CatDat

extensive

A category $\C$ is extensive when it has finite coproducts (denoted $+$) and for all objects $A,B \in \C$ the coproduct functor $$\C/A \times \C/B \to \C/(A+B),$$ which maps $(X \to A, Y \to B)$ to $(X+Y \to A+B)$, is an equivalence of categories. This is equivalent to the following three conditions:

1. Pullbacks along binary coproduct inclusions exist, i.e. for every morphism $T \to A + B$, the pullback $T \times_{A + B} A$ exists.
2. Binary coproducts are disjoint: The coproduct inclusions $A \rightarrow A + B \leftarrow B$ are monomorphisms, and their pullback $A \times_{A + B} B$ is the initial object $0$.
3. Binary coproducts are stable under pullbacks: For every morphism $T \to A + B$, if we define the pullbacks $T_A \coloneqq T \times_{A + B} A$ and $T_B \coloneqq T \times_{A + B} B$, then the canonical morphism $T_A + T_B \to T$ is an isomorphism.

• Dual property: coextensive
• Related properties: countably extensive, disjoint finite coproducts, distributive, infinitary extensive, pretopos
• nLab Link

Relevant implications

• cocomplete andextensive andlocally cartesian closed implies infinitary extensive
• coextensive andself-dual implies extensive
• cofiltered limits andextensive andterminal object implies cocartesian cofiltered limits
• countably extensive implies extensive
• disjoint finite coproducts andlocally cartesian closed implies extensive
• effective congruences andextensive implies mono-regular
• epi-regular andextensive andregular implies co-Malcev andeffective cocongruences
• extensive andfinite products implies distributive
• extensive andself-dual implies coextensive
• extensive implies disjoint finite coproducts andstrict initial object
• extensive implies finite coproducts
• pretopos is equivalent to Barr-exact andextensive

Examples

There are 28 categories with this property.

• category of combinatorial species
• category of compact Hausdorff spaces
• category of countable sets
• category of finite sets
• category of Hausdorff spaces
• category of Jónsson-Tarski algebras
• 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 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 sets with finite-to-one maps
• category of sheaves
• category of simplicial sets
• category of small categories
• category of smooth manifolds
• category of topological spaces
• category of uniform spaces
• category of Z-functors
• trivial category
• walking isomorphism

Counterexamples

There are 69 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 commutative algebras
• category of commutative monoids
• category of commutative rings
• category of countable groups
• 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 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 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 torsion abelian groups
• category of torsion-free abelian groups
• 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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