Cauchy complete

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

Relevant implications

• accessible implies Cauchy complete
• Cauchy complete andcokernel pairs andpreadditive implies cokernels
• Cauchy complete andessentially countable andlocally finite implies ℵ₁-accessible
• Cauchy complete andessentially finite implies cofiltered limits andcofiltered-limit-stable epimorphisms
• Cauchy complete andessentially finite implies filtered colimits andfiltered-colimit-stable monomorphisms
• Cauchy complete andessentially finite implies finitely accessible
• Cauchy complete andessentially small implies accessible
• Cauchy complete andessentially small implies coaccessible
• Cauchy complete andkernel pairs andpreadditive implies kernels
• coaccessible implies Cauchy complete
• coequalizers of kernel pairs implies Cauchy complete
• coequalizers implies Cauchy complete
• equalizers of cokernel pairs implies Cauchy complete
• equalizers implies Cauchy complete
• left cancellative implies Cauchy complete
• multi-cocomplete implies Cauchy complete
• multi-complete implies Cauchy complete
• one-way implies Cauchy complete
• pullbacks implies Cauchy complete
• pushouts implies Cauchy complete
• right cancellative implies Cauchy complete
• sequential colimits implies Cauchy complete
• sequential limits implies Cauchy complete
• ℵ₁-cofiltered limits implies Cauchy complete
• ℵ₁-filtered colimits implies Cauchy complete

Examples

There are 125 categories with this property.

• category of abelian groups
• category of abelian sheaves
• category of algebras
• category of Banach spaces with linear contractions
• category of cochain complexes of abelian groups
• category of combinatorial species
• category of commutative algebras
• category of commutative monoids
• category of commutative rings
• category of compact Hausdorff spaces
• category of connected sequences of sets
• category of countable groups
• category of countable sets
• category of countable-dimensional vector spaces
• category of empty-or-finite pairs of sets
• category of F(I)-sets
• category of fields
• category of fields of characteristic zero
• category of filtered vector spaces
• category of finite abelian groups
• category of finite groups
• category of finite ordered sets
• category of finite sets
• category of finite sets and bijections
• category of finite sets and injections
• category of finite sets and surjections
• category of finite Z-sets
• category of finite-dimensional vector spaces
• category of finite-dimensional vector spaces [countable field]
• category of finite-dimensional vector spaces [finite field]
• category of finite-dimensional vector spaces [uncountable field]
• category of finitely generated abelian groups
• category of finitely generated free abelian groups
• category of finitely generated projective modules over the ring of dual numbers
• category of free abelian groups
• category of graded abelian groups
• category of graded modules over a graded ring
• category of groups
• category of Hausdorff spaces
• category of Jónsson-Tarski algebras
• category of large families of abelian groups
• category of large families of sets
• category of large families of sets and bijections
• category of large families of sets which are mostly empty
• category of large families of sets which are mostly singletons
• category of large families of vector spaces which are mostly zero
• category of large vector spaces over a large field with a small basis
• category of left modules over a division ring
• category of left modules over a non-semisimple ring
• category of left modules over a ring
• category of locally ringed spaces
• category of long transfinite sequences of abelian groups
• category of M-sets
• category of measurable spaces
• category of metric spaces with continuous maps
• category of metric spaces with non-expansive maps
• category of metric spaces with ∞ allowed
• category of monoids
• category of non-empty sets
• category of pairs of sets
• category of partially ordered sets
• category of pointed sets
• category of pointed topological spaces
• category of preordered sets
• category of pseudo-metric spaces with non-expansive maps
• category of quivers
• category of quivers with finite components
• category of rings
• category of rngs
• category of schemes
• category of semigroups
• category of sequences of abelian groups
• category of sequences of sets
• category of set functions and commutative squares
• category of sets
• category of sets and bijections
• category of sets equipped with a binary relation
• category of sets equipped with a reflexive binary relation
• category of sets equipped with a symmetric binary relation
• category of sets equipped with a symmetric irreflexive binary relation
• category of sets equipped with a symmetric reflexive binary relation
• category of sets equipped with an irreflexive binary relation
• category of sets with a distinguished subset
• category of sets with finite-to-one maps
• category of sheaves
• category of simplicial sets
• category of small categories
• category of smooth manifolds
• category of topological spaces
• category of torsion abelian groups
• category of torsion-free abelian groups
• category of uniform spaces
• category of vector spaces
• category of Z-functors
• category of Z-sets
• cocompletion of a discrete–pair join
• delooping of a group
• delooping of a non-trivial finite group
• delooping of an infinite countable group
• delooping of an infinite uncountable group
• delooping of the additive monoid of natural numbers
• delooping of the additive monoid of ordinal numbers
• discrete category of sets
• discrete category on two objects
• dual of the category of sets
• dual of the category of topological spaces
• empty category
• forked commutative square
• indiscrete category of sets
• partially ordered collection of ordinal numbers
• partially ordered set of extended natural numbers
• partially ordered set of natural numbers
• preordered set of integers w.r.t. divisibility
• real interval [0,1]
• simplex category
• trivial category
• walking commutative square
• walking composable pair
• walking coreflexive pair
• walking fork
• walking isomorphism
• walking morphism
• walking parallel pair
• walking span
• walking splitting

Counterexamples

There are 8 categories without this property.

• category of coproducts of Euclidean spaces
• category of finite sets of cardinality a power of 3
• category of finite sets of even cardinality
• category of finite sets of odd cardinality
• category of finitely generated free modules over Z x Z
• category of partially ordered sets without isolated points
• category of sets and relations
• walking idempotent

Unknown

There are 0 categories for which the database has no information on whether they satisfy this property.

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