countable coproducts
A category has countable coproducts if it has coproducts for countable families of objects.
- Dual property: countable products
- Related properties: coproducts, finite coproducts
Relevant implications
coproducts implies finite coproducts and countable coproducts
[dualized] trivialcountable coproducts implies finite coproducts
[dualized] trivialcoequalizers and countable coproducts implies sequential colimits
[dualized] Mac Lane, V.2, Prop. 3. The proof can easily be adapted to this case. Namely, the limit of is the equalizer of two suitable endomorphisms of .finite coproducts and sequential colimits implies countable coproducts
[dualized] If is an infinite sequence of objects, then their product is the limit of the sequence .self-dual and countable products implies countable coproducts
trivial by self-dualityself-dual and countable coproducts implies countable products
trivial by self-duality
Examples
- category of abelian groups
- category of Banach spaces with linear contractions
- category of commutative rings
- category of free abelian groups
- category of groups
- category of left R-modules
- category of locally ringed spaces
- category of M-sets
- category of measurable spaces
- category of metric spaces with ∞ allowed
- category of metric spaces with continuous maps
- category of monoids
- category of pointed sets
- category of posets
- category of rings
- category of rngs
- category of schemes
- category of sets
- category of sets and relations
- category of simplicial sets
- category of small categories
- category of smooth manifolds
- category of topological spaces
- category of vector spaces
- category of Z-functors
- partial order [0,1]
- partial order of extended natural numbers
- partial order of ordinal numbers
- preorder of integers w.r.t. divisiblity
- trivial category
- walking isomorphism
- walking morphism
Counterexamples
- category of combinatorial species
- category of fields
- category of finite abelian groups
- category of finite orders
- 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 metric spaces with non-expansive maps
- category of non-empty sets
- 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
- walking parallel pair of morphisms
Unknown
For these categories the database has no info if they satisfy this property or not.
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