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 "locally 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 andpreadditive
- additive is equivalent to finite products andpreadditive
- cokernels andconormal andpreadditive implies effective cocongruences
- cokernels andkernels andpreadditive implies coquotients of cocongruences
- cokernels andkernels andpreadditive implies quotients of congruences
- cokernels andpreadditive implies coequalizers
- coquotients of cocongruences andcoregular andpreadditive implies kernels
- epi-regular andpreadditive implies conormal
- finite coproducts andpreadditive implies finite products
- finite products andpreadditive implies finite coproducts
- kernels andnormal andpreadditive implies effective congruences
- kernels andpreadditive implies equalizers
- mono-regular andpreadditive implies normal
- preadditive andquotients of congruences andregular implies cokernels
- preadditive implies locally essentially small andzero morphisms
Examples
There are 15 categories with this property.
- category of abelian groups
- category of abelian sheaves
- 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 torsion abelian groups
- category of torsion-free abelian groups
- category of vector spaces
- empty category
- trivial category
- walking idempotent
- walking isomorphism
- walking splitting
Counterexamples
There are 65 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 monoids
- category of commutative rings
- category of compact Hausdorff spaces
- category of countable groups
- category of countable sets
- category of fields
- category of finite groups
- category of finite ordered sets
- 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 Jónsson-Tarski algebras
- 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 pointed topological spaces
- category of posets
- category of prosets
- category of pseudo-metric spaces with non-expansive maps
- category of rings
- category of rngs
- category of schemes
- category of semigroups
- category of sets
- category of sets and relations
- category of sets with finite-to-one maps
- 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 countable 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
- poset [0,1]
- poset of extended natural numbers
- poset of natural numbers
- poset of ordinal numbers
- proset of integers w.r.t. divisibility
- simplex category
- walking commutative square
- walking composable pair
- walking coreflexive pair
- walking fork
- walking morphism
- walking parallel pair
- walking span
Unknown
There are 0 categories for which the database has no information on whether they satisfy this property.
—