Implication details
Claim: A category is co-Grothendieck abelian if and only if it is abelian and has a cogenerator and has exact cofiltered limits and has products.
Proof: This follows from the dual implication.
This implication has a dual.
Show 95 categories using this implication
- empty category
- trivial category
- discrete category on two objects
- category of abelian groups
- category of large families of abelian groups
- category of finitely generated abelian groups
- category of algebras
- delooping of the bicyclic monoid
- delooping of the double bicyclic monoid
- delooping of a group
- delooping of an infinite countable group
- delooping of a non-trivial finite group
- delooping of an infinite uncountable group
- delooping of the additive monoid of natural numbers
- delooping of the additive monoid of ordinal numbers
- category of Banach spaces with linear contractions
- category of sets equipped with a binary relation
- category of sets equipped with an irreflexive binary relation
- category of sets equipped with a reflexive binary relation
- category of sets equipped with a symmetric binary relation
- category of sets equipped with a symmetric irreflexive binary relation
- category of sets equipped with a symmetric reflexive binary relation
- category of commutative algebras
- category of commutative monoids
- category of commutative rings
- category of small categories
- category of cochain complexes of abelian groups
- category of compact Hausdorff spaces
- category of F(I)-sets
- category of finite sets and injections
- category of filtered vector spaces
- category of finite groups
- category of finite ordered sets
- category of finite sets
- category of finite-dimensional vector spaces
- category of finite-dimensional vector spaces [countable field]
- category of finite-dimensional vector spaces [finite field]
- category of finite-dimensional vector spaces [uncountable field]
- category of finitely generated free abelian groups
- category of groups
- category of countable groups
- category of finitely generated groups
- category of Hausdorff spaces
- category of Jónsson-Tarski algebras
- category of locally ringed spaces
- category of smooth manifolds
- category of metric spaces with continuous maps
- category of extended metric spaces
- category of monoids
- partially ordered set of extended natural numbers
- partially ordered collection of ordinal numbers
- category of partially ordered sets
- category of preordered sets
- category of finitely generated projective modules over the ring of dual numbers
- 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 sets and relations
- category of rings
- category of rngs
- category of schemes
- category of semigroups
- category of sequences of abelian groups
- category of pointed sets
- category of sets and bijections
- category of countable sets
- category of large families of sets
- category of large families of sets and bijections
- category of large families of sets which are mostly empty
- category of large families of sets which are mostly singletons
- category of sets with finite-to-one maps
- indiscrete category of sets
- category of topological spaces
- category of torsion-free abelian groups
- category of long transfinite sequences of abelian groups
- category of uniform spaces
- category of vector spaces
- category of countable-dimensional vector spaces
- category of large families of vector spaces which are mostly zero
- category of large vector spaces over a large field with a small basis
- category of Z-functors
- category of finite Z-sets
- preordered set of integers w.r.t. divisibility
- forked commutative square
- category of graded abelian groups
- category of graded modules over a graded ring
- real interval [0,1]
- walking commutative square
- walking composable pair
- walking fork
- walking isomorphism
- walking morphism
- walking parallel pair
- walking span
- walking splitting