Implication Details
Claim: If a category is Grothendieck abelian and is locally essentially small, then it is locally presentable.
Proof: See Deriving Auslander's formula, Cor. 5.2, or Sheafifiable homotopy model categories, Prop. 3.10.
Remark: The assumption that the category is locally essentially small is necessary (and is implicit in most of the literature), as the example for large shows (see here).
Show 20 categories using this implication
- category of large families of abelian groups
- simplex category
- category of coproducts of Euclidean spaces
- category of finite sets of even cardinality
- category of finite sets of odd cardinality
- category of finite sets of cardinality a power of 3
- category of finitely generated free modules over Z x Z
- category of partially ordered sets without isolated points
- discrete category of sets
- category of large families of sets which are mostly singletons
- category of abelian sheaves
- category of uniform spaces
- category of large vector spaces over a large field with a small basis
- cocompletion of a discrete–pair join
- forked commutative square
- walking coreflexive pair
- walking idempotent
- walking span
- dual of the category of sets
- dual of the category of topological spaces