CatDat

coextensive

A category $\C$ is coextensive when it has finite products and for all objects $A,B \in \C$ the product functor $A/\C \times B/\C \to (A \times B)/\C$ is an equivalence of categories. Equivalently (by using the characterization of extensive categories), pushouts along binary product projections exist, binary products are disjoint, and binary products are stable under pushouts. The prototypical example is the category of commutative rings.

• Dual property: extensive
• Related properties: codistributive, countably coextensive, disjoint finite products, infinitary coextensive

Relevant implications

• coextensive andcomplete andlocally cocartesian coclosed implies infinitary coextensive
• coextensive andcoregular andmono-regular implies Malcev andeffective congruences
• coextensive andeffective cocongruences implies epi-regular
• coextensive andfiltered colimits andinitial object implies cartesian filtered colimits
• coextensive andfinite coproducts implies codistributive
• coextensive andself-dual implies extensive
• coextensive implies disjoint finite products andstrict terminal object
• coextensive implies finite products
• countably coextensive implies coextensive
• disjoint finite products andlocally cocartesian coclosed implies coextensive
• extensive andself-dual implies coextensive

Examples

There are 6 categories with this property.

• category of commutative algebras
• category of commutative rings
• dual of the category of sets
• dual of the category of topological spaces
• trivial category
• walking isomorphism

Counterexamples

There are 91 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 monoids
• 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 Hausdorff spaces
• category of Jónsson-Tarski algebras
• 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 locally ringed spaces
• category of M-sets
• category of measurable spaces
• category of metric spaces with continuous maps
• 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 partially ordered sets
• category of pointed sets
• category of pointed topological spaces
• category of preordered sets
• category of pseudo-metric spaces with non-expansive maps
• category of rings
• category of rngs
• category of schemes
• category of semigroups
• category of sequences of abelian groups
• category of set functions and commutative squares
• category of sets
• category of sets and relations
• 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 torsion abelian groups
• category of torsion-free abelian groups
• category of uniform spaces
• category of vector spaces
• category of Z-functors
• 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
• 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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