CatDat

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] trivial
• finitely cocomplete is equivalent to   finite coproducts and  coequalizers

  [dualized] Mac Lane, V.2, Cor. 1
• self-dual and  finitely complete implies   finitely cocomplete

  trivial by self-duality
• self-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.

• category of metric spaces with continuous maps