CatDat

cocartesian coclosed

A category is cocartesian coclosed if its dual category is cartesian closed, i.e. if all finite coproducts and coexponentials $\mathrm{Coexp}(X,Y)$ exist, defined by the adjunction $\mathrm{Hom}(\mathrm{Coexp}[X,Y],T) \cong \mathrm{Hom}(Y,T \sqcup X)$.

• Dual property: cartesian closed
• Related properties: finite coproducts, locally cocartesian coclosed
• nLab Link

Relevant implications

• cartesian closed andself-dual implies cocartesian coclosed
• cocartesian coclosed andcofiltered limits implies cocartesian cofiltered limits
• cocartesian coclosed andcountable powers implies countable copowers
• cocartesian coclosed andcountable products implies countably codistributive
• cocartesian coclosed andfinite products implies codistributive
• cocartesian coclosed andpowers implies copowers
• cocartesian coclosed andproducts implies infinitary codistributive
• cocartesian coclosed andself-dual implies cartesian closed
• cocartesian coclosed andstrict initial object implies thin
• cocartesian coclosed andterminal object implies strict terminal object
• cocartesian coclosed implies finite coproducts
• cocomplete andessentially small andinfinitary codistributive andthin implies cocartesian coclosed
• initial object andlocally cocartesian coclosed implies cocartesian coclosed

Examples

There are 10 categories with this property.

• dual of the category of sets
• poset [0,1]
• poset of extended natural numbers
• poset of natural numbers
• poset of ordinal numbers
• trivial category
• walking commutative square
• walking composable pair
• walking isomorphism
• walking morphism

Counterexamples

There are 60 categories without this property.

• category of abelian groups
• category of abelian sheaves
• category of algebras
• category of Banach spaces with linear contractions
• category of combinatorial species
• category of commutative algebras
• category of commutative monoids
• category of commutative rings
• category of fields
• category of finite abelian groups
• category of finite groups
• category of finite ordered sets
• category of finite sets
• 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 M-sets
• 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 non-empty sets
• category of pairs of sets
• category of pointed sets
• category of pointed topological spaces
• category of posets
• category of prosets
• category of pseudo-metric spaces with non-expansive maps
• category of rings
• category of rngs
• category of schemes
• category of sets
• category of sets and relations
• category of sheaves
• category of simplicial sets
• category of small categories
• 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 countable group
• delooping of the additive monoid of natural numbers
• delooping of the additive monoid of ordinal numbers
• discrete category on two objects
• empty category
• proset of integers w.r.t. divisibility
• simplex category
• walking fork
• walking idempotent
• walking parallel pair
• walking span

Unknown

There are 0 categories for which the database has no information on whether they satisfy this property.

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