CatDat

exact filtered colimits

In a category $\mathcal{C}$, which we assume to have filtered colimits and finite limits, we say that filtered colimits are exact if for every finite category $\mathcal{I}$ the functor $\lim : [\mathcal{I}, \mathcal{C}] \to \mathcal{C}$ preserves filtered colimits. Equivalently, for every diagram $X : \mathcal{I} \times \mathcal{J} \to \mathcal{C}$, where $\mathcal{I}$ is finite and $\mathcal{J}$ is filtered, the canonical morphism $\mathrm{colim}_{j} \lim_{i} X(i,j) \to \lim_{i} \mathrm{colim}_j X(i,j)$ is an isomorphism.

• Related properties: filtered colimits, finitely complete
• nLab Link

Relevant implications

• coproducts anddistributive andexact filtered colimits implies infinitary distributive
• exact filtered colimits implies filtered colimits andfinitely complete
• Grothendieck abelian is equivalent to abelian andcoproducts andexact filtered colimits andgenerator
• locally finitely presentable implies exact filtered colimits

Examples

There are 28 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 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 rings
• category of rngs
• category of sets
• category of simplicial sets
• category of small categories
• category of vector spaces
• category of Z-functors
• poset of extended natural numbers
• trivial category
• walking commutative square
• walking composable pair
• walking isomorphism
• walking morphism

Counterexamples

There are 29 categories without this property.

• category of Banach spaces with linear contractions
• category of combinatorial species
• category of fields
• category of finite abelian groups
• category of finite orders
• 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 metric spaces with non-expansive maps
• category of metric spaces with ∞ allowed
• category of non-empty sets
• category of sets and relations
• category of smooth manifolds
• category of topological spaces
• delooping of a non-trivial finite group
• delooping of an infinite group
• delooping of the additive monoid of natural numbers
• delooping of the additive monoid of ordinal numbers
• discrete category on two objects
• empty category
• poset of natural numbers
• poset of ordinal numbers
• proset of integers w.r.t. divisibility
• walking fork
• walking parallel pair of morphisms
• walking span

Unknown

There are 8 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 Hausdorff spaces
• category of locally ringed spaces
• category of measurable spaces
• category of metric spaces with continuous maps
• category of schemes
• category of sheaves
• dual of the category of sets
• poset [0,1]