filtered colimits
A category has filtered colimits if it has colimits of diagrams indexed by filtered small categories. This is actually equivalent to having directed colimits.
- Dual property: cofiltered limits
- Related properties: cocomplete, directed colimits, sifted colimits
- nLab Link
Relevant implications
- Cauchy complete andfinite implies filtered colimits
- cofiltered limits andself-dual implies filtered colimits
- directed colimits is equivalent to filtered colimits
- exact filtered colimits implies filtered colimits andfinitely complete
- filtered colimits andfinite copowers implies copowers
- filtered colimits andfinite coproducts implies coproducts
- filtered colimits andpullbacks andreflexive coequalizers implies sifted colimits
- filtered colimits andself-dual implies cofiltered limits
- sifted colimits implies filtered colimits andreflexive coequalizers
- wide pushouts is equivalent to filtered colimits andpushouts
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.