CatDat

infinitary extensive

A category $\C$ is infinitary extensive when it has coproducts and for all families of objects $(A_i)_{i \in I}$ the coproduct functor $$\textstyle \prod_{i \in I} (\C/A_i) \to \C/(\coprod_{i \in I} A_i),$$ that maps a family of morphisms $(X_i \to A_i)_{i \in I}$ to their coproduct $\coprod_{i \in I} X_i \to \coprod_{i \in I} A_i$, is an equivalence of categories. This is equivalent to the following three conditions:

1. Pullbacks along coproduct inclusions exist, i.e. for every morphism $T \to \coprod_{i \in I} A_i$ and every $i \in I$ the pullback $T \times_{\coprod_{i \in I} A_i} A_i$ exists.
2. Coproducts are disjoint: Each coproduct inclusion $A_i \to \coprod_{i \in I} A_i$ is a monomorphism, and for $i \neq j$ the pullback $A_i \times_{\coprod_{i \in I} A_i} A_j$ is the initial object $0$.
3. Coproducts are stable under pullbacks: For every morphism $T \to \coprod_{i \in I} A_i$, if we define the pullbacks $T_i \coloneqq T \times_{\coprod_{i \in I} A_i} A_i$ for $i \in I$, then the canonical morphism $\coprod_{i \in I} T_i \to T$ is an isomorphism.

• Dual property: infinitary coextensive
• Related properties: countably extensive, disjoint coproducts, extensive, infinitary distributive
• 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 countably extensive

Examples

There are 20 categories with this property.

• category of Hausdorff spaces
• category of Jónsson-Tarski algebras
• category of locally ringed spaces
• category of M-sets
• category of metric spaces with continuous maps
• category of metric spaces with ∞ allowed
• category of pairs of sets
• category of partially ordered sets
• category of preordered sets
• category of schemes
• category of set functions and commutative squares
• category of sets
• category of sets with a distinguished subset
• 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 77 categories without this property.

• category of abelian groups
• category of abelian sheaves
• category of algebras
• category of Banach spaces with linear contractions
• category of cochain complexes of abelian groups
• category of combinatorial species
• category of commutative algebras
• category of commutative monoids
• category of commutative rings
• category of compact Hausdorff spaces
• category of countable groups
• category of countable sets
• category of fields
• category of filtered vector spaces
• 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 finite-dimensional vector spaces
• category of finite-dimensional vector spaces [countable field]
• category of finite-dimensional vector spaces [finite field]
• category of finite-dimensional vector spaces [uncountable field]
• category of finitely generated abelian groups
• category of free abelian groups
• category of graded abelian groups
• category of graded modules over a graded ring
• category of groups
• category of left modules over a division ring
• category of left modules over a non-semisimple ring
• category of left modules over a ring
• category of measurable spaces
• category of metric spaces with non-expansive maps
• category of monoids
• category of non-empty sets
• category of pointed sets
• category of pointed topological spaces
• category of pseudo-metric spaces with non-expansive maps
• category of rings
• category of rngs
• category of semigroups
• category of sequences of abelian groups
• category of sets and relations
• category of sets with finite-to-one maps
• category of smooth manifolds
• category of torsion abelian groups
• category of torsion-free abelian groups
• category of uniform spaces
• category of vector spaces
• delooping of a group
• delooping of a non-trivial finite group
• delooping of an infinite countable group
• delooping of an infinite uncountable 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
• dual of the category of topological spaces
• empty category
• forked commutative square
• partially ordered collection of ordinal numbers
• partially ordered set of extended natural numbers
• partially ordered set of natural numbers
• preordered set of integers w.r.t. divisibility
• real interval [0,1]
• simplex category
• walking commutative square
• walking composable pair
• walking coreflexive pair
• walking fork
• walking idempotent
• walking morphism
• walking parallel pair
• walking span
• walking splitting

Unknown

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

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