CatDat

Grothendieck topos

A Grothendieck topos is a category that is equivalent to the category of sheaves on a site.

• Related properties: elementary topos
• nLab Link

Relevant implications

• trivial implies   finitary algebraic and  Grothendieck topos and  split abelian and  self-dual and  essentially discrete and  essentially finite

  trivial
• Grothendieck topos is equivalent to   elementary topos and  coproducts and  generator and  locally essentially small

  Mac Lane & Moerdijk, Appendix, Prop. 4.4
• Grothendieck topos implies   locally presentable and  cogenerator

  For "locally presentable" see Prop. 3.4.16 in Handbook of Categorical Algebra Vol. 3. For "cogenerator" see the nLab.

Examples

• category of M-sets
• category of sets
• category of simplicial sets
• trivial category
• walking isomorphism

Counterexamples

• category of abelian groups
• category of Banach spaces with linear contractions
• category of combinatorial species
• category of commutative rings
• category of fields
• category of finite abelian groups
• category of finite orders
• category of finite sets
• category of finite sets and bijections
• category of finite sets and injections
• category of finite sets and surjections
• category of finitely generated abelian groups
• category of free abelian groups
• category of groups
• category of left R-modules
• category of locally ringed spaces
• category of measurable spaces
• category of metric spaces with ∞ allowed
• category of metric spaces with continuous maps
• category of metric spaces with non-expansive maps
• category of monoids
• category of non-empty sets
• category of pointed sets
• category of posets
• category of rings
• category of rngs
• category of schemes
• category of sets and relations
• category of small categories
• category of smooth manifolds
• category of topological spaces
• category of vector spaces
• category of Z-functors
• delooping of a non-trivial finite group
• delooping of an infinite group
• delooping of the additive monoid of natural numbers
• delooping of the additive monoid of ordinal numbers
• discrete category on two objects
• empty category
• partial order [0,1]
• partial order of extended natural numbers
• partial order of natural numbers
• partial order of ordinal numbers
• preorder of integers w.r.t. divisiblity
• walking morphism
• walking parallel pair of morphisms

Unknown

For these categories the database has no info if they satisfy this property or not.

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