CatDat

disjoint products

A category has disjoint products if it has products, the product projections $\prod_{i \in I} A_i \to A_i$ are epimorphisms, and the pushout of the projections $\prod_{i \in I} A_i \to A_i$ and $\prod_{i \in I} A_i \to A_j$ for $i \neq j$ exists and is given by the terminal object $1$.
This terminology does not seem to be common, but we have added it as a dual for the more commonly known property of having disjoint coproducts.

• Dual property: disjoint coproducts
• Related properties: disjoint finite products, infinitary coextensive, products

Relevant implications

• disjoint coproducts andself-dual implies disjoint products
• disjoint products andself-dual implies disjoint coproducts
• disjoint products is equivalent to disjoint finite products andproducts
• finitary algebraic andpointed implies disjoint products

Examples

There are 20 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 left modules over a division ring
• category of left modules over a ring
• category of monoids
• category of non-empty sets
• category of pointed sets
• category of rings
• category of rngs
• category of sets and relations
• category of vector spaces
• dual of the category of sets
• trivial category
• walking isomorphism

Counterexamples

There are 45 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 Hausdorff spaces
• 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 pairs of sets
• category of posets
• category of prosets
• category of schemes
• category of sets
• category of sheaves
• category of simplicial sets
• category of small categories
• category of smooth manifolds
• category of topological spaces
• category of Z-functors
• 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 [0,1]
• poset of extended natural numbers
• poset of natural numbers
• poset of ordinal numbers
• proset of integers w.r.t. divisibility
• walking commutative square
• walking composable pair
• walking fork
• walking morphism
• 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.

—