elementary topos
An elementary topos is a cartesian closed category with finite limits and a subobject classifier.
- Related properties: Grothendieck topos
- nLab Link
Relevant implications
elementary topos is equivalent to cartesian closed and finitely complete and subobject classifier
by definitionelementary topos implies finitely cocomplete and disjoint finite coproducts and epi-regular
Mac Lane & Moerdijk, Cor. IV.5.4, Cor. IV.10.5, Thm. 4.7.8.elementary topos and locally essentially small implies well-copowered
This follows from Mac Lane & Moerdijk, Theorem IV.7.8 (and Prop. I.3.1).Grothendieck topos is equivalent to elementary topos and coproducts and generator and locally essentially small
Mac Lane & Moerdijk, Appendix, Prop. 4.4
Examples
- category of combinatorial species
- category of finite sets
- 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 commutative rings
- category of fields
- category of finite abelian groups
- category of finite orders
- 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
- 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.