CatDat

binary coproducts

A category has binary coproducts if every pair $A,B$ of objects has a coproduct $A \sqcup B$.

• Dual property: binary products
• Related properties: finite coproducts
• nLab Link

Relevant implications

• finite coproducts is equivalent to   initial object and  binary coproducts

  [dualized] The non-trivial direction follows since finite products can be constructed recursively via $X_1 \times \cdots \times X_{n+1} = (X_1 \times \cdots \times X_n) \times X_{n+1}$.
• binary coproducts and  coequalizers implies   pushouts

  [dualized] The pullback of $f : X \to S$ and $g : Y \to S$ is the equalizer of $p_1 \circ f, \, p_2 \circ g : X \times Y \rightrightarrows S$.
• binary coproducts and  pushouts implies   coequalizers

  [dualized] The equalizer of $f,g : X \rightrightarrows Y$ is the pullback of $(f,g) : X \to Y \times Y$ with the diagonal $Y \to Y \times Y$.
• pushouts and  initial object implies   binary coproducts

  [dualized] If $1$ is a terminal object, then $X \times_1 Y = X \times Y$.
• binary coproducts and  inhabited implies   connected

  [dualized] For any two objects $A,B$ we have the zig-zag $A \to A \times B \to B$.
• groupoid and  binary coproducts and  inhabited implies   trivial

  [dualized] Let $\mathcal{C}$ be an inhabited groupoid with binary products. Then it is connected, so we may assume $\mathcal{C}=BG$ for a group $G$ with unique object $*$. But then $* \times * = *$, so there are $p,q \in G$ such that $G \to G \times G$, $x \mapsto (px,qx)$ is bijective. From here it is an easy exercise to deduce $G=1$.
• self-dual and  binary products implies   binary coproducts

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

  trivial by self-duality

Examples

• 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 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 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 smooth manifolds
• category of topological spaces
• category of vector spaces
• category of Z-functors
• empty category
• partial order [0,1]
• partial order of extended natural numbers
• partial order of natural numbers
• partial order of ordinal numbers
• preorder of integers w.r.t. divisiblity
• trivial category
• walking isomorphism
• walking morphism

Counterexamples

• category of fields
• category of finite orders
• category of finite sets and bijections
• category of finite sets and injections
• category of finite sets and surjections
• category of metric spaces with non-expansive maps
• category of non-empty sets
• 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
• 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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