CatDat

cocartesian cofiltered limits

In a category $\mathcal{C}$, which we assume to have cofiltered limits and finite coproducts, we say that cofiltered limits are cocartesian if for every finite set $I$ the coproduct functor $\coprod : \mathcal{C}^I \to \mathcal{C}$ preserves cofiltered limits. Equivalently, for every $X \in \mathcal{C}$ the functor $X \sqcup - : \mathcal{C} \to \mathcal{C}$ preserves cofiltered limits.
This is no standard terminology, its dual has been suggested in MO/510240. We have added it to the database since it clarifies the relationship between many related properties.

• Dual property: cartesian filtered colimits
• Related properties: cofiltered limits, finite coproducts

Relevant implications

• biproducts andcofiltered limits implies cocartesian cofiltered limits
• cartesian filtered colimits andself-dual implies cocartesian cofiltered limits
• cocartesian coclosed andcofiltered limits implies cocartesian cofiltered limits
• cocartesian cofiltered limits andcodistributive andcountable products implies countably codistributive
• cocartesian cofiltered limits andcodistributive andproducts implies infinitary codistributive
• cocartesian cofiltered limits andself-dual implies cartesian filtered colimits
• cocartesian cofiltered limits implies cofiltered limits andfinite coproducts
• cofiltered limits andextensive andterminal object implies cocartesian cofiltered limits

Examples

There are 33 categories with this property.

• category of abelian groups
• category of abelian sheaves
• category of Banach spaces with linear contractions
• category of commutative monoids
• category of Hausdorff spaces
• category of left modules over a division ring
• category of left modules over a ring
• category of locally ringed spaces
• category of M-sets
• category of measurable spaces
• category of metric spaces with ∞ allowed
• category of pairs of sets
• category of pointed sets
• category of pointed topological spaces
• category of posets
• category of prosets
• category of sets
• category of sheaves
• category of simplicial sets
• category of small categories
• category of topological spaces
• category of vector spaces
• category of Z-functors
• 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 37 categories without this property.

• category of algebras
• category of combinatorial species
• category of commutative algebras
• category of commutative rings
• 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 metric spaces with continuous maps
• category of metric spaces with non-expansive maps
• category of monoids
• category of non-empty sets
• category of pseudo-metric spaces with non-expansive maps
• category of rings
• category of rngs
• category of schemes
• category of sets and relations
• category of smooth manifolds
• 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

Unknown

There are 0 categories for which the database has no information on whether they satisfy this property.

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