cartesian closed
A category is cartesian closed if all finite products and exponentials exist, defined by the adjunction .
- Related properties: finite products
- nLab Link
Relevant implications
- cartesian closed andcopowers implies powers
- cartesian closed andcoproducts implies infinitary distributive
- cartesian closed andcountable copowers implies countable powers
- cartesian closed andfinite coproducts implies distributive
- cartesian closed andinitial object implies strict initial object
- cartesian closed andpointed implies trivial
- cartesian closed implies finite products
- complete andessentially small andinfinitary distributive andthin implies cartesian closed
- elementary topos is equivalent to cartesian closed andfinitely complete andsubobject classifier
- locally cartesian closed andterminal object implies cartesian closed
Examples
There are 18 categories with this property.
- category of combinatorial species
- category of finite sets
- category of M-sets
- category of non-empty sets
- category of pairs of sets
- category of posets
- category of prosets
- category of sets
- category of sheaves
- category of simplicial sets
- category of small categories
- poset [0,1]
- poset of extended natural numbers
- trivial category
- walking commutative square
- walking composable pair
- walking isomorphism
- walking morphism
Counterexamples
There are 48 categories without this property.
- category of abelian groups
- category of abelian sheaves
- category of algebras
- category of Banach spaces with linear contractions
- category of commutative algebras
- category of commutative monoids
- 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 Hausdorff spaces
- category of left modules over a division ring
- category of left modules over a ring
- category of locally ringed spaces
- category of measurable spaces
- category of metric spaces with continuous maps
- category of metric spaces with non-expansive maps
- category of metric spaces with ∞ allowed
- category of monoids
- category of pointed sets
- category of pointed topological spaces
- category of rings
- category of rngs
- category of schemes
- category of sets and relations
- 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
- poset of natural numbers
- poset of ordinal numbers
- proset of integers w.r.t. divisibility
- walking fork
- walking idempotent
- walking parallel pair
- walking span
Unknown
There is 1 category for which the database has no information on whether it satisfies this property. Please help us fill in the gaps by contributing to this project.