CatDat

infinitary extensive

A category $\mathcal{C}$ is infinitary extensive when it has coproducts and for all families of objects $(A_i)_{i \in I}$ the coproduct functor $\prod_{i \in I} \mathcal{C}/A_i \to \mathcal{C}/(\coprod_{i \in I} A_i)$ is an equivalence of categories. Equivalently, pullbacks of coproduct inclusions along arbitrary morphisms exist and coproducts are disjoint and stable under pullback.

• Dual property: infinitary coextensive
• Related properties: coproducts, disjoint coproducts, extensive
• nLab Link

Relevant implications

• cocomplete andextensive andlocally cartesian closed implies infinitary extensive
• finite products andinfinitary extensive implies infinitary distributive
• Grothendieck topos implies infinitary extensive
• infinitary coextensive andself-dual implies infinitary extensive
• infinitary extensive andself-dual implies infinitary coextensive
• infinitary extensive implies coproducts
• infinitary extensive implies extensive

Examples

There are 18 categories with this property.

• category of Hausdorff spaces
• category of locally ringed spaces
• category of M-sets
• category of measurable spaces
• category of metric spaces with continuous maps
• category of metric spaces with ∞ allowed
• category of pairs of sets
• category of posets
• category of prosets
• category of schemes
• category of sets
• category of sheaves
• category of simplicial sets
• category of small categories
• category of topological spaces
• category of Z-functors
• trivial category
• walking isomorphism

Counterexamples

There are 47 categories without this property.

• category of abelian groups
• category of abelian sheaves
• 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 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 groups
• category of left modules over a division ring
• category of left modules over a ring
• category of metric spaces with non-expansive maps
• category of monoids
• category of non-empty sets
• category of pointed sets
• category of rings
• category of rngs
• category of sets and relations
• category of smooth manifolds
• category of vector 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
• dual of the category of sets
• empty category
• poset [0,1]
• poset of extended natural numbers
• poset of natural numbers
• poset of ordinal numbers
• proset of integers w.r.t. divisibility
• walking commutative square
• walking composable pair
• walking fork
• walking morphism
• walking parallel pair of morphisms
• walking span

Unknown

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

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