CatDat

filtered colimits

A category has filtered colimits if it has colimits of diagrams indexed by small filtered categories. This is actually equivalent to having directed colimits.

• Dual property: cofiltered limits
• Related properties: cocomplete, directed colimits, filtered, sifted colimits, ℵ₁-filtered colimits
• nLab Link

Relevant implications

• biproducts andfiltered colimits andfiltered-colimit-stable monomorphisms andproducts implies CIP
• biproducts andfiltered colimits implies cartesian filtered colimits
• cartesian closed andfiltered colimits implies cartesian filtered colimits
• cartesian filtered colimits implies filtered colimits
• Cauchy complete andessentially finite implies filtered colimits
• coextensive andfiltered colimits andinitial object implies cartesian filtered colimits
• cofiltered limits andself-dual implies filtered colimits
• directed colimits is equivalent to filtered colimits
• exact filtered colimits implies filtered colimits
• filtered colimits andfinite copowers implies copowers
• filtered colimits andfinite coproducts implies coproducts
• filtered colimits andleft cancellative implies filtered-colimit-stable monomorphisms
• filtered colimits andleft cancellative implies sifted colimits
• filtered colimits andpullbacks andreflexive coequalizers implies sifted colimits
• filtered colimits andself-dual implies cofiltered limits
• filtered colimits andsubobject-trivial implies filtered-colimit-stable monomorphisms
• filtered colimits implies ℵ₁-filtered colimits
• filtered-colimit-stable monomorphisms implies filtered colimits
• finitely accessible implies filtered colimits
• sifted colimits implies filtered colimits
• wide pushouts is equivalent to filtered colimits andpushouts

Examples

There are 63 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 compact Hausdorff spaces
• category of fields
• category of filtered vector spaces
• category of finite sets and bijections
• category of finite sets and surjections
• category of groups
• category of Hausdorff spaces
• category of Jónsson-Tarski algebras
• 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 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 pseudo-metric spaces with non-expansive maps
• category of rings
• category of rngs
• category of semigroups
• category of sets
• category of sheaves
• category of simplicial sets
• category of small categories
• category of topological spaces
• category of torsion abelian groups
• category of torsion-free abelian groups
• category of vector spaces
• category of Z-functors
• delooping of a non-trivial finite group
• delooping of an infinite countable group
• delooping of an infinite uncountable group
• discrete category on two objects
• dual of the category of sets
• dual of the category of topological spaces
• 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 coreflexive pair
• walking fork
• walking isomorphism
• walking morphism
• walking parallel pair
• walking span
• walking splitting

Counterexamples

There are 23 categories without this property.

• category of combinatorial species
• category of countable groups
• category of countable sets
• category of finite abelian groups
• category of finite groups
• category of finite ordered sets
• category of finite sets
• category of finite sets and injections
• category of finite-dimensional vector spaces [countable field]
• category of finite-dimensional vector spaces [finite field]
• category of finite-dimensional vector spaces [uncountable field]
• category of finitely generated abelian groups
• category of free abelian groups
• category of metric spaces with continuous maps
• category of schemes
• category of sets and relations
• category of sets with finite-to-one maps
• category of smooth manifolds
• delooping of the additive monoid of natural numbers
• delooping of the additive monoid of ordinal numbers
• poset of natural numbers
• simplex category
• walking idempotent

Unknown

There are 0 categories for which the database has no information on whether they satisfy this property.

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