finite products
A category has finite products if it has products for finite families of objects. Equivalently, it has a terminal object and binary products.
- Dual property: finite coproducts
- Related properties: products
- nLab Link
Relevant implications
- additive is equivalent to finite products and preadditive
- cartesian closed implies finite products
- countable products implies finite products
- distributive implies finite coproducts and finite products
- essentially finite and finite products implies thin
- filtered limits and finite products implies products
- finite coproducts and preadditive implies finite products
- finite coproducts and self-dual implies finite products
- finite products and preadditive implies finite coproducts
- finite products and self-dual implies finite coproducts
- finite products and sequential limits implies countable products
- finite products is equivalent to binary products and terminal object
- finitely complete is equivalent to equalizers and finite products
- infinitary distributive implies coproducts and finite products
- products implies countable products and finite products
Examples
There are 38 categories with this property.
- 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
- partial order [0,1]
- partial order of extended natural numbers
- preorder of integers w.r.t. divisiblity
- trivial category
- walking isomorphism
- walking morphism
Counterexamples
There are 13 categories without this property.
- 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
- empty category
- partial order of natural numbers
- partial order of ordinal numbers
- walking parallel pair of morphisms
Unknown
There are 0 categories for which the database has no information on whether they satisfy this property.
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