CatDat

cartesian closed

A category is cartesian closed if all finite products and exponentials exist.

• Related properties: finite products
• nLab Link

Relevant implications

• pointed and  cartesian closed implies   trivial

  We have $X \cong X \times 1 \cong X \times 0 \cong 0$ for every object $X$.
• cartesian closed and  finite coproducts implies   distributive

  Each functor $A \times -$ is left adjoint and hence preserves finite coproducts (in fact, all colimits).
• cartesian closed and  coproducts implies   infinitary distributive

  Each functor $A \times -$ is left adjoint and hence preserves coproducts (in fact, all colimits).
• essentially small and  thin and  complete and  infinitary distributive implies   cartesian closed

  This is an application of the adjoint functor theorem. Specifically, if $P$ is a complete lattice in which $\sup_i \inf(t,x_i) = \inf(t, \sup_i y_i)$ always holds, then the functor $\int(t,-)$ is a left adjoint because it preserves all suprema.
• elementary topos is equivalent to   cartesian closed and  finitely complete and  subobject classifier

  by definition
• cartesian closed implies   finite products

  by definition
• cartesian closed and  initial object implies   strict initial object

  See the nLab.

Examples

• category of combinatorial species
• category of finite sets
• category of M-sets
• category of non-empty sets
• category of posets
• category of sets
• category of simplicial sets
• category of small categories
• partial order of extended natural numbers
• trivial category
• walking isomorphism
• walking morphism

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 pointed sets
• 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
• 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 of natural numbers
• partial order of ordinal numbers
• preorder of integers w.r.t. divisiblity
• walking parallel pair of morphisms

Unknown

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

• category of Z-functors
• partial order [0,1]