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Additive category
Definition
An additive category is a preadditive category (Preadditive category) with all finite biproducts.
Equivalently, a preadditive category is additive when it has a zero object and binary biproducts: a zero object is both initial and terminal (Initial object, terminal object, and zero object) and therefore supplies both the empty coproduct and the empty product (Limits of empty diagrams are terminal objects, and colimits of empty diagrams are initial objects); their canonical comparison is the unique endomorphism of the zero object and hence the identity, so this is an empty biproduct. Meanwhile, binary biproducts iterate using the canonical associativity and unitality isomorphisms (Biproducts are associative, commutative, and unital up to canonical isomorphism) to give every finite biproduct.
Depends on
Used by
- Additive categories are closed under passage to the opposite Corollary
- An additive category is an Ab-enriched category with a zero object and finite biproducts Corollary
- Filtered vector spaces can be additive with kernels and cokernels without being abelian Counterexample
- Topological abelian groups are additive but not abelian Counterexample
- Abelian category Definition
- Category with translation Definition
- Complexes, homotopies and contractibility in an additive category Definition
- Graded Grothendieck groups, shift action, and Cartan map Definition
- Split Grothendieck group of an additive category Definition
- Split Grothendieck rings of the type-A Soergel categories Definition
- Abelian groups, modules, and vector spaces are additive Example
- FALSE: the addition on an additive category is extra structure that must be chosen False statement
- Addition of roofs makes an additive localization Lemma
- Colimits of a graded additive functor equal right exactness plus coproduct preservation 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 modules with degree-zero maps form an abelian category Lemma
- Projective epimorphisms onto the simples generate every finite-length object Lemma
- Sum and product totalisations agree on finite diagonal double complexes Proposition
- A functor between additive categories is additive exactly when it preserves finite biproducts Theorem
- An additive category with all kernels and cokernels has all finite limits and colimits Theorem
- An additive category with kernels is idempotent complete Theorem
- An additive functor is left exact exactly when it preserves kernels Theorem
- An additive functor preserves finite biproducts Theorem
- Basic properties of the categorical trace Theorem
- Composition of morphisms between finite biproducts is matrix multiplication Theorem
- Exact adjoints induce adjoint operators on Grothendieck groups Theorem
- Morphisms between finite biproducts correspond to matrices Theorem
- Projective and simple classes are dual bases under splitting Theorem
- Projective Hom pairing descends and is graded sesquilinear Theorem
- The category of complexes in an additive category is additive Theorem
- The matrix category over a ring is additive Theorem
- Universal properties and functoriality of G0 and split K0 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
- Kiran S. Kedlaya, Solid modules over an ordinary ring, Definition 1.2.1 (standard reference, not scraped)
- The Stacks Project, Section 12.3, Definition 12.3.8 (standard reference, not scraped)
- Merlin Christ, Tobias Dyckerhoff, and Tashi Walde, Lax Additivity, Definition 2.1 (standard reference, not scraped)