cartesian filtered colimits
In a category , which we assume to have filtered colimits and finite products, we say that filtered colimits are cartesian if for every finite set the product functor preserves filtered colimits. Equivalently, for every the functor preserves filtered colimits.
This is no standard terminology, it has been suggested in MO/510240. We have added it to the database since it clarifies the relationship between many related properties.
- Dual property: cocartesian cofiltered limits
- Related properties: exact filtered colimits, filtered colimits, finite products
Relevant implications
- biproducts andfiltered colimits implies cartesian filtered colimits
- cartesian closed andfiltered colimits implies cartesian filtered colimits
- cartesian filtered colimits andcoproducts anddistributive implies infinitary distributive
- cartesian filtered colimits andcountable coproducts anddistributive implies countably distributive
- cartesian filtered colimits andself-dual implies cocartesian cofiltered limits
- cartesian filtered colimits andthin implies exact filtered colimits
- cartesian filtered colimits implies filtered colimits andfinite products
- cocartesian cofiltered limits andself-dual implies cartesian filtered colimits
- coextensive andfiltered colimits andinitial object implies cartesian filtered colimits
- exact filtered colimits implies cartesian filtered colimits
Examples
There are 36 categories with this property.
- category of abelian groups
- category of abelian sheaves
- category of algebras
- category of Banach spaces with linear contractions
- 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 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 posets
- category of prosets
- category of pseudo-metric spaces with non-expansive maps
- category of rings
- category of rngs
- category of sets
- category of sheaves
- category of simplicial sets
- category of small categories
- category of vector spaces
- category of Z-functors
- dual of the category of sets
- poset [0,1]
- poset of extended natural numbers
- trivial category
- walking commutative square
- walking composable pair
- walking isomorphism
- walking morphism
Counterexamples
There are 32 categories without this property.
- category of combinatorial species
- 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 pointed topological spaces
- category of sets and relations
- 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
- empty category
- poset of natural numbers
- poset of ordinal numbers
- proset of integers w.r.t. divisibility
- simplex category
- walking fork
- walking idempotent
- walking parallel pair
- walking span
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.