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
- additive is equivalent to finite coproducts and preadditive
- additive is equivalent to finite products and preadditive
- finite coproducts and preadditive implies finite products
- finite products and preadditive implies finite coproducts
- preadditive implies locally essentially small and zero morphisms
Examples
There are 9 categories with this property.
- 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
There are 42 categories without this property.
- 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
There are 0 categories for which the database has no information on whether they satisfy this property.
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