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...

Y} , the inverse image functor is a functor from the category of sheaves on Y to the category of sheaves on X. The direct image functor is the primary operation...

relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in...

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

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

_{X}} and the functors f ∗ {\displaystyle f_{*}} and Γ Y {\displaystyle \Gamma _{Y}} satisfy the hypotheses (since the direct image functor has an exact...

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...

inclusion functor Ab → Grp creates limits but does not preserve coproducts (the coproduct of two abelian groups being the direct sum). The forgetful functor Top...

the direct image with compact (or proper) support is an image functor for sheaves that extends the compactly supported global sections functor to the...

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! :...

section functor are contravariant right adjoints between the category of commutative rings and schemes, global Spec and the direct image functor for the...

The colimit of this functor is the same as the direct limit of the original direct system. A notion closely related to direct limits are the filtered...

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

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...

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

{\displaystyle {\text{Sh}}_{\text{Ab}}(X)} , by the definition of the direct image functor f ∗ {\displaystyle f_{*}} : S h A b ( X ) → f ∗ S h A b ( Y ) → Γ...

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

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

p_{*}\left(q^{*}{\mathcal {F}}\otimes ^{L}K\right)} where Rp* is the derived direct image functor and ⊗ L {\displaystyle \otimes ^{L}} is the derived tensor product...

a triple ( T , η , μ ) {\displaystyle (T,\eta ,\mu )} consisting of a functor T from a category to itself and two natural transformations η , μ {\displaystyle...

measure-theoretic pushforward of g: in essence, the transfer operator is the direct image functor in the category of measurable spaces. The left-adjoint of the Perron–Frobenius...

higher direct image of F {\displaystyle {\mathcal {F}}} under f, i.e., the derived functor of the direct image (also known as pushforward) functor f ∗ {\displaystyle...

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

Combinatorial species Exact functor Derived functor Dominant functor Enriched functor Kan extension of a functor Hom functor Product (category theory) Equaliser...

particularly important families of morphisms, whose images under a given simplicial set functor are called the face maps and degeneracy maps of that...

fixed object A, or the global sections functor on sheaves or the direct image functor. Their right derived functors are Extn(–,A), Extn(A,–), Hn(X, F) or...

especially in the field of representation theory, Schur functors (named after Issai Schur) are certain functors from the category of modules over a fixed commutative...

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

category of chain complexes of an abelian category, or the category of functors from a small category to an abelian category are abelian as well. These...

categorical sum. It follows that any functor which preserves limits will take terminal objects to terminal objects, and any functor which preserves colimits will...