CatDat

coproducts

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

• Dual property: products
• Related properties: cocomplete
• nLab Link

Relevant implications

• cartesian closed and  coproducts implies   infinitary distributive
• cocomplete is equivalent to   coequalizers and  coproducts
• coproducts and  distributive and  exact filtered colimits implies   infinitary distributive
• coproducts and  essentially small implies   thin
• coproducts and  self-dual implies   products
• coproducts implies   countable coproducts and  finite coproducts
• disjoint coproducts is equivalent to   coproducts and  disjoint finite coproducts
• filtered colimits and  finite coproducts implies   coproducts
• Grothendieck abelian is equivalent to   abelian and  coproducts and  exact filtered colimits and  generator
• Grothendieck topos is equivalent to   coproducts and  elementary topos and  generator and  locally essentially small
• infinitary distributive implies   coproducts and  finite products
• products and  self-dual implies   coproducts

Examples

There are 31 categories with this property.

• category of abelian groups
• category of Banach spaces with linear contractions
• category of commutative rings
• 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 monoids
• 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 topological spaces
• category of vector spaces
• category of Z-functors
• partial order [0,1]
• partial order of extended natural numbers
• partial order of ordinal numbers
• preorder of integers w.r.t. divisiblity
• trivial category
• walking isomorphism
• walking morphism

Counterexamples

There are 20 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 metric spaces with non-expansive maps
• category of non-empty sets
• 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
• partial order of natural 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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