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.
Show 28 categories using this implication
- category of algebras
- category of sets with a binary relation
- category of small categories
- category of compact Hausdorff spaces
- category of directed graphs
- 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 metric spaces with ∞ allowed
- category of monoids
- category of sets with a distinguished subset
- category of partially ordered sets
- category of preordered sets
- category of rings
- category of rngs
- 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
- preordered set of integers w.r.t. divisibility
- category of simplicial sets