In mathematics, certain functors may be derived to obtain other functors closely related to the original ones. This operation, while fairly abstract, unifies...

for fixed A in ModR. This is a left exact functor and thus has right derived functors RnT. The Ext functor is defined by Ext R n ⁡ ( A , B ) = ( R n T...

introduced to refine and in a certain sense to simplify the theory of derived functors defined on A. The construction proceeds on the basis that the objects...

so does CI, and the right derived functors of the inverse limit functor can thus be defined. The nth right derived functor is denoted R n lim ← : C I...

only defined on the level of derived categories, i.e., the functor is not obtained as the derived functor of some functor between abelian categories. If...

In mathematics, the Ext functors are the derived functors of the Hom functor. Along with the Tor functor, Ext is one of the core concepts of homological...

operations are six functors. Usually these are functors between derived categories and so are actually left and right derived functors. the direct image...

morphisms that satisfy properties generalising those of derived functors. A universal δ-functor is a δ-functor satisfying a specific universal property related...

particularly homological algebra, an exact functor is a functor that preserves short exact sequences. Exact functors are convenient for algebraic calculations...

total derived functor theorem of Quillen says that the total left derived functor LF: Ho(C) → Ho(D) is a left adjoint to the total right derived functor RG:...

formulate homology theories as derived functors on appropriate abelian categories, measuring the failure of an appropriate functor to be exact. One can describe...

computes the derived functors of the composition of two functors G ∘ F {\displaystyle G\circ F} , from knowledge of the derived functors of F {\displaystyle...

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 fundamental...

derived functor cohomology of an object and the homology of a chain complex since hypercohomology corresponds to the derived global sections functor R...

exact but not necessarily right exact. We may therefore form its right derived functors. Their values are abelian groups and they are denoted by H n ( G ,...

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

functors from the derived category of sheaves on X to abelian groups. In a broad sense of the word, "cohomology" is often used for the right derived functors...

adjoint functors, as does the unicity. The notation Rf! is an abuse of notation insofar as there is in general no functor f! whose derived functor would...

the groups Hi(X,E) for integers i are defined as the right derived functors of the functor E ↦ E(X). This makes it automatic that Hi(X,E) is zero for...

theory has yet to be developed.) The term "derived" is used in the same way as derived functor or derived category, in the sense that the category of...

mathematics, the Tor functors are the derived functors of the tensor product of modules over a ring. Along with the Ext functor, Tor is one of the central...

construct the resolutions needed to define sheaf cohomology (and other derived functors, such as sheaf Ext). There is a further group of related concepts applied...

the derived functors RiF(En) vanish for all i > 0 and n ≥ 0. Dually, a left resolution is acyclic with respect to a right exact functor if its derived functors...

additional structure of a "translation functor" and a class of "exact triangles". Prominent examples are the derived category of an abelian category, as...

derived functors of left exact functors. The étale cohomology groups Hi(F) of the sheaf F of abelian groups are defined as the right derived functors...

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...

the homotopy category (i.e., derived category). By definition, it is the left derived functor of the tensor product functor − ⊗ A − : M A × A M → R M {\displaystyle...

Ext functors, the derived functors of Hom-functors; if there is a bicommutant aspect, taken over the Steenrod algebra acting, it is only at a derived level...

In algebraic topology, a branch of mathematics, the Čech-to-derived functor spectral sequence is a spectral sequence that relates Čech cohomology of a...

as) a Kan extension from 1956 was in homological algebra to compute derived functors. In Categories for the Working Mathematician Saunders Mac Lane titled...