CatDat

exact cofiltered limits

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

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

• Dual property: exact filtered colimits
• Related properties: cocartesian cofiltered limits, cofiltered limits, cofiltered-limit-stable epimorphisms, finitely cocomplete
• nLab Link

Relevant implications

• cocartesian cofiltered limits andthin implies exact cofiltered limits
• exact cofiltered limits andself-dual implies exact filtered colimits
• exact cofiltered limits implies cocartesian cofiltered limits
• exact cofiltered limits implies cofiltered limits andfinitely cocomplete
• exact cofiltered limits implies cofiltered-limit-stable epimorphisms
• exact filtered colimits andself-dual implies exact cofiltered limits

Examples

There are 10 categories with this property.

• dual of the category of sets
• poset [0,1]
• poset of extended natural numbers
• poset of natural numbers
• poset of ordinal numbers
• trivial category
• walking commutative square
• walking composable pair
• walking isomorphism
• walking morphism

Counterexamples

There are 61 categories without this property.

• category of abelian groups
• category of algebras
• category of Banach spaces with linear contractions
• category of combinatorial species
• category of commutative algebras
• category of commutative monoids
• category of commutative rings
• 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 groups
• category of Hausdorff spaces
• category of left modules over a division ring
• category of left modules over a ring
• 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 pseudo-metric spaces with non-expansive maps
• category of rings
• category of rngs
• category of schemes
• category of sets
• category of sets and relations
• category of sets with finite-to-one maps
• category of sheaves
• category of simplicial sets
• category of small categories
• category of smooth manifolds
• category of topological spaces
• category of vector spaces
• category of Z-functors
• 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
• empty category
• proset of integers w.r.t. divisibility
• simplex category
• 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 abelian sheaves
• category of locally ringed spaces