CatDat

additive

A category is additive if it is preadditive and has finite products (equivalently, finite coproducts).

• Dual property: additive (self-dual)
• Related properties: preadditive, finite products
• nLab Link

Relevant implications

• additive is equivalent to   preadditive and  finite products

  by definition
• additive implies   disjoint finite coproducts

  If $f : T \to A + B$ is a morphism that factors through $A$ and $B$, then $p_B f = 0$ and $p_A f = 0$, so $f = 0$.
• abelian is equivalent to   additive and  equalizers and  coequalizers and  mono-regular and  epi-regular

  by definition
• additive and  pullbacks and  subobject classifier implies   trivial

  see MSE/4086192
• additive is equivalent to   preadditive and  finite coproducts

  [dualized] by definition

Examples

• category of abelian groups
• category of finite abelian groups
• category of finitely generated abelian groups
• category of free abelian groups
• category of left R-modules
• category of vector spaces
• trivial category
• walking isomorphism

Counterexamples

• category of Banach spaces with linear contractions
• category of combinatorial species
• 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 locally ringed spaces
• category of M-sets
• category of measurable spaces
• category of metric spaces with ∞ allowed
• category of metric spaces with continuous maps
• category of metric spaces with non-expansive maps
• category of monoids
• category of non-empty sets
• category of pointed sets
• category of posets
• category of rings
• category of rngs
• category of schemes
• category of sets
• category of sets and relations
• 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
• empty category
• partial order [0,1]
• partial order of extended natural numbers
• partial order of natural numbers
• partial order of ordinal numbers
• preorder of integers w.r.t. divisiblity
• walking morphism
• walking parallel pair of morphisms

Unknown

For these categories the database has no info if they satisfy this property or not.

—