CatDat

finite coproducts

A category has finite coproducts if it has coproducts for finite families of objects. Equivalently, it has an initial object and binary coproducts.

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

Relevant implications

• disjoint finite coproducts implies   finite coproducts

  by definition
• distributive implies   finite products and  finite coproducts

  by definition
• cartesian closed and  finite coproducts implies   distributive

  Each functor $A \times -$ is left adjoint and hence preserves finite coproducts (in fact, all colimits).
• preadditive and  finite coproducts implies   finite products

  Mac Lane, VIII.2., Theorem 2
• essentially finite and  finite coproducts implies   thin

  [dualized] Mac Lane, V.2, Prop. 3. The proof can easily be adapted to this case.
• finitely cocomplete is equivalent to   finite coproducts and  coequalizers

  [dualized] Mac Lane, V.2, Cor. 1
• coproducts implies   finite coproducts and  countable coproducts

  [dualized] trivial
• finite coproducts is equivalent to   initial object and  binary coproducts

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

  [dualized] trivial
• finite coproducts and  sequential colimits implies   countable coproducts

  [dualized] If $X_1,X_2,\dotsc$ is an infinite sequence of objects, then their product is the limit of the sequence $\cdots \to X_2 \times X_1 \to X_1$.
• preadditive and  finite products implies   finite coproducts

  [dualized] Mac Lane, VIII.2., Theorem 2
• additive is equivalent to   preadditive and  finite coproducts

  [dualized] by definition
• self-dual and  finite products implies   finite coproducts

  trivial by self-duality
• self-dual and  finite coproducts implies   finite 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
• 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
• empty category
• 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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