Left exact and right exact functors

Depends on

• Finite, small, and large limits and colimits; complete and cocomplete categories
• Preservation, reflection, and creation of limits and colimits; continuous and cocontinuous functors

Used by

• A right adjoint is left exact and a left adjoint is right exact Corollary
• Composition of Deligne kernels is balanced tensor product Corollary
• Exact finite tensor functors have projective right-module kernels Corollary
• Finite Eilenberg–Watts is a biequivalence Corollary
• Finite one-sided exactness is equivalent to the existence of the corresponding adjoint Corollary
• A coproduct-preserving left exact module functor is not tensor Counterexample
• The degree-zero projection is exact and cocontinuous but not a graded tensor functor Counterexample
• Additive cocontinuous module functors and their schematic category Definition
• An acyclic object for a left exact functor Definition
• An acyclic object for a right exact functor Definition
• Compactly supported global sections and their right derived functors Definition
• Exact functor between abelian categories Definition
• G-acyclic object for a left-exact functor Definition
• Left and right Nakayama functors by finite kernel calculus Definition
• Right hyperderived functor of a complex Definition
• The Deligne product of finite linear categories Definition
• An additive module functor is cocontinuous exactly when it is right exact and preserves coproducts Lemma
• Bilinear right exact functors are determined by their value on the regular modules Lemma
• Coherently shift-compatible functors and transformations form k-linear hom categories Lemma
• Colimits of a graded additive functor equal right exactness plus coproduct preservation Lemma
• Graded tensor functors are k-linear, right exact, coproduct preserving and shift-coherent Lemma
• Homogeneous free presentations prove the graded comparison is an isomorphism Lemma
• Nakayama kernels give well-defined adjoint functors Lemma
• Natural transformations of additive cocontinuous module functors are determined at the regular module Lemma
• The Hom functor of a small projective generator is exact, coproduct-preserving, and faithful Lemma
• The horseshoe construction stays short exact after applying a right exact functor Lemma
• The opposite Deligne product is the category of finite bimodules Lemma
• The left-to-right exact equivalence sends the identity to the Nakayama functor Proposition
• A left or right exact functor between abelian categories is automatically additive Theorem
• An equivalence between abelian categories is exact Theorem
• Categorical Eilenberg–Watts equivalences for finite linear categories Theorem
• Finite Deligne products exist via tensor-product algebras Theorem
• Finite Eilenberg–Watts for right exact linear functors Theorem
• Finite left exact functors are Hom functors with dual bimodule kernels Theorem
• Left exactness, right exactness, and exactness are characterized by short exact sequences Theorem
• The zero-th left derived functor of a right exact functor recovers the functor Theorem
• The zero-th right derived functor of a left exact functor recovers the functor Theorem

Dependency tree · two levels

7 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

• Emily Riehl, Category Theory in Context, 2nd ed., Definition 4.6.7 (standard reference, not scraped)