CatDat

filtered limits

A category has filtered limits if it has limits of diagrams indexed by cofiltered categories.

• Dual property: filtered colimits
• Related properties: complete
• nLab Link

Relevant implications

• complete implies   finitely complete and  filtered limits and  wide pullbacks and  connected limits

  trivial
• finite products and  filtered limits implies   products

  The product $\prod_{i \in I} X_i$ is the filtered limit of the finite partial products $\prod_{i \in E} X_i$ where $E$ ranges over the finite subsets of $I$.
• wide pullbacks is equivalent to   pullbacks and  filtered limits

  To prove $\Leftarrow$, a wide pullback can be constructed as a filtered limit of finite pullbacks, and finite pullbacks can be reduced to binary products (the empty-indexed pullback always exists). Conversely, assume that wide pullbacks exist in $\mathcal{C}$. For every object $A$ then the slice category $\mathcal{C} / A$ has wide pullbacks and a terminal object, hence is complete. Since a filtered limit can be finally reduced to such a slice, we are done.
• filtered limits implies   sequential limits

  trivial
• groupoid implies   self-dual and  mono-regular and  pullbacks and  filtered limits and  left cancellative and  well-powered

  easy
• self-dual and  filtered limits implies   filtered colimits

  trivial by self-duality
• self-dual and  filtered colimits implies   filtered limits

  trivial by self-duality

Examples

• category of abelian groups
• category of Banach spaces with linear contractions
• category of commutative rings
• category of fields
• category of finite sets and bijections
• category of finite sets and injections
• category of groups
• category of left R-modules
• category of locally ringed spaces
• category of M-sets
• category of measurable spaces
• category of metric spaces with ∞ allowed
• 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 topological spaces
• category of vector spaces
• category of Z-functors
• delooping of a non-trivial finite group
• delooping of an infinite group
• discrete category on two objects
• empty category
• partial order [0,1]
• partial order of extended natural numbers
• partial order of natural numbers
• partial order of ordinal numbers
• preorder of integers w.r.t. divisiblity
• trivial category
• walking isomorphism
• walking morphism
• walking parallel pair of morphisms

Counterexamples

• category of combinatorial species
• category of finite abelian groups
• category of finite orders
• category of finite sets
• category of finite sets and surjections
• category of finitely generated abelian groups
• category of free abelian groups
• category of metric spaces with continuous maps
• category of metric spaces with non-expansive maps
• category of non-empty sets
• category of schemes
• category of smooth manifolds
• delooping of the additive monoid of natural numbers

Unknown

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

• category of sets and relations
• delooping of the additive monoid of ordinal numbers