Implication details
Claim: If a category is cocomplete and has a generating collection and is locally essentially small and is well-copowered, then it is total.
Proof: Since the category is cocomplete and well-copowered, it is epi-cocomplete (meaning that it has wide pushouts of epimorphisms, even of non-small families of epimorphisms). The result then follows from Day, Thm. 1.
This implication has a dual.
Show 61 categories using this implication
- trivial category
- category of abelian groups
- category of algebras
- category of Banach spaces with linear contractions
- category of sets equipped with a 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 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 filtered vector spaces
- 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 ∞ allowed
- category of monoids
- category of sets with a distinguished subset
- partially ordered set of extended natural numbers
- partially ordered collection of ordinal numbers
- category of partially ordered sets
- category of preordered sets
- category of quivers
- 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 rings
- category of rngs
- category of semigroups
- category of sequences of abelian groups
- category of sequences of sets
- category of sets
- category of pointed sets
- category of set functions and commutative squares
- category of large families of sets which are mostly empty
- indiscrete category of sets
- category of pairs of sets
- category of sheaves
- category of abelian sheaves
- category of topological spaces
- category of pointed topological spaces
- category of torsion abelian groups
- category of torsion-free abelian groups
- category of uniform spaces
- category of vector spaces
- category of Z-sets
- preordered set of integers w.r.t. divisibility
- category of graded abelian groups
- category of graded modules over a graded ring
- real interval [0,1]
- category of simplicial sets
- walking commutative square
- walking composable pair
- walking isomorphism
- walking morphism