CatDat

exact filtered colimits

In a category $\C$, which we assume to have filtered colimits and finite limits, we say that filtered colimits are exact if the following equivalent conditions are satisfied:

1. For every finite category $\I$ the functor $\lim : [\I, \C] \to \C$ preserves filtered colimits.
2. For every small filtered category $\J$ the functor $\colim : [\J,\C] \to \C$ preserves finite limits.
3. For every diagram $X : \I \times \J \to \C$, where $\I$ is finite and $\J$ is small filtered, the canonical morphism $\colim_j \lim_i X(i,j) \to \lim_i \colim_j X(i,j)$ is an isomorphism.

• Dual property: exact cofiltered limits
• Related properties: cartesian filtered colimits, filtered colimits, finitely complete
• nLab Link

Relevant implications

• cartesian filtered colimits andthin implies exact filtered colimits
• exact cofiltered limits andself-dual implies exact filtered colimits
• exact filtered colimits andself-dual implies exact cofiltered limits
• exact filtered colimits implies cartesian filtered colimits
• exact filtered colimits implies filtered colimits andfinitely complete
• exact filtered colimits implies filtered-colimit-stable monomorphisms
• Grothendieck abelian is equivalent to abelian andcoproducts andexact filtered colimits andgenerator
• Grothendieck topos implies exact filtered colimits
• locally finitely presentable implies exact filtered colimits

Examples

There are 35 categories with this property.

• category of abelian groups
• category of abelian sheaves
• category of algebras
• category of commutative algebras
• category of commutative monoids
• category of commutative rings
• category of groups
• category of Jónsson-Tarski algebras
• category of left modules over a division ring
• category of left modules over a ring
• category of M-sets
• category of monoids
• category of pairs of sets
• category of pointed sets
• 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 torsion abelian groups
• category of torsion-free abelian groups
• category of vector spaces
• category of Z-functors
• poset [0,1]
• poset of extended natural numbers
• trivial category
• walking commutative square
• walking composable pair
• walking isomorphism
• walking morphism

Counterexamples

There are 43 categories without this property.

• category of Banach spaces with linear contractions
• category of combinatorial species
• category of compact Hausdorff spaces
• category of countable groups
• category of countable sets
• category of fields
• category of finite abelian groups
• category of finite groups
• category of finite ordered sets
• category of finite sets
• category of finite sets and bijections
• category of finite sets and injections
• category of finite sets and surjections
• category of finitely generated abelian groups
• category of free abelian groups
• category of Hausdorff spaces
• 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 non-empty sets
• category of pointed topological spaces
• category of sets and relations
• category of sets with finite-to-one maps
• category of smooth manifolds
• category of topological spaces
• delooping of a non-trivial finite group
• delooping of an infinite countable group
• delooping of the additive monoid of natural numbers
• delooping of the additive monoid of ordinal numbers
• discrete category on two objects
• dual of the category of sets
• empty category
• poset of natural numbers
• poset of ordinal numbers
• proset of integers w.r.t. divisibility
• simplex category
• walking coreflexive pair
• walking fork
• walking idempotent
• walking parallel pair
• walking span
• walking splitting

Unknown

There are 2 categories for which the database has no information on whether they satisfy this property. Please help us fill in the gaps by contributing to this project.

• category of locally ringed spaces
• category of schemes