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Biproduct
Definition
A finite family in a category with zero morphisms has a biproduct when both its coproduct and its product exist and the canonical morphism from the coproduct to the product is an isomorphism (Canonical morphism from a finite coproduct to a finite product).
When this happens, either object may be written
since the canonical comparison identifies them uniquely. For two objects one writes , and for the empty family one speaks of the empty biproduct.
Transporting the product structure across the canonical comparison to the coproduct object makes that comparison the identity. The resulting common object, injections, and projections form the associated biproduct diagram; this is the convention used whenever both kinds of structure maps are written on one displayed object.
Depends on
Used by
- Semiadditive category Definition
- Split short exact sequence in an abelian category Definition
- The Baer sum of extension classes Definition
- FALSE: finite products and finite coproducts already force biproducts False statement
- Bilinear right exact functors are determined by their value on the regular modules 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
- Projective epimorphisms onto the simples generate every finite-length object Lemma
- Biproducts are associative, commutative, and unital up to canonical isomorphism Proposition
- Sum and product totalisations agree on finite diagonal double complexes Proposition
- The empty biproduct is a zero object Proposition
- Zero and split triangles are distinguished Proposition
- A cartesian square over an epimorphism is also cocartesian Theorem
- A square is cartesian exactly when a short sequence is exact Theorem
- Biproduct data characterisation without addition Theorem
- Finite abelian categories admit finite-dimensional module models Theorem
- Intrinsic finite category hypotheses give a finite projective generator Theorem
- Kernels and cokernels of quasi-coherent modules Theorem
- Splitting lemma in an abelian category Theorem
- The category of complexes in an additive category is additive 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
- The Stacks Project, Section 12.3: Preadditive and additive categories (standard reference, not scraped)
- Saunders Mac Lane, Categories for the Working Mathematician, VIII.2 (standard reference, not scraped)