CatDat

disjoint finite coproducts

A category has disjoint finite coproducts if it has finite coproducts, for every pair of objects $A,B$ the coproduct inclusions $A \rightarrow A+B \leftarrow B$ are monomorphisms, and the pullback $A \times_{A + B} B$ exists and is given by the initial object $0$.

• Related properties: finite coproducts, disjoint coproducts
• nLab Link

Relevant implications

• disjoint coproducts is equivalent to   coproducts and  disjoint finite coproducts

  easy
• disjoint finite coproducts implies   finite coproducts

  by definition
• disjoint finite coproducts and  thin implies   trivial

  For every object $A$ the two inclusions $A \rightrightarrows A + A$ must be equal, so their equalizer is $A$, but also $0$ since the coproduct is disjoint. Hence $A = 0$.
• elementary topos implies   finitely cocomplete and  disjoint finite coproducts and  epi-regular

  Mac Lane & Moerdijk, Cor. IV.5.4, Cor. IV.10.5, Thm. 4.7.8.
• additive implies   disjoint finite coproducts

  If $f : T \to A + B$ is a morphism that factors through $A$ and $B$, then $p_B f = 0$ and $p_A f = 0$, so $f = 0$.

Examples

• category of abelian groups
• category of Banach spaces with linear contractions
• category of combinatorial species
• 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 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
• trivial category
• walking isomorphism

Counterexamples

• category of commutative rings
• 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
• category of rings
• 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 [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
• walking morphism
• 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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