CatDat

products

Given a family of objects $(A_i)_{i \in I}$, a product $\prod_{i \in I} A_i$ is defined as an object with morphisms $p_i : \prod_{i \in I} A_i \to A_i$ satisfying the following universal property: For every object $T$ and every family of morphisms $(f_i : T \to A_i)_{i \in I}$ there is a unique morphism $f : T \to \prod_{i \in I} A_i$ such that $p_i \circ f = f_i$ for all $i \in I$. This property refers to the existence of products.

• Dual property: coproducts
• Related properties: complete, finite products
• nLab Link

Relevant implications

• cofiltered limits andfinite products implies products
• cogenerating set andproducts andzero morphisms implies cogenerator
• complete is equivalent to equalizers andproducts
• coproducts andself-dual implies products
• disjoint products is equivalent to disjoint finite products andproducts
• essentially finite andfinite products implies products andthin
• essentially small andproducts implies thin
• infinitary codistributive implies finite coproducts andproducts
• infinitary coextensive implies products
• products andself-dual implies coproducts
• products implies countable products andfinite products

Examples

There are 40 categories with this property.

• category of abelian groups
• category of abelian sheaves
• category of algebras
• category of Banach spaces with linear contractions
• category of commutative algebras
• category of commutative monoids
• category of commutative rings
• category of groups
• category of Hausdorff spaces
• category of left modules over a division 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 ∞ allowed
• category of monoids
• category of non-empty sets
• category of pairs of sets
• category of pointed sets
• category of posets
• category of prosets
• category of rings
• category of rngs
• category of sets
• category of sets and relations
• category of sheaves
• category of simplicial sets
• category of small categories
• category of topological spaces
• category of vector spaces
• category of Z-functors
• dual of the category of sets
• poset [0,1]
• poset of extended natural numbers
• proset of integers w.r.t. divisibility
• trivial category
• walking commutative square
• walking composable pair
• walking isomorphism
• walking morphism

Counterexamples

There are 25 categories without this property.

• category of combinatorial species
• category of fields
• category of finite abelian groups
• category of finite orders
• category of finite sets
• category of finite sets and bijections
• category of finite sets and injections
• category of finite sets and surjections
• category of finitely generated abelian groups
• category of free abelian groups
• category of metric spaces with continuous maps
• category of metric spaces with non-expansive maps
• category of schemes
• category of smooth manifolds
• 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
• poset of natural numbers
• poset of ordinal numbers
• walking fork
• walking parallel pair of morphisms
• walking span

Unknown

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

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