k-linear categories and k-linear functors

Depends on

• Category, object, morphism, domain, codomain, identity, composition, and hom-collection
• Vector space over a field

Used by

• Composition of Deligne kernels is balanced tensor product Corollary
• The degree-zero projection is exact and cocontinuous but not a graded tensor functor Counterexample
• Absolutely simple objects Definition
• Coherently shift-compatible functors and natural transformations Definition
• Left and right Nakayama functors by finite kernel calculus Definition
• Locally finite k-linear abelian categories Definition
• The Deligne product of finite linear categories Definition
• Tilting objects and negligible morphisms in a root-of-unity category Definition
• The Deligne product of finite vector spaces is finite vector spaces Example
• 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
• Finite module duality is exact with commuting bimodule actions Lemma
• Finite vector-space copowers in a k-linear abelian category Lemma
• Finite-support families of finite-dimensional vector spaces are locally finite but not finite Lemma
• Graded tensor functors are k-linear, right exact, coproduct preserving and shift-coherent Lemma
• Homogeneous right multiplication reconstructs the graded kernel action Lemma
• Internal shifts are autoequivalences and commute with the graded tensor product Lemma
• Nakayama kernels give well-defined adjoint functors Lemma
• The opposite Deligne product is the category of finite bimodules Lemma
• Finite-dimensional module categories satisfy the intrinsic finiteness conditions Proposition
• The left-to-right exact equivalence sends the identity to the Nakayama functor Proposition
• Categorical Eilenberg–Watts equivalences for finite linear categories Theorem
• Exact adjoints induce adjoint operators on Grothendieck groups Theorem
• Finite abelian categories admit finite-dimensional module models 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
• Graded Eilenberg-Watts theorem with coherent shifts Theorem
• Intrinsic finite category hypotheses give a finite projective generator Theorem

Dependency tree · two levels

11 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

• Etingof, Gelaki, Nikshych, Ostrik, Tensor Categories, Definitions 1.2.2–1.2.3 (standard reference, not scraped)