binary products
A category has binary products if every pair of objects has a product .
- Dual property: binary coproducts
- Related properties: finite products
- nLab Link
Relevant implications
finite products is equivalent to terminal object and binary products
The non-trivial direction follows since finite products can be constructed recursively via .binary products and equalizers implies pullbacks
The pullback of and is the equalizer of .binary products and pullbacks implies equalizers
The equalizer of is the pullback of with the diagonal .pullbacks and terminal object implies binary products
If is a terminal object, then .binary products and inhabited implies connected
For any two objects we have the zig-zag .groupoid and binary products and inhabited implies trivial
Let be an inhabited groupoid with binary products. Then it is connected, so we may assume for a group with unique object . But then , so there are such that , is bijective. From here it is an easy exercise to deduce .self-dual and binary products implies binary coproducts
trivial by self-dualityself-dual and binary coproducts implies binary products
trivial by self-duality
Examples
- category of abelian groups
- category of Banach spaces with linear contractions
- category of combinatorial species
- category of commutative rings
- category of finite abelian groups
- category of finite orders
- category of finite sets
- category of finitely generated abelian groups
- category of free abelian groups
- category of groups
- category of left R-modules
- category of locally ringed spaces
- category of M-sets
- category of measurable spaces
- category of metric spaces with ∞ allowed
- 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 pointed sets
- category of posets
- category of rings
- category of rngs
- category of schemes
- category of sets
- category of sets and relations
- category of simplicial sets
- category of small categories
- category of smooth manifolds
- category of topological spaces
- category of vector spaces
- category of Z-functors
- empty category
- partial order [0,1]
- partial order of extended natural numbers
- partial order of natural numbers
- partial order of ordinal numbers
- preorder of integers w.r.t. divisiblity
- trivial category
- walking isomorphism
- walking morphism
Counterexamples
- category of fields
- category of finite sets and bijections
- category of finite sets and injections
- category of finite sets and surjections
- 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
- walking parallel pair of morphisms
Unknown
For these categories the database has no info if they satisfy this property or not.
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