Implication Details
Claim: If a category is Grothendieck abelian and is locally essentially small, then it has a cogenerator.
Proof: See Kashiwara-Schapira, Thm. 9.6.3.
Remark: The assumption that the category is locally essentially small is necessary (and is implicit in most of the literature), as the example shows (see here).
Show 21 categories using this implication
- category of abelian groups
- category of finitely generated abelian groups
- category of sets equipped with an irreflexive binary relation
- category of sets equipped with a symmetric irreflexive binary relation
- category of commutative monoids
- category of cochain complexes of abelian groups
- category of F(I)-sets
- category of finite sets and injections
- category of countable groups
- category of left modules over a ring
- category of left modules over a division ring
- category of left modules over a non-semisimple ring
- category of sequences of abelian groups
- category of sets and bijections
- category of large families of sets which are mostly empty
- category of sets with finite-to-one maps
- category of abelian sheaves
- category of torsion abelian groups
- category of long transfinite sequences of abelian groups
- category of graded abelian groups
- category of graded modules over a graded ring