Implication Details
Assumptions: Grothendieck abelian
Conclusions: abelian, coproducts, exact filtered colimits, generator
This is an equivalence.
Reason: This holds by definition.
Show 54 categories using this implication
- empty category
- trivial category
- discrete category on two objects
- category of abelian groups
- category of algebras
- category of finite sets and bijections
- category of Banach spaces with linear contractions
- category of commutative algebras
- category of commutative rings
- category of small categories
- category of compact Hausdorff spaces
- simplex category
- category of finite sets and surjections
- category of finite abelian groups
- category of finite groups
- category of finite ordered sets
- category of finite sets
- category of fields
- 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 non-expansive maps
- category of metric spaces with continuous maps
- category of metric spaces with ∞ allowed
- poset of natural numbers
- poset of extended natural numbers
- category of pseudo-metric spaces with non-expansive maps
- category of posets
- category of prosets
- category of left modules over a ring
- category of left modules over a division ring
- category of sets and relations
- category of rings
- category of schemes
- category of semigroups
- category of sets
- category of sets with finite-to-one maps
- category of pointed sets
- category of non-empty sets
- category of pairs of sets
- category of sheaves
- category of abelian sheaves
- category of combinatorial species
- category of torsion abelian groups
- category of torsion-free abelian groups
- category of vector spaces
- proset of integers w.r.t. divisibility
- category of simplicial sets
- walking fork
- walking isomorphism
- walking parallel pair
- walking splitting