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

• exact filtered colimits implies   filtered colimits and  finitely complete

  by definition
• distributive and  exact filtered colimits and  coproducts implies   infinitary distributive

  Each functor $A \times -$ preserves finite coproducts and filtered colimits, hence all coproducts.
• locally finitely presentable implies   exact filtered colimits

  Special case of Adamek-Rosicky, Prop. 1.59 with $\lambda = \aleph_0$.
• Grothendieck abelian is equivalent to   abelian and  coproducts and  generator and  exact filtered colimits

  by definition

Examples

• category of abelian groups
• category of commutative rings
• category of groups
• category of left R-modules
• category of M-sets
• category of monoids
• 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 vector spaces
• category of Z-functors
• partial order of extended natural numbers
• trivial category
• walking isomorphism
• walking morphism

Counterexamples

• 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 ∞ allowed
• category of metric spaces with non-expansive maps
• 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
• partial order of natural numbers
• partial order of ordinal numbers
• preorder of integers w.r.t. divisiblity
• walking parallel pair of morphisms

Unknown

For these categories the database has no info if they satisfy this property or not.

• category of locally ringed spaces
• category of measurable spaces
• category of metric spaces with continuous maps
• category of schemes
• partial order [0,1]