CatDat

cocomplete

A category is cocomplete when every small diagram in the category has a colimit.

• Dual property: complete
• nLab Link

Relevant implications

• essentially small and  thin and  complete implies   cocomplete

  The supremum of a subset in a (small) partial order is the infimum of the set of upper bounds.
• locally presentable implies   locally essentially small and  well-powered and  well-copowered and  complete and  cocomplete and  generator

  For the non-trivial conclusions see Adamek-Rosicky, Thm. 1.20, Cor. 1.28, Rem. 1.56, Thm. 1.58.
• cocomplete implies   finitely cocomplete and  filtered colimits and  wide pushouts and  connected colimits

  [dualized] trivial
• cocomplete is equivalent to   coproducts and  coequalizers

  [dualized] Mac Lane, V.2, Cor. 2
• wide pushouts and  initial object implies   cocomplete

  [dualized] See the nLab.
• essentially small and  thin and  cocomplete implies   complete

  [dualized] The supremum of a subset in a (small) partial order is the infimum of the set of upper bounds.
• self-dual and  complete implies   cocomplete

  trivial by self-duality
• self-dual and  cocomplete implies   complete

  trivial by self-duality

Examples

• category of abelian groups
• category of Banach spaces with linear contractions
• category of commutative rings
• 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 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 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
• 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.

• category of metric spaces with continuous maps