Cauchy complete category

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Idempotents
+-- {: .hide}
[[!include idempotents - contents]]
=--
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

The concept of _[[Cauchy completeness]]_, ordinarily thought of as applying to [[metric spaces]], was vastly generalized by [[Bill Lawvere]] in his influential paper _[Metric spaces, generalized logic, and closed categories](#Lawvere)_. It is now seen by [[category theory|category theorists]] as a concept of [[enriched category theory]], with close ties to the concept of [[Morita equivalence]] in the theory of [[modules]]. In category theory one also speaks of _[[idempotent completion|idempotent completeness]]_.

The basic idea is that the Cauchy [[completion]] of a [[category]] is the closure of a category under what are called "[[absolute limit]]s", i.e., those [[limit]]s that are preserved by any [[functor]] whatsoever. Equivalently, the Cauchy completion is the closure with respect to [[absolute colimit]]s. If $C$ is [[small category|small]], the Cauchy completion $\bar{C}$ of $C$ lies between $C$ and its "[[free cocompletion]]", aka [[presheaf category]]

$$C \hookrightarrow \bar{C} \hookrightarrow Set^{C^{op}}$$

and consists of the presheaves $F$ dubbed [[tiny object|tiny]] by Lawvere, meaning those presheaves which are [[connected object|connected]] and [[projective object|projective]]: the functor

$$hom_{Set^{C^{op}}}(F, -): Set^{C^{op}} \to Set$$

preserves small [[coproducts]] and [[coequalizers]]. All of these concepts generalize straightforwardly to the context of general $V$-[[enriched category theory|enriched categories]], where $V$ is a [[complete category|complete]], [[cocomplete category|cocomplete]], [[symmetric monoidal category|symmetric]] [[monoidal closed category]].

Lawvere defines a **point** of the Cauchy completion of a small $V$-category $C$ to be a $V$-enriched [[bimodule]] $p: \mathbf{1} \to C$ (in other words, a $V$-functor $\mathbf{1} \to V^{C^{op}}$) for which there is a bimodule $q: C \to \mathbf{1}$ right adjoint to $p$ (in the [[bicategory]] of enriched bimodules, see [[profunctor]]), where $\mathbf{1}$ is the unit $V$-category. Thus points of the Cauchy completion are certain $V$-enriched presheaves $p: C^{op} \to V$, and together form a $V$-category called the **Cauchy completion** whose [[hom-object|hom]]s are the presheaf homs. It is denoted $\bar{C}$.

As we will explain in more detail below, [[representable functor|representable presheaves]] belong to the Cauchy completion, and so the [[Yoneda embedding]] of $C$ factors through a full embedding

$$i: C \to \bar{C}$$

and we say the $V$-category $C$ is **Cauchy complete** if this embedding is an equivalence. We work through a few examples in the following section.

## In ordinary category theory
 {#InOrdinaryCategoryTheory}

### In terms of splitting of idempotents

+-- {: .num_defn #CauchyCompletionByRetractsOfRepresentablePresheaves}
###### Definition

For $C$ a [[small category]] write

$$
  \overline{C} \hookrightarrow [C^{op}, Set]
$$

for the [[full subcategory]] of the [[category of presheaves]] on $C$ on the
[[retract]]s of [[representable functors]].

This $\overline{C}$ is called the **Cauchy completion** of $C$.

=--

For instance ([BorceuxDejean, below theorem 1](#BorceuxDejean)).

+-- {: .num_theorem}
###### Theorem

The Cauchy completion $\overline{C}$ satisfies the following properties

1. $\overline{C}$ is a [[small category]];

1. $C$ is a [[full subcategory]] $C \hookrightarrow \overline{C}$;

1. every [[idempotent]] in $\overline{C}$ [[split idempotent|splits]];

1. the inclusion $C \hookrightarrow \overline{C}$ is an [[equivalence of categories]] precisely if already every [[idempotent]] in $C$ splits;

1. there is an [[equivalence of categories]]

   $$
     [C^{op}, Set] \simeq [\overline{C}^{op}, Set]
     \,.
   $$

=--

This appears for instance as ([BorceuxDejean, theorem 1](#BorceuxDejean)).

+-- {: .proof}
###### Proof

$\overline{C}$ is small because $[C^{op}, Set]$ is a [[well-powered category]]. It contains $C$ as a [[full subcategory]] because the [[Yoneda embedding]] is a [[full and faithful functor]]. Every idempotent splits in $\overline{C}$ because it does so in $[C^{op}, Set]$ and because the composite of two retractions is a retraction.

A retract of a representable $y(c) \in [C^{op}, Set]$ induces an idempotent on $y(c)$ and hence by the [[Yoneda lemma]] an idempotent on $c \in C$. If $C$ is already idempotent complete, this splits and produces a retraction of $c$ in $C$ and hence of $y(C)$ in $[C^{op}, Set]$. Since this is necessarily isomorphic to the original retraction, we find that every retract of the representable $y(c)$ is itself representable, therefore $C \simeq \overline{C}$ in this case.

=--

Note that this construction of the Cauchy completion via the Yoneda embedding, we do not take *all* retracts of representables, since this would produce an equivalent replete subcategory of $[C^{op}, Set]$ which is only *essentially small*. Instead, we use well-poweredness of $[C^{op}, Set]$ to provide a representative set of monomorphisms, amongst which we take the retracts to obtain a small full subcategory. For an alternative construction, see [[Karoubi envelope]].

### In terms of tiny objects
 {#InTermsOfTinyObjects}

+-- {: .num_prop #CauchyComplIsFullSubcatOnTinyObjects}
###### Proposition

The Cauchy completion $\overline{C}$ is equivalently the [[full subcategory]] of $[C^{op}, Set]$ on the [[tiny object]]s ("small projective objects").

=--

This appears for instance as ([BorceuxDejean, prop. 2](#BorceuxDejean)).

### In terms of absolute colimits
 {#InTermsOfAbsoluteColimits}

+-- {: .num_prop}
###### Proposition

The following conditions are equivalent for a [[small category]] $C$.

1. $C$ is Cauchy complete;

1. $C$ has all small [[absolute colimit]]s.

=--

### In terms of profunctors
 {#InOrdinaryCatTheoryByProfunctors}

We discuss Cauchy completion of [[small categories]] in terms of [[profunctor]]s.

Write $*$ for the terminal category (single object, single morphism). Let $C$ be a [[small category]]

+-- {: .num_remark}
###### Remark

The category $C$ is equivalent to the [[functor category]] out of the point

$$
  C \simeq Func(*,C)
  \,.
$$

Its [[category of presheaves]] is equivalent to the profunctor category

$$
  [C^{op}, Set] \simeq Profunc(*, C)
  \,.
$$

In these terms the [[Yoneda embedding]] $C \hookrightarrow [C^{op}, Set]$ is the canonical inclusion

$$
  Func(*,C) \hookrightarrow Profunc(*,C)
  \,.
$$

Accordingly the Cauchy completion, def. \ref{CauchyCompletionByRetractsOfRepresentablePresheaves} is a
[[full subcategory]] of the profunctor category

$$
  \overline{C} \hookrightarrow Profunc(*,C)
  \,.
$$

=--

+-- {: .num_prop #CuachyCompByLeftAdjointProfunctor}
###### Proposition

A [[profunctor]] $F : * ⇸ C$ belongs to $\overline{C}$ precisely if it has a [[right adjoint]] in [[Prof]].

=--

This appears for instance as ([BorceuxDejean, prop. 4](#BorceuxDejean)).

+-- {: .num_theorem}
###### Theorem

For a [[small category]] $C$, the following are equivalent

1. $C$ is Cauchy complete;

1. a profunctor $* ⇸ C$ has a [[right adjoint]] precisely if it is a functor;

1. for every [[small category]] $A$ a profunctor $A ⇸ C$ has a [[right adjoint]] precisely if it is a functor;

=--

This appears for instance as ([BorceuxDejean, theorem 2](#BorceuxDejean)).

### In terms of essential geometric morphisms
 {#InOrdinaryCatTheoryByEssGeomMorphisms}

In the context of [[topos theory]] we say, for $C$ [[small category]], that an [[adjoint triple]] of [[functor]]s

$$
  Set \stackrel{\overset{f_!}{\to}}{\stackrel{\overset{f^*}{\leftarrow}}{\underset{f_*}{\to}}}
  [C,Set]
$$

is an [[essential geometric morphism]] of [[topos]]es $f : Set \to [C,Set]$; or an **[[point of a topos|essential point]]** of $[C,Set]$.

By the [[adjoint functor theorem]] this is equivalently simply a single functor $f^* : [C, Set] \to Set$ that preserves all small [[limit]]s and [[colimit]]s. Write

$$
  Topos_{ess}(Set,[C,Set])
   \simeq
  LRFunc([C,Set], Set)
   \subset
  Func([C,Set], Set)
$$

for the [[full subcategory]] of the [[functor category]] on functors that have a [[left adjoint]] and a [[right adjoint]].

+-- {: .num_prop #CauchyCompletionByEssentialPoints}
###### Proposition

For $C$ a [[small category]] there is an [[equivalence of categories]]

$$
  \overline{C} \simeq Topos_{ess}(Set, [C,Set])^{op}
$$

of its Cauchy completion, def. \ref{CauchyCompletionByRetractsOfRepresentablePresheaves}, with the category of essential points of $[C,Set]$.

=--

+-- {: .proof}
###### Proof

We first exhibit a [[full subcategory|full inclusion]] $Topos_{ess}(Set,[C,Set])^{op} \hookrightarrow \overline{C}$.

So let $Set \stackrel{\overset{f_!}{\to}}{\stackrel{\overset{f^*}{\leftarrow}}{\underset{f_*}{\to}}} [C,Set]$ be an [[essential geometric morphism]]. Then because $f_!$ is [[left adjoint]] and thus preserves all small [[colimits]] and because every [[set]] $S \in Set$ is the colimit over itself of the singleton set we have that

$$
  f_! S \simeq \coprod_{s \in S} f_!(*)
$$

is fixed by a choice of [[copresheaf]]

$$
  F := f_!(*) \in [C, Set]
  \,.
$$

The $(f_! \dashv f^*)$-[[adjunction]] [[isomorphism]] then implies that for all $H \in [C,Set]$ we have

$$
  f^* H \simeq Set(*, f^* H) \simeq [C,Set](f_! *, H)
  \simeq [C,Set](F,H)
  \,.
$$

naturally in $H$, and hence that

$$
  f^*(-) \simeq [C,Set](F,-) : Set \to [C,Set]
  \,.
$$

By assumption this has a further right adjoint $f_\ast$ and hence preserves all [[colimits]]. By the discussion at [[tiny object]] it follows that $F \in [C,Set]$ is a tiny object. By prop. \ref{CauchyComplIsFullSubcatOnTinyObjects} this means that $F$ belongs to $\overline{C} \subset [C,Set]$.

A morphism $f \Rightarrow g$ between [[geometric morphisms]] $f,g : Set \to [C,Set]$ is a [[geometric transformation]], which is a [[natural transformation]] $f^* \Rightarrow g^*$, hence by the above a natural transformation $[C,Set](F,-) \Rightarrow [C,Set](G,-)$. By the [[Yoneda lemma]] these are in bijection with morphisms $G \to H$ in $[C,Set]$. This gives the full inclusion $Topos_{ess}(Set,[C,Set])^{op} \subset \overline{C}$.

The converse inclusion is now immediate by the same arguments: since the objects in $\overline{C}$ are precisely the [[tiny object]]s $F \in [C,Set]$ each of them corresponds to a functor $[C,Set](F,-) : [C,Set] \to Set$ that has a [[right adjoint]]. Since this generally also has a left adjoint, it is the [[inverse image]] of an essential geometric morphism $f : Set \to [C,Set]$.

=--

Write $Cat_{Cauchy} \hookrightarrow$ [[Cat]] for the [[full sub-2-category]]
of [[Cat]] on the Cauchy complete categories.

+-- {: .num_prop}
###### Proposition

The 2-category of Cauchy complete categories is a
[[coreflective subcategory|coreflective]] [[full sub-2-category]] of [[Topos]] with essential geometric morphisms

$$
  Cat_{Cauchy} \hookrightarrow Topos_{ess}
$$

exhibited by the [[2-adjunction]]

$$
  ([-,Set]
  \dashv
  Topos_{ess}(Set,-)
  )

  :
  Topos_{ess}
   \stackrel{\overset{Cat_{Cauchy}(-,Set)}{\hookleftarrow}}{\underset{Topos_{ess}(Set,-)}{\to}}
  Cat_{Cauchy}
  \,.
$$

=--

+-- {: .proof}
###### Proof

We first claim that when working with all categories
instead of just the Cauchy complete categories
there is a [[2-adjunction]]

$$
  ([-,Set]
  \dashv
  Topos_{ess}(Set,-)
  )
  :
  Topos_{ess}
   \stackrel{\overset{[-,Set]}{\leftarrow}}{\underset{Topos_{ess}(Set,-)}{\to}}
  Cat
  \,.
$$

This is exhibited by the following [[equivalence of categories|equivalence]]
of [[hom-categories]]

$$
  \begin{aligned}
    Func(C, Topos_{ess}(Set, E))
     & \simeq
    Func(C, LRFunc(E, Set))
    \\
     & \simeq
    LRFunc(E, Func(C,Set)) =: LRFunc(E, [C,Set])
    \\
     & \simeq
    Topos_{ess}([C,Set], E)
  \end{aligned}
$$

[[natural equivalence|natural]] in $C \in Cat$ and $E \in Topos$. Here

* the first equivalence is by definition of [[essential geometric morphism]];

* the second equivalence follows by observing that limits and colimits in presheaf categories are computed objectwise;

* the third equivalence is again the definition of essential geometric morphisms.

Now by prop. \ref{CauchyCompletionByEssentialPoints} we have that the components of the
[[unit of an adjunction|unit]] of this adjunction

$$
  C \to Topos_{ess}(Set,[C,Set])
$$

are [[equivalence of categories|equivalences]] precisely if $C$ is
Cauchy complete. This means that restricted along
$Cat_{Cauchy} \hookrightarrow Cat$ the adjunction exhibits a coreflective embedding.

=--

## In enriched category theory
 {#InEnrichedCategoryTheory}

We discuss Cauchy completion in $\mathcal{V}$-[[enriched category]] theory, for $\mathcal{V}$ a [[closed category|closed]] [[symmetric monoidal category]] with all [[limits]] and [[colimits]]. The discussion in ordinary category theory [above](InOrdinaryCategoryTheory) is the special case where $\mathcal{V} := $ [[Set]].

The key to the enriched version is the reformulation of ordinary Cauchy completion in terms of [[profunctors]] as discussed [above](InOrdinaryCatTheoryByProfunctors). These have an immediate generalization to enriched category theory, and so one takes this formulation as the definition.

As before, we have

+-- {: .num_remark}
###### Remark

Every small $\mathcal{V}$-category $C$ is equivalent to the $\mathcal{V}$-[[enriched functor category]]

$$
  C \simeq \mathcal{V}Func(I,C)
  \,,
$$

where $I$ is the $\mathcal{V}$-category with a single [[object]] $*$ and $I(*,*) = I_{\mathcal{V}}$, the tensor unit in $\mathcal{V}$.

Also,

$$
  [C^{op}, \mathcal{V}] \simeq \mathcal{V}Profunc(I, C)
$$

and the canonical $\mathcal{V}$-[[enriched functor]]

$$
  \mathcal{V}Func(I,C) \hookrightarrow \mathcal{V}Profunc(I,C)
$$

is the enriched [[Yoneda embedding]].

=--

+-- {: .num_defn}
###### Definition

For $C$ a small $\mathcal{V}$-[[enriched category]], the **Cauchy completion** of $C$ is the full $\mathcal{V}$-subcategory

$$
  \overline{C} \hookrightarrow \mathcal{V}Profunc(I,C)
$$

on those [[profunctor]]s with a [[right adjoint]] in $\mathcal{V}$[[Prof]].

=--

[Lack and Tendas](#LT2022) give criteria for a $\mathcal{V}$-enriched category to be Cauchy complete in Cor. 4.22 and Prop. 4.23 of their paper, based on assumptions stated in Def. 4.3.  Their results apply when $\mathcal{V}$:

* is a [[cocomplete category|cocomplete]] [[symmetric monoidal category|symmetric monoidal]] [[closed category|closed]] category
* has a [[zero object]] and finite [[biproducts]]
* has a strong [[separator]] consisting of [[dualizable objects]]
* has for all objects $A, B$ and morphisms $z : I \to A \otimes B$ some dualizable object $P$ and maps $x: P \to A$, $y: P^\ast \to A$ such that $z = (x \otimes y) i$ where $i: I \to P \otimes P^\ast$ is the unit.

## Examples

* When $\mathcal{V} = $ **[[Set]]**, a $\mathcal{V}$-category is an ordinary category. The Cauchy completion of an ordinary category is its _idempotent completion_, or [[Karoubi envelope]]. This also holds when $\mathcal{V} = \mathbf{Cat}$ or $\mathcal{V} = \mathbf{sSet}$, or more generally whenever $\mathcal{V}$ is a cartesian [[cosmos]] where the terminal object is [[tiny]] by Corollary 3.16 of [Lack and Tendas](#LT2022).

* When $\mathcal{V} = [0,\infty]$ is the extended nonnegative reals ordered by $\geq$ and with $+$ as monoidal product, $\mathcal{V}$-categories are generalized metric spaces. The Cauchy completion is the usual completion under _Cauchy nets_ or _Cauchy filters_.

* When $\mathcal{V} = $ **[[Ab]]** is the category of abelian groups, the Cauchy completion of $\mathcal{V}$-enriched category is its completion under _finite direct sums and idempotent splitting_.  This process of Cauchy completion can be accomplished by first taking the "finite coproduct completion", i.e. completing under finite direct sums, and then taking the [[Karoubi envelope]], i.e. splitting idempotents.

* More generally, when $\mathcal{V} = R \mathbf{Mod}$ is the category of modules of a commutative ring $R$, the Cauchy completion of $\mathcal{V}$-enriched category is its completion under _finite direct sums and idempotent splitting_.  This result, probably folklore for many decades, follows from Cor. 4.22 of [Lack and Tendas](#LT2022) (see above).

* When $\mathcal{V} = \mathbf{Ch}$ is the category of [[chain complexes]], a $\mathcal{V}$-category is a [[dg-category]].  Cauchy complete dg-categories are characterized by [Nikolić, Street, and Tendas](#NST2020).

* When $\mathcal{V} =$ **[[SupLat]]** is the category of [[suplattices]], a $\mathcal{V}$-category is a locally posetal, [[locally cocomplete bicategory]], i.e. a [[quantaloid]]. The Cauchy completion is some sort of completion under _arbitrary_ sums: it is large even if the original quantaloid is small, and its existence depends on the precise definition we choose of Cauchy completion.  See [Johnson](#J1989) for more details.

* When $\mathcal{V} =$ **[[Cat]]** is the category of [[small categories]], a $\mathcal{V}$-category is a [[2-category]]. Cauchy complete 2-categories are those for which the underlying category is Cauchy complete (see this [MathOverflow answer](https://mathoverflow.net/a/397943)).

* In the $\infty$-categorical context, we can consider enrichment in the $\infty$-category of [[spectra]]. The Cauchy completion of an $\infty$-category enriched in spectra is its completion under _all finite colimits_.

* Generalizing to bicategorical enrichment, we can construct from a [[site]] $(\mathcal{C}, J)$ a certain bicategory $\mathcal{W}$ such that the Cauchy complete, symmetric, skeletal $\mathcal{W}$-categories are just the sheaves on $(\mathcal{C}, J)$. Variations on this theme can yield $\mathcal{C}$-indexed categories, stacks, prestacks, or presheaves as Cauchy completions or sub-Cauchy completions for categories enriched in certain bicategories.

Now we look at two examples in more detail: metric spaces and ordinary categories.

### Metric spaces

We consider first the classical case of [[metric space]]s, but as redefined by Lawvere to mean a category enriched in the [[poset]] $V = ([0, \infty], \geq)$, with [[tensor product]] given by addition. So, to say $X$ is a [[Lawvere metric space]] means that with the set $X$ there is a distance function

$$d_X = hom_X: X \times X \to [0, \infty]$$

such that

$$d(x, y) + d(y, z) \geq d(x, z), \qquad 0 \geq d(x, x)$$

for all $x, y, z$ in $X$. (The associativity and identity axioms are here superfluous since $V$ is a poset.) A $V$-enriched functor $f: X \to Y$ here just means a function from $X$ to $Y$ such that

$$d_X(x, y) \geq d_Y(f(x), f(y))$$

for all $x, y$ in $X$ (again, preservation of composition and of identities is superfluous here), so that $V$-functors are [[short maps]] between metric spaces (Lipschitz maps with constant at most $1$). Finally, a $V$-enriched transformation $f \to g: X \to Y$ in this case boils down to an instance of a property: that

$$0 \geq d_Y(f(x), g(x))$$

for all $x$ in $X$. If $f, g$ are valued in $[0, \infty]$, this just means $f(x) \geq g(x)$ for all $x$.

A point of the Cauchy completion is an $X$-module $p: \mathbf{1} \to X$, i.e., an enriched functor or short map

$$p: X^{op} \to [0, \infty]$$

for which there is an $X$-module $q: X \to \mathbf{1}$ on the other side, an enriched functor

$$q: X \to [0, \infty]$$

that is [[right adjoint]] to $p$ in the sense of modules. This means there is a unit of the adjunction in the bicategory of modules:

$$(Id: \mathbf{1} \to \mathbf{1}) \to (\mathbf{1} \overset{p}{\to} X \overset{q}{\to} \mathbf{1})$$

and a counit:

$$(X \overset{q}{\to} \mathbf{1} \overset{p}{\to} X) \to (Id: X \to X)$$

Recall now that $Id_X: X \to X$ in the bicategory of modules is the unit bimodule $y_X: X \to V^{X^{op}}$ given by the enriched [[Yoneda embedding]], or in different words, $hom_X = d_X: X^{op} \times X \to [0, \infty]$. Recall also that module composition is defined by a [[coend]] formula for a tensor product. If one now tracks through the definitions, keeping in mind that we are in the very simple case of enrichment in a poset, the unit of the [[adjunction]] $p \dashv q$ boils down to having the property

$$0 \geq \int^{x \in X} q(x) + p(x) = \inf_{x \in X} q(x) + p(x)$$

and the counit boils down to having the property

$$p(x) + q(y) \geq d_X(x, y)$$

To better appreciate what these conditions mean, we point out that $p(x)$ should be thought of as the distance $d_{\bar{X}}(x, p)$ between $x$ and the "ideal point" $p$ in the Cauchy completion $\bar{X}$, and $q(x)$ should be thought of as the companion distance $d_{\bar{X}}(p, x)$. Thus the unit condition above would come down to saying that for every $\varepsilon \gt 0$ there exists $x \in X$ such that

$$d_{\bar{X}}(p, x) \lt \varepsilon, \, d_{\bar{X}}(x, p) \lt \varepsilon$$

and the counit condition imposes a necessary triangle inequality constraint on the distance functions $p$ and $q$, in order that we get an actual Lawvere metric space $\bar{X}$. If $p, p'$ are two points of the Cauchy completion thus defined, then their distance is defined by the usual formula for enriched presheaves:

$$d(p, p') = \int_{x \in X} \hom_{[0, \infty]}(p(x), p'(x)) = \sup_{x \in X} \max\{0, p'(x) - p(x)\}$$

It should be noted that even under the classical definition (where we impose symmetry $d(x, y) = d(y, x)$, separation $d(x, y) \gt 0$ for $x \neq y$, and finiteness $d(x,y) \lt \infty$), this provides an elegant alternative definition of Cauchy completion. In essence, all it is doing is taking the metric closure $\bar{X}$ of the embedding of $X$ into the already complete space of short maps:

$$y_X: X \to [0, \infty]^{X^{op}}: x \mapsto d_X(-, x)$$

The presheaf-hom definition of the distance formula for $\bar{X}$, being manifestly non-symmetric, is not the usual definition of distance in the classical symmetric case. However, if we first symmetrize the distance in $[0, \infty]$:

$$\sigma d(r, s) = d(r, s) + d(s, r) = \max\{0, s-r\} + \max\{0, r-s\} = |r - s|$$

or equivalently

$$\sigma d(r, s) = d(r, s) + d(s, r) = \max(\max\{0, s-r\}, \max\{0, r-s\}) = |r - s|$$

then we do retrieve the classical formula

$$d(p, p') = \int_{x \in X} \sigma d(p(x), p'(x)) = \sup_{x \in X} |p(x) - p'(x)|$$

In other words, the completion $\bar{X}$ of a symmetric metric space $X$ as a general (Lawvere) metric space is not necessarily the same as its completion $\sigma\bar{X}$ as a symmetric metric space, but $\sigma\bar{X}$ is the symmetrisation of $\bar{X}$.

### Ordinary ($Set$-enriched) categories

The analysis of Cauchy complete Lawvere metric spaces contains some of the seeds of what happens in other [[enriched category]] contexts; the case of ordinary [[small category|small categories]], where the enrichment is no longer in a mere [[poset]] but in [[Set]], reflects still more of the phenomena generally associated with Cauchy completions.

Let $C$ be a [[small category]] and let the [[module]] $p: \mathbf{1} \to C$ be a point of $\bar{C}$, with module $q: C \to \mathbf{1}$ as its [[right adjoint]] in the [[bicategory]] of modules. As [[functor]]s,

$$p: C^{op} \to Set, \, q: C \to Set$$

and the structure of the [[adjunction]] is given by unit and counit maps:

$$\eta: 1 \to \int^{c \in Ob(C)} q(c) \times p(c), \qquad \varepsilon_{c, d}: p(c) \times q(d) \to C(c, d)$$

As we said in the case of metric spaces, $p(c)$ and $q(c)$ measure "distances" = homs:

$$p(c) \cong Set^{C^{op}}(C(-, c), p), \qquad q(c) \cong Set^{C^{op}}(p, C(-, c))$$

The first [[isomorphism]] is an instance of the [[Yoneda lemma]], and the second can be seen as follows. The set $q(c)$ is the [[bimodule]] composite

$$(\mathbf{1} \overset{c}{\to} C \overset{q}{\to} \mathbf{1})$$

where $c$ is shorthand for the module $C(-, c): C^{op} \to Set$; this is just an instance of the [[Yoneda lemma]]:

$$q \circ_C C(-, c) \overset{def}{=} \int^{d \in C} q(d) \times C(d, c) \overset{Yoneda}{\cong} q(c).$$

Now using the adjunction $p \dashv q$, there are, for any set $S$, natural bijections

$$\frac{\frac{S \to q(c)}{S \to q \circ_C C(-, c)}}{\frac{p \circ_{\mathbf{1}} S \to C(-, c)}{p(-) \times S \to C(-, c)}}$$

and maps in the bottom line are in bijection with maps $S \to Set^{C^{op}}(p, C(-, c))$. Therefore we have a natural bijection

$$\frac{S \to q(c)}{S \to Set^{C^{op}}(p, C(-, c))}$$

and this proves $q(c) \cong Set^{C^{op}}(p, C(-, c))$.

With these identifications of $q(c)$ and $p(c)$, the unit of the adjunction $p \dashv q$ takes the form

$$\eta: 1 \to \int^{c} Set^{C^{op}}(p, C(-, c)) \times Set^{C^{op}}(C(-, c), p)$$

The [[coend]] above is a quotient of

$$\sum_c Set^{C^{op}}(p, C(-, c)) \times Set^{C^{op}}(C(-, c), p)$$

and hence the unit element $\eta$ is represented by a pair of [[natural transformation|transformation]]s

$$i: p \to C(-, c), \qquad \pi: C(-, c) \to p$$

for some $c$.

Given that, it is now not hard -- in fact it is fairly tautological -- to verify that on the basis of the triangular equation of the adjunction which says

$$(p \overset{p \circ \eta}{\to} p \circ q \circ p \overset{\varepsilon}{\to} p) = 1_p,$$

that

$$(p \overset{i}{\to} C(-, c) \overset{\pi}{\to} p) = 1_p$$

and so a point $p$ in the Cauchy completion $\bar{C}$ must be a [[retract]] of a [[representable functor|representable]] $C(-, c)$. Spelling this out a little more: the composite

$$C(-, c) \overset{\pi}{\to} p \overset{i}{\to} C(-, c)$$

is an [[idempotent]] represented by a morphism $e: c \to c$ in $C$ (by the Yoneda lemma), and this factorization through $p$ [[split idempotent|splits]] the idempotent $C(-, e)$ in $Set^{C^{op}}$.

Indeed, the claim is that modules $p: C^{op} \to Set$ in the Cauchy completion are precisely those presheaves on $C$ which arise as [[retract]]s of [[representable functor|representable]]s in $Set^{C^{op}}$, or in other words may be identified with objects of the [[split idempotent|idempotent-splitting]] completion of $C$ (aka the _[[Karoubi envelope]]_ of $C$). Therefore, in the $Set$-enriched case, the Cauchy completion _is_ the idempotent-splitting completion. In particular, representables themselves are points of the Cauchy completion.

Notice that in a finitely complete category (such as $Set$ or a [[presheaf]] category), idempotents $e: c \to c$ split automatically: just take the equalizer of the pair

$$c \stackrel{\overset{e}{\to}}{\underset{1}{\to}} c$$

For that matter, in any finitely cocomplete category, taking the [[coequalizer]] of the above pair would also split the idempotent. Indeed, we can say that idempotents split in a category iff all equalizers of such pairs exist, iff all coequalizers of such pairs exist.

Notice that if $C$ and $D$ are categories, then any functor $F: C \to D$ preserves [[retract]]s and therefore splittings of idempotents. Thus, the equalizers above are the sort of limits which are preserved by _any_ functor $F$ whatsoever. They are called **absolute limits** for that reason. For the same reason, the coequalizers above are **absolute colimits**: they are precisely the colimits preserved by any functor whatsoever.

Pursuing this a bit further: if $F: C^{op} \to Set$ is any functor, then (because idempotents split in $Set$) there is a unique extension $\bar{F}: \bar{C}^{op} \to Set$ of $F$. Therefore we have an equivalence

$$Set^{C^{op}} \simeq Set^{\bar{C}^{op}}$$

and we say that $C$ and $\bar{C}$ are **Morita equivalent**.

### Of posets

Every ordinary [[poset]] clearly is Cauchy complete, since the only idempotents are the [[identity morphism]]s. The [[internalization]] of this statement requires some extra assumptions:

+-- {: .num_prop}
###### Proposition

[[internalization|Internal]] to any [[regular category]] every [[poset]]
is Cauchy complete.

=--

This appears as ([Rosolini, prop. 2.1](#Rosolini)).

+-- {: .num_prop}
###### Proposition

[[internalization|Internal to]] any [[exact category]] the Cauchy completion of any [[preorder]] exists and is its [[poset reflection]].

=--

This appears as ([Rosolini, corollary. 2.3](#Rosolini)).

Moreover, the characterization of Cauchy completion by left adjoint profunctors requires the internal [[axiom of choice]]:

+-- {: .num_prop}
###### Proposition

In a given ambient context, the following are equivalent:

1. the [[axiom of choice]] holds;

1. every [[profunctor]] $F : A ⇸ C$ between [[posets]] is an ordinary functor when it has a [[right adjoint]].

=--

For instance ([BorceuxDejean, prop. 5](#BorceuxDejean)).

## Related concepts

* [[split idempotent]]

* [[Karoubi envelope]]

* [[free cocompletion]], [[free coproduct completion]]

* [[idempotent complete (infinity,1)-category]]

* [[free bicompletion]]

* [[reflexive completion]]

## References

The notion of Cauchy complete categories was introduced in

* {#Lawvere} [[Bill Lawvere]], _Metric spaces, generalized logic and closed categories_ Rend. Sem. Mat. Fis. Milano, 43:135–166 (1973)

  Reprints in Theory and Applications of Categories, No. 1 (2002) pp 1-37 ([tac](http://www.tac.mta.ca/tac/reprints/articles/1/tr1abs.html))

Surveys are in

* [[Francis Borceux]] and D. Dejean, _Cauchy completion in category theory_  Cahiers Topologie Géom. Différentielle Catégoriques, 27:133–146, (1986) ([numdam](http://www.numdam.org/item?id=CTGDC_1986__27_2_133_0))
  {#BorceuxDejean}

* A. Carboni and [[Ross Street]], _Order ideal in categories_ Pacific J. Math., 124:275–288, 1986.
 {#CarboniStreet}

Further references include for instance

* {#J1989} S. R. Johnson, _Small Cauchy completions_ , JPAA **62** (1989) pp.35-45.  ([web](https://www.sciencedirect.com/science/article/pii/0022404989900194))

* R. Walters, _Sheaves and Cauchy complete categories_ , Cahiers Top. Geom. Diff. Cat. 22 no. 3 (1981) 283-286 ([numdam](http://www.numdam.org/item?id=CTGDC_1981__22_3_283_0))

* R. Walters, _Sheaves on sites as Cauchy-complete categories, J. Pure Appl. Algebra 24 (1982) 95-102

* {#NST2020} Branko Nikolić, [[Ross Street]], [[Giacomo Tendas]], _Cauchy completeness for DG-categories_, Theory and Applications of Categories, 37(28):940-963, 2021 [link](http://www.tac.mtI found a link explaining that the category additionally has biproducts, given by the componentwise product of semilattices.a.ca/tac/volumes/37/28/37-28abs.html).

* {#LT2022} [[Stephen Lack]], [[Giacomo Tendas]], _Flat vs. filtered colimits in the enriched context_, Advances in Mathematics, 404(A):108381, 2022. [arXiv:2107.08612](https://arxiv.org/abs/2107.08612), [doi](https://doi.org/10.1016/j.aim.2022.108381).

Cauchy completion of [[internalization|internal]] [[prosets]] is discussed in

* G. Rosolini, _A note on Cauchy completeness for preorders_ ([pdf](http://www.disi.unige.it/person/RosoliniG/notccp.pdf))
 {#Rosolini}

[[!redirects Cauchy complete category]]
[[!redirects Cauchy complete categories]]
[[!redirects Cauchy-complete category]]
[[!redirects Cauchy-complete categories]]

[[!redirects idempotent complete category]]
[[!redirects idempotent complete categories]]
[[!redirects idempotent-complete category]]
[[!redirects idempotent-complete categories]]


Cauchy completion

# Cauchy completion

The *Cauchy completion* of an object $X$ is a [[universal]] way to make it "Cauchy complete".  There are different (overlapping) meanings of "Cauchy complete", corresponding to different notions of "Cauchy completion":

1. The Cauchy completion of a [[space]] is a [[complete space]].

1. The Cauchy completion of a [[category]] is the [[Karoubi envelope]] of the category.

1. The Cauchy completion of an [[enriched category]] is a [[Cauchy complete category]].

1. The Cauchy completion of an [[(infinity,1)-category]] is an [[idempotent complete (infinity,1)-category]].

category: disambiguation

[[!redirects Cauchy completion]]
[[!redirects Cauchy completions]]
[[!redirects Cauchy-completion]]
[[!redirects Cauchy-completions]]

[[!redirects Cauchy complete]]
[[!redirects Cauchy-complete]]
[[!redirects Cauchy completeness]]
[[!redirects Cauchy-completeness]]


Cauchy filter

# Cauchy filters
* table of contents
{: toc}

## Idea

A _Cauchy filter_ on a [[space]] $X$ is a [[proper filter]] on $X$ that contains  sets (meaning [[subsets]] of $X$) of arbitrarily small [[diameter]].

The precise definition depends on what sort of space $X$ is, up to	the full generality of a [[Cauchy space]].

## Definitions

+-- {: .num_defn #reals}
###### Definition

A __Cauchy filter__ on the [[real numbers]] is a [[proper filter]] $F$ with, for each strictly [[positive number]] $\delta \in \mathbb{R}$, a set $A \in F$ with, for each $x,y \in A$, $\vert y - x\vert \leq \delta$.
=--

Sometimes, one uses Cauchy filters on the rational numbers to construct the real numbers in the first place.

+-- {: .num_defn #rationals}
###### Definition

A __Cauchy filter__ on the [[rational numbers]] is a [[proper filter]] $F$ with, for each strictly [[positive number]] $\delta \in \mathbb{Q}$, a set $A \in F$ with, for each $x,y \in A$, $\vert y - x\vert \leq \delta$.
=--

In a [[metric space]], the [[diameter]] of a [[subset]] $A$ is the [[supremum]] of the [[distance]]s $d(x,y)$ for $x,y \in A$ (which is a [[lower real number]] in general).  However, we need not think precisely about these diameters; it is enough to characterise those sets with diameter at most $\delta$.

+-- {: .num_defn #metric}
###### Definition

A __Cauchy filter__ on a metric space is a [[proper filter]] $F$ with, for each strictly [[positive number]] $\delta$, a set $A \in F$ with, for each $x,y \in A$, $d(x,y) \leq \delta$.
=--

(It is actually sufficient to consider enough sufficiently small values of $\delta$, say [[rational number|rational]] $\delta$ or [[decimal fraction]] $\delta$.)

In a [[gauge space]], instead of a single number $\delta$ to estimate diameter, we use $\delta$ together with a gauging distance $d$.

+-- {: .num_defn #gauge}
###### Definition

A __Cauchy filter__ on a gauge space is a [[proper filter]] $F$ with, for each gauging distance $d$ and each strictly positive number $\delta$, a set $A \in F$ with, for each $x,y \in A$, $d(x,y) \leq \delta$.
=--

(It is actually sufficient to consider a base of gauging distances, as well as enough sufficiently small $\delta$.)

In a [[Booij premetric space]], we use the [[premetric]] to estimate diameter.

+-- {: .num_defn #metric}
###### Definition

A __Cauchy filter__ on a [[Booij premetric space]] is a [[proper filter]] $F$ with, for each strictly [[positive number]] $\delta$, a set $A \in F$ with, for each $x,y \in A$, $x \sim_\epsilon y$.
=--

In a [[topological group]], we use a [[neighbourhood]] of the [[identity element]] to estimate diameter.

+-- {: .num_defn #topgroup}
###### Definition

A __Cauchy filter__ on a topological group is a [[proper filter]] $F$ with, for each neighbourhood $U$ of the identity, a set $A \in F$ with, for each $x,y \in A$, $x^{-1} y \in U$ (or equivalently, for each $x,y \in A$, for some $n \in U$, $x n = y$).
=--

(It is sufficient to consider a [[neighbourhood base]] at the identity.  There is no difference between left and right even for nonabelian groups.)

In a [[uniform space]], we use an [[entourage]] $U$ to estimate diameter.

+-- {: .num_defn #uniform}
###### Definition

A __Cauchy filter__ on a uniform space is a [[proper filter]] $F$ with, for each entourage $U$, a set $A \in F$ with, for each $x,y \in A$, $x \approx_U y$ (that is, $(x,y) \in U$).
=--

(It is sufficient to consider a base of the uniformity.)

If you want to define uniformities in terms of [[uniform covers]]:

+-- {: .num_defn #ucover}
###### Definition

A __Cauchy filter__ on a uniform space is a [[proper filter]] $F$ with, for each uniform cover $U$, a set $A \in F$ with $A \in U$.
=--

(It is sufficient to consider a base of uniform covers.)

All of the above have non-symmetric versions: [[quasimetric spaces]], [[quasigauge spaces]], [[topological monoids]], [[quasiuniform spaces]].  The definition of Cauchy filter for these is the same, with these caveats:

* for a topological monoid, there is a difference between left and right in $x n = y$ in Definition \ref{topgroup}, giving left-Cauchy and right-Cauchy filters;
* there is no notion of quasiuniform cover to generalise Definition \ref{ucover}.

The most general context is that of a [[Cauchy space]], where the notion of Cauchy filter is axiomatic.

## Properties

Cauchy filters in all cases above have these properties:

* Every Cauchy filter is [[proper filter|proper]].
* The [[principal ultrafilter]] $U_x$ at any point $x$ is Cauchy.
* If $F$ is Cauchy, $G$ is [[proper filter|proper]], and $G$ refines $F$ ($G \supseteq F$), then $G$ is Cauchy.
* If $F$ and $G$ are Cauchy and their [[join]] $F \vee G$ is proper, then their [[intersection]] $F \cap G$ is Cauchy.

These conditions form the abstract definition of a [[Cauchy space]].

Furthermore, all of these have a notion of [[convergence]] given as follows:

*  A filter $F$ __converges__ to a point $x$ if $F \cap U_x$ is Cauchy.

In this way, every Cauchy space becomes a [[convergence space]], which agrees with the usual convergence on metric spaces, uniform spaces, etc.

A [[function]] $f\colon X \to Y$ between spaces is __[[Cauchy-continuous map|Cauchy-continuous]]__ if, for every Cauchy filter $F$ on $X$, the filter (generated by) $f(F)$ is Cauchy.  (These are the [[morphisms]] in the [[category]] of Cauchy spaces.)

## In nonstandard analysis

In [[nonstandard analysis]], the hyperpoints of a (quasi)uniform space have a relation of [[adequality]].  A proper filter $F$ is Cauchy iff its nonstandard extension $F^*$ contains a hyperset (a collection of hyperpoints) $A$ whose elements are all adequal.  So compared to the other definitions, a single $A$ of infinitesimal diameter suffices.

A hyperpoint $x$ is __[[finite hyperpoint|finite]]__ (or __limited__) if there is a proper filter $F$ (necessarily Cauchy) such that $F^*$ contains a hyperset whose elements are all adequal to $x$.  In the analogy between hyperpoints and [[ultrafilters]], the finite hyperpoints correspond to the Cauchy ultrafilters.

A map $f$ between (quasi)uniform spaces is Cauchy-continuous iff its nonstandard extension $f^*$ has the property that $f^*(x)$ and $f^*(y)$ are adequal whenever $x$ and $y$ are adequal and $x$ is finite.  (Compare that $f$ is [[uniformly continuous]] iff $f^*$ has this property regardless of whether $x$ is finite.)  Presumably one can define a [[Cauchy space]] in nonstandard analysis by specifying the finite hyperpoints and the relation of adequality only with these (although perhaps not every Cauchy space arises in this way).

[[!redirects Cauchy filter]]
[[!redirects Cauchy filters]]


Cauchy group theorem

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Statement

Let $G$ be a [[finite group]] with [[order of a group|order]] ${\vert G\vert} \in \mathbb{N}$.

+-- {: .num_theorem}
###### Theorem
**(Cauchy)**

If a [[prime number]] $p$ divides ${\vert G\vert}$, then equivalently

* $G$ has an element of [[order of an element|order]] $p$;

* $G$ has a [[subgroup]] of [[order of a group|order]] $p$.

=--

This result is not completely trivial. One route to this would go as follows: knowing that Sylow $p$-subgroups $H$ of $G$ exist (see [[class equation]] for a proof), any nontrivial element $h$ of $H$ would be of order $p^r$ for some $r \gt 0$, and then $h^{p^{r-1}}$ would be the desired element. Come to think of it, it's actually an immediate consequence of the theorem [here](/nlab/show/class+equation#ppower). But see [McKay](#McKay) for a snappier proof.

+-- {: .num_remark}
###### Remark
Cauchy had claimed a proof of his eponymous theorem in 1845, but in fact his proof had a gap. See [Meo](#Meo) for a historical discussion.
=--

## Related concepts

* [[Cauchy's theorems]]

## References

Name after [[Augustin Cauchy]].

* {#McKay}  James McKay, _Another proof of Cauchy's group theorem_, American Math. Monthly, 66 (1959), p. 119.

* {#Meo} M. Meo, _The mathematical life of Cauchy's group theorem_, Historia Mathematica Volume 31, Issue 2 (May 2004), 196–221. ([web](http://www.sciencedirect.com/science/article/pii/S031508600300003X))

[[!redirects Cauchy group theorem]]
[[!redirects Cauchy's group theorem]]
[[!redirects Cauchy's group theorem]]
[[!redirects Cauchy\'s group theorem]]
[[!redirects Cauchy's theorem (group theory)]]


Cauchy horizon

## Related concepts

* [[Cauchy surface]]

* [[event horizon]]

## References

* Wikipedia, _[Cauchy horizon](https://en.wikipedia.org/wiki/Cauchy_horizon)_

[[!redirects Cauchy horizons]]


Cauchy integral theorem

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Analysis
+-- {: .hide}
[[!include analysis - contents]]
=--
#### Complex geometry
+-- {: .hide}
[[!include complex geometry - contents]]
=--
=--
=--

# The Cauchy integral theorem
* table of contents
{: toc}

## Idea

Cauchy's integral theorem states that [[contour integrals]] of [[holomorphic functions]] in the [[complex plane]] $\mathbb{C}$ are invariant under [[homotopy]] of paths.  In particular, if a function is holomorphic on a [[simply connected space|simply connected]] [[subspace]] of $\mathbb{C}$, then its contour integral on a path depends only on the beginning and ending points of the path, and indeed can be given by subtracting the values there of an [[antiderivative]] (in accordance with the second [[Fundamental Theorem of Calculus]]).

## Statement

Let $D$ be an [[open subset]] of the [[complex plane]] $\mathbb{C}$, let $a$ and $b$ be two points in $D$, let $\gamma_1$ and $\gamma_2$ be two [[curves]] in $D$ from $a$ to $b$, let the region between them also lie entirely within $D$, and let $f$ be a [[holomorphic function]] on $D$. Then we have

$$ \int_{\gamma_1} f(z) \,\mathrm{d}z = \int_{\gamma_2} f(z) \,\mathrm{d}z .$$

In particular we have

$$ \int_{\gamma_1} f(z) \,\mathrm{d}z = 0 $$

if $a = b$ (because then $\gamma_2$ may be taken to be a constant); in other words, the contour integral of a holomorphic function is zero around any [[loop]] whose inside lies entirely within the function\'s domain.

## Related concepts

* [[Cauchy's integral formula]]

* [[Cauchy's theorems]]

* [[Goursat theorem]]

* [[Stokes theorem]]

## References

* Wikipedia, _[Cauchy integral theorem](https://en.wikipedia.org/wiki/Cauchy's_integral_theorem)_

category: analysis

[[!redirects Cauchy integral theorem]]
[[!redirects Cauchy's integral theorem]]
[[!redirects Cauchy’s integral theorem]]
[[!redirects Cauchy\'s integral theorem]]


Cauchy interlace theorem

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Linear algebra
+-- {: .hide}
[[!include homotopy - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Statement

For $n \in \mathbb{N}$, let $A \in Mat_{n+1 \times n +1}(\mathbb{C})$ be a [[hermitian matrix]] (for instance a [[symmetric matrix]] of all entries are [[real numbers]]).

Then for every [[principal submatrix]] $B \in Mat_{n \times n }(\mathbb{C})$, obtained by deleting the $i$th column and the $i$th row from $A$, for some $i \in \{1,\cdots, n+1\}$, the [[eigenvalues]]
$(b_1 \leq b_2 \leq \cdots b_n)$
of $B$ *interlace* the eigenvalues
$(a_1 \leq a_2 \leq \cdots a_{n+1})$ of $A$, in that

$$
  a_1
    \;\leq\;
  b_1
    \;\leq\;
  a_2
    \;\leq\;
  b_2
    \;\leq\;
  \cdots
    \;\leq\;
  b_n
    \;\leq\;
  a_{n+1}
  \,.
$$

## Related concepts

* [[principal submatrix]]

* [[Gershgorin circle theorem]]

## References

* Suk-Geun Hwang, *Cauchy's Interlace Theorem for Eigenvalues of Hermitian Matrices*, The American Mathematical Monthly Vol. 111, No. 2 (Feb., 2004), pp. 157-159 ([jstor:4145217](https://www.jstor.org/stable/4145217))

* Steve Fisk, *A very short proof of Cauchy's interlace theorem for eigenvalues of Hermitian matrices* ([arXiv:math/0502408](https://arxiv.org/abs/math/0502408))

See also:

* Wikipedia, *[Cauchy interlacing theorem](https://en.wikipedia.org/wiki/Min-max_theorem#Cauchy_interlacing_theorem)*

* Wikipedia, *[Poincaré separation theorem](https://en.wikipedia.org/wiki/Poincar%C3%A9_separation_theorem)*

[[!redirects Cauchy's interlace theorem]]


Cauchy principal value

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Integration theory
+--{: .hide}
[[!include integration theory - contents]]
=--
#### Functional analysis
+-- {: .hide}
[[!include functional analysis - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea ##

The _Cauchy principal value_ of a [[function]] which is [[integrable function|integrable]] on the [[complement]] of one point is, if it exists, the [[limit of a sequence|limit]] of the [[integrals]] of the function over subsets in the [[complement]] of this point as these integration [[domains]] tend to that point _symmetrically_ from all sides.

One also subsumes the case that the "point" is "at infinity", hence that the function is [[integrable function|integrable]] over every [[bounded subsets|bounded]] [[domain]]. In this case the Cauchy principal value is the [[limit of a sequence|limit]], if it exists, of the [[integrals]] of the function over bounded domains, as their bounds tend _symmetrically_ to infinity.

The operation of sending a [[compactly supported function|compactly supported]] [[smooth function]] ([[bump function]]) to Cauchy principal value of its pointwise product with a function $f$ that may be singular at the origin defines a [[distribution]], usually denoted $PV(f)$.

When the Cauchy principal value exists but the full [[integral]] does not (hence when the full integral "diverges") one may think of the Cauchy principal value as "exracting a finite value from a diverging quantity". This is similar to the _intuition_ of the early days of [[renormalization]] in [[perturbative quantum field theory]] ([[Schwinger-Tomonaga-Feynman-Dyson]]), but one has to be careful not to carry this analogy too far.

One point where the Cauchy principal value really does play a key role in [[perturbative quantum field theory]] is in the computation of [[Green functions]] ([[propagators]]) for the [[Klein-Gordon operator]] and the [[Dirac operator]]. See remark \ref{FeynmanPropagator} below and see at _[[Feynman propagator]]_ for more on this.

## Definition

### As an integral

+-- {: .num_defn #CauchyIntegralValueOfIntegralOverRealline}
###### Definition
**(Cauchy principal value of an integral over the real line)**

Let $f \colon \mathbb{R} \to \mathbb{R}$ be a [[function]] on the [[real line]] such that for every [[positive real number]] $\epsilon$ its [[restriction]] to $\mathbb{R}\setminus (-\epsilon, \epsilon)$ is [[integrable function|integrable]]. Then the _Cauchy principal value_ of $f$ is, if it exists, the [[limit of a sequence|limit]]

$$
  PV(f) \coloneqq \underset{\epsilon \to 0}{\lim}
  \underset{\mathbb{R} \setminus (-\epsilon, \epsilon)}{\int} f(x) \, d x
  \,.
$$

=--

### As a distribution

+-- {: .num_defn #CauchyPrincipalValueAsDistributionOnRealLine}
###### Definition
**(Cauchy principal value as distribution on the real line)**

Let $f \colon \mathbb{R} \to \mathbb{R}$ be a [[function]] on the [[real line]] such that for all [[bump functions]] $b \in C^\infty_{cp}(\mathbb{R})$ the Cauchy principal value of the pointwise product function $f b$ exists, in the sense of def. \ref{CauchyIntegralValueOfIntegralOverRealline}. Then this assignment

$$
  PV(f)
  \;\colon\;
  b \mapsto PV(f b)
$$

defines a [[distribution]] $PV(f) \in \mathcal{D}'(\mathbb{R})$.

=--

## Examples

### The principal value of $1/x$

+-- {: .num_example }
###### Example

Let $f \colon \mathbb{R} \to \mathbb{R}$ be an [[integrable function]] which is symmetric, in that $f(-x) = f(x)$ for all $x \in \mathbb{R}$. Then the principal value integral (def. \ref{CauchyIntegralValueOfIntegralOverRealline}) of $x \mapsto \frac{f(x)}{x}$ exists and is zero:

$$
  \underset{\epsilon \to 0}{\lim}
  \underset{\mathbb{R}\setminus (-\epsilon, \epsilon)}{\int}
  \frac{f(x)}{x} d x
  \; = \; 0
$$

This is because, by the symmetry of $f$ and the skew-symmetry of $x \mapsto 1/x$, the two contributions to the integral are equal up to a sign:

$$
  \int_{-\infty}^{-\epsilon} \frac{f(x)}{x} d x
  \;=\;
  -
  \int_{\epsilon}^\infty \frac{f(x)}{x} d x
  \,.
$$

=--

+-- {: .num_example #PrincipalValueOfInverseFunctionCharacteristicEquation}
###### Example

The principal value distribution $PV\left( \frac{1}{x}\right)$ (def. \ref{CauchyPrincipalValueAsDistributionOnRealLine}) solves the distributional equation

$$
  \label{DistributionalEquationxfOfxEqualsOne}
  x PV\left(\frac{1}{x}\right) = 1
  \phantom{AAA}
  \in \mathcal{D}'(\mathbb{R}^1)
  \,.
$$

Since the [[delta distribution]] $\delta \in \mathcal{D}'(\mathbb{R}^1)$ solves the equation

$$
  x \delta(x) = 0
  \phantom{AAA}
  \in \mathcal{D}'(\mathbb{r}^1)
$$

we have that more generally every [[linear combination]] of the form

$$
  \label{GeneralDistributionalSolutionToxfEqualsOne}
  F(x)
  \coloneqq
  PV(1/x) + c \delta(x)
  \phantom{AAA}
  \in \mathcal{D}'(\mathbb{R}^1)
$$

for $c \in \mathbb{C}$, is a distributional solution to $x F(x) =  1$.

The [[wave front set]] of these solutions is

\[
  WF\left(
    PV(1/x) + c \delta(x)
  \right)
  \;=\;
\begin{cases}
  \left\{
    (0,k) \;\vert\; k \in \mathbb{R}^\ast \setminus \{0\}
  \right\}, & c\not \in \{\pi i,-\pi i\},\\
  \left\{
  (0,k) \;\vert\;\mp k>0
  \right\}, & c=\pm\pi i.
\end{cases}
\]

=--

+-- {: .proof}
###### Proof

The first statement is immediate from the definition: For $b \in C^\infty_c(\mathbb{R}^1)$ any [[bump function]] we have that

$$
  \begin{aligned}
    \left\langle x PV\left(\frac{1}{x}\right), b \right\rangle
    & \coloneqq
    \underset{\epsilon \to 0}{\lim}
    \underset{\mathbb{R}^1 \setminus (-\epsilon, \epsilon)}{\int}
    \frac{x}{x}b(x) \, d x
    \\
    & =
    \int b(x)  d x
    \\
    & =
    \langle 1,b\rangle
  \end{aligned}
$$

Regarding the second statement: One computes that the Fourier transforms (with oscillation factor $-1$ and normalization factor 1) of $PV(1/x)$ and $\delta$ are given by $\xi\mapsto -\pi i \mathrm{sgn}(\xi)$ and $\xi \mapsto 1$, respectively. From this the statement immediately follows.

=--

+-- {: .num_prop}
###### Proposition

In fact (eq:GeneralDistributionalSolutionToxfEqualsOne) is the most general distributional solution to (eq:DistributionalEquationxfOfxEqualsOne).

=--

This follows by the characterization of [[extension of distributions]] to a point, see there at [this prop.](extension+of+distributions#SpaceOfPointExtensions) ([Hörmander 90, thm. 3.2.4](#HoermanderLPDO1))

+-- {: .num_defn #IntegrationAgainstInverseOfxWithImaginaryOffset}
###### Definition
**(integration against inverse variable with imaginary offset)**

Write

$$
  \tfrac{1}{x + i0^\pm}
  \;\in\;
  \mathcal{D}'(\mathbb{R})
$$

for the [[distribution]] which is the [[limit]] in $\mathcal{D}'(\mathbb{R})$ of the [[non-singular distributions]] which are given by the [[smooth functions]] $x \mapsto \tfrac{1}{x \pm i \epsilon}$ as the [[positive real number]] $\epsilon$ tends to zero:

$$
  \frac{1}{
     x + i 0^\pm
  }
  \;\coloneqq\;
  \underset{ { \epsilon \in (0,\infty) } \atop { \epsilon \to 0 }  }{\lim}
  \tfrac{1}{x \pm i \epsilon}
$$

hence the distribution which sends $b \in C^\infty(\mathbb{R}^1)$ to

$$
  b \mapsto
  \underset{\mathbb{R}}{\int}
    \frac{b(x)}{x \pm i \epsilon} \, d x
  \,.
$$

=--

+-- {: .num_prop #CauchyPrincipalValueEqualsIntegrationWithImaginaryOffsetPlusDelta}
###### Proposition
**([[Cauchy principal value]] equals integration with [[imaginary number|imaginary]] offset plus [[delta distribution]])**

The Cauchy principal value distribution $PV\left( \tfrac{1}{x}\right) \in \mathcal{D}'(\mathbb{R})$ (def. \ref{CauchyPrincipalValueAsDistributionOnRealLine}) is equal to the sum of the integration over $1/x$ with imaginary offset (def. \ref{IntegrationAgainstInverseOfxWithImaginaryOffset}) and a [[delta distribution]].

$$
 PV\left(\frac{1}{x}\right)
  \;=\;
 \frac{1}{x + i 0^\pm}
  \pm i \pi \delta
  \,.
$$
(Plemelj-Sochocki formula)

In particular, by prop. \ref{PrincipalValueOfInverseFunctionCharacteristicEquation} this means that $\tfrac{1}{x + i 0^\pm}$ solves the distributional equation

$$
  x \frac{1}{x + i 0^\pm}
  \;=\;
  1
  \phantom{AA}
  \in \mathcal{D}'(\mathbb{R}^1)
  \,.
$$

=--

+-- {: .proof}
###### Proof

Using that

$$
  \begin{aligned}
    \frac{1}{x \pm i \epsilon}
    & =
    \frac{
       x \mp i \epsilon
    }{
      (x + i \epsilon)(x - i \epsilon)
    }
    \\
    & =
    \frac{ x \mp i \epsilon }{(x^2 + \epsilon^2)}
  \end{aligned}
$$

we have for every [[bump function]] $b \in C^\infty_{cp}(\mathbb{R}^1)$

$$
  \begin{aligned}
    \underset{\epsilon \to 0}{\lim}
    \underset{\mathbb{R}^1}{\int}
    \frac{b(x)}{x \pm i \epsilon}
    d x
    & \;=\;
    \underset{
      (A)
    }{
    \underbrace{
    \underset{\epsilon \to 0}{\lim}
    \underset{\mathbb{R}^1}{\int}
    \frac{x^2}{x^2 + \epsilon^2} \frac{b(x)}{x}
    d x
    }
    }
    \mp i \pi
    \underset{(B)}{
    \underbrace{
    \underset{\epsilon \to 0}{\lim}
    \underset{\mathbb{R}^1}{\int}
    \frac{1}{\pi}
    \frac{\epsilon}{x^2 + \epsilon^2} b(x)
    \,
    d x
    }}
  \end{aligned}
$$

Since

$$
  \array{
     && \frac{x^2}{x^2 + \epsilon^2}
     \\
     & {}^{\mathllap{ { {\vert x \vert} \lt \epsilon } \atop { \epsilon \to 0 }  }}\swarrow
     &&
     \searrow^{\mathrlap{ {{\vert x\vert} \gt \epsilon} \atop { \epsilon \to 0 }  }}
     \\
     0 && && 1
  }
$$

it is plausible that $(A) = PV\left( \frac{b(x)}{x} \right)$, and similarly that $(B) = b(0)$. In detail:

$$
  \begin{aligned}
    (A)
    & =
    \underset{\epsilon \to 0}{\lim}
    \underset{\mathbb{R}^1}{\int}
    \frac{x}{x^2 + \epsilon^2} b(x)
    d x
    \\
    & =
    \underset{\epsilon \to 0}{\lim}
    \underset{\mathbb{R}^1}{\int}
     \frac{d}{d x}
     \left(
        \tfrac{1}{2}
        \ln(x^2 + \epsilon^2)
     \right)
      b(x)
    d x
    \\
    & =
    -\tfrac{1}{2}
    \underset{\epsilon \to 0}{\lim}
    \underset{\mathbb{R}^1}{\int}
     \ln(x^2 + \epsilon^2)
     \frac{d b}{d x}(x)
    d x
    \\
    & =
    -\tfrac{1}{2}
    \underset{\mathbb{R}^1}{\int}
     \ln(x^2)
     \frac{d b}{d x}(x)
    d x
    \\
    & =
    -
    \underset{\mathbb{R}^1}{\int}
     \ln({\vert x \vert})
     \frac{d b}{d x}(x)
    d x
    \\
    & =
    -
    \underset{\epsilon \to 0}{\lim}
    \underset{\mathbb{R}^1\setminus (-\epsilon, \epsilon)}{\int}
     \ln( {\vert x \vert} )
     \frac{d b}{d x}(x)
    d x
    \\
    & =
    \underset{\epsilon \to 0}{\lim}
    \underset{\mathbb{R}^1\setminus (-\epsilon, \epsilon)}{\int}
     \frac{1}{x}
     b(x)
    d x
    \\
    & =
    PV\left( \frac{b(x)}{x} \right)
  \end{aligned}
$$

and

$$
  \begin{aligned}
    (B)
    & =
    \tfrac{1}{\pi}
    \underset{\epsilon \to 0}{\lim}
    \underset{\mathbb{R}^1}{\int}
    \frac{\epsilon}{x^2 + \epsilon^2}
    b(x)
    \,
    d x
    \\
    & =
    \tfrac{1}{\pi}
    \underset{\epsilon \to 0}{\lim}
    \underset{\mathbb{R}^1}{\int}
    \left(
      \frac{d}{d x}
      \arctan\left( \frac{x}{\epsilon} \right)
    \right)
    b(x)
    \,
    d x
    \\
    & =
    -
    \tfrac{1}{\pi}
    \underset{\epsilon \to 0}{\lim}
    \underset{\mathbb{R}^1}{\int}
      \arctan\left( \frac{x}{\epsilon} \right)
    \frac{d b}{d x}(x)
    \,
    d x
    \\
    & =
    -
    \frac{1}{2}
    \underset{\mathbb{R}^1}{\int}
      sgn(x)
    \frac{d b}{d x}(x)
    \,
    d x
    \\
    & =
    b(0)
  \end{aligned}
$$

where we used that the [[derivative]] of the [[arctan]] function is $\frac{d}{ d x} \arctan(x) = 1/(1 + x^2)$ and that $\underset{\epsilon \to + \infty}{\lim} \arctan(x/\epsilon) = \tfrac{\pi}{2}sgn(x)$ is proportional to the [[sign function]].

=--

+-- {: .num_exmaple #FourierIntegralFormulaForStepFunction}
###### Example
**([[Fourier integral]] formula for [[step function]])**

The [[Heaviside distribution]] $\Theta \in \mathcal{D}'(\mathbb{R})$ is equivalently the following Cauchy principal value:

$$
  \begin{aligned}
    \Theta(x)
    & =
    \frac{1}{2\pi i} \int_{-\infty}^\infty \frac{e^{i \omega x}}{\omega - i 0^+}
    \\
    & \coloneqq
    \underset{ \epsilon \to 0^+}{\lim}
    \frac{1}{2 \pi i}
    \int_{-\infty}^\infty \frac{e^{i \omega x}}{\omega - i \epsilon} d\omega
    \,,
  \end{aligned}
$$

where the limit is taken over [[sequences]] of [[positive numbers|positive]] [[real numbers]] $\epsilon \in (-\infty,0)$ tending to zero.

=--

+-- {: .proof}
###### Proof

We may think of the [[integrand]] $\frac{e^{i \omega x}}{\omega - i \epsilon}$ uniquely extended to a [[holomorphic function]] on the [[complex plane]] and consider computing the given real [[line integral]] for fixed $\epsilon$ as a [[contour integral]] in the [[complex plane]].

If $x \in (0,\infty)$ is [[positive number|positive]], then the exponent

$$
  i \omega x = - Im(\omega) x + i Re(\omega) x
$$

has negative [[real part]] for _positive_ [[imaginary part]] of $\omega$. This means that the [[line integral]] equals the complex [[contour integral]] over a contour $C_+ \subset \mathbb{C}$ closing in the [[upper half plane]]. Since $i \epsilon$ has positive [[imaginary part]] by construction, this contour does encircle the [[pole]] of the [[integrand]] $\frac{e^{i \omega x}}{\omega - i \epsilon}$ at $\omega = i \epsilon$. Hence by the [[Cauchy integral formula]] in the case $x \gt 0$ one gets

$$
  \begin{aligned}
    \underset{\epsilon \to 0^+}{\lim}
    \frac{1}{2 \pi i}
    \int_{-\infty}^\infty \frac{e^{i \omega x}}{\omega - i \epsilon} d\omega
    & =
    \underset{\epsilon \to 0^+}{\lim}
    \frac{1}{2 \pi i}
    \oint_{C_+} \frac{e^{i \omega x}}{\omega - i \epsilon} d \omega
    \\
    &
    =
    \underset{\epsilon \to 0^+}{\lim}
    \left(e^{i \omega x}\vert_{\omega = i \epsilon}\right)
    \\
    & =
    \underset{\epsilon \to 0^+}{\lim}
    e^{- \epsilon x}
    \\
    & =
    e^0 = 1
  \end{aligned}
  \,.
$$

Conversely, for $x \lt 0$ the real part of the integrand decays as the _[[negative number|negative]]_ imaginary part increases, and hence in this case the given line integral equals the contour integral for a contour $C_- \subset \mathbb{C}$ closing in the lower half plane. Since the integrand has no pole in the lower half plane, in this case the [[Cauchy integral formula]] says that this integral is zero.

=--

Conversely, by the [[Fourier inversion theorem]], the [[Fourier transform]] of the [[Heaviside distribution]] is the Cauchy principal value as in prop. \ref{CauchyPrincipalValueEqualsIntegrationWithImaginaryOffsetPlusDelta}:

+-- {: .num_example #RelationToFourierTransformOfHeavisideDistribution}
###### Example
**(relation to [[Fourier transform]] of [[Heaviside distribution]] / [[Schwinger parameter|Schwinger parameterization]])**

$$
  \begin{aligned}
    \widehat \Theta(x)
    & =
    \int_0^\infty e^{i k x} \,  dk
    \\
    & =
    i \frac{1}{x + i 0^+}
  \end{aligned}
$$

Here the second equality is also known as _complex [[Schwinger parameter|Schwinger parameterization]]_.

=--

+-- {: .proof}
###### Proof

As [[generalized functions]] consider the [[limit of a sequence|limit]] with a decaying component:

$$
  \begin{aligned}
    \int_0^\infty e^{i k x} \,  dk
    & =
    \underset{\epsilon \to 0^+}{\lim}
    \int_0^\infty e^{i k x - \epsilon k} \,  dk
    \\
    & =
    -
    \underset{\epsilon \to 0^+}{\lim}
    \frac{1}{ i x - \epsilon}
    \\
    & =
    i \frac{1}{x + i 0^+}
  \end{aligned}
$$

=--

### The principal value of $1/(q(x) + m^2)$
 {#PrincipalValueOfPowerOfAQuadraticForm}

Let $q \colon \mathbb{R}^{n} \to \mathbb{R}$ be a non-degenerate real [[quadratic form]]
[[analytic continuation|analytically continued]] to a real quadratic form

$$
  q \;\colon\; \mathbb{C}^n \longrightarrow \mathbb{C}
  \,.
$$

Write $\Delta$ for the [[determinant]] of $q$

Write $q^\ast$ for the induced quadratic form on [[dual vector space]]. Notice that $q$ (and hence $a^\ast$) are assumed non-degenerate but need not necessarily be positive or negative definite.

+-- {: .num_prop #FourierTransformOfPrincipalValueOfPowerOfQuadraticForm}
###### Proposition
**([[Fourier transform of distributions|Fourier transform]] of principal value of power of [[quadratic form]])**

Let $m \in \mathbb{R}$ be any [[real number]], and $\kappa \in \mathbb{C}$ any [[complex]] number. Then the [[Fourier transform of distributions]] of $1/(q + m^2 + i 0^+)^\kappa$ is

$$
  \widehat
  {
    \left(
      \frac{1}{q + m^2 + i0^+}
    \right)
  }
  \;=\;
  \frac{
    2^{1- \kappa} (\sqrt{2\pi})^{n} m^{n/2-\kappa}
  }
  {
    \Gamma(\kappa) \sqrt{\Delta}
  }
  \frac{
    K_{n/2 - \kappa}\left( m \sqrt{q^\ast - i 0^+} \right)
  }
  {
    \left(\sqrt{q^\ast - i0^+ }\right)^{n/2 - \kappa}
  }
  \,,
$$

where

1. $\Gamma$ deotes the [[Gamma function]]

1. $K_{\nu}$ denotes the [[modified Bessel function]].

Notice that $K_\nu(a)$ diverges for $a \to 0$ as $a^{-\nu}$ ([DLMF 10.30.2](http://dlmf.nist.gov/10.30#E2)).

=--

([Gel'fand-Shilov 66, III 2.8 (8) and (9), p 289](#GelfandShilov66))

+-- {: .num_example #FeynmanPropagator}
###### Example
**([[Feynman propagator]])**

Let $q \coloneqq \eta^{-1}$ be the dual [[Minkowski metric]] in [[dimension]] $p+1$. Then

$$
  \Delta_F(x)
  \;\propto\;
  \widehat{
    \frac{1}{
      -\eta^{-1}(k,k) - m^2 + i0^+
    }
  }
$$

is the [[Feynman propagator]] for the [[Klein-Gordon equation]] on [[Minkowski spacetime]]. In this case prop. \ref{FourierTransformOfPrincipalValueOfPowerOfQuadraticForm} implies that its [[singular support]] is the [[light cone]] $\{x \in \mathbb{R}^{p,1} \;\vert\; \eta(x,x) = 0\}$.

=--

### The Fourier transform of $\delta(q + m^2)$
 {#FourierTransformOfDeltaOfTheMassShell}

Let $q \colon \mathbb{R}^{n} \to \mathbb{R}$ be a non-degenerate real [[quadratic form]]
[[analytic continuation|analytically continued]] to a real quadratic form

$$
  q \;\colon\; \mathbb{C}^n \longrightarrow \mathbb{C}
  \,.
$$

Write $\Delta$ for the [[determinant]] of $q$. Write $t \in \mathbb{N}$ for the number of negative eigenvalues.

Write $q^\ast$ for the induced quadratic form on [[dual vector space]]. Notice that $q$ (and hence $a^\ast$) are assumed non-degenerate but need not necessarily be positive or negative definite.

+-- {: .num_prop #FourierTransformOfDeltaDistributionappliedToMassShell}
###### Proposition
**([[Fourier transform]] of [[delta distribution]] applied to [[mass shell]])**

Let $m \in \mathbb{R}$, then the [[Fourier transform of distributions]] of the [[delta distribution]] $\delta$ applied to the "mass shell" $q + m^2$ is

$$
  \widehat{
    \delta(q + m^2)
  }
  \;=\;
  - \frac{i}{\sqrt{{\vert\Delta\vert}}}
  \left(
    e^{i \pi t /2 }
    \frac{
      K_{n/2-1}
      \left(
        m \sqrt{ q^\ast + i0^+ }
      \right)
    }{
      \left(\sqrt{q^\ast + i0^+}\right)^{n/2 - 1}
    }
    \;-\;
    e^{-i \pi t /2 }
    \frac{
      K_{n/2-1}
      \left(
        m \sqrt{ q^\ast - i0^+ }
      \right)
    }{
      \left(\sqrt{q^\ast - i0^+}\right)^{n/2 - 1}
    }
  \right)
  \,,
$$

where $K_\nu$ denotes the [[modified Bessel function]] of order $\nu$.

Notice that $K_\nu(a)$ diverges for $a \to 0$ as $a^{-\nu}$ ([DLMF 10.30.2](http://dlmf.nist.gov/10.30#E2)).

=--

([Gel'fand-Shilov 66, III 2.11 (7), p 294](#GelfandShilov66))

+-- {: .num_example #CausalPropagator}
###### Example
**([[causal propagator]])**

Let $q \coloneqq \eta^{-1}$ be the dual [[Minkowski metric]] in [[dimension]] $p+1$. Then

$$
  \Delta_S(x)
  \;\propto\;
  \widehat{
     \delta(-\eta(k,k) - m^2)
     \,
     sgn(k_0)
  }
$$

is the [[causal propagator]] for the [[Klein-Gordon equation]] on [[Minkowski spacetime]]. In this case prop. \ref{FourierTransformOfDeltaDistributionappliedToMassShell} implies that its [[singular support]] is the [[light cone]] $\{x \in \mathbb{R}^{p,1} \;\vert\; \eta(x,x) = 0\}$.

=--

## Related concepts

* [[zeta function regularization]]

## References ##

Named after [[Augustin Cauchy]]

* [[Ram Kanwal]], section 8.3 of _Linear Integral Equations_ Birkhäuser 1997

Detailed discussion of relation to [[Bessel functions]] is in

* {#GelfandShilov66} [[I. M. Gel'fand]], G. E. Shilov, _Generalized functions_, 1–5 , Acad. Press  (1966–1968) transl. from И. М. Гельфанд, Г. Е. Шилов Обобщенные функции, вып. 1-3, М.:Физматгиз, 1958; 1: Обобщенные функции и действия над ними, 2: Пространства основных обобщенных функций, 3: Некоторые вопросы теории дифференциальных уравнений

References on homogeneous distributions

* {#HoermanderLPDO1} [[Lars Hörmander]], _The Analysis of Linear Partial Differential Operators I_ (Springer, 1990, 2nd ed.)

See also

* Wikipedia, _[Cauchy principal value](http://en.wikipedia.org/wiki/Cauchy_principal_value)_

* Wikipedia, _[Hadamard principal value](http://en.wikipedia.org/wiki/Hadamard_finite_part_integral)_

[[!redirects Cauchy principal values]]

[[!redirects principal value]]
[[!redirects principal vaues]]


Cauchy real number

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Analysis
+-- {: .hide}
[[!include analysis - contents]]
=--
#### Constructivism, Realizability, Computability
+-- {: .hide}
[[!include constructivism - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

A Cauchy real number is a [[real number]] that is given as the limit of a [[Cauchy sequence]] of [[rational numbers]].  One may use this idea as a *definition* of the general concept of real number. This is due to [[Georg Cantor]] in 1872, the same year that [[Richard Dedekind]] developed [[Dedekind cuts]] as a definition of the same concept.

## Definitions

If we simply want a construction of the [[real line]] $\mathbb{R}$ for the purposes of [[classical mathematics]], then we can use Cantor\'s original version.  If we wish to use weak [[foundations]] or [[internalisation|internalise]] the Cauchy real line, then there are subtler alternatives.

### Classical version

Putting Cantor\'s definition in modern terminology, $\mathbb{R}$ is the [[quotient set]] of the set of Cauchy sequences of rational numbers, with two sequences considered equivalent if their difference converges to zero.  Although the notion of Cauchy sequence (and convergence, for that matter) is best known in the context of [[metric spaces]] (which cannot be defined in general without having, at least implicitly, constructed $\mathbb{R}$ already), it is easy to state the definitions in elementary terms.  (We can also view these as Cauchy sequences in a [[Booij premetric space]].)  If there is any tricky point, it is that the requirements made for all $\epsilon \gt 0$ need be made only for rational $\epsilon$. (We could also treat $\mathbb{Q}$ as a [[uniform space]] or even a [[Cauchy space]], although again to write down the definitions of those structures still requires one to handle the $\epsilon$s.)

To be explicit:  A __Cauchy real number__ $x$ is an [[infinite sequence]] $(x_0,x_1,x_2,\ldots)$ of [[rational numbers]] such that, for every positive rational number $\epsilon$, there exists a [[natural number]] $\alpha$ such that
$$ {|x_i - x_j|} \leq \epsilon $$
holds whenever $i,j \geq \alpha$.  Two Cauchy real numbers $x,y$ are considered __equal__ if, for every positive rational number $\epsilon$, there exists a natural number $\alpha$ such that
$$ {|x_i - y_i|} \leq \epsilon $$
holds whenever $i \geq \alpha$.  It is easy to prove that this is an [[equivalence relation]] on the set of Cauchy sequences, so the set $\mathbb{R}$ of real numbers is a quotient set.

This definition comes in two steps: one to identify the Cauchy sequences from among the infinite sequences, another to identify equivalent sequences.  Actually, we can do this in one step by placing a [[partial equivalence relation]] on the set of *all* infinite sequences of rational numbers.  Two sequences $x,y$ are considered equivalent if, for every positive rational number $\epsilon$, there exists a natural number $\alpha$ such that
$$ {|x_i - y_j|} \leq \epsilon $$
holds whenever $i,j \geq \alpha$.  It is immediate that a sequence $x$ is Cauchy if and only if it equivalent to itself, and it is easy to prove that two Cauchy sequences $x,y$ are equivalent if and only if they are equal as real numbers using the definition above.  Thus, we can construct $\mathbb{R}$ immediately as a [[subquotient]] of the [[function set]] $\mathbb{Q}^{\mathbb{N}}$.

### Moduli of convergence

In weak foundations, we sometimes want to be a little more strict about how a sequence is Cauchy and how the difference of two such sequences converges to zero.  We do this by requiring explicit moduli of convergence.  These moduli can always be constructed using (for example) either [[countable choice]] or the principle of [[excluded middle]], but the requirement makes a difference in (for example) a [[localic topos]] over any non-[[discrete space]].

(Since the term 'Cauchy real number' is used ambiguously in the constructive literature, we can identify Cantor\'s classical definition above as a __Cantor real number__ or a __classical Cauchy real number__.)

In general, a __[[modulus of convergence]]__ may be any function $\alpha$ from the positive rational numbers to the natural numbers such that, for every natural number $n$, there is a positive rational number $\epsilon$ such that $\alpha(\epsilon) \geq n$.

A __modulated Cauchy real number__ $x$ is an infinite sequence $(x_0,x_1,x_2,\ldots)$ of rational numbers such that there exists a modulus $\alpha$ such that
$$ {|x_i - x_j|} \leq \epsilon $$
holds whenever $i,j \geq \alpha(\epsilon)$.  Two modulated Cauchy real numbers $x,y$ are considered __equal__ if there exists a modulus $\alpha$ such that
$$ {|x_i - y_i|} \leq \epsilon $$
holds whenever $i \geq \alpha(\epsilon)$.  Again we can combine these conditions into a single partial equivalence relation: that there exists a modulus $\alpha$ such that
$$ {|x_i - y_j|} \leq \epsilon $$
holds whenever $i,j \geq \alpha(\epsilon)$.

Some variations are often met.  A modulus $\alpha$ may be extended to all rational numbers, although the conditions above can only be required for positive $\epsilon$.  Alternatively, it is enough to define $\alpha$ (and require the conditions) for $\epsilon$ of specific form; common choices are $1/n$ and $1/2^n$ for $n$ a natural number, which work well with $\alpha(\epsilon) = 1/\epsilon$ and $\alpha(\epsilon) = -\log_2\epsilon$, respectively.  (The important criterion is to use a [[detachable set]] of positive rational numbers of which zero is a limit point; then $\alpha$ can be extended from this set to all positive rational numbers by rounding down.)

It is also possible to fix a specific modulus $\alpha$ ahead of time.  Then we need to treat each index separately, in this way:  An __$\alpha$-regular Cauchy real number__ $x$ is an infinite sequence $(x_0,x_1,x_2,\ldots)$ of rational numbers such that
$$ {|x_i - x_j|} \leq \delta + \epsilon $$
holds whenever $i \geq \alpha(\delta)$ and $j \geq \alpha(\epsilon)$.  Two $\alpha$-regular Cauchy real numbers are considered __equal__ if
$$ {|x_i - y_i|} \leq 2\epsilon $$
holds whenever $i \geq \alpha(\epsilon)$.  Once more, we can combine these conditions into a single partial equivalence relation: that
$$ {|x_i - y_j|} \leq \delta + \epsilon $$
holds whenever $i \geq \alpha(\delta)$ and $j \geq \alpha(\epsilon)$.

### Cauchy approximations

The modulated Cauchy real numbers could also be constructed through [[Cauchy approximations]] rather than sequences of rational numbers with a modulus of convergence. Let $\mathbb{Q}$ be the [[rational numbers]] $\mathbb{Q}$ and let

$$\mathbb{Q}_{+} \coloneqq \{x \in \mathbb{Q} \vert 0 \lt x\}$$

be the positive rational numbers.

#### As an initial algebra

Let us define a *modulated Cauchy algebra* to be a [[set]] $A$ with a function $\iota \in A^{C(\mathbb{Q})}$, where $C(\mathbb{Q})$ is the set of [[Cauchy approximation]]s in $\mathbb{Q}$ and $A^{C(\mathbb{Q})}$ is the set of functions with domain $C(\mathbb{Q})$ and codomain $A$, such that
$$\forall a \in C(\mathbb{Q}). \forall b \in C(\mathbb{Q}). \left( \forall \epsilon \in \mathbb{Q}_{+}. \vert a_\epsilon - b_\epsilon \vert \lt 4 \epsilon \right) \implies (\iota(a) = \iota(b))$$

A *modulated Cauchy algebra homomorphism* is a function $f:B^A$ between modulated Cauchy algebras $A$ and $B$ such that

$$\forall a \in C(\mathbb{Q}). f(\iota_A(a)) = \iota_B(a)$$

The *category of modulated Cauchy algebras* is the category $MCAlg$ whose objects $Ob(MCAlg)$ are modulated Cauchy algebras and whose morphisms $Mor(MCAlg)$ are modulated Cauchy algebra homomorphisms. The set of **modulated Cauchy real numbers**, denoted $\mathbb{R}_C$, is defined as the initial object in the category of modulated Cauchy algebras.

#### As a quotient inductive type

The __modulated Cauchy real numbers__ $\mathbb{R}_C$ are a [[quotient inductive type]] inductively generated by the following:

* a function $\iota: C(\mathbb{Q}) \to \mathbb{R}_C$, where $C(\mathbb{Q})$ is the type of [[Cauchy approximation]]s in $\mathbb{Q}$.

* a dependent family of terms
$$a:C(\mathbb{Q}), b:C(\mathbb{Q}) \vdash eq_{\mathbb{R}_C}(a, b): \left( \prod_{\epsilon:\mathbb{Q}_{+}} \vert a_\epsilon - b_\epsilon \vert \lt 4 \epsilon \right) \to (\iota(a) =_{\mathbb{R}_C} \iota(b))$$

* a family of dependent terms
$$a:\mathbb{R}_C, b:\mathbb{R}_C \vdash \tau(a, b): isProp(a =_{\mathbb{R}_C} b)$$

This is the approach of [Booij20](#Booij20)

### Generalisations

While requiring a modulus of convergence, even fixing the modulus of convergence, may be more restrictive, it is also possible to use a potentially more lax definition.

One way is to use [[multivalued functions]] from the natural numbers.  A __multivalued Cauchy real number__ $x$ is an [[entire relation]] between natural numbers and rational numbers such that, for every positive rational number $\epsilon$, there exists a natural number $\alpha$ such that, whenever $i, j \geq \alpha$,
$$ {|a - b|} \leq \epsilon $$
holds for some $a,b$ such that $x_{i,a}$ and $x_{j,b}$ hold.  Two multivalued Cauchy real numbers are considered __equal__ if, for every positive rational number $\epsilon$, there exists a natural number $\alpha$ such that, whenever $i \geq \alpha$,
$$ {|a - b|} \leq \epsilon $$
holds for some $a,b$ such that $x_{i,a}$ and $y_{i,b}$ hold. The partial equivalence relation subsuming both conditions is that, for every positive rational number $\epsilon$, there exists a natural number $\alpha$ such that, whenever $i,j \geq \alpha$,
$$ {|a - b|} \leq \epsilon $$
holds for some $a,b$ such that $x_{i,a}$ and $y_{j,b}$.

Another way is to use [[nets]] (also called 'generalised sequences').  A __generalised Cauchy real number__ $x$ is a net $(x_\nu)_{\nu\colon D}$ (where $D$ is any [[directed set]] of indices) of rational numbers such that, for every positive rational number $\epsilon$, there exists an index $\alpha$ in $D$ such that
$$ {|x_i - x_j|} \leq \epsilon $$
holds whenever $i,j \geq \alpha$ in $D$.  Two generalised Cauchy real numbers $x,y$ are considered __equal__ if, for every positive rational number $\epsilon$, there exists an index $\alpha$ for $x$ and an index $\beta$ for $y$ such that
$$ {|x_i - y_j|} \leq \epsilon $$
holds whenever $i \geq \alpha$ and $j \geq \beta$.  Actually, this is already a partial equivalence relation on all nets which subsumes both conditions.

Requiring a modulus of convergence would make no difference for either of these, since we would expect such a modulus to also to be either multivalued or given by a net, and these are easy to construct explicitly (even in constructive	mathematics).

(Since the term 'Cauchy real number' logically applies just as well to these generalisations, we may speak of __sequential real numbers__ for the classical or modulated version.)

### Relations between these definitions

Classically, all of these definitions are equivalent.  In fact, to prove their equivalence, we need only the following equivalent principles (cf. [Booij 2020](#Booij20)):

{#DedekindCauchyEquivalence}
* Every [[Dedekind real number]] is a [[modulated Cauchy real number]]

* For every [[Dedekind real number]] there exists a signed-digit [[radix representation]] of the real number

* For every [[Dedekind real number]] there exists a [[locator]]

* For every [[Dedekind real number]] there exists [[Dedekind cut structure]]

* For every [[Dedekind real number]] there exists a [[Cauchy sequence]] of [[rational numbers]] whose limit is the real number.

If we don\'t accept any of the above, then we still have these results:

+-- {: .un_theorem}
###### Theorems
(constructive).

Given any modulus of convergence $\alpha$, every $\alpha$-regular Cauchy real number is a modulated Cauchy real number.  Furthermore, any two $\alpha$-regular real numbers are equal as $\alpha$-regular real numbers if and only if they are equal as modulated Cauchy real numbers.  Conversely, every modulated Cauchy real number is equal (as a modulated Cauchy real number) to some $\alpha$-regular Cauchy real number.

Every modulated Cauchy real number is a classical Cauchy real number, and any two equal modulated Cauchy real numbers are equal as classical Cauchy real numbers.

Every classical Cauchy real number is a multivalued Cauchy real number, and any two equal classical Cauchy real numbers are equal as multivalued Cauchy real numbers.

Every multivalued Cauchy real number becomes a generalised Cauchy real number whose index set $D$ comes equipped with a surjection to the natural numbers.  Furthermore, any two multivalued Cauchy real numbers are equal as multivalued Cauchy real numbers if and only if they are equal as generalised Cauchy real numbers.  Conversely, any generalised Cauchy real number is equal (as a generalised Cauchy real number) to some multivalued Cauchy real number.

Every generalised Cauchy real number becomes a [[Dedekind real number]] in the usual way, defining lower and upper sets in terms of the order relation on generalised Cauchy real numbers.  Furthermore, any two generalised Cauchy real numbers are equal as generalised Cauchy real numbers if and only if they are equal as Dedekind cuts.  Conversely, any Dedekind cut is equal (as a Dedekind cut) to some generalised Cauchy real number.
=--

Hence even in [[constructive mathematics]], there are only three notions that we need consider: modulated Cauchy real numbers, classical Cauchy real numbers, and Dedekind real numbers; and there is a [[function]] from each of these to the next.

Just to be explicit, here are the missing converses:

+-- {: .un_theorem}
###### Axioms
Every classical Cauchy real number is modulated, and any two equal Cauchy real numbers are equal as modulated Cauchy real numbers.

Every multivalued Cauchy real number is equal (as a multivalued Cauchy real number) to some classical Cauchy real number, and two classical Cauchy real numbers are equal if they are equal as multivalued Cauchy real numbers.
=--

Most practitioners of both [[constructive mathematics]] and [[topos theory]] want to use the Dedekind real numbers.  Without [[excluded middle]] or [[countable choice]], the classical Cauchy real numbers are not very well behaved.  The modulated Cauchy real numbers, however, do have their good points; for example, the [[fundamental theorem of algebra]] is simplest for them.  They also make sense in [[predicative mathematics]] with [[function sets]], whereas multivalued Cauchy reals  require [[fullness]] and the Dedekind reals require [[powersets]] for their definitions.

On the other hand, even the (modulated) Cauchy real numbers are not necessarily Cauchy complete, i.e. a Cauchy sequence (even a modulated one) of Cauchy real numbers need not converge to another Cauchy real number (though it always does converge to a Dedekind real number, since the Dedekind real numbers *are* always Cauchy complete).  The problem is that without enough countable choice, we cannot lift a (modulated) Cauchy sequence of (modulated Cauchy) real numbers to a Cauchy sequence of Cauchy sequences in order to "diagonalize" it; a countermodel is constructed by [Lubarsky](#Lubarsky).

For non-modulated Cauchy sequences and reals, there are additional problems even if we assume representatives are already chosen.  A modulated Cauchy sequence of modulated Cauchy sequences does converge to a modulated Cauchy sequence.  Moreover, a *modulated* Cauchy sequence of *classical* Cauchy sequences, and a *classical* Cauchy sequence of *modulated* Cauchy sequences, both necessarily converge to a *classical* Cauchy sequence.  But the results need not be modulated, and a classical Cauchy sequence of classical Cauchy sequences need not converge to a classical Cauchy sequence.  In fact, [Lubarsky](#Lubarsky) shows that:

* The generic classical Cauchy sequence, in the [[classifying topos]] of classical Cauchy sequences, is not modulated.  Thus, we cannot prove that every classical Cauchy sequence is modulated.  Hence, we cannot prove that a classical Cauchy sequence of modulated Cauchy sequences has a modulated limit (since a classical Cauchy sequence of rationals can be regarded as a classical Cauchy sequence of (constant) modulated Cauchy sequences).

* The generic classical Cauchy sequence of classical Cauchy sequences, in the classifying topos of such, does not converge to a classical Cauchy sequence.

* The generic modulated Cauchy sequence of classical Cauchy sequences, in the classifying topos of such, does not converge to a modulated Cauchy sequence.

(Actually, Lubarsky writes using [[Heyting-valued sets]] on topological spaces rather than classifying toposes of propositional [[geometric theories]], but it seems almost certain to me that his results can be rephrased as the above.)

### Using locators

Let $F$ be an [[Archimedean ordered field]]. Every element of $F$ satisfies the axioms of a [[Dedekind cut]] with respect to the [[strict order]] [[relation]] of $F$, and is thus a [[real number]].

The Dedekind real numbers $\mathbb{R}_D$ are the [[terminal]] [[Archimedean ordered field]], i.e. for any other Archimedean ordered field $F$, there exists a unique [[strictly monotonic]] [[field homomorphism]] $u_F:F \to \mathbb{R}_C$. Since every [[ring homomorphism]] between [[fields]] is an [[injection]], the Dedekind real numbers being the terminal Archimedean ordered field $F$ implies that every other Archimedean ordered field is a [[subfield]] of $\mathbb{R}_D$ via the unique ring homomorphism $u_F:F \to \mathbb{R}_D$:

$$\forall F.\mathrm{isAOField}(F) \Rightarrow (F \subseteq \mathbb{R}_D)$$

In this sense, the Dedekind real numbers is the collection of *all* real numbers.

An element $x \in F$ is a **Cauchy real number** if [[existential quantifier|there exists]] a [[locator]] of $x$. The **Cauchy real numbers** are the set of Dedekind real numbers $x \in \mathbb{R}_D$ where there exist a locator

$$\mathbb{R}_C = \{x \in \mathbb{R}_D \vert \exists p \in \mathrm{locator}(x)\}$$

Alternatively, if one doesn't have the [[Dedekind real numbers]], then one can still characterize the Cauchy real numbers by its universal property. The **Cauchy real numbers** $\mathbb{R}_C$ is the [[terminal object|terminal]] Archimedean ordered field where for every element $x \in \mathbb{R}_C$ there exists a locator of $x$: i.e. for any other Archimedean ordered field $F$ where for every element $x \in F$ there exists a locator of $x$, there exists a unique [[strictly monotonic]] [[field homomorphism]] $u_F:F \to \mathbb{R}_C$.

Since every [[ring homomorphism]] between [[fields]] is an [[injection]], the Cauchy real numbers being the terminal Archimedean ordered field $F$ where for every element $x \in F$ there exists a locator of $x$ implies that every other such Archimedean ordered field is a [[subfield]] of $\mathbb{R}_C$ via the unique ring homomorphism $u_F:F \to \mathbb{R}_C$:

$$\forall F.\left(\mathrm{isAOField}(F) \wedge \forall x \in F.\exists p \in \mathrm{locator}(x)\right) \Rightarrow (F \subseteq \mathbb{R}_C)$$

In this sense, the Cauchy real numbers is the collection of *all* real numbers for which there exist a locator.

Either way, this definition makes sense in any [[Heyting category]] with [[exponential objects]] and a [[natural numbers object]], even when the Heyting category is not a [[pretopos]] and the Cauchy real numbers cannot be constructed as a [[quotient set]]. In [[classical mathematics]], one can prove that the [[Dedekind real numbers]] is a [[discrete field]], which implies that the Cauchy real numbers and the Dedekind real numbers coincide.

In [[dependent type theory]], the property that there exists a locator of $x$ is rendered as the [[bracket type]] of the [[type]] of [[locators]] of $x$.

In addition, in [[constructive mathematics]], the Cauchy real numbers $\mathbb{R}_C$ is not to be confused with the set of real numbers
$$\mathbb{R}^\mathcal{L} = \bigcup_{x \in \mathbb{R}_D} \mathrm{locator}(x)$$
which have as [[structure]] a [[locator]]. That the two notions coincide is equivalent to the [[WLPO]], and is not provable in general.

## Algebraic closure

The [[algebraic closure]] of the modulated Cauchy real numbers is the modulated Cauchy [[complex numbers]]; i.e. the [[fundamental theorem of algebra]] is true for the modulated Cauchy real numbers.

## Generalisations

Although the notion of [[metric space]] doesn\'t make sense until we know what real numbers are, once we have these, we can recognise that the rational numbers form a metric space $\mathbb{Q}$ and the real numbers were constructed from them in a way that makes reference only to the metric-space structure of $\mathbb{Q}$.  Thus, this procedure may be generalised to any metric space to produce its [[complete metric space|completion]].

We can also interpret $\mathbb{Q}$ as a [[uniform space]] or even as a [[Cauchy space]] and define analogous notions of completion for these.  However, these require us (in general) to use generalised Cauchy sequences, that is Cauchy [[nets]], even in classical mathematics.  (Of course, without excluded middle or countable choice, we should use nets even for metric spaces.)

## Motivation

Suppose that we wish to approximate a real number $x$ by common fractions with arbitrarily large denominators.  That is, given any natural number $i$, we wish to find [[integers]] $a_i$ such that $x_i \coloneqq a_i/i$ is within $1/i$ of $x$.  Then the sequence of values $(0,x_1,x_2,x_3,\ldots)$ is an $\alpha$-regular Cauchy real with $\alpha(1/i) \coloneqq i$ as modulus of convergence.  (You can extend $\alpha$ to every positive rational number by $\alpha(\epsilon) \coloneqq \lfloor{1/\epsilon}\rfloor$, but that is not important.)  Conversely, given an $\alpha$-regular Cauchy real $(x_0,x_1,x_2,\ldots)$ for this modulus $\alpha$, we can round each $x_i$ (for $i \gt 0$) to the nearest rational number with denominator $i$ (which can be done using only rational arithmetic) to produce an equal $\alpha$-regular Cauchy real.

We might instead want to approximate $x$ by arbitrarily long decimal fractions.  That is, given any natural number $i$, we wish to find integers $a_i$ such that $x_i \coloneqq a_i/10^i$ is within $1/10^i$ of $x$.  Now the sequence of values $(x_0,x_1,x_2,\ldots)$ is an $\alpha$-regular Cauchy real with $\alpha(1/10^i) \coloneqq i$ as modulus of convergence.  Conversely, we can round off the values of any $\alpha$-regular Cauchy real as before.  Of course, there is nothing special about the base $10$ here; for theoretical purposes, base $2$ is popular.

Thus, to *define* a real number to be an $\alpha$-regular Cauchy real (for any of these choices of $\alpha$) is to make into a definition our intuition that we can round real numbers in this way.

Note we may be rounding up or down, regardless of which is nearer.  For example, in approximating $e = \exp\,1$ by decimal fractions, we might get ($2,2.7,2.71,\ldots)$ or $(3,2.7,2.72,\ldots)$, but we might also get $(2,2.8,2.71,\ldots)$.  To choose to always round down, towards zero, or towards the nearer approximant (with a rule for $0.5$) requires an application of [[excluded middle]] (or at least the [[lesser limited principle of omniscience]]).

Even for Dedekind reals in [[neutral constructive mathematics]], we can always approximate a real number in this way up to any given $i$.  Choice is needed only to make infinitely many approximations at once.  Trying to avoid this can motivate multivalued Cauchy real numbers.

## See also

* [[generalized Cauchy real number]]

* [[HoTT book real number]]

* [[Dedekind real number]]

* [[Eudoxus real number]]

* [[prealgebra real number]]

* [[locator]]

## References
 {#References}

See also the references at *[[real number]]* and *[[constructive analysis]]*.

Original articles:

*  [[Georg Cantor]]; 1872; _[Ueber die Ausdehnung eines Satzes aus der Theorie der trigonometrischen Reihen](http://www.maths.tcd.ie/pub/HistMath/People/Cantor/Ausdehnung/)_; Section 1.

Dicussion in [[constructive analysis]]:

* {#Bishop67} [[Errett Bishop]]: *[[Foundations of Constructive Analysis]]*, McGraw-Hill (1967)

* {#BishopBridges85} [[Errett Bishop]], [[Douglas Bridges]]: *[[Constructive Analysis]]*, Grundlehren der mathematischen Wissenschaften **279**, Springer (1985) [[doi:10.1007/978-3-642-61667-9](https://doi.org/10.1007/978-3-642-61667-9)]

* [[Fred Richman]], [[Douglas Bridges]], Peter Schuster, *A weak countable choice principle*. Proceedings of the American Mathematical Society 128(9):2749-2752, March 2000. [[doi:10.1090/S0002-9939-00-05327-2](http://dx.doi.org/10.1090/S0002-9939-00-05327-2)]

{#ReferencesCauchyCompleteness} The proof that Cauchy sequences (with a modulus of convergence) as such (i.e. not passing to their [[equivalence classes]]) are [[sequentially Cauchy complete]]:

* [Bishop (1967, p. 27)](#Bishop67)

* [Bishop & Bridges (1985, p. 29)](#BishopBridges85)

See also

* [[Fred Richman]], *The fundamental theorem of algebra: a constructive development without choice*. Pacific Journal of Mathematics **196** 1 (2000) 213–230 [[doi:10.2140/pjm.2000.196.213](http://dx.doi.org/10.2140/pjm.2000.196.213), [pdf](https://msp.org/pjm/2000/196-1/pjm-v196-n1-p10-p.pdf)]

On why the (modulated) Cauchy real numbers, hence after passage to [[equivalence classes]] of Cauchy sequences, are not [[sequentially Cauchy complete]] (without the [[axiom of countable choice]]):

* {#Lubarsky} [[Robert Lubarsky]], _On the Cauchy Completeness of the Constructive Cauchy Reals_, Electronic Notes in Theoretical Computer Science **167** (2007) 225-254 [[doi:10.1016/j.entcs.2006.09.012](https://doi.org/10.1016/j.entcs.2006.09.012)]

See also

* [[Auke Booij]], *Extensional constructive real analysis via locators*, ([abs:1805.06781](https://arxiv.org/abs/1805.06781))

A [[constructive mathematics|constructive]] algebraic proof of the [[fundamental theorem of algebra]] for the [[modulated Cauchy real numbers]] without countable choice:

* Wim Ruitenberg, Constructing Roots of Polynomials over the Complex Numbers, Computational Aspects of Lie Group Representations and Related Topics, CWI Tract, Vol. 84, Centre for Mathematics and Computer Science, Amsterdam, 1991, pp. 107–128. ([pdf](https://www.mscsnet.mu.edu/~wim/publica/roots_new.pdf))

Implementation of [[Errett Bishop|Bishop]]-style [[Cauchy real numbers]] in [[Agda]]:

* {#Lundfall15} [[Martin Lundfall]], *Formalizing real numbers in Agda* (2015) [<a href="https://wcl.cs.rpi.edu/pilots/library/papers/TAGGED/4211-Lundfall%20(2015)%20-%20Formalizing%20Real%20Numbers%20in%20Agda.pdf">pdf</a>, [[Lundfall-RealNumbersInAgda.pdf:file]], [github](https://github.com/MrChico/Reals-in-agda)]

* {#Murray22} [[Zachary Murray]], *Constructive Analysis in the Agda Proof Assistant* [[arXiv:2205.08354](https://arxiv.org/abs/2205.08354), [github](https://github.com/z-murray/honours-project-constructive-analysis-in-agda)]

review:

* [[Zachary Murray]], *Constructive Real Numbers in the Agda Proof Assistant*, [talk at *CQTS*](Center+for+Quantum+and+Topological+Systems#MurrayFeb2023), (Feb 2023) [video:[YT](https://www.youtube.com/watch?v=7Q_sjfyPJqU)]

A novel construction principle of the type of real numbers, as a [[higher inductive-inductive type]] in [[univalence axiom|univalent]] [[homotopy type theory]], not reliant on representatives by sequences of [[rational numbers]], and with provable [[Cauchy completeness]] even without extra [[axiom of countable choice]] is laid out (cf. *[[HoTT book real numbers]]*) in

* [[Univalent Foundations Project]]: Chapter 11 of:  *[[Homotopy Type Theory – Univalent Foundations of Mathematics]]* (2013) [[web](http://homotopytypetheory.org/book/), [pdf](http://hottheory.files.wordpress.com/2013/03/hott-online-323-g28e4374.pdf)]

but has not yet been implemented in any [[proof assistant]],

Exposition and review:

* [[Andrej Bauer]], *The real numbers in homotopy type theory*, talk at *[Computability and Complexity in Analysis](http://cca-net.de/cca2016/)*, Faro (2016) [[pdf](https://math.andrej.com/wp-content/uploads/2016/06/hott-reals-cca2016.pdf), [[Bauer-RealsInHoTT.pdf:file]]]

* {#Booij20} [[Auke Booij]], *Analysis in Univalent Type Theory* (2020) [[etheses:10411](http://etheses.bham.ac.uk/id/eprint/10411), [pdf](https://etheses.bham.ac.uk/id/eprint/10411/7/Booij2020PhD.pdf), [[Booij-AnalysisInUF.pdf:file]]]

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