filtered colimits
A category has filtered colimits if it has colimits of diagrams indexed by filtered categories.
- Dual property: filtered limits
- Related properties: cocomplete
- nLab Link
Relevant implications
exact filtered colimits implies filtered colimits and finitely complete
by definitioncocomplete implies finitely cocomplete and filtered colimits and wide pushouts and connected colimits
[dualized] trivialfinite coproducts and filtered colimits implies coproducts
[dualized] The product is the filtered limit of the finite partial products where ranges over the finite subsets of .wide pushouts is equivalent to pushouts and filtered colimits
[dualized] To prove , a wide pullback can be constructed as a filtered limit of finite pullbacks, and finite pullbacks can be reduced to binary products (the empty-indexed pullback always exists). Conversely, assume that wide pullbacks exist in . For every object then the slice category has wide pullbacks and a terminal object, hence is complete. Since a filtered limit can be finally reduced to such a slice, we are done.filtered colimits implies sequential colimits
[dualized] trivialgroupoid implies self-dual and epi-regular and pushouts and filtered colimits and right cancellative and well-copowered
[dualized] easyself-dual and filtered limits implies filtered colimits
trivial by self-dualityself-dual and filtered colimits implies filtered limits
trivial by self-duality
Examples
- category of abelian groups
- category of Banach spaces with linear contractions
- category of commutative rings
- category of fields
- category of finite sets and bijections
- category of finite sets and surjections
- 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 metric spaces with non-expansive maps
- category of monoids
- category of non-empty sets
- 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
- delooping of a non-trivial finite group
- delooping of an infinite group
- discrete category on two objects
- empty category
- 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
- walking parallel pair of morphisms
Counterexamples
- 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 smooth manifolds
- delooping of the additive monoid of natural numbers
- partial order of natural numbers
Unknown
For these categories the database has no info if they satisfy this property or not.