directed colimits
A category has directed colimits if it has colimits of diagrams indexed by directed (small) posets. This is actually equivalent to having filtered colimits. Directed colimits are (somewhat confusingly) also known as inverse limits.
- Dual property: directed limits
- Related properties: cocomplete, filtered colimits
- nLab Link
Relevant implications
- directed colimits andself-dual implies directed limits
- directed colimits is equivalent to filtered colimits
- directed colimits implies sequential colimits
- directed limits andself-dual implies directed colimits
- groupoid implies directed colimits andepi-regular andpushouts andright cancellative andself-dual andwell-copowered
Examples
There are 53 categories with this property.
- category of abelian groups
- category of abelian sheaves
- category of algebras
- category of Banach spaces with linear contractions
- category of commutative algebras
- category of commutative monoids
- category of commutative rings
- category of fields
- category of finite sets and bijections
- category of finite sets and surjections
- 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 rings
- category of rngs
- category of sets
- category of sheaves
- category of simplicial sets
- category of small categories
- category of topological spaces
- category of vector spaces
- category of Z-functors
- delooping of a non-trivial finite group
- delooping of an infinite group
- discrete category on two objects
- dual of the category of sets
- empty category
- poset [0,1]
- poset of extended natural numbers
- poset of ordinal numbers
- proset of integers w.r.t. divisibility
- trivial category
- walking commutative square
- walking composable pair
- walking fork
- walking isomorphism
- walking morphism
- walking parallel pair
- walking span
Counterexamples
There are 13 categories without this property.
- category of combinatorial species
- category of finite abelian groups
- category of finite orders
- category of finite sets
- category of finite sets and injections
- category of finitely generated abelian groups
- category of free abelian groups
- category of sets and relations
- category of smooth manifolds
- delooping of the additive monoid of natural numbers
- delooping of the additive monoid of ordinal numbers
- poset of natural numbers
- walking idempotent
Unknown
There is 1 category for which the database has no information on whether it satisfies this property. Please help us fill in the gaps by contributing to this project.