Preadditive category

Depends on

• Category, object, morphism, domain, codomain, identity, composition, and hom-collection

Used by

• Additive category Definition
• Additive functor Definition
• The idempotent completion of a preadditive category Definition
• A preadditive category with two objects and a nonzero hom-group Example
• FALSE: a preadditive category with a zero object must have binary biproducts False statement
• A small product of preadditive categories is preadditive Proposition
• Additive functors and natural transformations form a preadditive category Proposition
• An additive functor preserves zero morphisms Proposition
• In a preadditive category with a zero object, the zero morphism is the neutral element of each hom-group Proposition
• A one-object preadditive category is the same thing as a ring Theorem
• A semiadditive category is preadditive exactly when every morphism has an additive inverse Theorem
• In a preadditive category with a zero object, a morphism is monic exactly when its kernel is zero Theorem
• In a preadditive category, a finite product is automatically a biproduct Theorem
• In a preadditive category, an object is initial exactly when it is terminal Theorem
• In a preadditive category, the equalizer of a parallel pair is the kernel of their difference Theorem
• The hom-bifunctor of a preadditive category takes values in abelian groups Theorem
• The opposite of a preadditive category is preadditive Theorem

Dependency tree · two levels

3 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 12.3.1 (standard reference, not scraped)