Implication details
Claim: If a category has cofiltered limits and is extensive and has a terminal object, then it has cocartesian cofiltered limits.
Proof: Let be an extensive category with cofiltered limits and a terminal object. Then the coproduct functor is an equivalence. The forgetful functor creates connected limits, and hence preserves cofiltered limits. For every the functor also preserves cofiltered limits. The composition of these functors is and therefore also preserves cofiltered limits.
This implication has a dual.
Show 30 categories using this implication
- 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 small categories
- category of compact Hausdorff spaces
- category of F(I)-sets
- category of groups
- category of Hausdorff spaces
- category of Jónsson-Tarski algebras
- category of locally ringed spaces
- category of M-sets
- category of measurable spaces
- category of extended metric spaces
- category of sets with a distinguished subset
- category of partially ordered sets
- category of preordered sets
- category of quivers
- category of sequences of sets
- category of sets
- category of set functions and commutative squares
- category of large families of sets
- category of pairs of sets
- category of sheaves
- category of topological spaces
- category of uniform spaces
- category of Z-functors
- category of Z-sets
- preordered set of integers w.r.t. divisibility
- category of simplicial sets