Implication Details
Assumptions: Grothendieck topos
Conclusions: cogenerator, exact filtered colimits, infinitary extensive, locally presentable
Proof: A Grothendieck topos is locally presentable by Prop. 3.4.16 in Handbook of Categorical Algebra Vol. 3, has a cogenerator (see nLab) and is infinitary extensive by Giraud's Theorem. To show that it has exact filtered colimits, first observe that this is clearly true in every presheaf topos (since has the property). Every Grothendieck topos is a full reflective subcategory of a presheaf topos such that the reflector preserves finite limits (nLab), so we conclude with Lemma 3 here.
Show 26 categories using this implication
- category of Banach spaces with linear contractions
- category of commutative monoids
- category of compact Hausdorff spaces
- category of countable groups
- category of finite groups
- category of finite sets and bijections
- category of finite sets and injections
- category of finitely generated abelian groups
- category of groups
- category of Hausdorff spaces
- category of Jónsson-Tarski algebras
- category of M-sets
- category of measurable spaces
- category of monoids
- category of pairs of sets
- category of rngs
- category of sets
- category of sheaves
- category of simplicial sets
- category of topological spaces
- empty category
- trivial category
- walking idempotent
- walking isomorphism
- dual of the category of sets
- dual of the category of topological spaces