Implication Details
Claim: A category is an elementary topos if and only if it is cartesian closed and is finitely complete and has a subobject classifier.
Proof: This holds by definition.
Show 74 categories using this implication
- empty category
- trivial category
- discrete category on two objects
- category of abelian groups
- category of finitely generated abelian groups
- delooping of a group
- delooping of an infinite countable group
- delooping of a non-trivial finite group
- delooping of an infinite uncountable group
- delooping of the additive monoid of natural numbers
- category of Banach spaces with linear contractions
- category of commutative monoids
- category of small categories
- category of cochain complexes of abelian groups
- category of compact Hausdorff spaces
- simplex category
- category of finite sets and injections
- category of finite abelian groups
- category of finite groups
- category of finite ordered sets
- category of finite sets
- category of finite-dimensional vector spaces
- category of finite-dimensional vector spaces [countable field]
- category of finite-dimensional vector spaces [finite field]
- category of finite-dimensional vector spaces [uncountable field]
- category of free abelian groups
- category of groups
- category of countable groups
- category of Hausdorff spaces
- category of Jónsson-Tarski algebras
- category of locally ringed spaces
- category of M-sets
- category of smooth manifolds
- category of measurable spaces
- category of metric spaces with non-expansive maps
- category of metric spaces with continuous maps
- category of metric spaces with ∞ allowed
- category of monoids
- category of pseudo-metric spaces with non-expansive maps
- category of partially ordered sets
- category of preordered sets
- category of left modules over a ring
- category of left modules over a division ring
- category of left modules over a non-semisimple ring
- category of sets and relations
- category of schemes
- category of semigroups
- category of sequences of abelian groups
- category of sets
- category of pointed sets
- category of set functions and commutative squares
- category of countable sets
- category of non-empty sets
- category of pairs of sets
- category of sheaves
- category of abelian sheaves
- category of combinatorial species
- category of topological spaces
- category of torsion abelian groups
- category of uniform spaces
- category of vector spaces
- category of Z-functors
- preordered set of integers w.r.t. divisibility
- forked commutative square
- category of graded abelian groups
- category of graded modules over a graded ring
- category of simplicial sets
- walking coreflexive pair
- walking idempotent
- walking isomorphism
- walking morphism
- walking splitting
- dual of the category of sets
- dual of the category of topological spaces