CatDat

biproducts

A category has biproducts when it has zero morphisms, finite products (denoted $\times$), finite coproducts (denoted $\oplus$), and for every finite family of objects $A_1,\dotsc,A_n$ the canonical morphism

$\mu : A_1 \oplus \cdots \oplus A_n \to A_1 \times \cdots \times A_n$

$A \xrightarrow{\Delta} A \times A \xrightarrow{f \times g} B \times B \xrightarrow{\mu^{-1}} B \oplus B \xrightarrow{\nabla} B$

• Dual property: biproducts (self-dual)
• Related properties: finite coproducts, finite products, zero morphisms
• nLab Link

Relevant implications

• additive implies biproducts
• biproducts andfinitely cocomplete andpointed implies counital
• biproducts andfinitely complete andpointed implies unital
• biproducts implies disjoint finite coproducts
• biproducts implies disjoint finite products
• biproducts implies finite coproducts andfinite products andzero morphisms

Examples

There are 12 categories with this property.

• category of abelian groups
• category of abelian sheaves
• category of commutative monoids
• category of finite abelian groups
• category of finitely generated abelian groups
• category of free abelian groups
• category of left modules over a division ring
• category of left modules over a ring
• category of sets and relations
• category of vector spaces
• trivial category
• walking isomorphism

Counterexamples

There are 53 categories without this property.

• category of algebras
• category of Banach spaces with linear contractions
• category of combinatorial species
• category of commutative algebras
• category of commutative rings
• category of fields
• category of finite orders
• category of finite sets
• category of finite sets and bijections
• category of finite sets and injections
• category of finite sets and surjections
• category of groups
• category of Hausdorff spaces
• category of locally ringed spaces
• category of M-sets
• category of measurable spaces
• category of metric spaces with continuous maps
• category of metric spaces with non-expansive maps
• category of metric spaces with ∞ allowed
• category of monoids
• category of non-empty sets
• category of pairs of sets
• category of pointed sets
• category of posets
• category of prosets
• category of rings
• category of rngs
• category of schemes
• category of sets
• category of sheaves
• category of simplicial sets
• category of small categories
• category of smooth manifolds
• category of topological spaces
• category of Z-functors
• delooping of a non-trivial finite group
• delooping of an infinite group
• delooping of the additive monoid of natural numbers
• delooping of the additive monoid of ordinal numbers
• discrete category on two objects
• dual of the category of sets
• empty category
• poset [0,1]
• poset of extended natural numbers
• poset of natural numbers
• poset of ordinal numbers
• proset of integers w.r.t. divisibility
• walking commutative square
• walking composable pair
• walking fork
• walking morphism
• walking parallel pair of morphisms
• walking span

Unknown

There are 0 categories for which the database has no information on whether they satisfy this property.

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