finitely cocomplete
A category is finitely cocomplete when every finite diagram has a colimit.
- Dual property: finitely complete
- Related properties: cocomplete
- nLab Link
Relevant implications
elementary topos implies finitely cocomplete and disjoint finite coproducts and epi-regular
Mac Lane & Moerdijk, Cor. IV.5.4, Cor. IV.10.5, Thm. 4.7.8.cocomplete implies finitely cocomplete and filtered colimits and wide pushouts and connected colimits
[dualized] trivialfinitely cocomplete is equivalent to finite coproducts and coequalizers
[dualized] Mac Lane, V.2, Cor. 1self-dual and finitely complete implies finitely cocomplete
trivial by self-dualityself-dual and finitely cocomplete implies finitely complete
trivial by self-duality
Examples
- category of abelian groups
- category of Banach spaces with linear contractions
- category of combinatorial species
- category of commutative rings
- category of finite abelian groups
- category of finite sets
- category of finitely generated 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 monoids
- category of pointed sets
- category of posets
- category of rings
- category of rngs
- category of sets
- category of simplicial sets
- category of small categories
- 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 natural numbers
- partial order of ordinal numbers
- preorder of integers w.r.t. divisiblity
- trivial category
- walking isomorphism
- walking morphism
Counterexamples
- category of fields
- category of finite orders
- category of finite sets and bijections
- category of finite sets and injections
- category of finite sets and surjections
- category of free abelian groups
- category of metric spaces with non-expansive maps
- category of non-empty sets
- category of schemes
- category of sets and relations
- category of smooth manifolds
- 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
- walking parallel pair of morphisms
Unknown
For these categories the database has no info if they satisfy this property or not.