Implication Details
Claim: If a category is a Grothendieck topos, then it has a cogenerator and has exact filtered colimits and is infinitary extensive and is 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 34 categories using this implication
- empty category
- trivial category
- category of finitely generated 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 monoids
- category of commutative rings
- category of small categories
- category of compact Hausdorff spaces
- category of finite sets and injections
- category of finite groups
- category of groups
- category of countable groups
- category of Hausdorff spaces
- category of Jónsson-Tarski algebras
- category of M-sets
- category of measurable spaces
- category of monoids
- category of rings
- category of rngs
- category of semigroups
- category of sets
- category of set functions and commutative squares
- category of sets with finite-to-one maps
- category of pairs of sets
- category of sheaves
- category of topological spaces
- category of simplicial sets
- walking idempotent
- walking isomorphism
- dual of the category of sets
- dual of the category of topological spaces