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

• disjoint coproducts is equivalent to   coproducts and  disjoint finite coproducts

  easy
• infinitary distributive implies   finite products and  coproducts

  by definition
• cartesian closed and  coproducts implies   infinitary distributive

  Each functor $A \times -$ is left adjoint and hence preserves coproducts (in fact, all colimits).
• distributive and  exact filtered colimits and  coproducts implies   infinitary distributive

  Each functor $A \times -$ preserves finite coproducts and filtered colimits, hence all coproducts.
• Grothendieck topos is equivalent to   elementary topos and  coproducts and  generator and  locally essentially small

  Mac Lane & Moerdijk, Appendix, Prop. 4.4
• Grothendieck abelian is equivalent to   abelian and  coproducts and  generator and  exact filtered colimits

  by definition
• essentially small and  coproducts implies   thin

  [dualized] Mac Lane, V.2, Prop. 3. The proof works for any category with products.
• cocomplete is equivalent to   coproducts and  coequalizers

  [dualized] Mac Lane, V.2, Cor. 2
• coproducts implies   finite coproducts and  countable coproducts

  [dualized] trivial
• finite coproducts and  filtered colimits implies   coproducts

  [dualized] The product $\prod_{i \in I} X_i$ is the filtered limit of the finite partial products $\prod_{i \in E} X_i$ where $E$ ranges over the finite subsets of $I$.
• self-dual and  products implies   coproducts

  trivial by self-duality
• self-dual and  coproducts implies   products

  trivial by self-duality

Examples

• 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

• 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

For these categories the database has no info if they satisfy this property or not.

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