preadditive
A category is preadditive when it is locally essentially small* and each hom-set carries the structure of an abelian group such that the composition is bilinear. Notice that "preadditive" is an extra structure. The property here just says that some preadditive structure exists.
*We demand this instead of the more common "locall small" to ensure that preadditive categories are invariant under equivalences of categories.
- Dual property: preadditive (self-dual)
- Related properties: additive
- nLab Link
Relevant implications
preadditive implies locally essentially small and zero morphisms
trivialpreadditive and finite coproducts implies finite products
Mac Lane, VIII.2., Theorem 2additive is equivalent to preadditive and finite products
by definitionpreadditive and finite products implies finite coproducts
[dualized] Mac Lane, VIII.2., Theorem 2additive 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
- empty category
- 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
- 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.
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