors in question are direct image f∗ : Sh(X) → Sh(Y) inverse image f∗ : Sh(Y) → Sh(X) direct image with compact support f! :...

are six functors. Usually these are functors between derived categories and so are actually left and right derived functors. the direct image f ∗ {\displaystyle...

is a fundamental result in category theory. It is an abstract result on functors of the type morphisms into a fixed object. It is a vast generalisation...

Note that an exact functor, because it preserves both kernels and cokernels, preserves all images and coimages. Exact functors are most useful in the...

quantifier can be understood as the left adjoint of a functor between power sets, the inverse image functor of a function between sets; likewise, the universal...

statements are more delicate since for an open immersion j, the inverse image functor j ∗ {\displaystyle j^{*}} does not usually admit a left adjoint...

the functor and its "inverse" is not necessarily the identity mapping. Instead it is sufficient that each object be naturally isomorphic to its image under...

the category of contravariant functors from D {\displaystyle D} to the category of sets; such a contravariant functor is frequently called a presheaf...

is adjoint to the push-forward; the composition operator is the inverse image functor. Since the domain considered here is that of Borel functions, the...

contravariant functor P : Set → Set which maps each set to its power set and each function to its inverse image map. To represent this functor we need a pair...

{\displaystyle F} and G {\displaystyle G} are inverse to each other, the corresponding monad is the identity functor. In general, adjunctions are not equivalences—they...

in an interval I, it has an inverse function, which is a real function with domain f(I) and image I. This is how inverse trigonometric functions are defined...

mathematics, the inverse function of a function f (also called the inverse of f) is a function that undoes the operation of f. The inverse of f exists if...

dualities can be done using derived categories and certain direct and inverse image functors of sheaves (with respect to the classical analytical topology on...

transformation that is an isomorphism (i.e., admits the inverse). nerve 1.  The nerve functor N is the functor from Cat to sSet given by N ( C ) n = Hom C a t...

(I)} This is called the inverse image functor. In the motivating example, this is just precompositition, so given a functor F I ∈ Hom _ ( I o p , A )...

mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition...