Right-(Left-)derived Functors

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Fix an abelian category A, let J be a Δ-subcategory of K(A), let DJ be the corresponding derived category, and let

Q = QJ : J → DJ

be the canonical Δ-functor. For any Δ-functors F and G from J to another Δ-category E, or from DJ to E, Hom(F, G) will denote the abelian group of Δ-functor morphisms from F to G.

A Δ-functor F : J → E is right-derivable if there exists a Δ-functor

RF : DJ → E

and a morphism of Δ-functors

ζ : F → RF ◦ Q

such that for every Δ-functor G : DJ → E the composed map

Hom(RF, G) →natural Hom(RF ◦ Q, G ◦ Q) →via ζ Hom(F, G ◦ Q)

is an isomorphism, (the map “via ζ” is an isomorphism). The Δ-functor F is left-derivable if there exists a Δ-functor

LF : DJ → E

and a morphism of Δ-functors

ζ : LF ◦ Q → F

such that for every Δ-functor G : DJ → E the composed map

Hom(G, LF) →natural Hom(G ◦ Q, LF ◦ Q) →via ζ Hom(G ◦ Q, F)

is an isomorphism (the map “via ζ” is an isomorphism).

Such a pair (RF, ζ) and (LF, ζ) are called the right-derived and left-derived functors of F respectively. Composition with Q gives an embedding of Δ-functor categories

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