Grothendieck abelian
A category is Grothendieck abelian if it is abelian, has coproducts (and is therefore cocomplete), a generator, and filtered colimits commute with finite limits. These categories play an important role in homological algebra.
- Related properties: abelian, cocomplete, generator, exact filtered colimits
- nLab Link
Relevant implications
Grothendieck abelian is equivalent to abelian and coproducts and generator and exact filtered colimits
by definitionGrothendieck abelian implies locally presentable
See Deriving Auslander's formula, Cor. 5.2, or Sheafifiable homotopy model categories, Prop. 3.10.Grothendieck abelian implies cogenerator
Kashiwara-Schapira, Thm. 9.6.3Grothendieck abelian and self-dual implies trivial
This follows since the dual of a non-trivial Grothendieck abelian category cannot be Grothendieck abelian. See Peter Freyd, Abelian categories, p. 116.
Examples
- category of 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 abelian groups
- 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 finitely generated abelian groups
- category of free abelian groups
- 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.
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