complete Segal space

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#### $(\infty,1)$-Category theory
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#Contents#
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## Idea

A _complete Segal space_ is a model for an [[internal category in an (∞,1)-category]] in [[∞Grpd]], with the latter [[presentable (∞,1)-category|presented]] by [[sSet]]/[[Top]]. So complete Segal spaces present [[(∞,1)-categories]].  They are also called _Rezk categories_ after [[Charles Rezk]].

More in detail, a complete Segal space $X$ is

* for each $n \in \mathbb{N}$ a [[Kan complex]] $X_n$, thought of as the _space of composable sequences of $n$-morphisms and their composites_;

* forming a [[simplicial object]] $X_\bullet$ in [[sSet]] (a [[bisimplicial set]]);

such that

1. there is a [[composition]] operation well defined up to [[coherence|coherent]] [[homotopy]]: exibited by the [[Segal maps]]

   $$
     X_k \to X_1 \times_{X_0} \cdots \times_{X_0} X_1
   $$

   (into the iterated [[homotopy pullback]] of the [[∞-groupoid]] of 1-morphisms over the $\infty$-groupoid of objects) being [[homotopy equivalences]]

   (so far this defines a _[[Segal space]]_);

1. the notion of [[equivalence]] in $X_\bullet$ is compatible with that in the ambient [[∞Grpd]] ("completeness"): the sub-simplicial object $Core(X_\bullet)$ on the invertible morphisms in each degree is homotopy constant: it has all face and degeneracy maps being [[homotopy equivalences]].

   (this says that if a morphism is an equivalence under the explicit composition operation then it is already a morphism in $X_0$ ).

## Definition
 {#Definition}

We first discuss

* [Complete Segal spaces](#CompleteSegalSpaces)

as such, and then the more general notion of

* [Complete Segal space objects](#CompleteSegalSpaceObjects)

internal to a suitable model category/$(\infty,1)$-category $\mathcal{C}$ -- this reduces to the previous notion for $\mathcal{C} = sSet_{Quillen}$.

### Complete Segal spaces
 {#CompleteSegalSpaces}

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

A **[[Segal space]]** is a [[simplicial object]] in [[simplicial sets]]

$$
  X \in [\Delta^{op}, sSet]
$$

such that

* it is fibrant in the [[Reedy model structure]] $[\Delta^{op}, sSet_{Quillen}]_{Reedy}$;

* it is a [[local object]] with respect to the [[spine]] inclusions $\{Sp[n] \hookrightarrow \Delta[n]\}_{n \in \mathbb{N}}$;

  equivalently: for all $n \in \mathbb{N}$ the [[Segal map]]

  $$
    X_n \to X_1 \times_{X_0} \cdots \times_{X_0} X_1
  $$

  is a [[weak homotopy equivalence]] (of [[Kan complexes]], in fact).

=--

([Rezk, 4.1](#Rezk)).

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

For $X$ a Segal space, its **[[homotopy category]]** $Ho(X)$ is the
[[Ho(Top)]]-[[enriched category]] whose [[objects]] are the vertices of $X_0$

$$
  Obj(X) = (X_0)_0
$$

and for $x,y \in Obj(X)$ the [[hom object]] is the [[homotopy type]] of the [[homotopy fiber product]]

$$
  Ho(X)(x,y) := \pi_0 \Big(\{x\} \times_{X_0} X_1 \times_{X_0} \{y\}\Big)
  \,.
$$

The [[composition]]

$$
  Ho_X(x,y) \times Ho_X(y,z) \to Ho_X(x,z)
$$

is the (uniquely defined) action of the [[infinity-anafunctor]]

$$
  X_1 \times_{X_0} X_1
  \underoverset{\simeq}{(d_2, d_0)}{\leftarrow}
  X_2
  \stackrel{d_1}{\to}
  X_1
$$

on these connected components.

=--

([Rezk, 5.3](#Rezk))

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

For $X$ a Segal space, write

$$
  X_{hoequ} \hookrightarrow X_1
$$

for the inclusion of the connected components of those vertices that become [[isomorphisms]] in the homotopy category, def. \ref{HomotopyCategory}.

=--

([Rezk, 5.7](#Rezk))

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

A Segal space $X$ is called a **complete Segal space** if

$$
  s_0 : X_0 \to X_{hoequ}
$$

is a weak equivalence.

=--

([Rezk, 6.](#Rezk))

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

This condition is equivalent to $X$ being a [[local object]] with respect to the morphism $N(\{0 \stackrel{\simeq}{\to} 1\}) \to *$. This is discussed [below](#CharacterizationOfCompleteness).

=--

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

The completeness condition may also be thought of as _[[univalence]]_. See there for more.

=--

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

There is a Segal completion functor given in ([Rezk, 14.](#Rezk)).

=--

### Complete Segal space objects
 {#CompleteSegalSpaceObjects}

(...)

## Properties

### Characterization of Completeness
 {#CharacterizationOfCompleteness}

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

A Segal space $X$ is a complete Segal space precisely if it is a [[local object]] with respect to the morphism $N(0 \stackrel{\simeq}{\to} 1) \to *$, hence precisely if with respect to the canonical [[sSet]]-[[enriched category|enriched]] [[hom objects]] we have that

$$
  X_0
   \simeq
  [\Delta^{op}, sSet](*, X)
    \to
  [\Delta^{op}, sSet](N(0 \stackrel{\simeq}{\to} 1), X)
$$

is a weak equivalence.

=--

([Rezk, theorem 6.2](#Rezk))

### Model category structure
 {#ModelCategoryStructure}

The category $[\Delta^{op}, sSet]$ of [[simplicial presheaves]] on the [[simplex category]] ([[bisimplicial sets]]) supports a [[model category]] structure whose fibrant objects are precisely the complete Segal spaces: the _[[model structure for complete Segal spaces]]_. This presents the [[(∞,1)-category of (∞,1)-categories]].

### Relation to simplicial localization
 {#RelationToSimplicialLocalization}

Given $\mathcal{C}$ a [[category with weak equivalences]] $\mathcal{W} \subset Mor(\mathcal{C})$ or more generally a "[[relative category]]", then there is canonically a complete Segal space associated with it by the "relative nerve" construction, def. \ref{RelativeNerve} followed by fibrant replacement, and hence by the [model structure](#ModelCategoryStructure) this determines an [[(∞,1)-category]] $N_{Rezk}(\mathcal{C},\mathcal{W})$ .

On the other hand classical [[simplicial localization]]-theory provides several ways (e.g. [[hammock localization]]) to turn $(\mathcal{C}, \mathcal{W})$ into an [[(∞,1)-category]] $\mathcal{C}[\mathcal{W}]^{-1}$ which universally turns the elements in $\mathcal{W}$ into [[homotopy equivalences]].

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

These constructions are compatible in that there is an [[equivalence of (∞,1)-categories]]

$$
  N_{Rezk}(\mathcal{C},\mathcal{W})
  \simeq
  \mathcal{C}[\mathcal{W}^{-1}]
  \,.
$$

=--

For [[simplicial model categories]] this is ([Rezk, theorem 8.3](#{#Rezk}). For general [[model categories]] this is ([Bergner 07, theorem 6.2](#Bergner07)). For the fully general case this follows from results by [[Clark Barwick]], [[Daniel Kan]] and [[Bertrand Toën]] as pointed out by [[Chris Schommer-Pries]] [here on MathOverflow](http://mathoverflow.net/a/93139/381).

### Model categories for presheaves

There is a notion of [[right/left fibration of complete Segal spaces]] analogous to [[right/left Kan fibrations]] for [[quasi-categories]].

## Examples
 {#Examples}

We discuss some examples. For more and more basic examples see also at _[Segal space -- Examples](Segal+space#Examples)_.

### Ordinary categories as complete Segal spaces
 {#OrdinaryCategoriesAsCompleteSegalSpaces}

We discuss how an ordinary [[small category]] is naturally regarded as a complete Segal space.
([Rezk, 3.5](#Rezk))

#### Preliminaries

We need the following basic ingredients.

Write $(-)^{(-)} : Cat^{op} \times Cat \to Cat$ for the [[internal hom]] in [[Cat]], sending two categories $A$, $X$ to the [[functor category]] $X^A = Func(A,X)$.

By the discussion at [[nerve]] we have a canonical functor

$$
  \Delta \hookrightarrow Cat
$$

including the [[simplex category]] into [[Cat]] by regarding the [[simplex]] $\Delta[n]$ as the category generated from $n$ consecutive morphisms.

The [[nerve]] itself is then then functor

$$
  N : Cat \to sSet
$$

to [[sSet]] sending a category $C$ to

$$
  N(C) : k \mapsto C^{\Delta[k]}
  \,.
$$

Its restriction along $Grpd \hookrightarrow Cat$ to [[groupoids]] lands in [[Kan complexes]] $KanCplx \hookrightarrow $ [[sSet]].

The [[core]] operation is the functor

$$
  Core : Cat \to Grpd
$$

[[right adjoint]] to the inclusion of [[Grpd]] into [[Cat]]. It sends a category to the groupoid obtained by discarding all non-invertible morphisms.

#### The construction

Let $C$ be a [[small category]]. Define

$$
  \mathbf{C} \in [\Delta^{op}, sSet]
$$

by

$$
  \mathbf{C}_k := N(Core(C^{\Delta[k]}))
  \,.
$$

In degree 0 this is the the [[core]] of $C$ itself. In degree 1 it is the groupoid $\mathbf{C}_1$ underlying the [[arrow category]] of $C$.

One sees that the [[source]] and [[target]] functors $s, t : C^{\Delta[1]} \to C$ are [[isofibrations]] and hence their image under [[core]] and [[nerve]] are [[Kan fibrations]]. Therefore it follows that the [[homotopy pullback]] (see there) $\mathbf{C}_1 \times_{\mathbf{C}_0} \cdots \times_{\mathbf{C}_0} \mathbf{C}_1$ is given already be the ordinary [[pullback]] in the [[1-category]] [[Grpd]]. Using this, it is immediate that for all $k$ the functors

$$
  Core(C^{\Delta[k]}) \to Core(C^{\Delta[1]}) \times_{Core(C)} \cdots \times_{Core(C)} Core(C^{\Delta[1]})
$$

are [[isomorphisms]], and so in particular

$$
  \mathbf{C}_k
   \to
  \mathbf{C}_1 \times_{\mathbf{C}_0} \cdots \times_{\mathbf{C}_0} \mathbf{C}_1
$$

is an equivalence.

It is clear that the [[composition]] operation in the complete Segal space defined this way "is" the composition in $C$. In particular the morphisms that are invertible under this composition are precisely those that are already invertible in $C$. Therefore we have the core simplicial object

$$
  Core(\mathbf{C}) : k \mapsto N(Core(C)^{\Delta[k]})
   = N(Core(C))^{\Delta[k]}
  \,,
$$

where, note, now we _first_ take the core of $C$ and then form morphism categories.

This simplicial Kan complex has in each positive degree a [[path space object]] for the [[Kan complex]] $N(Core(C))$.

Notably (since $\Delta[k]$ is [[weak homotopy equivalence|weak homotopy equivalent]] to the point) it follows that indeed all the face and degeneracy maps are weak homotopy equivalences.

So for every category $C$, the simplicial object $\mathbf{C}$ constructed as above is a complete Segal space. This construction extends to a functor $Cat \to completeSegalSpace$ and this is homotopy full and faithful.

#### Properties of the inclusion
 {#PropertiesOfTheInclusionOfCategories}

Write

$$
 Sing_J : Cat \to [\Delta^{op}, sSet]
$$

for the functor just defined

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

For $C$ and $D$ two categories, there are [[natural isomorphisms]]

$$
  Sing_J(C \times D)
  \simeq
  Sing_J(C) \times Sing_J(D)
$$

and

$$
  Sing_J(D^C) \simeq (Sing_J D)^{Sing_J C}
  \,.
$$

A functor $f : C \to D$ is an [[equivalence of categories]] precisely if $Sing_J(f)$ is an equivalence in the [[Reedy model structure]] $[\Delta^{op}, sSet]_{Reedy}$ (hence is degreewise a [[weak homotopy equivalence]] of Kan complexes).

=--

This appears as ([Rezk, theorem 3.7](#Rezk)).

### Relative and Model categories as complete Segal spaces
 {#RelativeCategoryAsCompleteSegalSpace}

Let $C$ be a category with a class $W \subset Mor(C)$ of [[category with weak equivalences|weak equivalences]].  For instance, $C$ could be a [[model category]] or (much) more generally a "[[relative category]]".  Then the [above](#OrdinaryCategoriesAsCompleteSegalSpaces) construction has the following evident variant.

+-- {: .num_prop #RelativeNerve}
###### Definition

Let $N(C,W) \in [\Delta^{op}, sSet]$ be given by
$$ N(C,W) : n \mapsto N(Core_W(C^{\Delta[n]})) \,, $$
where now $Core_W(-)$ denotes the [[subcategory]] on those [[natural transformations]] whose components are [[weak equivalences]] in $C$.

=--

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

The typical [[model category]] is not a [[small category]] with respect to the base choice of [[universe]]. In this case $N(C,W)$ will be a "[[large set|large]]" bisimplicial set. In other words, one needs to employ some [[universe enlargement]] to interpret this definition.

=--

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

If $C$ is a model category, then $Core_W(C^{\Delta[n]})$ is the subcategory of weak equivalences in any of the standard [[model structures on functors]] on $C^{\Delta[n]}$. By a [classical fact](/nlab/show/%28infinity%2C1%29-categorical+hom-space#SpacesOfEquivalences) discssed at _[[(∞,1)-categorical hom-space]]_, its [[nerve]] is a model for the [[core]] of the corresponding [[(∞,1)-category of (∞,1)-functors]].

=--

The bisimplicial set $N(C,W)$ is not, in general, a complete Segal space.  It does, however, represent the same [[(∞,1)-category]] as the [[simplicial localization]] of $C$ at $W$; see [this MO question](http://mathoverflow.net/questions/92916/does-the-classification-diagram-localize-a-category-with-weak-equivalences/93139).

We can, of course, always reflect $N(C,W)$ into a complete Segal space by passing to a [[fibrant replacement]] in the [[model structure for complete Segal spaces]].  But something better is true here: it suffices to make a _[[Reedy model structure|Reedy]] fibrant_ replacement (which does not change the [[homotopy type]] of the simplicial sets $N(Core_W(C^{\Delta[n]}))$, but only "arranges them more nicely").

+-- {: .num_prop}
###### Proposition
Any [[Reedy model structure|Reedy fibrant replacement]] of $N(C,W)$ is a complete Segal space.
=--

This is ([Rezk, theorem 8.3](#Rezk)).

### Quasi-categories as complete Segal spaces
 {#QuasiCategoriesAsCompleteSegal}

The formula $\mathcal{C} \mapsto k \mapsto Core(\mathcal{C}^{\Delta[k]})$ also defines a relative functor from quasi-categories to complete segal spaces,
which has a one-sided inverse $X \mapsto X_{\bullet,0}$. This is the $\Gamma$ appearing in [Joyal & Tierney (2007), proposition 4.10](#JoyalTierney07).

However, to get a right Quillen functor, we need to use a different model of Core.

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

Write

$$
  \Delta_J : \Delta \to sSet
$$

for the [[cosimplicial object|cosimplicial]] [[simplicial set]] that sends $[n]$ to the [[nerve]] of the [[codiscrete groupoid]] on $n+1$ objects

$$
  \Delta_J[n] = N(0 \stackrel{\simeq}{\to} \cdots \stackrel{\simeq}{\to} n)
  \,.
$$

Write

$$
 Sing_J : sSet \to [\Delta^{op}, sSet]
$$

for the functor given by

$$
  Sing_J(X)_n = Hom_{sSet}(\Delta[n] \times \Delta_J[\bullet], X)
  \,.
$$

=--

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

For $X \in sSet$ a [[quasi-category]]/[[inner Kan complex]], $Sing_J(X)$ is a complete Segal space.

=--

See at _[[model structure for dendroidal complete Segal spaces]]_ the section _[Quasi-operads to dendroidal complete Segal spaces](model+structure+for+dendroidal+complete+Segal+spaces#QuasiOperadsToDendroidalCompleteSegal)_

## Related concepts

* [[semi-Segal space]], [[Segal space]]

* [[model structure for complete Segal spaces]]

* [[higher complete Segal space]], [[dendroidal complete Segal space]]

* [[Rezk completion]]

* [[directed homotopy type theory]]

* [[table - models for (infinity,1)-operads]]

## References

### General

Complete Segal spaces were originally defined in

* {#Rezk} [[Charles Rezk]], _A model for the homotopy theory of homotopy theory_ , Trans. Amer. Math. Soc., 353(3), 973-1007 ([arXiv:math/9811037](https://arxiv.org/abs/math/9811037))

The relation to [[quasi-categories]] is discussed in

* {#JoyalTierney07} [[Andre Joyal]], [[Myles Tierney]], _Quasi-categories vs. Segal spaces_, in *Categories in Algebra, Geometry and Mathematical Physics*, Contemporary Mathematics **431** (2007) [[arXiv:math/0607820](http://arxiv.org/abs/math/0607820), [doi:10.1090/conm/431](https://doi.org/10.1090/conm/431)]

Further discussion of the relation to [[simplicial localization]] is in

* {#Bergner07} [[Julia Bergner]], _Complete Segal spaces arising from simplicial categories_ ([arXiv:0704.1624](http://arxiv.org/abs/0704.1624))

A survey of the definition and its relation to equivalent definitions is in section 4 of

* [[Julia Bergner]], _A survey of $(\infty, 1)$-categories_ ([arXiv](http://arxiv.org/abs/math.AT/0610239)).

See also pages 29 to 31 of

* [[Jacob Lurie]], _[[On the Classification of Topological Field Theories]]_

For literature on the variants and refinements see at _[[Theta space]]_ and _[[n-fold complete Segal space]]_.

Related MathOverflow discussion includes

* [Does the classification diagram localize a category with weak equivalences?](http://mathoverflow.net/q/92916/381)

### Groupoidal version
 {#ReferencesGroupoidalVersion}

The groupoidal version of complete Segal spaces (that modelling just [[groupoid objects in an (∞,1)-category]] instead of general [[category objects in an (∞,1)-category]]) is discussed in

* [[Julia Bergner]], _Adding inverses to diagrams encoding algebraic structures_, Homology, Homotopy Appl. 10(2), 2008, 149-174 ([arXiv:math/0610291](http://arxiv.org/abs/math/0610291))

* [[Julia Bergner]], _Adding inverses to diagrams II: Invertible homotopy theories are spaces_, Homology, Homotopy and Applications, vol. 10(1) 2008 ([pdf](http://www.math.ucr.edu/~jbergner/Groupoid.pdf))

[[!redirects complete Segal spaces]]
[[!redirects Rezk category]]
[[!redirects Rezk categories]]

[[!redirects Segal completeness]]

[[!redirects Segal completion]]
[[!redirects Segal completions]]


complete small category

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### Context
#### Category theory
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#### Limits and colimits
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# Complete small categories
* table of contents
{: toc}

## Definition

A **complete small category** (or **small complete category**) is a [[category]] which is both [[small category|small]] (has only a [[set]] of [[objects]] and [[morphisms]]) and [[complete category|complete]] (has a [[limit]] of every small [[diagram]]).

(Note that the phrase "[[small-complete category]]" is sometimes used to mean merely a complete category, i.e. one which is "complete relative to small diagrams".  Therefore, "complete small category" is a safer term for our concept.)

## In classical logic

In the presence of [[classical logic]], complete small categories reduce to [[complete lattices]] by the following theorem due to [Freyd 64, p. 78](#Freyd64).
.

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

If (in some [[universe]] $U$) a small [[category]] $D$ has [[products]] of families of objects whose size is at least that of its [[hom-set|set of morphisms]], then $D$ is a [[preorder]].  In particular, any complete small category is a [[preorder]].

=--

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

Let $x$, $y$ be any two objects, and suppose (contrary to $D$ being a preorder) that there are at least two different morphisms $x \,\underoverset{s}{r}{\rightrightarrows}\, y$. Then the set of morphisms

$$x \to \prod_{f \in Mor(D)} y$$

has [[cardinality]] at least $2^{|Mor(D)|} \gt {|Mor(D)|}$, which is a contradiction.

=--

This basic fact has profound implications for instance for the [[adjoint functor theorem]] and the existence of [[Kan extensions]]. See there for more.

(More generally, we only need to assume the existence of [[powers]], not arbitrary products.)

## In Grothendieck toposes

The [[internal logic]] of a [[topos]] is not, in general, [[classical logic|classical]], so the above proof does not apply when considering [[internal categories]] in a topos.  However, in a [[Grothendieck topos]], one can show that the conclusion still holds by essentially reducing the question to the analogous one in [[Set]].  A brief description of the argument can be found in the answer to [this question](http://mathoverflow.net/questions/43433/small-complete-categories-in-a-grothendieck-topos).

## In more general toposes and constructive mathematics

However, it is possible to have non-preorder complete small categories in non-Grothendieck topoi.  In particular, the [[effective topos]], along with most other [[realizability topos|realizability toposes]], does contain such internal categories, arising for example from the [[modest sets]] or [[partial equivalence relations]].  See [Hyland88](#Hyland88) and [HRR90](#HRR90) for details.

It follows that, in [[constructive mathematics]], it is impossible to prove even that every complete small category in $Set$ or in a Grothendieck topos must be a preorder.

There is a subtlety, however, because in the absence of the [[axiom of choice]], there is a difference between a category being "strongly complete" in the sense of being equipped with limit-assigning [[functors]], and being "weakly complete" in the sense that there merely *exists* a limit for every diagram.  The complete small categories in realizability toposes are only weakly complete, which categorically means that only the [[stack]] completions of their corresponding [[indexed categories]] are complete in the relevant indexed-category sense.

It seems to be unknown whether there exists a (necessarily non-Grothendieck) topos containing a *strongly* complete small category.  However, one of the complete small categories in the effective topos does become strongly complete when the ambient context is restricted to the subcategory of [[assemblies]] rather than the entire effective topos.

## Modelling impredicative type theory

Complete small categories have applications to the modeling of impredicative [[polymorphism]].  Suppose that there is a full subcategory $\mathbf C$ of $\mathbf{Set}$ that is small and complete.  Then we can interpret an impredicatively-quantified type as
\[\llbracket \forall X.T \rrbracket = \prod_{A\in \mathbf{C}} \llbracket T\rrbracket_{A/X}\]
although there are more elaborate formulations that use a subset of this product.

At least at first glance, this requires $\mathbf{C}$ to be *strongly* complete.  Hence it works in the category of assemblies, but not the entire realizability topos.

An alternative approach is to suppose that there is a full [[replete subcategory]] $\mathcal{C}$ of $\mathbf{Set}$ which is complete in the sense of being closed under set-indexed products and equalizers, yet [[essentially small]], i.e. with a small skeleton $\mathbf C$. (Being replete, $\mathcal{C}$ will not itself be small.)  Such a category can be obtained as the repletion of any *weakly* complete *actually* small subcategory, such as exist in realizability toposes.  See [RS04](#RS04) and [MS09](#MS09).  The interpretation of the impredicative $\forall$ is then achieved by a product over the sets in the small skeleton $\mathbf C$. However, to eliminate terms of type $\forall$, one must pick an isomorphism to a set in the skeleton, and be confident that the choice does not matter, and for this reason a relational parametricity is built in to the interpretation of $\forall$. In more detail:
\[\llbracket \forall X.T \rrbracket = \{\pi\in \prod_{A\in \mathbf{C}} \llbracket T\rrbracket_{A/X} | \forall R\subseteq A\times B. \mathcal{R}_R(
\pi(A),\pi(B))\}
\]
where $\mathcal{R}_R$ is a binary logical relation, suitably defined, and
\[
\llbracket (t:\forall X.T)(U)\rrbracket =
gpd(i)(\llbracket t\rrbracket_D)
\]
where $i\colon D \to \llbracket U\rrbracket$ is a bijection with a set in the skeleton $\mathbf C$, and we are using the fact that for any bijection $j\colon C\to D$, there is a functorial action $gpd(j)\llbracket T\rrbracket_{C/X}\to \llbracket T\rrbracket_{D/X}.$
For discussion about whether this works, see the nForum thread for this page.

Note that to model polymorphic simple type theory such as [[System F]], it is not necessary for this purpose that $\mathbf{C}$ be *complete*; all it needs is to have products indexed by its own set of *objects*.  Freyd's theorem does not apply in this case, since there might be fewer objects than morphisms; and indeed there are plenty of categories even classically that have products indexed by their set of objects, e.g. any one-object category.  It is true, however, that (assuming classical logic) no *full subcategory of $Set$* can have products indexed by its own set of objects; this (and a bit more) was shown in [Reynolds 84](#Reynolds84), again by reducing to Cantor's diagonal argument.  On the other hand, to break this argument we do not have to move to realizability; Grothendieck toposes suffice, as shown in [Pitts 87](#Pitts87) via the [[Yoneda embedding]] of a syntactic model into its topos of presheaves.

In contrast, something more like a true complete small category may be necessary to model an impredicative dependent type theory such as the [[calculus of constructions]] with a proof-relevant impredicative sort (traditionally called "Set"), where the impredicative universe admits [[Pi-types]] with *arbitrary* domain.

## Related concepts

* [[adjoint functor theorem]]

## References

Theorem \ref{CompleteSmallCategoriesArePosets}  appears as exercise D in ch.3 (p.78) in:

* {#Freyd64} [[Peter Freyd]], *Abelian Categories*, Harper & Row New York 1964. (Reprinted with author's comment as [TAC reprints no. 3 (2003)](http://www.tac.mta.ca/tac/reprints/articles/3/tr3abs.html))

Reviewed as Thm. 2.1 in:

* [[Michael Shulman]], _Set theory for category theory_ ([arXiv:0810.1279](https://arxiv.org/abs/0810.1279))

The original reference for the complete small categories in the effective topos is:

* {#Hyland88} [[Martin Hyland]], _A small complete category_, Annals of Pure and Applied Logic 40 (1988) ([pdf](https://webdpmms.maths.cam.ac.uk/~martin/Research/Oldpapers/smallcomplete88.pdf))

It was expanded further in the following paper, which discusses the strong/weak completeness issue and the relation to stacks in more detail:

* {#HRR90} Hyland, J. M., Robinson, E. P. and Rosolini, G. (1990), The Discrete Objects in the Effective Topos. Proceedings of the London Mathematical Society, s3-60: 1-36. [doi:10.1112/plms/s3-60.1.1](https://doi.org/10.1112/plms/s3-60.1.1)

There was much related activity around that time, by [[Peter Freyd]], [[Eugenio Moggi]], [[Andrew Pitts]], [[John Reynolds]], Guiseppe Rosolini, and others, including:

* {#Reynolds84} [[John Reynolds]], Polymorphism is not set-theoretic. In: Kahn G., MacQueen D.B., Plotkin G. (eds) Semantics of Data Types. SDT 1984. Lecture Notes in Computer Science, vol 173. Springer, Berlin, Heidelberg

* {#Pitts87} [[Andy Pitts]], Polymorphism is Set Theoretic, Constructively. In Category Theory and Computer Science, Proceedings, Edinburgh 1987, Lecture Notes in Computer Science Vol. 283 (Springer-Verlag, Berlin, 1987), pp 12-39, [PDF](https://www.cl.cam.ac.uk/~amp12/papers/polist/polist.pdf)

A more recent survey of this type of model for polymorphic type theory is in

* Andrea Asperti and Simone Martini, Categorical models of polymorphism, Information and Computation, Volume 99, Issue 1, 1992, Pages 1-79, [DOI](https://doi.org/10.1016/0890-5401(92)90024-A.)

The idea of using a replete complete subcategory which is essentially small in [[constructive set theory|IZF]] is discussed in

* {#RS04} Using Synthetic Domain Theory to Prove Operational Properties of a Polymorphic Programming Language Based on Strictness", Guiseppe Rosolini and [[Alex Simpson]], unpublished, 2004. [ResearchGate](https://www.researchgate.net/profile/Giuseppe_Rosolini/publication/237887060_Using_Synthetic_Domain_Theory_to_Prove_Operational_Properties_of_a_Polymorphic_Programming_Language_Based_on_Strictness/links/0deec53356d020dc85000000/Using-Synthetic-Domain-Theory-to-Prove-Operational-Properties-of-a-Polymorphic-Programming-Language-Based-on-Strictness.pdf)

* {#MS09} Relational parametricity for computational effects, Rasmus Møgelberg and [[Alex Simpson]], Logical Methods in Computer Science, Vol 5 (3:7), 2009. [arxiv](https://arxiv.org/abs/0906.5488).

The interplay with colimits is discussed in

* [[Dusko Pavlovic|Duško Pavlović]], _On completeness and cocompleteness in and around small categories_ , APAL **74** (1995) pp.121-152.

[[!redirects complete small category]]
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[[!redirects small complete categories]]

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complete space

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

# Complete spaces
* table of contents
{: toc}

## Idea

A [[space]] (with "space" taken in a sense relevant to the field of [[topology]]) is _complete_ (or _Cauchy-complete_) if every [[sequence]], [[net]], or [[filter]] that should converge really does [[convergence|converge]].  We identify the sequences, nets, or filters that "should" converge as the _[[Cauchy sequence|Cauchy]]_ ones.

A space that is not complete has "gaps" that may be filled to form its _completion_; it is rather natural to make the space (or equivalently its underlying topological space) [[Hausdorff space|Hausdorff]] at the same time.  Forming the completion of a Hausdorff space is an important example of [[completion]] in the general abstract sense.

## Definitions

A space (which may be a [[metric space]], a [[Cauchy space]], or anything in between) is __Cauchy-complete__, or simply __complete__, if every [[Cauchy filter]] converges, equivalently if every [[Cauchy net]] converges.  A space is __[[sequentially Cauchy-complete]]__ if every [[Cauchy sequence]] converges.  Note that a sequentially complete metric space must be complete (in [[classical mathematics]]), but this does not hold for more general spaces (nor even for metric spaces in [[constructive mathematics]]).

A space is __topologically complete__ if its [[underlying topological space]] is completely [[metrizable space|metrizable]].  There are various other notions related to this.  See [[topologically complete space]].

## Completion

The set $\mathcal{C}X$ of Cauchy filters on a space $X$ may generally be given the same sort of structure as $X$ itself has, and this space will be complete.  Exactly how to do this depends on what structure $X$ is supposed to have, of course, and one can make the general statement false by requiring something artificial as the structure in question, most extremely the structure of being a specific non-complete space.  But it works for most natural categories of spaces.

The general idea is this: every point in $X$ generates a principal [[ultrafilter]] (consisting of those sets to which the point belongs), so there is a natural map from $X$ to $\mathcal{C}X$.  Furthermore, this map is a morphism of the appropriate structure, which in particular makes it [[Cauchy-continuous map|Cauchy-continuous]] (preserving Cauchy filters) and continuous (preserving limits).  So all of the limits in $X$ still exist in $\mathcal{C}X$, but now each Cauchy filter in $X$ (having become both a Cauchy filter in $\mathcal{C}$ and a point in $\mathcal{C}$) has a limit as well.  The additional Cauchy filters based on the additional points in $\mathcal{C}X$ will also have a limit in $\mathcal{C}X$, essentially because $\mathcal{C}$ is a [[monad]] (so a Cauchy filter of Cauchy filters folds into a single Cauchy filter).

There is a problem that $\mathcal{C}X$ is rather larger than necessary; for example, all of the filters that converge to a given point in $X$ (not just the free ultrafilter at that point) exist in $\mathcal{C}X$ and converge to one another.  But you can take a [[quotient space|quotient]] of $\mathcal{C}X$ to make it Hausdorff, obtaining the __Hausdorff completion__ of $X$.  In case $X$ was not Hausdorff to begin with, one can sometimes also force the quotient to leave in just as much redundancy as $X$ has but no more, obtain a straight __completion__ of $X$.  But really, it\'s most natural to make the space Hausdorff at the same time.

+-- {: .standout}
Details to come, if I get around to it.
=--

We have a picture like this, where $X$ is the original space, $\mathcal{H}$ gives a Hausdorff quotient, and an overline indicates completion:
$$ \array {
  &                 & \overline{X} \\
  & \hookrightarrow &              & ↠ \\
X &                 &              &                 & \mathcal{H}\overline{X} \cong \overline{\mathcal{H}X} \\
  & ↠        &              & \hookrightarrow \\
  &                 & \mathcal{H}X
} $$
(Here the arrows are drawn horizontally to put styles on them; they should all be diagonal in the only possible way.)

At least if $X$ is a [[metric space]], then we can also construct its completion as a [[locale]], the [[localic completion]], whose spatial part is the above space, but which in [[constructive mathematics]] may not be spatial.  This is useful to have even if $X$ is already complete.

## Properties

### Relation to compact spaces

A [[compact space]] is necessarily complete.  A space is called __precompact__ if its completion is compact.  For metric spaces (or even [[uniform spaces]]), there is a natural notion of a [[totally bounded space]]; in [[classical mathematics]], we have the theorem that a space is totally bounded if and only if it is precompact.  Similarly, a space is compact if and only if it is both complete and totally bounded (or in [[constructive mathematics]], both complete and precompact).  Thus the purely topological property of compactness is the conjunction of the nontopological properties of completeness and total boundedness.

In some [[constructive analysis|constructive approaches to analysis]] (including most of Brouwer\'s school and some of Bishop\'s school), 'complete and totally bounded' is taken as the *definition* of 'compact', because it holds of examples such as the [[unit interval]] that fail to be compact (in the usual sense) without the [[fan theorem]].  However, in this case, compactness is no longer a [[topological property]]; see [[Bishop-compact space]] for more information.  This is reconciled somewhat with the theory of [[localic completion]], in which a uniform space is totally bounded if and only if its localic completion is compact (in the usual sense).

### Relation to Baire spaces

Every complete metric space is a [[Baire space]].  Since being a Baire space is a [[topological property]], it follows that every [[topologically complete space]] is a Baire space.

There is also a topological property of [[Cech-complete space|Čech-completeness]] that is related  to this; in particular, a metric space is Čech-complete if and only if it is complete, and every Čech-complete space is a Baire space.  In general, we have these proper implications: topologically complete → Čech-complete → Baire.

It is not the case that a complete uniform space (meaning complete with respect to its uniformity) is a Baire space. See MathOverflow [here](https://mathoverflow.net/questions/212308/baire-category-theorem-for-complete-uniform-spaces) for some examples.

## Generalization

When [[Bill Lawvere]] interpreted (in [Lawvere 1973](#Lawvere1973)) [[metric spaces]] as certain [[enriched categories]], he found that a metric space was complete if and only if every [[adjunction]] of [[bimodules]] over the enriched category is induced by an [[enriched functor]].  Accordingly, this becomes the notion of [[Cauchy-complete category]].  (Note that one *must* say 'Cauchy' here, since this is *weaker* than being a [[complete category]], which is based on an incompatible analogy.)

## Related concepts

* a complete [[normed vector space]] is a _[[Banach space]]_

* [[completion of a space]]

## References

*  {#Ark1977} A.V. Arkhangel′skii (1977).  _Complete space_.  Matematicheskaya entsiklopediya.  [Updated English version](https://www.encyclopediaofmath.org/index.php/Complete_space).

*  {#Lawvere1973} [[Bill Lawvere]] (1973).  _Metric spaces, generalized logic and closed categories_.  Reprinted in [[TAC]], 1986.  [Web](http://www.tac.mta.ca/tac/reprints/articles/1/tr1abs.html).

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complete spread

## Idea

A complete spread over a [[locale]] $X$ is a [[map of locales]] $L\to X$
that is in the same relation to an [[etale map]] of [[locales]] $L\to X$
as a [[cosheaf]] of sets over $X$ is to a [[sheaf]] of sets over $X$.

## Definition

The original definition of complete spreads
is due to R. H. Fox in 1957 and is slightly different
from the definition for [[locales]] presented below.

A __spread__ over a [[locale]] $X$ is a [[locally connected locale]] $L$
together with a [[map of locales]] $l\colon L\to X$
such that the connected components of opens $l^*U$ for all opens $U$ in $X$
form a [[base]] for the [[locale]] $L$.

See Proposition 4.3 in [Funk](#Funk) for other equivalent
characterizations of spreads.

Recall that the [[category of elements]]
of the [[cosheaf of connected components]] of $l\colon L\to X$
has as objects pairs $(U,c)$, where $U$ is an open in $X$
and $c$ is a connected component of the open $l^*U$ in $L$.

A __complete spread__ over a [[locale]] $X$ is a spread $l\colon L\to X$
such that the [[unit]] of the adjunction between
[[locally connected locales]] over $X$ and [[cosheaves]] of sets over $X$
given by the [[display locale]] functor and the [[cosheaf of connected components]] functor is an isomorphism.

## Properties

Complete spreads form precisely the [[essential image]]
of the [[display locale]] functor,
which therefore becomes an equivalence
between the category of [[cosheaves]] of sets on a [[locale]] $X$
and the category of [[complete spreads]] over $X$.

The inverse functor is given by the [[cosheaf of connected components]] construction.

## References

* {#Funk} [[Jonathon Funk]], _The display locale of a cosheaf_.

* [[Marta Bunge]], [[Jonathon Funk]], _[[Singular Coverings of Toposes]]_.

[[!redirects spread]]
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complete theory

[[!redirects Complete theory ]]
[[!redirects complete ]]
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### Context
#### Model theory
+-- {: .hide}
[[!include model theory - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

**Deductive completeness** is maximality condition on a [[logical theory]] $\mathbb{T}$ saying that the deductive closure of $\mathbb{T}$ is as large as possible within the bounds of consistency.

## Definition (in classical model theory)

Let $\mathbb{T}$ be a consistent logical theory over signature $\Sigma$. $\mathbb{T}$ is called _complete_ if for any sentence $\sigma$ over $\Sigma$ either $\mathbb{T}\vdash \sigma$ or $\mathbb{T}\cup\sigma$ is inconsistent.

## The case of geometric logic

A [[geometric theory]] $\mathbb{T}$ over a signature $\Sigma$ is called _complete_ if for any geometric sentence $\sigma$ over $\Sigma$ is either $\mathbb{T}$-provably equivalent to $\top$ or to $\bot$, but not both.

+-- {: .num_theorem}
###### Proposition
A geometric theory $\mathbb{T}$ is complete iff the [[classifying topos]] of $\mathbb{T}$ is [[two-valued topos|two-valued]].
=--

This occurs as remark 2.5 in Caramello ([2012](#Ca12)).

Remark: a first-order theory $\mathbb{T}$ is complete in the sense of classical model theory iff its [[coherent logic|Morleyization]] is complete in the sense of geometric logic.

## Related entries

* [[Gödel's incompleteness theorem]]

* [[compactness theorem]]

* [[model complete theory]]

* [[substructure completeness]]

* [[two-valued topos]]

## References

* {#Ca12}[[Olivia Caramello]], _Atomic toposes and countable categoricity_ , Appl. Cat. Struc. **20** no. 4 (2012) pp.379-391. ([arXiv:0811.3547](http://arxiv.org/abs/0811.3547))

* C. C. Chang, [[Jerome Keisler|H. J. Keisler]], _Model theory_ , North-Holland Amsterdam 1973.


complete topological vector space

{:mynumdef: .num_defn style="border:solid #cccccc;border-width:2px 1px;padding:0 1em;margin:0 1em;"}

# Idea

As [[TVS|topological vector spaces]] are [[uniform spaces]], it is appropriate to discuss [[complete uniform space|completeness]].  As with a uniform space, a topological vector space is _complete_ if it has no holes: everything that should be there actually is there.

Where this gets interesting is in the question as to what _should_ be there.  To determine this, one has to have some method of discovering holes.  This is usually done by means of [[Cauchy nets]], but in a given application it may not be necessary that all holes be filled and this leads to weaker notions of completeness.

The more general notions make sense for arbitrary [[topological vector spaces]] but some more refined notions are only used for [[locally convex topological vector spaces]].

# Definitions

In strict order of decreasing strength, we have the following notions of completeness.

+-- {: mynumdef #WeakCplt}
###### Definition
A [[locally convex topological vector space]] is **weakly complete** if it is complete for its [[weak topology]].
=--

+-- {: mynumdef #BCplt}
###### Definition
A [[locally convex topological vector space]] is **$B$-complete** or is a **Ptak space** if a subspace of its [[continuous dual]] is [[weakly closed]] whenever its intersection with any [[equicontinuous]] subset is weakly closed (in the subset).
=--

+-- {: mynumdef #BCplt}
###### Definition
A [[locally convex topological vector space]] is **$B_r$-complete** if a dense subspace of its [[continuous dual]] is [[weakly closed]] whenever its intersection with any [[equicontinuous]] subset is weakly closed (in the subset).
=--

+-- {: mynumdef #Cplt}
###### Definition
A [[topological vector space]] is **complete** if every Cauchy [[net]] converges.
=--

+-- {: mynumdef #QuasiCplt}
###### Definition
A [[topological vector space]] is **quasi-complete** if every [[bornology|bounded]], [[closed]] subset is complete.
=--

+-- {: mynumdef #SeqCplt}
###### Definition
A [[topological vector space]] is **sequentially complete** or **semi-complete** if every [[Cauchy sequence]] [[converges]].
=--

+-- {: mynumdef #LocCplt}
###### Definition
A [[locally convex topological vector space]] is **locally complete** if for $B \subseteq E$ a [[bounded]], [[closed]], [[absolutely convex]] subset then its norm space, $E_B$, is a Banach space.
=--

# Properties
##Sequentially Complete versus Locally Complete
Sequentially complete implies locally complete because every locally Cauchy sequence is a Cauchy sequence. The inverse implication "locally complete" $\Rightarrow$ sequentially complete is true for example in [[metrizable]] locally convex topological vector spaces, but not in general: A Cauchy sequence will not be locally Cauchy in general.

The problem to precisly characterize the spaces in which every convergent sequence is locally convergent is an open problem according to Köthe volume 1 (which is quite old, so it could have been solved in the meantime).

# References

See [[functional analysis bibliography]]

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completely decomposable topology > history

see *[[cd-structure]]*


completely distributive category

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## Definition
 {#Definition}

For [[locally small category]] $\mathcal{K}$ denote by $\mathcal{PK}$ the [[full subcategory]] of $[\mathcal{K}^{op},Set]$ on [[small colimits]] of [[representables]]. The [[category]] $\mathcal{K}$ is [[cocomplete category|cocomplete]] iff the [[Yoneda embedding]] $Y \colon \mathcal{K}\to\mathcal{PK}$ has a [[left adjoint]] $colim \colon \mathcal{PK}\to\mathcal{K}$. If $colim$ has a *further* left adjoint $W \colon \mathcal{K}\to\mathcal{PK}$ and $\mathcal{K}$ is moreover [[complete category|complete]], then it is called *completely distributive*.

## Examples

* Every [[totally distributive category]], in particular every [[presheaf category]] $[C^{op},Set]$ for small $C$ is completely distributive.

## Related pages

* [[totally distributive category]]

* [[completely distributive lattice]]

* Completely distributive categories are an example of [[continuous algebras]] for a [[lax-idempotent 2-monad]]. (But the condition of being completely distributive seems  to be stronger since it also requires completeness.)

## References

* [[Francisco Marmolejo]], [[Bob Rosebrugh]], and [[Richard Wood]], *Completely and totally distributive categories I*, JPAA 216 no. 8-9 (2012).  [PDF](http://www.mta.ca/~rrosebru/articles/ctd.pdf)

[[!redirects completely distributive categories]]


completely distributive lattice

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## Definition

A **completely distributive lattice** is a [[complete lattice]] $L$ in which arbitrary [[joins]] and arbitrary meets [[distributive lattice|distribute]] over each other.

More formally: given a complete lattice $L$ and functions $p: J \to I$ and $f: J \to L$, we have

$$\bigwedge_{i \in I} \bigvee_{j \in p^{-1}(i)} f(j) \geq \bigvee_{sections\; s: I \to J} \bigwedge_{i \in I} f(s(i))$$

where "section" means [[section]] of $p$. Complete distributivity states that this inequality is an equality, for all $f, p$. The same statement then holds upon switching $\bigwedge$ and $\bigvee$, i.e., complete distributivity is a [[duality|self-dual]] property.

## Properties

### Algebraic lattices

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

The category of [[Alexandroff locales]] is equivalent to that of completely distributive [[algebraic lattice|algebraic lattices]].
=--

This appears as remark 4.3 in ([Caramello 2011](#Caramello11)).

### Completely distributive Boolean algebras

+-- {: .num_lemma}
###### Lemma
A complete totally ordered poset is completely distributive.
=--

+-- {: .proof}
###### Proof
(Note: this uses the [[axiom of choice]].) Suppose we had a strict inequality

$$\bigwedge_{i \in I} \bigvee_{j \in p^{-1}(i)} f(j) \gt \bigvee_{sections\; s: I \to J} \bigwedge_{i \in I} f(s(i)).$$

Denote the left side by $x$ and the right by $y$. Either there is no element $z$ strictly between $x$ and $y$, or there is. In the former case, we have for each $i$ that $\bigvee_{j \in p^{-1}(i)} f(j) \geq x$, and so (using trichotomy) we have $f(j) \gt y$ for some $j \in p^{-1}(i)$. Choosing such a $j$ for each $i$, we obtain a section $s$ with $f(s(i)) \gt y$ for all $y$, whence $f(s(i)) \geq x$ for this case, so that $\bigvee_{sections\; s: I \to J} \bigwedge_{i \in I} f(s(i)) \geq x \gt y$, contradiction. If there is $z$ with $x \gt z \gt y$, we argue similarly to obtain a section $s$ with $f(s(i)) \gt z$ for all $i$, whence $\bigwedge_{i \in I} f(s(i)) \geq z$, and we obtain a contradiction as before with $z$ replacing $x$.
=--

+-- {: .num_prop}
###### Proposition
A complete Boolean algebra $B$ is completely distributive iff it is atomic (a [[CABA]]), i.e., is a [[power set]] as a Boolean algebra.
=--

+-- {: .proof}
###### Proof
For a complete atomic Boolean algebra $B$, it is classical that the canonical map $B \to P(atoms(B))$, sending each $b \in B$ to the set of [[atoms]] below it, is an isomorphism. Such [[power sets]] are products of copies of $\mathbf{2} = \{0 \leq 1\}$, which is completely distributive by the lemma, and [[products]] of completely distributive lattices are completely distributive.

In the other direction, suppose $B$ is completely distributive. Take $p = \pi_2: \mathbf{2} \times B \to B$, and $\alpha: \mathbf{2} \times B \to B$ by $\alpha(0, b) \coloneqq \neg b$ and $\alpha(1, b) \coloneqq b$. Sections of $p$ correspond to functions $g: B \to \mathbf{2}$, and so complete distributivity gives

$$1 = \bigwedge_{b \in B} (b \vee \neg b) = \bigvee_{g: B \to \mathbf{2}} \bigwedge_{b \in B} \alpha(g(b), b).$$

Write $a(g)$ as abbreviation for $\bigwedge_{b \in B} \alpha(g(b), b)$, we have

$$b = b \wedge 1 = b \wedge \bigvee_{g: B \to \mathbf{2}} a(g) = \bigvee_{g: B \to \mathbf{2}} b \wedge a(g)$$

so $b \wedge a(g) \neq 0$ for some $g$ if $b \neq 0$. Notice that $b \wedge a(g) \neq 0$ implies $g(b) = 1$, from which we infer two things:

* Whenever $b \leq a(g)$ with $b \neq 0$, then $a(g) \leq \alpha(g(b), b) = \alpha(1, b) = b$; therefore $a(g)$ is an atom whenever $a(g) \neq 0$;

* Provided that $b \wedge a(g) \neq 0$, the preceding point gives that $a(g)$ is an atom below $b \wedge a(g) \leq b$.

The last point shows every element $b$ has an atom $a(g)$ below it, so that $B$ is atomic, as was to be shown.
=--

### Constructive complete distributivity

A complete lattice $A$ is called **constructively completely distributive** (CCD) if the join-assigning morphism $D A \to A$ has a left adjoint, with $D A$ the poset of [[downsets]].

_Constructive complete distributivity is equivalent to complete distributivity if and only if the [[axiom of choice]] holds_ ([Wood&Fawcett (1990)](#WoodFawcett)).

Constructively completely distributive lattices are an example of [[continuous algebras]] for a [[lax-idempotent 2-monad]].

### Remark

Completely distributive lattices correspond to tight [[Galois connections]] (Raney 1953). This generalizes to a correspondence between totally distributive toposes and [[essential geometric morphism|essential localizations]] (Lucyshyn-Wright 2011).

CCD lattices are precisely the [[nuclear objects]] in the category of complete lattices.

The (bi-) category $\mathfrak{CCD}$ with CCD lattices and sup-preserving maps is the [[Karoubi envelope|idempotent splitting]] of the (bi-) category of relations $\mathfrak{Rel}$. This plays an important role in _[[domain]]-theoretical_ approaches to the semantics of [[linear logic]].

## Related concepts

* [[continuous lattice]]

* [[totally distributive category]]

## Links

* wikipedia-entry: [completely distributive lattice](http://en.wikipedia.org/wiki/Completely_distributive_lattice)

## References

* [[Olivia Caramello]], _A topos-theoretic approach to Stone-type dualities_  , ms. 2011. ([arXiv:1103.3493](http://arxiv.org/abs/1103.3493))
 {#Caramello11}

* {#WoodFawcett} [[Barry Fawcett]], [[Richard J. Wood]], *Constructive complete distributivity I* , Math. Proc. Camb. Phil. Soc. **107** (1990) 81-89 [[doi:10.1017/S0305004100068377](https://doi.org/10.1017/S0305004100068377)]

* R. Guitart, J. Riguet, _Enveloppe Karoubienne de Catégories de Kleisli_ , Cah. Top. Geom. Diff. Cat. **XXXIII** no.3 (1992) pp.261-266. ([pdf](http://archive.numdam.org/article/CTGDC_1992__33_3_261_0.pdf))

* {#LW11}[[Rory Lucyshyn-Wright|R. Lucyshyn-Wright]], _Totally Distributive Toposes_ , arXiv.1108.4032 (2011). ([pdf](http://arxiv.org/pdf/1108.4032v3))

* G. N. Raney, _Tight Galois Connections and Complete Distributivity_ , Trans.Amer.Math.Soc **97** (1960) pp.418-426. ([pdf](http://www.ams.org/journals/tran/1960-097-03/S0002-9947-1960-0120171-3/S0002-9947-1960-0120171-3.pdf))

* R. Rosebrugh, R. J. Wood, _Constructive complete distributivity IV_ , App. Cat. Struc. **2** (1994) pp.119-144. ([preprint](http://www.mta.ca/~rrosebru/articles/ccd4.pdf))

* [[Isar Stubbe|I. Stubbe]], _Towards "Dynamic Domains": Totally Continuous Complete Q-Categories_ , Theo. Comp. Sci. **373** no.1-2 (2007) pp.142-160.

[[!redirects completely distributive lattices]]
[[!redirects constructive completely distributive lattice]]
[[!redirects constructively completely distributive lattice]]
[[!redirects constructively completely distributive lattices]]
[[!redirects constructive completely distributive lattices]]
[[!redirects CCD lattice]]


completely prime filter

\tableofcontents

## Definition

Recall that a [[filter]] $F$ on a [[lattice]] $L$ is called __[[prime filter|prime]]__ if $\bot \notin F$ and, whenever $x \vee y \in F$, then $x \in F$ or $y \in F$.  In other words, for every [[finite set|finite index set]] $I$, $x_k \in F$ for some $k$ whenever $\bigvee_{i\colon I} x_i \in F$.

We now generalise from finitary [[joins]] to arbitrary joins:  A filter $F$ on a [[complete lattice]] $L$ is __completely prime__ if, for any index set $I$ whatsoever, $x_k \in F$ for some $k$ whenever $\bigvee_{i\colon I} x_i \in F$.  Equivalently, a completely prime filter is given by a simultaneous [[suplattice]] and lattice [[homomorphism]] from $L$ to the lattice $TV$ of [[truth values]] (which is classically the [[boolean domain]] $\mathbb{2}$).

In particular, if $L$ is a [[frame]], then a completely prime filter of $L$ is given by a frame homomorphism from $L$ to $TV$.  Thinking of $L$ as a [[locale]], this is the same as a [[point of a locale|point]] of $L$.

## See also

* [[locale]]

* [[locale of real numbers]]

[[!redirects completely prime filter]]
[[!redirects completely prime filters]]


completeness theorem

# Contents
* toc
{: toc}

## Idea

A formal system is *complete* or *semantically complete* when all its [[tautologies]] are [[theorems]].  Since tautologies are defined by reference to "all models", this can depend on the class of models under consideration.

## Examples

* [[Kurt Gödel]] showed that [[classical mathematics|classical]] [[first-order logic]] is complete, that is, any first-order sentence, $\phi$, which is true under any interpretation, is provable from the axioms of first-order logic. This is generally known as **the completeness theorem**.  Here the "interpretations" are the models in [[sets]], assigning a set to every sort and a binary [[truth value]] $\{\bot,\top\}$ to every proposition.

* If in an [[infinitary logic]], infinitely many quantifiers may appear in sentences, then completeness does not hold.

* In [[constructive mathematics]], Gödel's proof is not available, and (perhaps depending on definitions) constructive logic may not be complete with respect to ordinary models in sets.  However, there are still completeness theorems with respect to models in arbitrary [[categories]], which generally proceed by way of construction of a [[syntactic category|syntactic]] model in which the true statements are precisely the theorems.

## Related entries

* [[Deligne completeness theorem]]

* [[incompleteness theorem]]

[[!redirects Gödel-Henkin completeness theorem]]
[[!redirects Gödel completeness theorem]]


completing the square

Given a [[field]] $K$ whose [[characteristic]] is not 2, let $p\in K[x]$ be a [[polynomial]] over $K$ with value $p = \tfrac{1}{2} a x^2 - b x$. Then

$$
  \tfrac{1}{2} a x^2 - b x
  =
  \tfrac{1}{2} a(x - \tfrac{b}{a})^2 - \tfrac{b^2}{2a}
$$

## Related concepts

* [[quadratic form]]

* [[Gaussian integral]]

* [[quadratic function]]

##  References

* Wikipedia, _[Completing the square](https://en.wikipedia.org/wiki/Completing_the_square)_


completion

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

# Completions
* table of contents
{:toc}

## Idea

The general idea of completion is that given an object of some sort $C$, we construct from it a larger object, containing $C$ as a subobject, which moreover has certain properties that $C$ might have lacked.  This larger object is then called the _completion_ of $C$ with respect to these properties.

## Completion and idempotence

Outside of category theory (see below), "completion" is usually used mostly for *idempotent* operations.  That is, the completion of a thing with respect to some property is itself *complete* with respect to that property, and the completion process doesn't change an object that is already complete.  From the perspective of [[stuff, structure, property]] this makes sense because we complete with respect to a *property*.  Namely, according to the yoga of SSP, given some ambient category $K$, the subcategory of objects having a property $P$ should be a *full* subcategory, whose inclusion [[functor]] forgets that property.  A "completion" operation is then usually a left (or, occasionally, right) [[adjoint functor|adjoint]] to that inclusion functor, making the category of objects with property $P$ [[reflective subcategory|reflective]] (or coreflective).  Note that a [[full and faithful functor]] with a left adjoint is always [[monadic functor|monadic]] and the monad is [[idempotent monad|idempotent]].

By contrast, when we add *structure* to objects, i.e. we consider an adjoint to a forgetful functor which is faithful but not necessarily full, we usually do not use the word "completion" but rather the word "free".  For example, the category of complete [[metric spaces]] is a full reflective subcategory of the category of metric spaces, and the reflection is called "completion."  By contrast, the forgetful functor from the category of groups to the category of sets has a left adjoint, but we call that left adjoint the "free group on a set" rather than the "completion of a set into a group."

Additionally, we tend to use the term 'completion' only for a [[faithful functor|faithful]] [[reflector]].  Note that the reflector is faithful if and only if the unit of the reflection is [[monomorphism|monic]]; we might even want to require it to be a [[regular monomorphism]].  For example, the left adjoint of the forgetful functor from [[abelian groups]] to groups is called 'abelianisation' and may even be called 'free abelian group on a group' but is not normally called 'abelian completion' or anything like that. (The tendency in such cases is rather to resort to suffixes like "-ification" and "-ization": sheafification, abelianization, localization.) This fits with the intuition that a completion "adds more stuff" to the object we started with in order to "make it complete".

## List of completions

* [[complete space|Cauchy completion]] of a [[metric space]], or more generally a [[uniform space]]

* [[Dedekind completion]] of a [[linear order]] (or sometimes a more general [[preorder]] or [[quasiorder]])

* [[completion of a ring]]

* [[MacNeille completion]] of a [[partially ordered set]]

* [[profinite completion of a group|profinite completion]] of a discrete group

* [[Grothendieck group]] of a cancellative commutative [[monoid]]; [[group completion]] of a cancellative monoid

* More generally, [[group completion]] of a monoid, which may end up being the [[trivial group]].

* Field of fractions of an [[integral domain]]

* The [[Stone-Cech compactification]] of a [[Tychonoff space]]

* [[Moore closure]] operators (monads) on posets or preorders, e.g., the subgroup generated by a subset of a group, the ideal generated by a subset of a ring, etc.

+-- {: .query}
David: It's not clear to me when in the 'List of completions' we have examples of enriched category completions: Cauchy completion of a metric space (yes) of a uniform space (no?), Dedekind completion of a linear order (yes for 2-enriched categories?).

[[Mike Shulman]]: Although orders are 2-enriched categories, Dedekind completion of an order is not a categorical completion, at least not in the sense of adding limits or colimits.  That would be the construction of downsets or ideals.

Cauchy completion of a metric space is, of course, an instance of Cauchy completion of enriched categories.  I believe that Cauchy completion of a uniform space is actually also an instance of a general categorical notion of Cauchy completion, but in the more general setting of an [[equipment]] (namely, the equipment of sets and filters).  See "Categorical interpretation" at [[uniform space]] for a too-brief summary of this point of view.

[[David Corfield]]: so is there are clear-cut distinction of level in all these cases, where we can separate completions of (enriched) categories (equipment) from completions of sets (with properties and structures)?

[[Mike Shulman]]: I think the reason that completion of metric spaces and uniform spaces feels more like the completion of sets (than categories) is that they are basically (enriched) (0,1)-categories.  If the question is where to separate this list from the one below, I would put completions of anything enriched in a preorder up here, and completions of things enriched non-posetally below.
=--

## Free completion and lax-idempotence

The general contrast between "completion" for adding a property and "free" for adding structure applies to operations on [[category|categories]] as well.  For instance, since [[split idempotents]] are preserved by any functor, the 2-category of categories with split idempotents is a full sub-2-category of [[Cat]].  Therefore, it makes perfect sense to call the left adjoint of its inclusion a "completion," in this case the (Set-enriched) [[Cauchy completion]] or [[Karoubi envelope]].  By contrast, the forgetful functor from the 2-category of [[monoidal categories]] to $Cat$ forgets structure, rather than properties, so its left adjoint should be called a "free" construction rather than a "completion."

However, in this case there is an intermediate notion between properties and structure, namely [[property-like structure]].  Intuitively, property-like structure is "structure which, when it exists, is essentially unique" or "a property which is not necessarily preserved by morphisms."  The canonical example is the existence of [[limits]] and [[colimits]] in a category: when they exist they are unique up to unique canonical isomorphism (so that "having limits" is intuitively a property of a category), but not every functor preserves limits, so the forgetful functor from  categories-with-limits to categories is not full.  (Left) adjoints to functors which forget property-like structure are usually called *free completions*.  The monads they induce are not idempotent, but they are often [[lax-idempotent monad|lax-idempotent]] or colax-idempotent.  (For example, when we freely add limits to a category that already had limits, the old limits are no longer limits and the new ones take their place.)

Property-like structure is most common in category theory, but it does occur elsewhere as well.  For instance, a [[monoid]] is a [[semigroup]] with property-like structure: a semigroup has at most one identity element, but that identity element is not necessarily preserved by semigroup homomorphisms.  Thus the adjoining of a new formal identity element to a semigroup (which is not an idempotent operation) might be called its "free completion into a monoid".  Notice that if the original semigroup is already a monoid, then its free completion into a monoid will be a *different* monoid (with a new identity); this shows the distinction between free completion and mere completion.

## List of completions in category theory

### Idempotent completions

The following completions add a *property* in the strictest sense, and as such are idempotent.

* [[Cauchy complete category|Cauchy completion]] of an enriched category: addition of all absolute limits.
* [[exact completion]] of a [[regular category]] (the "ex/reg" completion)
* [[extensive category|extensive]] completion
* [[pretopos]] completion of a [[coherent category]]

### Non-idempotent completions

These completions add a property-like structure, are often lax-idempotent or colax-idempotent.

* [[free cocompletion]]: the [[presheaf category]] of a [[small category]] is its free cocompletion under all small colimits.
* [[free completion]]: the dual (the opposite of the free cocompletion of the opposite).
* Ind-completion: the "free cocompletion under [[filtered colimits]]," consisting of all [[ind-object|ind-objects]].
* Pro-completion: the dual (free completion under cofiltered limits, the opposite of the Ind-completion of the opposite), consisting of all [[pro-objects]].
* [[ideal completion]]: the free cocompletion under filtered colimits of monomorphisms.

Note that these completions take small categories to large categories.  Free completions and cocompletions of large categories can be obtained using categories of [[accessible functor|accessible presheaves]].  However, if we only want to add limits and/or colimits for a given *set* of diagram shapes, the free completion process will preserve smallness.  For example we have:

* the free completion under finite products or coproducts
* the free completions under finite limits or colimits

All these sorts of completions work just as well in an [[enriched category|enriched]] setting, as long as the enriching category $V$ is nice enough ([[locally presentable category|locally presentable]] is certainly enough).

Some other non-idempotent completions:

* The [[reg/lex completion]] and [[ex/lex completion]] of a category with finite limits.

## Non-unique completions
{#nonunique}

The completions above are mostly (all?) [[the|unique in the relevant sense]]: unique up to unique isomorphism for objects of [[groupoids]] (or [[categories]]), more generally with a [[contractible space|contractible]] $\infty$-[[infinity-groupoid|groupoid]] of completions.

However, there are other sorts of completions in mathematics, such as:

*  'The' [[algebraic closure]] of a [[field]] is unique up to isomorphism, but the isomorphism is not unique. This is an example of an [[injective hull]].
*  Every [[model theory|model]] of a [[first-order theory]] may be interpreted as giving a completion of that theory (but not one with a recursively enumerable axiomatisation); every statement that is true in the model is an axiom of the complete theory.  However, this is far from unique.
*  An [[ultrapower]] of a structure may be viewed as an elementary completion (see [[elementary embedding]]), as described by [[Terry Tao]] in [a blog post](http://terrytao.wordpress.com/2010/11/27/nonstandard-analysis-as-a-completion-of-standard-analysis/).

## Related concepts

* [[completion monad]]

* [[completion of a ring]], [[completion of a module]]

* [[analytic completion]]

* [[free completion]] (of a [[small category]])

## References

* {#Sullivan70} [[Dennis Sullivan]], pp.9 in _Localization, Periodicity and Galois Symmetry_ (The 1970 MIT notes) edited by [[Andrew Ranicki]], K-Monographs in Mathematics, Dordrecht: Springer ([pdf](http://www.maths.ed.ac.uk/~aar/surgery/gtop.pdf))

* [[Marta Bunge]], _Tightly Bounded Completions_ , TAC **28** no. 8 (2013) pp.213-240. ([pdf](http://www.tac.mta.ca/tac/volumes/28/8/28-08.pdf))

[[!redirects completion]]
[[!redirects completions]]


completion monad

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

#Contents#
* table of contents
{:toc}

## Idea

The _completion monad_ is a [[monad (in computer science)]] which is used for _exact_ computations with [[real numbers]], in the sense of [[constructive analysis]]. It has been used for [[certified programming]] with guaranteed exactness of real number approximations.

## References

The completion monad was introduced in

* [[Russel O'Connor]], _A Monadic, Functional Implementation of Real Numbers_ MSCS, 17(1):129-159, 2007 ([arXiv:cs/0605058](http://arxiv.org/abs/cs/0605058))

and implemented in [[Coq]] in

* [[Russel O'Connor]], _Certified Exact Transcendental Real Number Computation in Coq_ In TPHOLs 2008, volume 5170 of LNCS, pages 246261, 2008.

For more see

* Robbert Krebbers, [[Bas Spitters]], _Type classes for efficient exact real arithmetic in Coq_ ([arXiv:1106.3448](http://arxiv.org/abs/1106.3448/))


completion of a group

> Not to be confused with *[[group completion]]*.

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

#Contents#
* table of comtents
{:toc}

## Idea

(...)

## Definition

\begin{definition}\label{RCompleteGroup}
**($R$-complete group)**
\linebreak
For $R$ a [[solid ring]], a [[group]] $G$ is called *$R$-complete* if it has a [[central series]]

$$
  G
  \,=\,
  G_0
    \supset
  G_1
    \supset
  \cdots
    \supset
  G_k
  \,=\,
  1
$$

(i.e. such that under [[commutator subgroups]] $[-,-]$ we have $[G, G_i] \subset G_{i+1}$)

such that each [[quotient group]] $G_j/G_{j+1}$ (which is an [[abelian group]] by the previous property) carries the [[mathematical structure|structure]] of an $R$-[[module]].
\end{definition}

\begin{proposition}\label{CompletionOfAGroup}
**(completion of a group)**
\linebreak
Given a [[solid ring]] $R$ and any group $G$, there is a [[universal construction]] of an $R$-complete group $G \xrightarrow{\;\;} G_{\widehat{R}}$ -- the *$R$-completion* of $G$ -- given by the [[limit]]

$$
  G_{\widehat{R}}
  \;\simeq\;
  \underset{
    \underset{
      { G \to C \in }
      \atop
      { Grp^{G/}_{/R CompGrp} }
    }{\longleftarrow}
  }{\lim}
  \;C\;
$$

over the [[codomain]] [[functor]] from the [[comma category]] of $R$-complete groups under $G$.
\end{proposition}

More generally:

\begin{definition}
**(fiberwise $R$-complete group)**
\linebreak
A [[short exact sequence]] of [[groups]]

\[
  \label{AShortExactSequenceOfGroups}
  1
  \to
  N
  \longrightarrow
  \widehat{G}
  \longrightarrow
  G
  \to
  1
\]

is *fiberwise* or *relative* $R$-complete if $N$ is so (in the sense of Def. \ref{RCompleteGroup}).
\end{definition}

\begin{proposition}\label{FiberwiseCompletionOfAGroup}
**(fiberwise completion of a group)**
\linebreak
Given a [[solid ring]] $R$ and any [[short exact sequence]] of [[groups]] (eq:AShortExactSequenceOfGroups), there is a [[universal construction]] of a fiberwise $R$-complete exact sequence
$$
 \array{
  1
  &\to&
  N
  &\longrightarrow&
  \widehat{G}
  &\longrightarrow&
  G
  &\to&
  1
  \\
  &&
  \big\downarrow
  &&
  \big\downarrow
  &&
  \big\downarrow
  \mathrlap{{}^{=}}
  \\
  1
  &\to&
  N_{\widehat{R}}
  &\longrightarrow&
  \widehat{G}_{\widehat{R}_G}
  &\longrightarrow&
  G
  &\to&
  1
 }
$$
\end{proposition}

([Bousfield & Kan 1971, §2](#BousfieldKan71))

## Properties

### Special cases

\begin{example}\label{ExampleProfiniteCompletion}
**([[profinite completion]])**
\linebreak
For $R = \mathbb{Z}/p$ (a [[cyclic group]] of [[prime number|prime]] [[order of a group|order]]) and applied to [[finitely generated groups]], $R$-completion coincides with $p$-[[profinite completion of a group|profinite completion]].
\end{example}

\begin{example}\label{ExampleMalcevCompletion}
**([[Malcev completion]])**
\linebreak
For $R = \mathbb{Q}$ (the [[rational numbers]]) and applied to [[nilpotent groups]], $R$-completion coincides with [[Malcev completion]].
\end{example}

(e.g. [Bousfield & Kan 1972, p. 99](#BousfieldKan72))

## Related concepts

* [[completion of a ring]]

* [[completion of a space]]

  * [[p-completion]], [[p-adic homotopy theory]]

  * [[rationalization]], [[rational homotopy theory]]

* [[solid ring]]

## References

Original articles:

* {#BousfieldKan71} [[Aldridge Bousfield]], [[Daniel Kan]], §2 of: *Localization and completion in homotopy theory*, Bull. Amer. Math. Soc. **77** 6 (1971) 1006-1010 [[doi:10.1090/S0002-9904-1971-12837-9](https://doi.org/10.1090/S0002-9904-1971-12837-9), [pdf](https://www.ams.org/journals/bull/1971-77-06/S0002-9904-1971-12837-9/S0002-9904-1971-12837-9.pdf)]

* {#BousfieldKan72} [[Aldridge Bousfield]], [[Daniel Kan]], Chapter IV of: _[[Homotopy limits, completions and localizations]]_, Lecture Notes in Mathematics, **304** Springer (1972) [[doi:10.1007/978-3-540-38117-4](https://doi.org/10.1007/978-3-540-38117-4)]

Further developments:

* [[Carles Casacuberta]], [[An Descheemaeker]], *Relative group completions*, Journal of Algebra, **285** 2 (2005) 451-469 [[doi:10.1016/j.jalgebra.2004.10.023](https://doi.org/10.1016/j.jalgebra.2004.10.023)]

[[!redirects completions of a group]]
[[!redirects completion of groups]]
[[!redirects completions of groups]]

[[!redirects R-completion of a group]]
[[!redirects R-completions of a group]]
[[!redirects R-completion of groups]]
[[!redirects R-completions of groups]]

[[!redirects complete group]]
[[!redirects complete groups]]

[[!redirects R-complete group]]
[[!redirects R-complete groups]]


completion of a module

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Algebra
+-- {: .hide}
[[!include higher algebra - contents]]
=--
#### Topology
+-- {: .hide}
[[!include topology - contents]]
=--
#### Formal geometry
+-- {: .hide}
[[!include formal geometry -- contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Directly analogous to the concept of [[completion of a ring]] is the _completion of a module_ over that ring.

In particular the [[formal completion]] or _adic completion_ of a ring $A$ at an ideal $\mathfrak{a}$ has a corresponding analog for modules. Where the adic completion $A_{\mathfrak{a}}^\wedge$ of the ring itself has the geometric interpretation of forming the [[formal neighbourhood]] $Spf(A_{\mathfrak{a}}^\wedge)$ of [[spectrum of a commutative ring|ring spectra]] $Spec(A/\mathfrak{a}) \hookrightarrow Spec(A)$, so under the interpretation (see [here](module#RelationToVectorBundlesInIntroduction)) of $A$-[[modules]] as [[bundles]] over $Spec(A)$, the $\mathfrak{a}$-adic completion $N_{\mathfrak{a}}^\wedge$ of an $A$-module $N$ has the interpretation of being the restriction of that bundle to that formal neighbourhood.

## Definition

### For modules over a commutative ring

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

For $A$ a [[commutative ring]], $\mathfrak{a} \subset A$ an ideal in $A$ and for $N$ an $A$-[[module]], then the _$\mathfrak{a}$-adic completion_ or _formal completion at $\mathfrak{a}$_ of $N$ is the [[filtered limit]]

$$
  N^{\wedge}_{\mathfrak{a}}
  \coloneqq
  \underset{\leftarrow}{\lim}_n
  N/(\mathfrak{a}^n N)
$$

of [[quotients]] of $N$ by the submodules induced by all powers of the ideal.

=--

There is a canonical projection map $N \longrightarrow N^\wedge_{\mathfrak{a}}$. Its [[kernel]] is sometimes called the $\mathfrak{a}$-[[adic residual]].

### For $\infty$-modules over an $E_\infty$-ring

Let $A$  be an [[E-∞ ring]] and $\mathfrak{a} \subset \pi_0 A$ a [[generators and relations|finitely generated]] ideal of its underlying [[commutative ring]].

#### Torsion approximation
 {#TorsionApproximation}

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

An $A$-[[∞-module]] $N$ is an _$\mathfrak{a}$-torsion module_ if for all elements $n \in \pi_k N$ and all elements $a \in \mathfrak{a}$ there is $k \in \mathbb{N}$ such that $a^k n = 0$.

=--

([Lurie "Completions", def. 4.1.3](#LurieCompletions)).

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

The [[full sub-(∞,1)-category]]

$$
  A Mod_{\mathfrak{a}tor}
  \hookrightarrow
  A Mod
$$

is [[reflective sub-(∞,1)-category|co-reflective]] and the co-reflector $\Pi_{\mathfrak{a}}$ -- the _[[torsion approximation]]_ -- is [[smashing localization|smashing]].

=--

([Lurie "Completions", prop. 4.1.12](#LurieCompletions)).

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

For $N \in A Mod_{\leq 0}$ then [[torsion approximation]], prop. \ref{TorsionApproximation}, intuced a [[monomorphism]] on $\pi_0$

$$
  \pi_0 \Pi_{\mathfrak{a}} N \hookrightarrow \pi_0 N
$$

including the $\mathfrak{a}$-nilpotent elements of $\pi_0 N$.

=--

([Lurie "Completions", prop. 4.1.18](#LurieCompletions)).

#### Localization
 {#Localization}

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

An $A$-[[∞-module]] $N$ is an _$\mathfrak{a}$-local module_ if for every $\mathfrak{a}$-torsion module $T$ (def. \ref{TorsionInfinityModule}), the [[derived hom space]]

$$
 Hom_A(T,N) \simeq \ast
$$

is contractible.

=--

([Lurie "Completions", def. 4.1.9](#LurieCompletions)).

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

For $\mathfrak{a} = (a)$ generated from a single element, then the [localization of an (∞,1)-ring](localization+of+a+commutative+ring#ForEInfinityRings)-map $A \to A[a^{-1}]$ is given by the [[(∞,1)-colimit]] over the sequence of right-multiplication with $a$

$$
  A[a^{-1}]
  \simeq
  \underset{\rightarrow}{\lim}
  (
    A \stackrel{\cdot a}{\longrightarrow} A
    \stackrel{\cdot a}{\longrightarrow} A
    \stackrel{\cdot a}{\longrightarrow}  \cdots
  )
  \,.
$$

=--

([Lurie "Completions", remark 4.1.11](#LurieCompletions))

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

The [[full sub-(∞,1)-category]]

$$
  A Mod_{\mathfrak{a}loc}
  \hookrightarrow
  A Mod
$$

of [[∞-modules]] [[localization of a module|local]] away from $\mathfrak{a}$ is [[reflective sub-(∞,1)-category|reflective]]. The reflector

$$
  \Pi_{\mathfrak{a}dR} \colon A Mod \longrightarrow A Mod_{\mathfrak{a}loc}
$$

is called _[[localization of a module|localization]]_.

=--

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

There is a [[natural transformation|natural]] [[homotopy fiber sequence]]

$$
  ʃ_{\mathfrak{a}} \longrightarrow id \longrightarrow ʃ_{\mathfrak{a}dR}
$$

relating $\mathfrak{a}$-torsion approximation on the left with $\mathfrak{a}$-[[localization of a module|localization]] on the right.

=--

#### Completion

+-- {: .num_defn #InfinityCompletion}
###### Definition/Proposition

An [[∞-module]] $N$ over $A$ is _$\mathfrak{a}$-complete_ if for all $\mathfrak{a}$-local $\infty$-modules $L$ (def. \ref{LocalInfinityModule}) then $Hom_A(L,N) \simeq \ast$.

The [[full sub-(∞,1)-category]]

$$
  A Mod_{\mathfrak{a}comp}
  \hookrightarrow
  A Mod
$$

of the [[(∞,1)-category of ∞-modules]] on the $\mathbb{a}$-complete ones is a [[reflective sub-(∞,1)-category]]. The reflector

$$
  \flat_{\mathfrak{a}} \colon A Mod \longrightarrow A Mod_{\mathfrak{a}comp} \hookrightarrow A Mod
$$

$$
  N \mapsto N^{\wedge}_{\mathfrak{a}}
$$

is called _$\mathfrak{a}$-completion_.

=--

([Lurie "Completions", def. 4.2.1, lemma 4.2.2](#LurieCompletions)).

Definition \ref{InfinityCompletion} relates to the traditional definition, def. \ref{TraditionalDefinition}, as follows

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

Let $N$ a homotopically discrete [[∞-module]] over the [[E-∞ ring]] $A$ which is a [[Noetherian module]] in that all its submodules are finitely [[generators and relations|finitely generated]]. Then the $\mathfrak{a}$-completion of $N$ in the sense of def. \ref{InfinityCompletion} coincides with the traditional definition def. \ref{TraditionalDefinition}.

=--

([Lurie "Completions", prop. 4.3.6](#LurieCompletions))

## Properties

### General properties

The [[full sub-(∞,1)-category]] $A Mod_{\mathfrak{a} comp}$ is a [[locally presentable (∞,1)-category]].

([Lurie "Completions", prop. 4.1.17](#LurieCompletions))

### Monoidalness
 {#Monoidalness}

We discuss how both $\mathfrak{a}$-completion $\flat_{\mathfrak{a}}$ and $\mathfrak{a}$-[[torsion approximation]] $\Pi_{\mathfrak{a}}$ on $A Mod$ are [[monoidal (∞,1)-functors]] with respect to the [[smash product of spectra]] over $A$.

Let $A$  be an [[E-∞ ring]] and $\mathfrak{a} \subset \pi_0 A$ a [[generators and relations|finitely generated]] ideal of its underlying [[commutative ring]].

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

The completion reflection $\flat_{\mathfrak{a}}$, def. \ref{InfinityCompletion}, is a [[monoidal (∞,1)-functor]].

=--

([Lurie "Completions", remark 4.2.6](#LurieCompletions)).

For the torsion approximation functor $\Pi_{\mathfrak{a}}$ one gets something slightly weaker, it preserves "monoids without unit":

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

The [[full sub-(∞,1)-category]] of $\mathfrak{a}$-torsion modules, def. \ref{TorsionInfinityModule}, is [[reflective sub-(∞,1)-category|co-reflective]]

$$
  A Mod_{\mathfrak{a}tor}
  \stackrel{\hookrightarrow}{\underset{\Pi_{\mathfrak{a}}}{\longleftarrow}}
  A Mod
 \,.
$$

Moreover, the coreflector $\Pi_{\mathfrak{a}}$ is "[[smashing localization|smashing]]", in that there is $V \in A Mod$ such that $\Pi_{\mathfrak{a}}(-) \simeq V \wedge (-)$ is given by the [[smash product]] with $V$. If $\mathfrak{a} = (\{x_i\}_i) $ then $V$ is the [[tensor product]] $V =\underset{i}{\otimes} V_i$ over all the [[homotopy fibers]]

$$
  \Omega (A[x_i^{-1}]/A) \longrightarrow A \longrightarrow A[x_i^{-1}]
  \,.
$$

=--

([Lurie "Completions", prop. 4.1.12](#LurieCompletions)).

From the general properties of [[smashing localization]] it follows that

+-- {: .num_cor}
###### Corollary

The coreflection $\Pi_{\mathfrak{a}} \colon A Mod \to A Mod$

1. preserves small [[(∞,1)-colimits]];

1. is a "[[monoidal (∞,1)-functor]]" except possibly for preservation of units.

=--

See also ([Lurie "Completions", cor. 4.1.16](#LurieCompletions)).

### Relation to localization

The [[homotopy cofiber]] of $\mathfrak{a}$-completion $\Pi_{\mathfrak{a}}$ is [[localization of a module|localization]] away from $\mathfrak{a}$, in that there is a [[homotopy fiber sequence]]

$$
  (-)_{\mathfrak{a}}^{\wedge} \to id \to (-)[\mathfrak{a}^{-1}]
$$

with the completion functor of def. \ref{InfinityCompletion} on the left and the localization functor of prop. \ref{LocalizationAwayByColimit} on the right.

([Lurie "Completions", example 4.1.14, remark 4.1.20](#LurieCompletions))

### Relation of formal completion to torsion approximation
 {#RelationToTorsionApproximation}

For suitable ideals $\mathfrak{a}\subset A$ of a commutative ring $A$ or more generally of an [[E-∞ ring]], then the [[derived functor]] of $\mathfrak{a}$-adic [[completion of a module|completion of A-modules]] forms together with $\mathfrak{a}$-[[torsion approximation]] an [[adjoint modality]] on the
[[(∞,1)-category of modules]] over $A$. See at _[[fracture square]]_ for details.

[[!include arithmetic cohesion -- table]]

## References

Discussion in the context of [[higher algebra]] is in

* {#LurieCompletions} [[Jacob Lurie]], section 4 of _[[Proper Morphisms, Completions, and the Grothendieck Existence Theorem]]_

Discussion of formal completion of [[(infinity,1)-modules]] in terms of [[totalization]] of [[Amitsur complexes]] is in

* {#Carlsson07} [[Gunnar Carlsson]], _Derived completions in stable homotopy theory_, Journal of Pure and Applied Algebra Volume 212, Issue 3, March 2008, Pages 550–577 ([arXiv:0707.2585](http://arxiv.org/abs/0707.2585))

[[!redirects completions of a module]]

[[!redirects completion of modules]]
[[!redirects completions of modules]]

[[!redirects formal completion of a module]]


completion of a ring

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Algebra
+-- {: .hide}
[[!include higher algebra - contents]]
=--
#### Topology
+-- {: .hide}
[[!include topology - contents]]
=--
#### Formal geometry
+-- {: .hide}
[[!include formal geometry -- contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Ideas

### General

Quite generally, the _completion of a ring_ is a [[completion]] of a [[topological ring]] to a [[complete space|complete]] [[topological ring]], when possible, for instance of a [[normed ring]] to a [[Banach ring]].

### Formal/adic completion and formal neighbourhoods

A special case of ring completion is the _formal completion_ or _adic completion_ of a [[commutative ring]] $R$, which is its topological completion with respect to the [[adic topology]] induced by a [[maximal ideal]] $I\subset R$ ([Sullivan 05, definition 1.3](#Sullivan05)).

The underlying ring $\widehat R_I$ of this formal completion is the [[limit]]

$$
  \widehat R_I \coloneqq \underset{\leftarrow}{\lim} (R/I^n)
$$

(formed in the [[category]] [[CRing]] of [[commutative rings]]) of the [[quotients]] of $R$ by all the [[powers]] of this ideal, $I$ ([Sullivan 05, proposition 1.13](#Sullivan05)). Notice that this may be considered purely algebraically.

In words, this limit construction says that the elements of $R_I$ are sequences of elements in $R$ which "successively add smaller and smaller elements, as seen by the ideal $I$". This is as for [[formal power series]] rings, which  are indeed the archetypical example of formal completions, see example \ref{FormalPowerSeries} below.

Generally, the [[Isbell duality|dual]] [[geometry|geometric]] meaning of formal ring completion is in [[formal geometry]]: the proper [[spectrum (geometry)|geometric spectrum]] of a formally completed ring is known as a _[[formal spectrum]]_ $Spf(R,I)$. Geometrically this is the [[formal neighbourhood]] of the [[spectrum of a commutative ring|spectrum]] $Spec(R/I)$ inside $Spec(R)$.

$$
  Spec(R/I)
  \hookrightarrow
  Spf(\widehat R_I)
  \hookrightarrow
  Spec(R)
  \,.
$$

### Derived completion functor

The [[derived functor]] of adic completion was originally discussed in ([Greenless-May 92](#GreenleesMay92) ("Greenlees-May duality"). For discussion of its relation to derived [[torsion subgroup]] functor see ([Porta-Shaul-Yekutieli 10](#PortaShaulYekutieli10)) and see at _[fracture theorem -- Arithmetic fracturing for chain complexes](http://ncatlab.org/nlab/show/fracture+theorem#CompletionAndTorsionOnDerivedCategories)_.

## Examples
 {#Examples}

The archetypical example which most clearly exhibits the [[geometry|geometric]] meaning of formal completions of rings is the following

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

For $R$ any [[ring]] and $R[x]$ the [[polynomial ring]] with [[coefficients]] in $R$, then the formal completion of $R[x]$ at the ideal $(x)$ generated by the free generator $x$ is the ring of [[formal power series]] $R[ [x] ]$.

If $R$ is a [[field]], then geometrically $Spec(R[x])$ is the [[affine line]] in [[algebraic geometry]]/[[arithmetic geometry]] over $R$, while $Spf(R[ [x] ])$ is the [[formal disk]] inside the affine line around the origin.

=--

More generally:

\begin{example}\label{CompletionOfNoetherianRingByPowerSeries}
**([[completion of a ring|completion]] of [[Noetherian ring]] by [[power series]])** \linebreak
For $R$ a [[Noetherian ring]] and $I = (a_1, \cdots, a_n)$ a finitely generated [[ideal]], the completion of $R$ at $I$ is [[isomorphism|isomorphic]] to the [[power series ring]] over $R$ on the $a_i$, in that

$$
  R\hat{_{I}}
  \;\simeq\;
  R
  [ [ x_1, \cdots, x_n  ] ]
  /
  \big(
    x_1 - a_1, \cdots, x_n - a_n
  \big)
  \,.
$$

\end{example}

(e.g. [[The Stacks Project]] [Lemma 0316](https://stacks.math.columbia.edu/tag/0316), [Buchholtz 08, Sec. 6.4](Atiyah-Segal+completion+theorem#Buchholtz08))

The key class of example of completions in [[non-archimedean analytic geometry]] is the following.

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

The [[p-adic integers]] are the completion of the ring of [[integers]] at the [[prime ideal]]  $(p) \subset \mathbb{Z}$.
Similarly the [[p-adic rational numbers]] are the completion of the [[rational numbers]] $(p)$, and the [[p-adic complex numbers]] are the completion of the [[complex numbers]] at $(p)$.

=--

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

In view of the example \ref{FormalPowerSeries} one sees that the $p$-adic numbers in \ref{pAdicNumbers} are in fact [[analogy|analogous]] to [[formal power series]] rings, hence that they behave like [[function rings]] on [[formal disks]] in _some_ kind of geometry.

This [[analogy]] is part of what is known as the [[function field analogy]], which says that the ring of [[integers]] $\mathbb{Z}$ behaves like the  would-be "ring of polynomials $\mathbb{F}_1[x]$ over [[F1]]".

=--

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

The [[Atiyah-Segal completion theorem]] states that for $G$ a [[topological group]] and $X$ a [[G-space]], then the [[topological K-theory]] ring $K(X\sslash G)$ of the [[homotopy quotient]] $X \sslash G$ (of the [[Borel construction]]) is the completion of the $G$-[[equivariant K-theory]] ring of $X$.

So this says that the "very naive" equivariant K-theory embodied by $K(X  \sslash G)$ is an infinitesimal approximation to the genuine equivariant K-theory.

=--

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

The phenonemon of example \ref{AtiyahSegalTheorem} appears for other [[generalized cohomology theories]] too, such as [[complex cobordism]] ([GreenleesMay 97](#GreenMay97)). For [[complex oriented cohomology theories]] it says that the [[formal group]] assigned by these to $\ast  \sslash U(1)$ is to be thought of as the formal completion of a more globally defined something.  One case where this "something" has been understood in some detail is that where the cohomology theory is [[elliptic cohomology]]. In that case the analog of equivariant K-theory is _[[equivariant elliptic cohomology]]_, see there for more details.

=--

## Properties

### Relation of formal completion to torsion approximation
 {#RelationToTorsionApproximation}

For suitable ideals $\mathfrak{a}\subset A$ of a commutative ring $A$, then the [[derived functor]] of $\mathfrak{a}$-adic [[completion of a module|completion of A-modules]] forms together with $\mathfrak{a}$-[[torsion approximation]] an [[adjoint modality]] on the
[[(∞,1)-category of modules]] over $A$. See at _[arithmetic fracturing for chain complexes](fracture+theorem#CompletionAndTorsionOnDerivedCategories)_ for details.

### As modality in arithmetic cohesion

[[!include arithmetic cohesion -- table]]

## Related concepts

* [[analytic completion]]

* [[localization of a ring]], [[localization of a commutative ring]]

* [[completion of a module]]

* [[formal scheme]]

## References

A classical account is in section 1 of

* {#Sullivan05} [[Dennis Sullivan]], _Geometric topology: localization, periodicity and Galois symmetry_, volume 8 of K- Monographs in Mathematics. Springer, Dordrecht, 2005. The 1970 MIT notes, Edited and with a preface
by [[Andrew Ranicki]] ([pdf](http://www.maths.ed.ac.uk/~aar/books/gtop.pdf))

Brief surveys include

* Wikipedia, _[Completion (ring theory)](https://en.wikipedia.org/wiki/Completion_%28ring_theory%29)_

* {#Bridges11} Brett Bridges, _An introduction to ring completions_, lecture notes 2011 ([pdf](http://www.math.uwo.ca/~srankin/courses/4123/2011/brett_presentation.pdf))

Discussion of the [[derived functor]] of adic completion ("Greenless-May duality") is in

* {#GreenleesMay92} [[John Greenlees]], [[Peter May]], _Derived functors of I-adic completion and local homology_, J. Algebra 149 (1992), 438–453 ([pdf](http://math.uchicago.edu/~may/PAPERS/73.pdf))

* Leovigildo Alonso, Ana Jeremías, [[Joseph Lipman]], _Local Homology and Cohomology on Schemes_ ([arXiv:alg-geom/9503025](http://arxiv.org/abs/alg-geom/9503025))

* {#PortaShaulYekutieli10} Marco Porta, Liran Shaul, [[Amnon Yekutieli]], _On the Homology of Completion and Torsion_ ([arXiv:1010.4386](http://arxiv.org/abs/1010.4386))

for more on this see also at _[fracture theorem -- Arithmetic fracturing for chain complexes](http://ncatlab.org/nlab/show/fracture+theorem#CompletionAndTorsionOnDerivedCategories)_

Discussion of interrelation between completion and [[etale morphisms]] is in

* Leovigildo Alonso, Ana Jeremias, Marta Perez, _Local structure theorems for smooth maps of formal schemes_ ([arXiv:math/0605115](http://arxiv.org/abs/math/0605115))

Discussion of formal completion of [[(infinity,1)-modules]] in terms of [[totalization]] of [[Amitsur complexes]] is in

* {#Carlsson07} [[Gunnar Carlsson]], _Derived completions in stable homotopy theory_ ([arXiv:0707.2585](http://arxiv.org/abs/0707.2585))

Completion for complex cobordism theory is in

* {#GreenMay97} [[John Greenlees]], [[Peter May]], _Localization and Completion Theorems for MU-Module Spectra_, Annals of Mathematics Second Series, Vol. 146, No. 3 (Nov., 1997), pp. 509-544

[[!redirects completion of a ring]]
[[!redirects completion of rings]]
[[!redirects completions of a ring]]
[[!redirects completions of rings]]
[[!redirects completion (ring theory)]]

[[!redirects completion of a commutative ring]]
[[!redirects completion of commutative rings]]

[[!redirects formal completion of a ring]]
[[!redirects formal completion of rings]]

[[!redirects complete ring]]
[[!redirects complete rings]]


completion of a space

* [[Cauchy complete space|Cauchy completion]]

* [[p-adic completion]], [[p-adic homotopy theory]]

* [[rationalization]], [[rational homotopy theory]]

## Related concepts

* [[completion of a group]]

category: disambiguation

[[!redirects completion of spaces]]


completion theorem for complex cobordism cohomology

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

As with the [[Atiyah-Segal completion theorem]] for [[equivariant K-theory]], similar results have been developed for [[equivariant complex cobordism cohomology theory]] ([Löffler 74](#Loffler74), see [Greenlees-May, theorem 1.1](#Greenlees-May)), and more generally for [[MU]]-[[module spectrum|modules]].

For instance, for certain classes of [[groups]], $G$, the [[cohomology ring]] $MU(B G)$ is the [[completion of a ring|completion]] of $MU_G(\ast)$ at its [[augmentation ideal]].

[[!include Segal completion -- table]]

## References

The proof is due to

* {#Loffler74} Peter Löffler, _Bordismengruppen unitärer Torusmannigfaltigkeiten_, Manuscripta Math. 12 (1974), 307-327 ([doi:10.1007/BF01171078](https://doi.org/10.1007/BF01171078))

* G. Comezaña, [[Peter May]], _A completion theorem in complex cobordism. In J. P. May, et al. Equivariant homotopy and cohomology theory. NSF-CBMS Regional Conference Series in
Mathematics No. 91. 1996

Detailed review is in

* {#GreenleesMay} [[John Greenlees]], [[Peter May]], _Localization and Completion Theorems for MU-Module Spectra_, ([pdf](http://www.math.uchicago.edu/~may/PAPERS/gmjuly.pdf))


complex

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

#Contents#
* table of contents
{:toc}

## Definition

In a [[additive category]] [[category with translation|with translation]] $T : C \to C$ a **complex** is a [[differential object]]

$$
  d_X : X \to T X
$$

such that

$$
  X \stackrel{d_X}{\to} T X \stackrel{T d_X}{\to} T T X
$$

is the [[zero morphism]].

## Examples

* A complex in the category $Gr(C)$ of [[graded object]]s in an [[additive category]] $C$ is called a [[chain complex]].

* For $d_X : X \to T X$ a complex, the shifted [[differential object]] $ d_{T X} : T X \stackrel{-T(d_X)}{\to}  T T X$ is again a complex.

## Related concepts

* [[chain complex]]

## References

For instance section 11 of

* [[Masaki Kashiwara]], [[Pierre Schapira]], _[[Categories and Sheaves]]_,

[[!redirects complexes]]