Additive functor

Depends on

• Preadditive category
• Covariant functor, identity functor, composite functor, and contravariant functor

Used by

• The degree-zero projection is exact and cocontinuous but not a graded tensor functor Counterexample
• Abelian subcategory and exact embedding Definition
• Additive cocontinuous module functors and their schematic category Definition
• Coherently shift-compatible functors and natural transformations Definition
• Cohomological delta functor Definition
• Exact functor between abelian categories Definition
• Exact functor between triangulated categories Definition
• Homological delta functor Definition
• Homological functor on a triangulated category Definition
• Left derived objects relative to supplied projective resolution data Definition
• Left total derived functor on the bounded above derived category Definition
• Right derived objects relative to supplied injective resolution data Definition
• Right total derived functor on the bounded below derived category Definition
• FALSE: an additive functor commutes with homology False statement
• FALSE: every additive functor has L₀ naturally isomorphic to itself False statement
• Colimits of a graded additive functor equal right exactness plus coproduct preservation Lemma
• F(A) is a (B,A)-bimodule for every additive functor F Lemma
• Homogeneous right multiplication reconstructs the graded kernel action Lemma
• Natural transformations of additive cocontinuous module functors are determined at the regular module Lemma
• Pushforward along a closed immersion preserves sheaf cohomology Lemma
• The Hom functor of a small projective generator is exact, coproduct-preserving, and faithful Lemma
• The induced cohomology map is independent of the chosen injective comparison extension Lemma
• The induced homology map is independent of the chosen comparison lift Lemma
• Additive functors and natural transformations form a preadditive category Proposition
• Additive functors preserve chosen Gaussian cancellations Proposition
• An additive functor applies degreewise to complexes and chain maps Proposition
• An additive functor preserves zero morphisms Proposition
• An additive functor preserves finite biproducts Theorem
• Finite Eilenberg–Watts for right exact linear functors Theorem
• Homology is an additive functor Theorem
• Left derived functors relative to supplied data are additive functors Theorem
• Right derived functors relative to supplied data are additive functors Theorem
• The idempotent completion is idempotent complete and its inclusion is fully faithful and universal Theorem
• Universal properties and functoriality of G0 and split K0 Theorem

Dependency tree · two levels

4 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• The Stacks Project, Section 12.7, Additive functors (standard reference, not scraped)