CatDat

Cauchy complete

A category is Cauchy complete if every idempotent splits. That is, every idempotent endomorphism $e : X \to X$ (that is, $e^2 = e$) may be written as $e = i \circ p$ for some morphisms $p : X \to Y$ and $i : Y \to X$ with $p \circ i = \mathrm{id}_Y$. Equivalently, the pair $e,\mathrm{id}_X : X \rightrightarrows X$ has an equalizer (or coequalizer).

• Dual property: Cauchy complete (self-dual)
• Related properties: finitely complete
• nLab Link

Relevant implications

• equalizers implies   Cauchy complete

  If $e : X \to X$ is an idempotent, then the equalizer of $e, \mathrm{id}_X : X \rightrightarrows X$ provides a splitting of $e$.
• left cancellative implies   Cauchy complete

  Any idempotent monomorphism must be the identity and therefore splits.
• coequalizers implies   Cauchy complete

  [dualized] If $e : X \to X$ is an idempotent, then the equalizer of $e, \mathrm{id}_X : X \rightrightarrows X$ provides a splitting of $e$.
• right cancellative implies   Cauchy complete

  [dualized] Any idempotent monomorphism must be the identity and therefore splits.

Examples

• category of abelian groups
• category of Banach spaces with linear contractions
• category of combinatorial species
• category of commutative rings
• 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 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 metric spaces with non-expansive maps
• category of monoids
• category of non-empty sets
• category of pointed sets
• category of posets
• category of rings
• category of rngs
• category of schemes
• category of sets
• category of simplicial sets
• category of small categories
• category of smooth manifolds
• category of topological spaces
• category of vector spaces
• category of Z-functors
• 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
• trivial category
• walking isomorphism
• walking morphism
• walking parallel pair of morphisms

Counterexamples

• category of sets and relations

Unknown

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

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