Implication details
Claim: A category is an elementary cotopos if and only if it is cocartesian coclosed and is finitely cocomplete and has a quotient object classifier.
Proof: This follows from the dual implication.
This implication has a dual.
Show 66 categories using this implication
- empty category
- trivial category
- discrete category on two objects
- category of abelian groups
- category of large families of abelian groups
- category of finitely generated abelian groups
- delooping of the bicyclic monoid
- delooping of the double bicyclic monoid
- 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
- delooping of the additive monoid of ordinal numbers
- category of Banach spaces with linear contractions
- category of sets equipped with a binary relation
- category of sets equipped with an irreflexive binary relation
- category of sets equipped with a reflexive binary relation
- category of sets equipped with a symmetric binary relation
- category of sets equipped with a symmetric irreflexive binary relation
- category of sets equipped with a symmetric reflexive binary relation
- category of commutative algebras
- category of commutative rings
- category of cochain complexes of abelian groups
- category of F(I)-sets
- category of finite sets and injections
- category of filtered vector spaces
- category of finite abelian groups
- category of finite groups
- category of finite sets of even cardinality
- category of fields
- category of fields of characteristic zero
- category of finitely generated free abelian groups
- category of finitely generated free modules over Z x Z
- category of groups
- category of finitely generated groups
- category of sets with a distinguished subset
- partially ordered set of natural numbers
- partially ordered collection of ordinal numbers
- category of left modules over a non-semisimple ring
- category of sets and relations
- category of sequences of abelian groups
- category of pointed sets
- category of sets and bijections
- category of large families of sets and bijections
- category of large families of sets which are mostly empty
- category of sets with finite-to-one maps
- indiscrete category of sets
- category of abelian sheaves
- category of torsion abelian groups
- category of torsion-free abelian groups
- category of long transfinite sequences of abelian groups
- category of Z-functors
- category of finite Z-sets
- preordered set of integers w.r.t. divisibility
- category of graded abelian groups
- category of ω-sets
- real interval [0,1]
- walking commutative square
- walking composable pair
- walking fork
- walking idempotent
- walking isomorphism
- walking morphism
- walking span
- walking splitting