condensed E-infinity ring

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## Idea

By a *condensed $E_\infty$ ring* one will generally mean an [[E-infinity ring|$E_\infty$]] in [[condensed mathematics]], hence [[internalization|internal to]] [[condensed infinity-groupoid|condensed $\infty$-groupoids]].

## Definition

A **condensed $E_\infty$-ring** is a [[(infinity,1)-sheaf]] of [[E-infinity-rings|$E_\infty$-rings]] on the [[pro-étale (infinity,1)-site|pro-étale (\infty,1)-site]] of the point, small relative to a [[universe]] $\mathcal{U}$.

## See also

* [[E-infinity ring]]

* [[condensed mathematics]]

* [[condensed infinity-groupoid]]

* [[condensed spectrum]]

* [[condensed module spectrum]]

[[!redirects condensed E-infinity rings]]


condensed infinity-groupoid

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## Idea

The original motivation behind condensed infinity-groupoids is to create well-behaved categories of mathematical structures such as condensed [[HZ]]-[[module spectra]] in which one could do derived analytic geometry using category-theoretic methods without resorting to not-so-well behaved categories of topological spaces.

However, [[David Corfield]] offers another motivation for condensed infinity-groupoids in the blog post [Pyknoticity vs Cohesiveness](https://golem.ph.utexas.edu/category/2020/03/pyknoticity_versus_cohesivenes.html#c057721) on the nCafé, as a way to extend [[cohesion]] to the $p$-adic case (see at [[condensed cohesion]]):

> Mathematics rather than physics is the subject of chapter 5 of my book, where I’m presenting [[cohesive homotopy type theory|cohesive HoTT]] as a means to gain some kind of conceptual traction over the vast terrain that is modern geometry. However I’m aware that there are some apparent limitations, problems with ‘$p$-adic’ forms of cohesion, cohesion in algebraic geometry, and so on. In the briefest note (p. 158) I mention the closely related pyknotic and condensed approaches of, respectively, (Barwick and Haine) and (Clausen and Scholze).

## Definition

A **condensed infinity-groupoid** or **condensed anima** is a [[hypercomplete (infinity,1)-topos | hypercomplete]] [[(infinity,1)-sheaf]] of [[infinity-groupoids]] on the [[pro-étale (infinity,1)-site]] of the point, small relative to a universe $\mathcal{U}$.

Equivalently, it is an [[(infinity,1)-functor]]

$$T:ProfiniteSpaces^\op \longrightarrow \infty Grpd_\mathcal{U}$$

from the [[opposite category|opposite]] [[(infinity,1)-category]] of [[profinite spaces]] to the [[(infinity,1)-category]] [[Infinity-Grpd]] of [[infinity-groupoids]] which are small relative to a [[universe]] $\mathcal{U}$, such that the natural maps

$$T(\mathbb{0}) \to \mathbb{1}$$

and

$$T(S + S^{'}) \to T(S) \times T(S^{'})$$

are [[homotopy equivalences]] for any profinite space $S$ and $S^{'}$, and the natural map

$$T(S) \to  \operatorname{lim}_{\Delta} T(S_\bullet)$$

is a [[homotopy equivalence]] for any [[hypercover]] of profinite spaces $S_\bullet \to S$.

When restricted to the site of [[extremally disconnected topological space | extremally disconnected]] compact Hausdorff spaces, hyperdescent is automatic, so the simplest equivalent definition of a condensed infinity-groupoid is simply a finite-product-preserving [[(infinity,1)-functor]] $T:ExtrDisc^\op \longrightarrow \infty Grpd_\mathcal{U}$, see ([Barwick-Haine, corollary 2.4.4](#BarwickHaine19)).

## Properties

The [[(infinity,1)-category]]
$\mathrm{Cond}\infty\mathrm{Grpd}$ of condensed infinity-groupoids is a locally small [[locally cartesian closed (infinity,1)-category|locally cartesian closed]] [[(infinity,1)-pretopos]].

[[Peter Scholze]] speculates in [this comment on the nCafé](https://golem.ph.utexas.edu/category/2020/03/pyknoticity_versus_cohesivenes.html#c057721) that $\mathrm{Cond}\infty\mathrm{Grpd}$ seem to form a [[predicative elementary (infinity,1)-topos]]. If true, then $\mathrm{Cond}\infty\mathrm{Grpd}$ is an example of an [[elementary (infinity,1)-topos]] which is not a [[Grothendieck (infinity,1)-topos]].

## Terminology

In the model of [[infinity-groupoids]] as [[simplicial sets]]/[[Kan complexes]], condensed infinity-groupoids are sometimes referred to as **condensed anima**.

## Related concepts

* [[condensed mathematics]]
* [[condensed set]]
* [[condensed spectrum]]
* [[condensed E-infinity ring]]
* [[condensed module spectrum]]

## References

* {#Corfield20} [[David Corfield]], _Modal homotopy type theory_, Oxford University Press 2020 ([ISBN: 9780198853404](https://global.oup.com/academic/product/modal-homotopy-type-theory-9780198853404))

* {#BarwickHaine19} [[Clark Barwick]], [[Peter Haine]], _Pyknotic objects, I. Basic notions_, ([arXiv:1904.09966](https://arxiv.org/abs/1904.09966)

See [[condensed mathematics]] and [[infinity-groupoid#Anima]].

[[!redirects condensed infinity-groupoids]]
[[!redirects condensed anima]]
[[!redirects condensed ∞-groupoids]]
[[!redirects condensed ∞-groupoid]]


condensed local contractibility

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## Idea
 {#Idea}

The typical [[gros topos|gros]] -[[toposes]] and -[[(∞,1)-toposes|$\infty$-toposes]] considered in [[arithmetic geometry|arithmetic]]/[[algebraic geometry]] are far from being [[cohesive toposes|cohesive]] over [[Set]] and even farther from being [[cohesive (infinity,1)-topos|$\infty$-cohesive]] over [[Infinity-Grpd|$Grpd_\infty$]], respectively: The extra [[left adjoint]] to their [[terminal geometric morphism]] would mean that [[varieties]]/[[schemes]] were [[locally connected topos|locally connected]] or even [[locally infinity-connected (infinity,1)-topos|locally $\infty$-connected]], which they are not.

(This is in contrast to the situation in [[differential geometry]], where  [[smooth manifolds]], [[formal smooth manifolds]], and [[supermanifolds]], etc.,  generate the [[cohesive (infinity,1)-topos|cohesive $\infty$-toposes]] over [[Infinity-Grpd|$Grpd_\infty$]] of [[smooth infinity-groupoids|smooth $\infty$-groupoids]], [[formal smooth infinity-groupoids|formal smooth $\infty$-groupoids]] and [[super formal smooth infinity-groupoid|super formal smooth $\infty$-groupoids]], etc.)

But [[cohesion]] may usefully be considered over non-trivial [[base toposes]] or [[base (infinity,1)-topos|base $\infty$-toposes]]. For example, the [[slice (infinity,1)-topos|slice $\infty$-topos]] of [[global equivariant homotopy theory]] over the archetypical $G$-[[orbi-singularity]] is cohesive over proper $G$-[[equivariant homotopy theory]] (see at *[[cohesion of global- over G-equivariant homotopy theory]]*).

{#AlmostCohesionInIntroduction} Analogously, the [[gros topos]] of [[pro-étale topos|pro-étale]] [[arithmetic geometry|arithmetic]]/[[algebraic geometry]], while not cohesive over [[sets]], has at least an extra [[left adjoint]] over the base topos of [[condensed sets]].
This and the analogous statement for the respective [[hypercomplete (infinity,1)-topos|hypercomplete $\infty$-toposes]] is claimed in [Scholze 2020](#Scholze2020), see also [Scholze a](#Scholzea).

In view of the terminology of "[[condensed mathematics]]" one might call this local contractibility of pro-étale arithmetic geometry relative to condensed sets: _condensed local contractibility_.

## Statement
 {#Statement}

> under construction

\begin{definition}
Given

* $k$ a separably closed field,

* $\kappa$ an uncountable [[strong limit cardinal]]

write

* $Sch^{\leq \kappa}_{/Spec(k)}$ for the category of $\kappa$-small [[schemes]] over $Spec(k)$,

* $Sch^{\leq \kappa}_{/_{pet} Spec(k)}$ for its [[full subcategory]] of those that are [[pro-étale morphism of schemes|pro-étale]] over $Spec(k)$,

  [[equivalence of categories|equivalently]] this is the category of $\kappa$-small [[profinite sets]],

both regarded as [[sites]] via the [[pro-étale site|pro-étale topology]], and write

\[
  \label{TheGeometricMorphism}
  Sh\big( Sch^{\leq \kappa}_{/Spec(k)}\big)
  \underoverset
    {\underset{\Gamma}{\longrightarrow}}
    {\overset{LConst}{\longleftarrow}}
    {\;\; \bot \;\;}
  Sh\big( Sch^{\leq \kappa}_{/_{pet}Spec(k)}\big)
\]

for the canonical [[geometric morphism]] between their [[sheaf toposes]].
\end{definition}

\begin{remark}
\label{TerminologyOfCondensedSets}
  The [[base topos]] in (eq:TheGeometricMorphism) is also called the category of $\kappa$-*[[condensed sets]]*, or of $\kappa$-[[pyknotic sets]] (with $\kappa$ often omitted in the latter case).
\end{remark}

The following seems to be claimed:
\begin{prop}
  The geometric morphism (eq:TheGeometricMorphism) has the following properties:

* $LConst$ is [[fully faithful functor|fully faithful]];

* $\Gamma$ is a [[left adjoint]]

and

* $LConst$ is a [[right adjoint]].

\end{prop}

The first two claims are fairly standard, the last one would make the topos of pro-étale schemes be [[locally connected topos|locally connected]] and hence *almost* [[cohesive topos|cohesive]] [[base topos|over]] $\kappa$-[[condensed sets]]/$k$-[[pyknotic sets]] (by Rem. \ref{TerminologyOfCondensedSets}).

## Structures in condensed local contractibility
 {#StructuredInCondensedCohesion}

The point of axiomatic [[cohesion]] is that it formally implies the presence of structures of the form of key phenomena known in traditional [[differential topology]], but now [[internalization|internal]] into any given [[cohesive topos]]/[[cohesive (infinity,1)-topos|$\infty$-topos]]. Hence to the extent that [[pro-étale topos|pro-étale]] [[arithmetic geometry]] is cohesive over the [[pro-étale topos|pro-étale]] [[base topos]] it inherits these structures.

There would be much to be discussed here. The following lists some first observations with links to further commentary.

### Differential cohomology diagram

Every [[spectrum object]] in a [[locally infinity-connected (infinity,1)-topos|locally $\infty$-connected $\infty$-topos]] sits in the center of a *[[differential cohomology diagram]]*, which may be understood as decomposing the object  into its various [[modality|modal]] aspects, such that, at least for [[smooth infinity-groupoids|smooth $\infty$-groupoids]], it exhibits each object as representing a [[Whitehead-generalized cohomology theory|Whitehead-generalized]] [[differential cohomology theory]].

Since the construction of the differential cohomology hexagon does not require the extra left adjoint to [[preserved limit|preserve]] [[finite products]], it should exist in [[pro-étale topos|pro-étale]] [[arithmetic geometry]]. Commentary in this direction is in [Scholze b](#Scholzeb).

## References

The idea of the possibility of pro-étale local contractibility is voiced in discussion here:

* [[Marc Hoyois]], [[Urs Schreiber]], *étale ∞-site* (around [nForum comment 43431](https://nforum.ncatlab.org/discussion/5473/etale-site/?Focus=43431#Comment_43431)), November 2013

The claim of cohesion-except-for-product-preservation of gros [[pro-étale toposes]] over the corresponding pro-étale base topos appears in:

* {#Scholze2020} [[Peter Scholze]], *[Re: Pyknoticity versus Cohesiveness](https://golem.ph.utexas.edu/category/2020/03/pyknoticity_versus_cohesivenes.html#c057631)*, March 2020

Related remarks and discussion:

* {#Scholzea} [[Peter Scholze]], _Answer to 'What is the precise relationship between pyknoticity and cohesiveness?'_, April 2020 ([MO:https://mathoverflow.net/a/356836](https://mathoverflow.net/a/356836/447)).

* {#Scholzeb} [[Peter Scholze]], _Answer to 'Cohesion relative to a pyknotic/condensed base'_, Feb 2021 ([MO:a/384900](https://mathoverflow.net/a/384900/381)).

* {#Scholzec} [[Peter Scholze]], _Answer to 'Condensed / pyknotic approach to orbifolds?'_, Feb 2021 ([MO:a/384898](https://mathoverflow.net/a/384898/381))

More concrete discussion, phrased in terms of [[fractured (infinity,1)-topos|fractured $(\infty,1)$-toposes]]:

* [[Qi Zhu]], *Fractured Structure on Condensed Anima*, MSc thesis, Bonn (2023) [[pdf](https://1429cecd-24a0-4223-8b7c-1ebf47aa12e2.filesusr.com/ugd/8e912a_fe1c2f8e90094d0fa065fb8129687963.pdf), [[QiZhu-FracturedCondensed.pdf:file]]]

exposition:

* [[Qi Zhu]], *Fractured structure on condensed spaces*, talk notes (2023) [[pdf](https://1429cecd-24a0-4223-8b7c-1ebf47aa12e2.filesusr.com/ugd/8e912a_8d2a90176ed14969a877e77ec1b787da.pdf), [[QiZhuFracturedCondensed.pdf:file]]]

* [[Nima Rasekh]], *What is a topological structure?*, talk notes (April 2023) [[pdf](https://guests.mpim-bonn.mpg.de/rasekh/Condensed.pdf), [[Rasekh-TopologicalStructure.pdf:file]]]

[[!redirects condensed cohesion]]


condensed mathematics

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## Idea
 {#Idea}

**Condensed mathematics** aims to provide a more convenient framework in which to treat [[algebra|algebraic]] objects which are equipped with a [[topological space|topology]], such as [[topological abelian groups]] or [[topological vector spaces]]. Related aims are to turn [[functional analysis]] into a branch of [[commutative algebra]], and various types of [[analytic geometry]] into [[algebraic geometry]].

For instance, the [[category]] of [[topological abelian groups]] is not [[abelian category|abelian]], which means for instance that it lacks a good theory of [[exact sequences]].
But the category of [[condensed abelian groups]] is [[abelian category|abelian]] and it contains topological abelian groups as a [[full subcategory]] (under appropriate conditions, see [ScholzeLCM](#ScholzeLCM), Proposition 1.7).

According to [[Peter Scholze]] in [this comment on the nCafé](https://golem.ph.utexas.edu/category/2020/03/pyknoticity_versus_cohesivenes.html#c057780), current expositions of condensed mathematics rely heavily on the [[axiom of choice]] and choice-like axioms such as the [[presentation axiom]], so it is unknown how much they still hold in the context of more [[constructive mathematics]] without choice.

## Concepts

### Condensed objects

* [[condensed set]]

* [[condensed abelian group]]

* [[condensed ring]]

* [[condensed module]]

#### Solid objects

* [[solid abelian group]]

* [[solid module]]

* [[solidification functor]]

* [[solid spectrum]]

#### Liquid objects

* [[liquid vector space]]

* [[liquid module]]

#### Other

* [[trace-class map]]

* [[nuclear module]]

* [[analytic ring]]

### Higher condensed objects

* [[condensed infinity-groupoid]]

* [[condensed object in an (infinity,1)-category]]

* [[condensed spectrum]]

* [[condensed E-infinity ring]]

* [[condensed module spectrum]]

## Applications

The theory of solid modules was used to formulate the correct theory of $\ell$-adic sheaves for the geometrization of the [[local Langlands correspondence]] by Fargues and Scholze ([FarguesScholze21](#FarguesScholze21)). It has also been used by Mann to formalize the [[six operations]] for [[rigid analytic geometry]] ([Mann22](#Mann22)).

Condensed mathematics has also been used in [ClausenScholze22](#ClausenScholze22) to provide new "analysis-free" proofs of the finiteness of [[coherent cohomology]], [[Serre duality]], [[GAGA]], and the [[Hirzebruch-Riemann-Roch theorem]] in [[complex analysis]].

## Related entries

* [[bornological topos]]

* [[pyknotic set]]

* [[analytic geometry]]

## References

* {#ScholzeLCM} [[Peter Scholze]], _Lectures on condensed mathematics_, [pdf](https://www.math.uni-bonn.de/people/scholze/Condensed.pdf)

* {#ScholzeLAG} [[Peter Scholze]], _Lectures on analytic geometry_, [pdf](https://www.math.uni-bonn.de/people/scholze/Analytic.pdf)

* {#ClausenScholze22}[[Dustin Clausen]], [[Peter Scholze]], _Condensed Mathematics and Complex Geometry_, [pdf](https://people.mpim-bonn.mpg.de/scholze/Complex.pdf)

* {#DustinScholze20}[[Dustin Clausen]], [[Peter Scholze]], _Masterclass in condensed mathematics,_ [YouTube playlist](https://www.youtube.com/playlist?list=PLAMniZX5MiiLXPrD4mpZ-O9oiwhev-5Uq), [website](https://www.math.ku.dk/english/calendar/events/condensed-mathematics/) (including pdf notes)

[[formal proof|Formalization]]/[[certified programming|verification]] in [[proof assistants]] ([[Lean]]):

* [[Peter Scholze]], *[Liquid tensor experiment](https://xenaproject.wordpress.com/2020/12/05/liquid-tensor-experiment/)*, December 2020

* [[Peter Scholze]], *[Half a year of Liquid Tensor Experiment: Amazing developments](https://xenaproject.wordpress.com/2021/06/05/half-a-year-of-the-liquid-tensor-experiment-amazing-developments/)*, June 2021

Applications of condensed mathematics include the following:

* {#FarguesScholze21} [[Laurent Fargues]], [[Peter Scholze]], _Geometrization of the local Langlands correspondence_ ([arXiv:2102.13459](https://arxiv.org/abs/2102.13459))

* {#Mann22} Lucas Mann, _A p-Adic 6-Functor Formalism in Rigid-Analytic Geometry_ ([arXiv:2206.02022](https://arxiv.org/abs/2206.02022))

On condensed [[fractured (infinity,1)-topos|fractured $\infty$-topos]]-[[structure]] (cf. [[condensed local contractibility]]):

* [[Qi Zhu]], *Fractured Structure on Condensed Anima*, MSc thesis, Bonn (2023) [[pdf](https://1429cecd-24a0-4223-8b7c-1ebf47aa12e2.filesusr.com/ugd/8e912a_fe1c2f8e90094d0fa065fb8129687963.pdf), [[QiZhu-FracturedCondensed.pdf:file]]]

with exposition in

* [[Qi Zhu]], *Fractured structure on condensed spaces*, talk notes (2023) [[pdf](https://1429cecd-24a0-4223-8b7c-1ebf47aa12e2.filesusr.com/ugd/8e912a_8d2a90176ed14969a877e77ec1b787da.pdf), [[QiZhuFracturedCondensed.pdf:file]]]

* [[Nima Rasekh]], *What is a topological structure?*, talk notes (April 2023) [[pdf](https://guests.mpim-bonn.mpg.de/rasekh/Condensed.pdf), [[Rasekh-TopologicalStructure.pdf:file]]]


condensed module spectrum

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## Idea

By a *condensed module spectrum* one should generally mean a *[[module spectrum]]* in the context of [[condensed mathematics]], hence a [[condensed object in an (infinity,1)-category|condensed object]] [[internalization|internal to]] [[module spectra]].

## Definition

Given an E-infinity ring $R$, a **condensed $R$-module spectrum** is a [[(infinity,1)-sheaf]] of $R$-[[module spectra]] on the [[pro-étale (infinity,1)-site]] of the point, small relative to a universe $\mathcal{U}$.

## See also

* [[module spectrum]]

* [[condensed mathematics]]

* [[condensed infinity-groupoid]]

* [[condensed spectrum]]

* [[condensed E-infinity ring]]

[[!redirects condensed module spectra]]


condensed object in an (infinity,1)-category

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## Definition

Given an [[(infinity,1)-category]] $\mathcal{C}$, a **condensed object** in $\mathcal{C}$ is a [[(infinity,1)-sheaf]] of objects in $\mathcal{C}$ on the [[pro-étale (infinity,1)-site]] of the point, small relative to a universe $\mathcal{U}$.

## See also

* [[internalization]]

* [[condensed mathematics]]
* [[condensed set]]
* [[condensed abelian group]]
* [[condensed infinity-groupoid]]
* [[condensed spectrum]]
* [[condensed E-infinity ring]]
* [[condensed module spectrum]]

[[!redirects condensed objects in an (infinity,1)-category]]

[[!redirects condensed object]]
[[!redirects condensed objects]]


condensed ring

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## Definition

A [[ring object]] [[internalization|in]] the [[category]] $CondSet$ of [[condensed sets]].

## Related concepts

* [[condensed set]]

* [[condensed abelian group]]

* [[condensed E-infinity ring]]

* [[pre-analytic ring]]

## References

* {#ScholzeLCM} [[Peter Scholze]], _Lectures on condensed mathematics_, [pdf](https://www.math.uni-bonn.de/people/scholze/Condensed.pdf)

[[!redirects condensed rings]]


condensed set

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## Idea

Condensed sets are basic objects in [[condensed mathematics]], whose aim is to provide a convenient setting
in the framework for working with [[algebra|algebraic]] objects that are equipped with sort of a [[topological space|topology]]. A related alternative is provided by [[pyknotic sets]].

> Topological spaces formalize the idea of spaces with a notion of "nearness" of points. However, they fail to handle the idea of "points that are infinitely near, but distinct" in a useful way. Condensed sets handle this idea in a useful way. ([Scholze 21](#Scholze21))

For instance, $\mathbb{R}/\mathbb{Q}$ and $\ell^2(\mathbb{N})/\ell^1(\mathbb{N})$ are indiscrete as topological spaces but retain structure as condensed sets.

Condensed sets contain information about limits of images of any set, $A$, along the [[ultrafilters]] of $A$.

## Definition

\begin{definition}

A _condensed set_ is a [[sheaf]] of [[sets]] on the [[pro-étale site]] of a point — in other words, on the [[category]] of [[profinite spaces]] with finite jointly [[surjective]] families of maps as [[covers]] — that is the [[colimit]] of a [[small diagram]] of [[representables]] (a [[small sheaf]]).

That is, a condensed set is a [[functor]]
$$
  ProfiniteSet^op
  \longrightarrow
  Set
$$
such that the natural maps
$$
  T(\emptyset)
  \longrightarrow
  *
$$
and
$$
  T(S\sqcup S')
  \longrightarrow
  T(S) \times T(S')
$$
are [[bijections]] for any [[profinite sets]] $S$ and $S'$,
whereas the natural [[fork]]
$$
  T(S)\to T(S')
  \rightrightarrows
  T(S'\times_S S')
$$
is an [[equalizer]] for any [[surjection]] of [[profinite sets]] $S'\to S$.

\end{definition}

[Scholze, p.7](#ScholzeLCM) modifies this definition to deal with size issues:

For any uncountable [[strong limit cardinal]] $\kappa$, the category of _$\kappa$-condensed sets_ $CondSet_\kappa$ is the [[category of sheaves]] on the [[site]] of profinite sets of [[cardinality]] less than $\kappa$, with finite jointly surjective families of maps as covers.

The category of condensed sets $CondSet$ is then the (large) [[colimit]] of the category of $\kappa$-condensed sets along the [[filtered category|filtered]] [[poset]] of all uncountable [[strong limit cardinals]] $\kappa$, hence is the category of [[small sheaves]].

Another possible way to deal with set theoretic issue is presented in [Analytic Stacks](#ClausenScholze23).

The category of **light profinite sets** is the full subcategory of $ProfiniteSet$ consisting of countable sequential limits of finite sets.
The category of **light condensed sets** $LightCondSet$ is the [[category of sheaves]] on the [[site]] of light profinite sets with finite jointly surjective families of maps as covers.

## Equivalence of sites

Several different [[sites]] can be used to define [[condensed sets]], and, more generally, condensed [[∞-groupoids]]:

* [[compact Hausdorff spaces]];

* [[Stone spaces]] (compact Hasdorff [[totally disconnected spaces]]);

* [[Stonean spaces]] (compact Hausdorff [[extremally disconnected spaces]]).

In all three cases, morphisms are given by continuous maps and covering families are given by finite families of jointly surjective continuous maps.

The equivalence of sites is established in [Yamazaki](#Yamazaki). See also [Proposition 2.3, 2.7](#ScholzeLCM).

The category of _light_ profinite sets is also equivalent to the opposite of the category of countable [[Boolean algebras]] ([Clausen-Scholze 2023](#ClausenScholze23), Lecture 2).

## Properties

Condensed sets form a [[locally small]], [[well-powered category|well-powered]], [[locally cartesian closed]] [infinitary-pretopos](pretopos#infinitary) $CondSet$, that is neither a [[Grothendieck topos]] nor an [[elementary topos]] -- since it lacks both a small [[separator]] (indeed, it is not even [[total category|total]]) and a [[subobject classifier]]. It has a large [[separator]] of finitely presentable [[projective object|projectives]], and hence is algebraically exact. ([Campbell 20](#Campbell20))

According to [[Peter Scholze]] in [this comment on the nCafé](https://golem.ph.utexas.edu/category/2020/03/pyknoticity_versus_cohesivenes.html#c057798) and [[Mike Shulman]] in [this comment on the nCafé](https://golem.ph.utexas.edu/category/2020/03/pyknoticity_versus_cohesivenes.html#c057792), condensed sets satisfy external [[CoSHEP]] and [[WISC]], but internal CoSHEP fails.

See [Proposition 1.7](#ScholzeLCM) for the following proposition.

\begin{proposition}
The [[forgetful functor]]
from the category of [[topological spaces]]
to κ-condensed sets is a [[faithful functor]].
It becomes [[fully faithful]] when restricted
to κ-[[compactly generated spaces]].

This functor admits a left adjoint,
which sends a condensed set $T$
to the [[topological space]] given by the underlying set $T(*)$ of $T$
equipped with the [[quotient topology]] induced by the map
$$\coprod_{S\to T}S\to T(*),$$
where $S$ runs over all (κ-small) [[profinite sets]] mapping into $T$.
The [[counit]] of this adjunction coincides with the counit $X^{cg}\to X$
of the adjunction between (κ-small) [[compactly generated spaces]]
and [[topological spaces]].
\end{proposition}

The left adjoint exists also for topological groups and other algebraic
structures, but in this case, the underlying set is not $T(*)$.

For the category of _light_ condensed sets, there is a functor that reflects back onto the category of [[sequential topological spaces]]

## Related concepts

* [[pyknotic set]]

* [[pro-étale site]]

* [[profinite set]]

* [[condensed infinity-groupoid]]

* [[condensed abelian group]]

* [[condensed mathematics]]

## References

* {#ScholzeLCM} [[Peter Scholze]], _Lectures on condensed mathematics_, [pdf](https://www.math.uni-bonn.de/people/scholze/Condensed.pdf)
* {#ScholzeLAG} [[Peter Scholze]], _Lectures on analytic geometry_, [pdf](https://www.math.uni-bonn.de/people/scholze/Analytic.pdf)
* {#DustinScholze20}[[Dustin Clausen]], [[Peter Scholze]], _Masterclass in condensed mathematics,_ [YouTube playlist](https://www.youtube.com/playlist?list=PLAMniZX5MiiLXPrD4mpZ-O9oiwhev-5Uq), [website](https://www.math.ku.dk/english/calendar/events/condensed-mathematics/) (including pdf notes)
* {#Scholze21} [[Peter Scholze]], Math Stack Exchange [answer](https://math.stackexchange.com/a/4199337/368788).
* {#Campbell20} [[Alexander Campbell]], _How nice is the category of condensed sets?_, [talk abstract](http://web.science.mq.edu.au/groups/coact/seminar/cgi-bin/abstract.cgi?talkid=1645).
* {#ClausenScholze23}[[Dustin Clausen]], [[Peter Scholze]], _Analytic Stacks,_ [YouTube playlist](https://www.youtube.com/playlist?list=PLx5f8IelFRgGmu6gmL-Kf_Rl_6Mm7juZO), [website](https://indico.math.cnrs.fr/event/10345/)

The equivalence of various sites for condensed sets is established in

* {#Yamazaki} Koji Yamazaki, _Condensed Sets on Compact Hausdorff Spaces_, [arXiv:2211.13855](https://arxiv.org/abs/2211.13855).

[[!redirects condensed sets]]
[[!redirects CondSet]]


condensed spectrum

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* table of contents
{:toc}

## Idea

By a condensed spectrum one should generally mean a *[[spectrum]]* in the context of [[condensed mathematics]], hence a [[condensed object in an (infinity,1)-category|condensed object]] [[internalization|internal to]] [[spectra]].

A very special case of this general notion has almost been considered in [Scholze (2019), Footnote 12](#ScholzeLCM): The condensed simplicial abelian groups alluded to there there may be understood, under the [[stable Dold-Kan correspondence]], to represent (the condensed version of) the particular case of [[connective spectrum|connective]] [[HZ|$H\mathbb{Z}$]]-[[module spectra]].

## Definition

A **condensed spectrum** is a [[hypercomplete object | hypercomplete]] [[(infinity,1)-sheaf]] of [[spectra]] on the [[pro-étale (infinity,1)-site]] of the point, small relative to a universe $\mathcal{U}$.

## See also

* [[spectrum]]

* [[condensed mathematics]]

* [[condensed infinity-groupoid]]

* [[condensed E-infinity ring]]

* [[condensed module spectrum]]

* [[condensed abelian group]]

## References

A discussion of condensed spectra in the literature seems not to be available yet. For general background on [[condensed mathematics]] see:

* {#ScholzeLCM} [[Peter Scholze]], _Lectures on condensed mathematics_ (2019) $[$[pdf](https://www.math.uni-bonn.de/people/scholze/Condensed.pdf)$]$

[[!redirects condensed spectra]]


conditional convergence

> This entry is about conditional convergence of [[series]] in [[real analysis]] and [[functional analysis]]. For conditional convergence of [[spectral sequences]] in [[homological algebra]] and [[stable homotopy theory]], see *[[conditional convergence of spectral sequences]]*.

---

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\tableofcontents

## Definition

### For series in the real numbers

In [[real analysis]], given a sequence $a:\mathbb{N} \to \mathbb{R}$ in the [[real numbers]], the series $\sum_{n = 0}^\infty a_n$ is **conditionally convergent** if the [[sequence]] of [[partial sums]] is a [[Cauchy sequence]], but the [[sequence]] of [[partial sums]] of the series $\sum_{n = 0}^\infty \vert a_n \vert$ is not Cauchy.

### For series in Banach spaces

In [[functional analysis]], given a sequence $a:\mathbb{N} \to B$ in a [[Banach space]] $B$, the series $\sum_{n = 0}^\infty a_n$ is **conditionally convergent** if the [[sequence]] of [[partial sums]] is a [[Cauchy sequence]] in $B$, but the [[sequence]] of [[partial sums]] of the series $\sum_{n = 0}^\infty \Vert a_n \Vert$ is not Cauchy in the real numbers.

## See also

* [[absolute convergence]]
* [[Banach space]]
* [[Riemann series theorem]]

## External links

* Wikipedia, *[Conditional convergence](https://en.wikipedia.org/wiki/Conditional_convergence)*

[[!redirects converges conditionally]]
[[!redirects conditionally converges]]
[[!redirects conditionally convergent]]

[[!redirects conditionally convergent series]]


conditional convergence of spectral sequences

> This entry is about conditional convergence of [[spectral sequences]] in [[homological algebra]] and [[stable homotopy theory]]. For conditional convergence of [[series]] in [[real analysis]] and [[functional analysis]], see *[[conditional convergence]]*.

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{:toc}

## Idea

In the context of [[algebraic topology]], _conditional convergence_ refers to a more general kind of convergence of [[spectral sequences]] than actual ("strong") convergence ([def.](spectral+sequence#Convergence)). It is the right concept of convergence for spectral sequences that are not concentrated in the first or third quadrant.

Notably an $E$-[[Adams spectral sequence]]  is in general only conditionally convergent (to the $E$-[[nilpotent completion]]), unless finiteness conditions are imposed

## Definition

Given a [[spectral sequence]] [induced](exact+couple#SpectralSequencesFromExactCouples) from an unrolled [[exact couple]] of the form

$$
  \array{
     \cdots
       &\stackrel{}{\longrightarrow}&
     \mathcal{D}^{3,\bullet}
       &\stackrel{i_2}{\longrightarrow}&
     \mathcal{D}^{2,\bullet}
       &\stackrel{i_1}{\longrightarrow} &
     \mathcal{D}^{1,\bullet}
       &\stackrel{i_0}{\longrightarrow}&
     \mathcal{D}^{0,\bullet}
     \\
     &&
     \downarrow^{\mathrlap{}}  &{}_{\mathllap{k_2}}\nwarrow &
     {}^{\mathllap{j_2}}\downarrow &{}_{\mathllap{k_1}}\nwarrow &
     {}^{\mathllap{j_1}}\downarrow &{}_{\mathllap{k_0}}\nwarrow
     & {}_{\mathllap{j_0}}\downarrow
     \\
     && \mathcal{E}^{3,\bullet} && \mathcal{E}^{2,\bullet}
     && \mathcal{E}^{1,\bullet} && \mathcal{E}^{0,\bullet}
   }
$$

then it is said to **converge conditionally** to $\mathcal{D}^{0,\bullet}$ if, when regarding $\{\mathcal{D}^{s,\bullet}\}$ as a  [[filtering]] for $\mathcal{D}^{0,\bullet}$ it is _exhaustive_ and _complete_, in that:

1. $\underset{\longleftarrow}{\lim}_s \mathcal{D}^{s,\bullet} = 0$ (the [[limit]] over the filtering vanishes);

1. $\underset{\longleftarrow}{\lim}^1_s \mathcal{D}^{s,\bullet} = 0$ (the [[lim^1]] over the filtering vanishes).

([Boardman 99, def. 5.10](#Boardman99), see also [Rognes 12, def. 2.20](#Rognes12))

The point is that: Given a spectral sequence $\{E_r^{\bullet,\bullet}, d_r\}_r$ as above, that converges conditionally, then sufficient condition that it converges strongly ([def.](spectral+sequence#Convergence)) is that also the [[lim^1]] over its pages vanishes:

$$
  \underset{\longleftarrow}{\lim}^1_r E_r = 0
$$

([Boardman 99, theorem 7.3](#Boardman99), see also [Rognes 12, theorem 2.24](#Rognes12))

## References

Due to

* {#Boardman99} [[Michael Boardman]], _Conditionally convergent spectral sequences_, 1999 ([pdf](http://www.uio.no/studier/emner/matnat/math/MAT9580/v12/undervisningsmateriale/boardman-conditionally-1999.pdf))

Lecture notes include

* {#Rognes12} [[John Rognes]], p 44 of _The Adams spectral sequence_, 2012 ([pdf](http://folk.uio.no/rognes/papers/notes.050612.pdf))

[[!redirects conditionally convergent spectral sequence]]


conditional expectation

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* table of contents
{: toc}

## Idea

In [[probability theory]] a _conditional expectation value_ or _conditional expectation_, for short, is like an _[[expectation value]]_ of some [[random variable]]/[[observable]], but _conditioned_ on the assumption that a certain [[event (in probability theory)|event]] is assumed to have occured.

More technically: If $(\Omega,\mathfrak{A},P)$ is a [[probability space]], the *conditional expectation* $E[X|\Sigma]$ of a (measurable) random variable $X$ with respect to some sub-$\sigma$-algebra $\Sigma\subseteq \mathfrak{A}$ is some measurable random variable which is a ''coarsened'' version of $X$. We can think of $E[X|\Sigma]$ as a random variable with the same [[domain]] but which is measured with a sigma algebra containing only restricted information on the original [[event (in probability theory)|event]] since to some events in $\mathfrak{A}$ has been assigned probability $1$ or $0$ in a consistent way.

## Conditional expectation relative to a random variable

Let $(\Omega,\mathfrak{A},P)$ be a [[probability space]], let $Y$ be a measurable function into a [[measure space]] $(U,\Sigma,P^Y)$ equipped with the [[pushforward measure]] induced by $Y$, let $X:(\Omega,\mathfrak{A},P)\to(\mathbb{R},\mathcal{B}(\mathbb{R}), \lambda)$ be a real-valued [[random variable]].

Then for $X$ and $Y$ there exists a essentially unique (two sets are defined to be equivalent if their difference is a set of measure $0$) [[integrable function]] $g=:E[X|Y]$ such that the following [[commuting diagram|diagram commutes]]:

$$
  \array{
   (\Omega,\mathfrak{A},P)&
    \stackrel{Y}{\to}&
    (\U, \Sigma, P^Y)
    \\
    \downarrow^{\mathrlap{X}}
    &&
   \swarrow_{\mathrlap{g=:E[X|Y]}}
  \\
   (\mathbb{R},\mathcal{B}(\mathbb{R}),\lambda)
 }
$$

where $g:y\mapsto E[X|Y=y]$. Here ''commutes'' shall mean that

(1) $g$ is $\Sigma$-measurable.

(2) the [[integrals]] over $X$ and $g\circ Y$ are equal.

In this case $g=E[X|Y]$ is called a version of the **conditional expectation of $X$ provided $Y$**.

In more detail (2) is equivalent to that for all $B\in \Sigma$ we have

$$\int_{Y^{-1}(B)}X(\omega)d P(\omega)=\int_B g(u)d P^Y (u)$$

and to

$$\int_{Y^{-1}(B)}X(\omega)d P(\omega)=\int_{Y^{-1}(B)}(g\circ Y)(\omega)d P (\omega)$$

(The equivalence of the last two formulas is given since we always have $\int_B g(u)d P^Y (u)=\int_{Y^{-1}(B)} (g\circ Y)(\omega)d P (\omega)$ by the substitution rule.)

Note that it does *not* follow from the preceding definition that the conditional expectation exists. This is a consequence of the [[Radon-Nikodym theorem]] as will be shown in the following section. (Note that the argument of the theorem applies to the definition of the conditional expectation by random variables if we consider the [[pushforward measure]] as given by a sub-$\sigma$-algebra of the original one. In this sense $E[X|Y]$ is a ''coarsened version'' of $X$ factored by the information (i.e. the $\sigma$-algebra) given by $Y$.)

## Conditional expectation relative to a sub-$\sigma$-algebra

Note that by construction of the pushforward-measure it suffices to define the conditional expectation only for the case where $\Sigma:=\mathfrak{S}\subseteq \mathfrak{A}$ is a sub-$\sigma$-algebra.

(Note that we loose information with the notation $P^Y$; e.g $P^{id}_\mathfrak{A}$ is different from $P^{id}_\mathfrak{S}$)

The diagram

$$\array{
(\Omega,\mathfrak{A},P)&
\stackrel{id}{\to}&
(\Omega, \mathfrak{S}, P^{id})
\\
\downarrow^X&&
\swarrow^{Z=:E[X|\mathfrak{S}]}
\\
(\mathbb{R},\mathcal{B}(\mathbb{R}),\lambda)
}$$

is commutative (in our sense) iff

(a) $Z$ is $\mathfrak{S}$-measurable

(b) $\int_A Z d P=\int_A X d P$, $\forall A\in \mathfrak{S}$

We hence can write the conditional expectation as the equivalence class

$$E[X|\mathfrak{S}]=\{Z\in L^1 (\Omega, F,P)|\int_A ZdP=\int_A XdP\;\forall A\in \mathfrak{S}\}$$

An element of this class is also called a *version*.

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

$E[X|\mathfrak{S}]$ exists and is unique almost surely.

=--

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

Existence: By

$$Q(A):=\int_A X(\omega)P(d\omega)$$

$A\in \mathfrak{A}$ is defined a measure $Q$ on $(\Omega,\mathfrak{A},P)$ (if $X\ge 0$; if not consider the positive part $X^+$ and the negative part $X^-$ of $X=X^+ -X^-$ separate and use linearity of the integral). Let $P|_{\mathfrak{S}}$ be the restriction of $P$ to $\mathfrak{S}$. Then

$$Q\lt\lt P|_{\mathfrak{S}}$$

meaning: $P|_{\mathfrak{S}}(M)=0\Rightarrow Q(M)=0$ for all $M\in\mathfrak{S}$. This is the condition of the [[Radon-Nikodym derivative|theorem of Radon-Nikodym]] (the other condition of the theorem that $P|_{\mathfrak{S}}$ is $\sigma$-finite is satisfied since $P$ is a probability measure). The theorem implies that $Q$ has a density w.r.t $P|_{\mathfrak{S}}$ which is $E[X|\mathfrak{S}]$.

Uniqueness: If $g$ and $g^\prime$ are candidates, by linearity the integral over their difference is zero.
=--

## Conditional probability

From elementary probability theory we know that $P(A)=E[1_A]$.

For $A\in \mathfrak{S}$ we call $P(A|\mathfrak{S}):=E[1_A|\mathfrak{S}]$ the *conditional probability of $A$ provided $B$*.

## Conditional distribution, Conditional density

## Integral kernel, Stochastic kernel

In probability theory and statistics, a stochastic kernel is the transition function of a stochastic process. In a discrete time process with continuous probability distributions, it is the same thing as the kernel of the integral operator that advances the probability density function.

### Integral kernel

An *integral transform* $T$ is an assignation of the form

$$(Tf)(u)=\int K(t,u)f(t)dt$$

where the function of two variables $K( \dots ,\cdots)$ is called *integral kernel of the transform $T$*.

### Stochastic kernel

Let $(\Omega_1,\mathfrak{A}_1)$ be a measure space, let $(\Omega_2,\mathfrak{A}_2)$ be a measurable space.

A map $Q: \Omega_1\times \mathfrak{A}_2$ satisfying

(1) $Q(-, A):\Omega_1\to [0,1]$ is $\mathfrak{A}_1$ measurable $\forall A_2\in \mathfrak{A}_2$

(2) $Q(\omega,-):\mathfrak{A}_2\to [0,1]$ is a probability measure on $(\Omega_2,\mathfrak{A}_2)$, $\forall \omega_1\in \Omega_1$

is called a *stochastic kernel* or *transition kernel* (or *Markov kernel* - which we avoid since it is confusing) *from $(\Omega_1,\mathfrak{A}_1)$ to $(\Omega_2,\mathfrak{A}_2)$.

Then $Q$ induces a function between the classes of measures on $(\Omega_1, \mathfrak{A}_1)$ and on $(\Omega_2, \mathfrak{A}_2)$

$$\overline{Q}:
\begin{cases}
M(\Omega_1, \mathfrak{A}_1)&
\to&
M(\Omega_2, \mathfrak{A}_2)
\\
\mu&
\mapsto&
(A\mapsto \int_{\Omega_1} Q(-, A) d\mu)
\end{cases}$$

If $\mu$ is a probability measure, then so is $\overline{Q}(\mu)$. The symbol $Q(\omega, A)$ is sometimes written as $Q(A|\omega)$ in optical proximity to a conditional probability.

The stochastic kernel is hence in particular an integral kernel.

In a discrete stochastic process (see below) the transition function is a stochastic kernel (more precisely it is the function $\overline{Q}$ induced by a kernel $Q$).

#### Coupling (Koppelung)

Let $(\Omega_1,\mathfrak{A}_1, P_1)$ be a probability space, let $(\Omega_2,\mathfrak{A}_2)$ be a measure space, let $Q:\Omega_1\times \mathfrak{A}_2\to [0,1]$ be a stochastic kernel from $(\Omega_1,\mathfrak{A}_1, P_1)$ to $(\Omega_2,\mathfrak{A}_2)$.

Then by

$$P(A):=\int_{\Omega_1}(\int_{\Omega_2} 1_A (\omega_1,\omega_2 Q(\omega_1,\omega_2))P_1(d \omega_1)$$

is defined a probability measure on $\mathfrak{A}_1\otimes\mathfrak{A}_2$ which is called *coupling*. $P=:P\otimes Q$ is unique with the property

$$P(A_1\times A_2)=\int_{A_1} Q(\omega_1, A_2) P_1(d\omega_1)$$

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

Let (with the above settings) $Y:\Omega_1\to \Omega_2$ be $(\mathfrak{A}_1,\mathfrak{A}_2)$-measurable, let $X$ be a $d$-dimensional random vector.

Then there exists a stochastic kernel from $(\Omega_1, \mathfrak{A}_1)$ to $(\mathbb{R}^d,\mathcal{B}(\mathbb{R})^d)$ such that

$$P^{X,Y}=P^Y\otimes Q$$

and $Q$ is (a version of) the conditional distribution of $X$ provided $Y$, i.e.

$$Q(y,-)=P^X(-|Y=y)$$
=--

This theorem says that that $Q$ (more precisely $y\mapsto Q(y,-)$) fits in the diagram

$$\array{
(\Omega_1,\mathfrak{A}_1,P)&
\stackrel{Y}{\to}&
(\Omega_2,\mathfrak{A}_2, P^Y)
\\
\downarrow^X&&
\swarrow^{Q}
\\
(\mathbb{R},\mathcal{B}(\mathbb{R}),\lambda)
}$$

and $E[X|Y]=Q$.

#### Discrete case

In the discrete case, i.e. if $\Omega_1$ and $\Omega_2$ are finite- or enumerable sets, it is possible to reconstruct $Q$ by just considering one-element sets in $\mathfrak{A}_2$ and the related probabilities

$$p_{ij}:= Q(i,\{j\})$$

called *transition probabilities* encoding $Q$ assemble to a (perhaps countably infinite) matrix $M$ called *transition matrix of $Q$ resp. of $\overline{Q}$*. Note that $p_{ij}$ is the probability of the transition of the *state* (aka. *elementary event* or *one-element event*) $i$ to the *event* $\{j\}$ (which in this case happens to have only one element, too). We have $\sum_i p_{ij}=1$ forall $i\in \Omega_1$.

If $\rho:=(p_i)_{i\in \Omega_1}$ is a counting density on $\Omega_1$, then

$$pM=(\sum_{i\in \Omega} p_i p_{ij})_{j\in \Omega_2}$$

is a counting density on $\Omega_2$.

The conditional expectation plays a defining role in the theory of *martingales* which are *stochastic processes* such that the conditional expectation of the next value (provided the previous values) equals the present realized value.

### Stochastic processes

The terminology of *stochastic processes* is a special interpretation of some aspects of [[infinitary combinatorics]] in terms of [[probability theory]].

Let $I$ be a [[total order]] (i.e. transitive, antisymmetric, and total).

A *stochastic process* is a diagram $X_I: I\to \mathcal{R}$ where $\mathcal{R}$ is the class of random variables such that $X_I(i)=:X_i:(\Omega_i, \mathfrak{F}_i, P_i)\to (S_i, \mathfrak{S}_i)$ is a random variable. Often one considers the case where all $(S_i, \mathfrak{S}_i)=(S, \mathfrak{S})$ are equal; in this case $S$ is called *state space of the process $X_I$*.

If all $\Omega_i=\Omega$ are equal and the class of $\sigma$-algebras $(\mathfrak{A}_i)_{i\in I}$ is *filtered* i.e.

$$\mathfrak{F}_i\subseteq \mathfrak{F}_j\;;iff\;; i\le j$$

and all $X_l$ are $\mathfrak{F}_l$ measurable, the process is called *adapted process*.

For example the *natural filtration* where $\mathfrak{F}_i=\sigma(\{X^{-1}_l(A), l\le i, A\in \mathfrak{S}\})$ gives an adapted process.

In terms of a diagram we have for $i\le j$

$$\array{
(\Omega_j,\mathfrak{A}_j,P_j)&
\stackrel{f}{\to}&
(\Omega_i,\mathfrak{A}_i,P_i)
\\
\downarrow^{X_j}&&
\swarrow^{\omega_i\mapsto Q(\omega_i,-)}
\\
(\mathbb{R},\mathcal{B}(\mathbb{R}),\lambda)
}$$

and $\overline{Q}:(\Omega_i,\mathfrak{A}_i,P_i)\to(\Omega_j,\mathfrak{A}_j,P_j)$ where $Q:\Omega_i\times\mathfrak{A}_j\to [0,1]$ is the transition probability for the passage from state $i$ to state $j$.

### Martingale

An adapted stochastic process with the natural filtration in discrete time is called a *martingale* if all $E[X_i]\lt \infty$ and $\forall i\le j, E[X_j|\mathfrak{A}_i]=X_i$.

$$\array{
(\Omega_j,\mathfrak{A}_j,P_j)&
\stackrel{f}{\to}&
(\Omega_i,\mathfrak{A}_i,P_i)
\\
\downarrow^{X_j}&&
\swarrow^{E[X_j|\mathfrak{A}_i]=X_i}
\\
(\mathbb{R},\mathcal{B}(\mathbb{R}),\lambda)
}$$
[[martingale]]

(...)

### Markow Process

An adapted stochastic process satisfying

$$P(X_t|\mathfrak{A}_s)=P(X_t|X_s)\;;\forall s\le t$$

is called a *Markow process*.

## Chapman-Kolmogorow Equation

For a Markow process the Chapman-Kolmogorow equation encodes the statement that the transition probabilities of the process form a [[semigroup]].

If in the notation from above $(P_t:\Omega\times\mathfrak{A}\to [0,1])_t$ is a family of stochastic kernels $(\Omega,\mathfrak{A})\to(\Omega,\mathfrak{A})$ such that all $P_t(\omega,-):\mathfrak{A}\to [0,1]$ are probabilities, then $(P_t)_t$ is called *transition semigroup* if

$$\overline P_t (P_s(\omega,A))=P_{s+t} (\omega, A)$$

where

$$\overline P_t: P_s(\omega,-)\mapsto (A\mapsto\int_\Omega P_t (y,A) P_s(\omega,-)(d_y))$$

## In quantum probability theory
 {#InQuantumProbability}

In the dual algebraic formulation of [[probability theory]] known as _[[noncommutative probability theory]]_ or _[[quantum probability theory]]_, where the concept of _[[expectation value]]_ is primitive (while that of the corresponding [[probability space]] (if it exists) is a derived concept), the concept of conditional expection appears as follows (e.g. [Redei-Summers 06, section 7.3](#RedeiSummers06)):

Let $(\mathcal{A},\langle -\rangle)$ be a [[quantum probability space]], hence a [[complex numbers|complex]] [[star algebra]] $\mathcal{A}$ of [[quantum observables]], and a [[state on a star-algebra]] $\langle -\rangle \;\colon\; \mathcal{A} \to \mathbb{C}$.

This means that for $A \in \mathcal{A}$ any [[observable]], its _[[expectation value]]_ in the given [[state on a star-algebra|state]] is

$$
  \mathbb{E}(A) \;\coloneqq\;  \langle A \rangle \in \mathbb{C}
  \,.
$$

More generally, if $P \in \mathcal{A}$ is a [[real part|real]] [[idempotent]]/[[projector]]

$$
  \label{RealIdempotent}
  P^\ast = P
  \,,
  \phantom{AAA}
  P P = P
$$

thought of as an event, then for any observable $A \in \mathcal{A}$ the [[conditional expectation value]] of $A$, conditioned on the observation of $P$, is

$$
  \label{ConditionalExpectation}
  \mathbb{E}(A \vert P)
  \;\coloneqq\;
  \frac{
    \left \langle P A P \right\rangle
  }{
    \left\langle P \right\rangle
  }
  \,.
$$

## Related concepts

* [[Bayesian reasoning]]

* [[wavefunction collapse]]

## References

See also

* Wikipedia, _[Conditional expectation](https://en.wikipedia.org
/wiki/Conditional_expectation)_

Discussion form the point of view of [[quantum probability]] is in

* {#RedeiSummers06} [[Miklos Redei]], [[Stephen Summers]], section 7.3 of _Quantum Probability Theory_ ([arXiv:quant-ph/0601158](https://arxiv.org/abs/quant-ph/0601158))

category: probability

[[!redirects conditional probability]]
[[!redirects conditional probabilities]]

[[!redirects conditional expectation value]]
[[!redirects conditional expectation values]]


Conduché functor

[[!redirects Conduche functor]]

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#### Higher category theory
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# Contents
* table of contents
{: toc}

## Idea

A **Conduché functor**, also called a **Conduché fibration** or an **exponentiable functor**, is a [[functor]] which is an [[exponentiable morphism]] in [[Cat]].  (In accordance with [[Stigler's law of eponymy]], the notion was actually defined in [Giraud 64](#Giraud64) before [Conduché 72](#Conduche72).)  This turns out to be equivalent to a certain "factorization lifting" property which includes both [[Grothendieck fibrations]] and opfibrations.

So, roughly speaking, a functor $p\colon E\to B$ is a **strict Conduché functor** if given any morphism $\alpha$ in $E$, and any way of factoring its image in $B$, we can lift that factorization back up to a factorization of $\alpha$, in a way that is unique up to isomorphism.   There is also a weak version of this idea.

## Failure of local cartesian closedness in Cat

As is evident from the fact that such functors have a name, not every functor is exponentiable in [[Cat]].  In particular, although $Cat$ is [[cartesian closed category|cartesian closed]], it is not [[locally cartesian closed category|locally cartesian closed]].

It is easy to write down examples of [[colimits]] in $Cat$ that are not preserved by [[pullback]] (as they would be if pullback had a [[right adjoint]]).  For instance, let $\mathbf{2}$ denote the [[walking arrow]], i.e. the [[ordinal]] $2$ regarded as a [[category]], $1$ the [[terminal category]], and $\mathbf{3} = \mathbf{2} \sqcup_1 \mathbf{2}$ the ordinal $3 = (a \to b \to c)$ regarded as a category.  Then the [[pushout]] square
$$\array{1 & \overset{}{\to} & \mathbf{2}\\
  \downarrow && \downarrow\\
  \mathbf{2}& \underset{}{\to} & \mathbf{3}}$$
in the [[slice category]] $Cat/\mathbf{3}$ pulls back along the inclusion $\mathbf{2}\to \mathbf{3}$ of the arrow $(a\to c)$ to the square
$$\array{0 & \overset{}{\to} & 1\\
  \downarrow && \downarrow\\
  1& \underset{}{\to} & \mathbf{2}}$$
which is certainly not a pushout.

One way to describe the problem is that the pushout has "created new morphisms" that didn't exist before.  But another way to describe the problem is that the inclusion $\mathbf{2}\to\mathbf{3}$ fails to notice that the morphism $(a\to c)$ acquires a new factorization in $\mathbf{3}$ which it didn't have in $\mathbf{2}$.  Conduché's observation was that this latter failure is really the only problem that can prevent a functor from being exponentiable.

## Definition

A [[functor]] $p\colon E\to B$ is a **strict Conduché functor** if for any [[morphism]] $\alpha\colon a\to b$ in $E$ and any factorization $p a \overset{\beta}{\to} c \overset{\gamma}{\to} p b$ of $p \alpha$ in $B$, we have:

1. there exists a factorization $a \overset{\tilde{\beta}}{\to} d \overset{\tilde{\gamma}}{\to} b$ of $\alpha$ in $E$ such that $p \tilde{\beta} = \beta$ and $p \tilde{\gamma} = \gamma$, and

1. any two such factorizations in $E$ are connected by a [[zigzag]] of commuting morphisms which map to $id_c$ in $B$.

(Here, 'commuting morphism' means a morphism $d \to d'$ in $E$ such that the pair of triangles in

$$\array{
 & & d & \stackrel{\tilde{\gamma}}{\to} & b \\
 & ^\mathllap{\tilde{\beta}} \nearrow & \downarrow & \nearrow^\mathrlap{\tilde{\gamma}'} & \\
a & \underset{\tilde{\beta}'}{\to} & d' & &
}$$

[[commuting diagram|commute]].)

The theorem is then that the following are equivalent:

* $p$ is a strict Conduché functor.
* $p$ is [[exponentiable morphism|exponentiable]] in the 1-category $Cat$.
* $p$ is exponentiable in the [[strict 2-category]] $Cat$.

By "exponentiable in the strict 2-category $Cat$" we mean that pullback along $p$ has a strict right 2-adjoint (i.e. a $Cat$-enriched right adjoint).  Of course, this implies ordinary exponentiability in the 1-category $Cat$, while the converse follows via an argument involving [[cotensors]] with $\mathbf{2}$ in $Cat$.

For exponentiability in the weak [[2-category]] $Cat$, in the sense of pullback having a weak/pseudo 2-adjoint, we can simply weaken the condition.  We say that $p\colon E\to B$ is a **(weak) Conduché functor** if for any morphism $\alpha\colon a\to b$ in $E$ and any factorization $p a \overset{\beta}{\to} c \overset{\gamma}{\to} p b$ of $p \alpha$ in $B$, we have:

1. there exists a factorization $a \overset{\tilde{\beta}}{\to} d \overset{\tilde{\gamma}}{\to} b$ of $\alpha$ in $E$, and an isomorphism $p d \cong c$, such that modulo this isomorphism $p \tilde{\beta} = \beta$ and $p \tilde{\gamma} = \gamma$, and

1. any two such factorizations in $E$ are connected by a zigzag of commuting morphisms which map to isomorphisms in $B$.

A functor can then be shown to be a weak Conduché functor if and only if it is exponentiable in the weak sense in $Cat$.

A Conduché functor is **discrete** if each factorisation is unique (equivalently, if it reflects identities). Discrete Conduché functors generalise [[discrete fibrations]] and [[discrete opfibrations]].

## Conduché functors and 2-functors to Prof

The Conduché criterion can be reformulated in a more conceptual way by analogy with [[Grothendieck fibrations]].  We first observe that to give a functor $p\colon E\to B$ is essentially the same as to give a [[normal lax functor|normal]] [[lax 2-functor]] $B\to Prof$ from $B$ to [[Prof]], the 2-category of [[profunctors]].  The latter is also known as a [[displayed category]]; see there for more on this correspondence.

Specifically, given a functor $p$, we define $B\to Prof$ as follows.  Each object $b\in B$ is sent to the fiber category $p^{-1}(b)$ of objects lying over $b$ and morphism lying over $1_b$.  And each morphism $f\colon a\to b$ in $B$ to the profunctor $H_f\colon p^{-1}(a) ⇸ p^{-1}(b)$ for which $H_f(x,y)$ is the set of arrows $x\to y$ in $E$ lying over $f$.  The lax structure maps $H_f \otimes H_g \to H_{g f}$ are given by composition in $E$.  The converse construction of a functor $p$ from a normal lax 2-functor into $Prof$ is an evident generalization of the [[Grothendieck construction]].  Now we can say that:

* $p$ is a fibration iff the corresponding functor $B\to Prof$ factors through a pseudo 2-functor landing in $Cat^{op}$, via the contravariant inclusion $Cat^{op}\to Prof$.
* Similarly, $p$ is an opfibration iff $B\to Prof$ factors through a pseudo 2-functor landing in $Cat$ via the covariant inclusion $Cat \to Prof$.
* The functor $B\to Prof$ factors through a *lax* 2-functor landing in $Cat^{op}$ iff $p$ admits all "weakly cartesian" liftings, and dually.
* Finally, $p$ is a (strict) Conduché functor iff the functor $B\to Prof$ is itself a pseudo 2-functor (though it may not land in $Cat$ or $Cat^{op}$).  This can be seen by comparing the definition of the tensor product of profunctors with the explicit description in terms of unique factorizations above.

Thus Conduché functors into $B$ correspond to pseudofunctors from $B$, regarded as a locally discrete bicategory, to the bicategory $Prof$.  However, morphisms between Conduché functors over $B$ do not correspond to pseudonatural transformations between such pseudofunctors.  To get the correct transformations, we must instead regard $B$ as a vertically discrete [[double category]], and $Prof$ as a pseudo double category with profunctors horizontally and functors vertically; then pseudo double functors $B\to Prof$ again correspond to Conduché functors into $B$, and *vertical* double transformations between them correspond to functors between Conduché functors into $B$.

More generally, the slice category $Cat/B$ is equivalent to the hom-category $Dbl_{normal,lax}(B,Prof)$, with its full subcategory consisting of Conduché functors corresponding to the pseudo double functors.

## Higher-categorical versions

Non-strict Conduché functors and [[Street fibrations]] may be equivalently characterized by an "up-to-iso" version of the above constructions using [[essential fibers]].

[Ayala and Francis](#AyalaFrancis) prove an analogous characterization of exponentiable [[(∞,1)-functors]].  The [[(∞,1)-category theory|(∞,1)-categorical context]] eliminates the "level-shifting" in the characterization via $Prof$ (i.e. the presence of a [[bicategory]] [[Prof]] when discussing only exponentiable 1-functors).  Thus, there is an [[(∞,1)-category]] [[(∞,1)Prof]] such that exponentiable $(\infty,1)$-functors into an $(\infty,1)$-category $B$ correspond to $(\infty,1)$-functors $B\to (\infty,1)Prof$.

As in the 1-categorical case, ordinary $(\infty,1)$-transformations between functors $B\to (\infty,1)Prof$ do not give the correct maps between exponentiable $(\infty,1)$-functors over $B$; we need to instead regard $(\infty,1)Prof$ as a sort of "$(\infty,1)$-double category".  Ayala and Francis consider only the vertically-invertible fragment of this $(\infty,1)$-double category, which can be represented as a functor from an $\infty$-groupoid to an $(\infty,1)$-category (a sort of [[proarrow equipment]] with all 2-cells and all 1-cells in the domain invertible); this is what they call a "flagged" $(\infty,1)$-category and is also what is represented by a non-complete [[Segal space]].  Of course, restricting to the vertically-invertible fragment of $(\infty,1)Prof$ also restricts what it classifies to the $\infty$-groupoid of exponentiable $(\infty,1)$-functors over $B$ rather than the whole $(\infty,1)$-category thereof.,

## In a double category - completeness
Functors satisfying a Conduché-like condition are central to completeness/cocompleteness in [[pseudo-double categories]]. In short, in 1-category theory, $A$ is complete iff for any small $M$ and locally small $C$, $K: M\to C$ a functor, $T : A\to M$, $T$ has a pointwise right Kan extension along $K$; in pseudo double categories, this is too strong, and we must restrict to double functors $K: M\to C$ satisfying a Conduché condition.

A pseudo-double category is *unitary* if the vertical identity arrows are strict identities, so $1_A\circ f = f = f\circ 1_B$ on-the-nose. We here assume all double categories are unitary; this is not a severe restriction in practice.

Call a lax functor $F: \mathbb{A}\to\mathbb{B}$ between pseudo-double categories *unitary* if it it preserves identity vertical arrows on-the-nose and the comparison map $1_{FA}\to F(1_A)$ is the identity; therefore it is only really lax from the point of view of composition. A unitary lax functor $1\to\mathbb{A}$ is automatically strict, as $1$ has no nontrivial compositions, whereas a lax functor $1\to\mathbb{A}$ is an object endowed with a vertical monad.

Note that if we are working in a 2-category of lax functors, and we want to define the limit of a functor $F : \mathbb{A}\to\mathbb{B}$ as the right Kan extension of $F$ along $!_{\mathbb{A}}:\mathbb{A}\to 1$, we probably want to consider the initial extension $1\to \mathbb{B}$ along unitary lax functors rather than lax functors, so that our answer is just an object of $\mathbb{B}$ and a cocone rather than a vertical monad in $\mathbb{B}$. Thus we will here consider unitary Kan extensions.

The correct definition of "complete" for a pseudo-double category $\mathbb{A}$ is that it admits pointwise unitary lax right Kan extensions of any lax double functor $S: \mathbb{I}\to\mathbb{A}$ ($\mathbb{I}$ small) along all lax double functor along a lax double functor $R: \mathbb{I}\to\mathbb{J}$, where $R$ satisfies the "right Conduché condition: that the laxity 2-cell for vertical composition in $R$,
\begin{centre}
\begin{tikzcd}
	{\mathbb{I}_a\times_{\mathbb{I}_o}\mathbb{I}_a} & {\mathbb{J}_a\times_{\mathbb{J}_o}\mathbb{J}_a} \\
	{\mathbb{I}_a} & {\mathbb{J}_a}
	\arrow[""{name=0, anchor=center, inner sep=0}, "{\circ_\mathbb{I}}"', from=1-1, to=2-1]
	\arrow[""{name=1, anchor=center, inner sep=0}, "{\circ_\mathbb{J}}", from=1-2, to=2-2]
	\arrow["{R\times R}", from=1-1, to=1-2]
	\arrow["R"', from=2-1, to=2-2]
	\arrow["\alpha", shorten <=9pt, shorten >=9pt, Rightarrow, from=1, to=0]
\end{tikzcd}
\end{centre}
is [[exact square|exact]].
Here, $\mathbb{I}_o$ is the category of objects and horizontal morphisms of $\mathbb{I}$, and $\mathbb{I}_a$ is the category of arrows and 2-cells.

The situation for cocompleteness is dual, replacing lax by colax everywhere.

For more on this see [Grandis and Paré](#GrandisParé07), Theorem 5.2.

## Examples

* The above considerations show that any Grothendieck fibration *or* opfibration is a (strict) Conduché functor, while any [[Street fibration]] or opfibration is a non-strict Conduché functor.

* If $\mathbf{2}$ denotes the [[interval category]], then any [[normal lax functor]] out of $\mathbf{2}$ is necessarily pseudo, since there are no composable pairs of nonidentity arrows in $\mathbf{2}$.  It follows that, as pointed out by Jean Benabou, *any* functor with codomain $\mathbf{2}$ is a Conduché functor.  Note that functors with codomain $\mathbf{2}$ can also be identified with [[profunctors]], the two fiber categories being the source and target of the corresponding profunctor.

* As with [[exponentiable morphisms]] in any category, Conduché functors are closed under composition.

## References

* {#Giraud64} [[J. Giraud]], _Méthode de la descente_, Bull. Math. Soc. Mémoire **2** (1964). ([numdam](http://www.numdam.org/numdam-bin/item?id=MSMF_1964__2__R3_0))

* {#Conduche72} [[F. Conduché]], _Au sujet de l'existence d'adjoints à droite aux foncteurs 'image reciproque' dans la catégorie des catégories_ , C. R. Acad. Sci. Paris **275** Série A (1972) pp.891-894. ([gallica](http://gallica.bnf.fr/ark:/12148/bpt6k56191352/f19.image#))

* {#GrandisParé07} [[Marco Grandis]], [[Robert Paré]] _Lax Kan extensions for double categories (on weak double categories, part IV)_ , Cahiers de topologie et géométrie différentielle catégoriques, tome 48, no 3 (2007), p. 163-199

The definitions and proofs of the above theorems, along with the 2-categorical generalization (Conduché considered only the 1-categorical case) can also be found in

* [[Peter Johnstone]], "Fibrations and partial products in a 2-category", *Appl. Categ. Structures* 1 (1993), 141--179

A description of the characterization in terms of lax normal functors can be found in

* [[Ross Street]], \'Powerful functors\', [pdf](http://www.math.mq.edu.au/~street/Pow.fun.pdf)

Discrete Conduché functors are considered in

* {#BN00}[[Marta Bunge|M. Bunge]], [[Susan Niefield|S. Niefield]], _Exponentiability and single universes_ , JPAA **148** (2000) pp.217-250.

* [[Peter Johnstone]], _A Note on Discrete Conduché Fibrations_ , TAC **5** no.1 (1999) pp.1-11. ([pdf](http://www.tac.mta.ca/tac/volumes/1999/n1/n1.pdf))

An analogue of Conduché functors for [[∞-categories]], classified by maps into an ∞-category version of [[Prof]], is studied in

* {#AyalaFrancis} [[David Ayala]] and [[John Francis]], _Fibrations of ∞-categories_, [arxiv](https://arxiv.org/abs/1702.02681)

[[!redirects Conduché fibration]]
[[!redirects Conduchè fibration]]
[[!redirects Conduche fibration]]

[[!redirects Conduché functor]]
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[[!redirects Conduché functors]]
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[[!redirects exponentiable functors]]

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[[!redirects strictly powerful functors]]

[[!redirects Conduché ∞-fibration]]
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[[!redirects discrete exponentiable functors]]


conductor

A conductor is a modulus (in the sense of number theory) associated to an abelian extension of number fields and reflects its ramification data. It has been introduced in the work of Weber at the end of 19th century. It is one of the central notions in [[class field theory]]. A later version is due [[Emil Artin]].

* wikipedia [conductor (class field theory)](http://en.wikipedia.org/wiki/Conductor_%28class_field_theory%29), [conductor-discriminant formula](http://en.wikipedia.org/wiki/Conductor-discriminant_formula)
* Serge Lang, _Algebraic number theory_, GTM 110, Springer 1970, 2000

[[!redirects conductors]]


cone

> This entry is mostly about cones in [[homotopy theory]] and [[category theory]]. For more geometric cones see at _[[cone (Riemannian geometry)]]_.

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#### Category theory
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#### Limits and colimits
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# Contents
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## Idea

In [[homotopy theory]], the _cone_ of a [[space]] $X$ is the space obtained by taking the $X$-shaped [[cylinder object|cylinder]] $X \times I$, where $I$ may be an [[interval object]], and squashing one end down to a point.  The eponymous example is where $X$ is the [[circle]], i.e. the [[topological space]] $S^1$, and $I$ is the standard interval $[0,1]$.  Then the [[cartesian product]] $X \times I$ really is a [[cylinder]], and the cone of $X$ is likewise a cone.

This notion also makes sense when $X$ is a [[category]], if $I$ is taken to be the [[interval category]] $\{ 0 \to 1 \}$, i.e. the [[ordinal]] $\mathbf{2}$.  Note that since the interval category is directed, this gives two different kinds of cone, depending on which end we squash down to a point.

Another, perhaps more common, meaning of 'cone' in category theory is that of a _cone over (or under) a [[diagram]]_.  This is just a [[diagram]] over the cone category, as above.  Explicitly, a cone over $F\colon J \to C$ is an [[object]] $c$ in $C$ equipped with a [[morphism]] from $c$ to each vertex of $F$, such that every *new* triangle arising in this way [[commutative triangle|commutes]].  A cone which is [[universal property|universal]] is a [[limit]].

In [[category theory]], the word _[[cocone]]_ is sometimes used for the case when we squash the other end of the interval; thus $c$ is equipped with a morphism *to* $c$ *from* each vertex of $F$ (but $c$ itself still belongs to $C$).  A cocone in this sense which is universal is a [[colimit]].  However, one should beware that in homotopy theory, the word [[mapping cocone|cocone]] is used for a different dualization.

This definition generalizes to [[higher category theory]]. In particular in [[(∞,1)-category theory]] a cone over an [[∞-groupoid]] is essentially a cone in the sense of [[homotopy theory]].

## Definition

### In homotopy theory

If $X$ is a space, then the cone of $X$ is the [[homotopy pushout]]
of the identity on $X$ along the unique map to the point:

$$
\array{
  X & \to & X \\
  \downarrow & & \downarrow \\
  * & \to & cone(X)
}\,.
$$

This homotopy pushout can be computed as the ordinary [[pushout]] $cone(X) := X\times I \amalg_X *$

$$
  \array{
    X &\stackrel{d_1}{\to} & X \times I
    \\
    \downarrow && \downarrow
    \\
    * &\to& cone(X)
  }
  \,.
$$

If $X$ is a [[simplicial set]], then the cone of $X$ is the [[join of simplicial sets|join]] of $X$ with the point.

The [[mapping cone]] (q.v.) of a morphism $f \colon X \to Y$ is then the pushout along $f$ of the inclusion $X \to cone(X)$.

### As a monad

In contexts where [[intervals]] $I$ can be treated as [[monoid]] objects, the cone construction as quotient of a cylinder with one end identified with a point,

$$C(X) = I \times X/(0 \times X) \sim p,$$

carries a structure of [[monad]] $C$. In such cases, the monoid has a multiplicative [[identity]] $1$ and an absorbing element $0$, where multiplication by $0$ is the constant map at $0$. In that case, a $C$-algebra consists of an object $X$ together with

* An [[action]] of the monoid, $a: I \times X \to X$.

* A constant or basepoint $x_0 \colon 1 \to X$

such that $a(0, x) = x_0$ for all $x$. This equation can be expressed in any category $\mathbf{C}$ with finite products and a suitable interval object $I$ as monoid (for example, $Top$, where $I = [0, 1]$ is a monoid under real multiplication, or under $min$ as multiplication). Under some reasonable assumptions (e.g., if the $\mathbf{C}$ has quotients, and these are preserved by the functor $I \times -$), the category of $C$-algebras will be monadic over $\mathbf{C}$ and the free $C$-algebra on $X$ will be $C(X)$ as described above. The category of $C$-algebras will also be monadic over the category of pointed $\mathbf{C}$-objects, $1 \downarrow \mathbf{C}$.

These observations apply for example to $Top$, and also to $Cat$ where the [[interval category]] $\mathbf{2}$ is a monoid in $Cat$ under the $min$ operation (see below).

If in addition the underlying category $\mathbf{C}$ is [[cartesian closed category|cartesian closed]], or more generally if $I$ is [[exponentiable object|exponentiable]], the monad $C$ on pointed $\mathbf{C}$-objects also has a [[right adjoint]] $P$ which can be regarded as a [[path space]] construction $P$, where we have a pullback

$$\array{
P(X) & \to & 1 \\
\downarrow & & \downarrow \\
X^I & \stackrel{eval_0}{\to} & X.
}$$

For general abstract reasons, the right adjoint $P$ carries a comonad structure whereby $C$-algebras are equivalent to $P$-coalgebras. Considered over the category of simplicial sets, this is closely connected to [[decalage]].

### In category theory

If $C$ is a category, then the cone of $C$ is the [[cocomma category]] of the identity on $C$ and the unique map to the [[terminal category]]:

$$
\array{
  C & \to & C \\
  \downarrow & \Rightarrow  & \downarrow \\
  * & \to & cone(C)
}\,.
$$

Again, this may be computed as a pushout:

$$
  \array{
    C &\stackrel{d_1}{\to} & C \times \mathbf{2}
    \\
    \downarrow && \downarrow
    \\
    * &\to& cone(C)
  }
  \,.
$$

The cone of $C$ may equivalently be thought of, or defined, as the result of adjoining a new [[initial object]] to $C$.

### Cones over a diagram
 {#ConesOverADiagram}

A cone *in* a category $C$ is given by a category $J$ together with a [[functor]] $cone(J) \to C$.  By the [[universal property]] of the cocomma category, to give such a functor is to give an object $c$ of $C$, a functor $F \colon J \to C$, and a [[natural transformation]]
$$T: \Delta(c) \to F$$
where $\Delta(c):J\to C$ denotes the [[constant functor]] at the object $c$.  Such a transformation is called a _cone over the diagram_ $F$.

In other words, a cone consists of morphisms (called the **components** of the cone)

$$
  T_j
  \;\colon\;
  c \longrightarrow F(j)
  \,,
$$

one for each object $j$ of $J$, which are compatible with all the morphisms $F(f): F(j) \to F(k)$ of the diagram, in the sense that each diagram
$$
  \array{
    {}&{}&c&{}&{}
    \\
    {}&
    \mathllap{\scriptsize{T_j}}\swarrow
      &{}&
    \searrow\mathrlap{\scriptsize{T_k}} &{}
    \\
    F(j)
      &{}&
    \stackrel{F(f)}{\longrightarrow} &{}& F(k)
   \\
}
$$
commutes.

It's called a *cone* because one pictures $c$ as sitting at the vertex, and the diagram itself as forming the base of the cone.

A *[[morphism]]* if such cones is a [[natural transformation]] $\alpha\colon \Delta(c)\to\Delta(c')$ such that the diagram
$$
  \array{
    \Delta(c)
     &{}&\overset{\alpha}{\longrightarrow} &{}&
    \Delta(c')
    \\
     {}&
      \mathllap{\scriptsize{T}}\searrow
     &{}&
      \swarrow\mathrlap{\scriptsize{T'}} &{}
    \\
    {}&{}&F&{}&{}
}
$$

[[commuting diagram|commutes]]. Note that naturality of any such $\alpha$ implies that for all $i,j\in J$, $\alpha_i=\alpha_j$, so that $\alpha=\Delta(\phi)$ for some $\phi \colon c \to c'$ in $C$.  The single component $\phi$ itself is often referred to as the cone morphism.

An equivalent definition of a cone morphism $\phi : T \to T'$ says that all component diagrams
$$
  \array{
    c &{}&
   \overset{\phi}{\longrightarrow}
   &{}& c'
   \\
   {}&
   \mathllap{\scriptsize{T_j}}\searrow
   &{}&
   \swarrow\mathrlap{\scriptsize{T'_j}} &{}
   \\
   {}&{}&F(j)&{}&{}
}
$$

commute.

Cones and their morhisms over a given [[diagram]] $J$ clearly form a [[category]]. The [[terminal object]] in this category, if it exists, is the *[[limit]]* of the diagram (see [there](limit#InTermsOfUniversalCones)).

A *[[cocone]]* in $C$ is precisely a cone in the [[opposite category]] $C^{op}$.

### Over a diagram in an $(\infty,1)$-category

For $F : D \to C$ a [[diagram]] of [[(∞,1)-categories]], i.e. an [[(∞,1)-functor]], the $(\infty,1)$-category of $(\infty,1)$-cones over $F$ is the [[over quasi-category]] denoted $C_{/F}$. Its objects are cones over $F$. Its [[k-morphism]]s are $k$-homotopies between cones. The [[limit in a quasi-category|(∞,1)-categorical limit]] over $F$ is, if it exists, the [[terminal object]] in $C_{/F}$.

## See also

* [[cone type]]

These are shaped like the homotopy-theoretic cone, so maybe there is a deeper relationship:

* [[positive cone]] (in an [[ordered group]], such as an [[operator algebra]]),
* [[future cone]] (of an [[event]] in a [[Lorentzian manifold]], such as [[spacetime]]),
* [[convex cone]] (in a [[vector space]]).

[[!redirects cone]]
[[!redirects cones]]
[[!redirects cocone]]
[[!redirects cocones]]

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cone (Riemannian geometry)

> For other, related, concepts of a similar name see at _[[cone]]_.

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## Idea

In ([[pseudo-Riemannian geometry|pseudo-]])[[Riemannian geometry]], a _cone_ is a part of a  (pseudo-)[[Riemannian manifold]] where the [[metric tensor]] is locally of the form $d s^2 = d r^2 + r^2 d s^2_1$. The point that would correspond to $r = 0$ is the "conical singularity".

## Examples

### Spherical cones
 {#SphericalCones}

The metric cone on the [[round sphere]] is simply [[Euclidean space]]

$$
  C(S^n) \simeq \mathbb{R}^{n+1} \setminus \{0\}
$$

and hence may in fact be continued non-singularly also at the cone tip.

For $G$ a [[finite group]] with a [[free action|free]] [[action]] on the [[round sphere]] $S^n$, the [[quotient space]] $S^n/G$ exists as a [[Riemannian manifold]]. The metric cone $C(S^n/G)$ on this is singular at the origin as soon as $G$ is not the [[trivial group]].

If here $G = \mathbb{Z}_n$ is a [[cyclic group]] one says that this cone is obtained from flat [[Euclidean space]] by introducing a "deficit angle".

If one passes beyond [[smooth manifolds]] to [[orbifolds]], then the cone tip in $C(S^n/G)$ may be included. The result is the [[orbifold]] $\mathbb{R}^{n+1}\sslash G$ which is the [[homotopy quotient]] of [[Euclidean space]] by the linear $G$-[[action]] ($G$-[[representation]]).

Such conical singularities appear for instance in the [[far-horizon geometry]] of [[BPS state|BPS]] [[black branes]]. Special cases are [[ADE-singularities]].

### $G_2$-manifolds
 {#ExamplesG2Manifolds}

* Three examples of cones admitted by [[simply-connected]] [[G2-manifolds]] seem to be known: cones on $\mathbb{C}P^3$, $SU(3)/U(1)\times U(1)$ and $S^3 \times S^3$. ([Atiyah-Witten 01](#AtiyahWitten01))

* The [[metric cone over complex projective 3-space]] carries the [[mathematical structure|structure]] of  a [[G2-manifold]] whose [[Riemannian metric]] is [[invariant]] under the canonical [[Sp(2)]] [[action]] by left-[[matrix multiplication]] on homomogeneous coordinates in $\mathbb{H}^2 \simeq_{\mathbb{R}} \mathbb{C}^4 \to \mathbb{C}P^3$ ([Byant-Salamon 89](metric+cone+over+complex+projective+3-space#BryantSalamon89), see also [Acharya-Bryant-Salamon 20](metric+cone+over+complex+projective+3-space#AcharyaBryantSalamon20)).

## Related concepts

* [[conifold]]

* [[piecewise flat spacetime]]

* [[Sasakian manifold]]

* [[conical space]]

* [[M-theory on G2-manifolds]]

## References

### General

Discussion in the context of [[3-manifolds]] and [[orbifolds]]:

* Daryl Cooper, Craig Hodgson, Steve Kerckhoff, _Three-dimensional Orbifolds and Cone-Manifolds_,  MSJ Memoirs Volume 5, 2000 ([pdf](https://web.math.ucsb.edu/~cooper/37.pdf), [euclid:1389985812](https://projecteuclid.org/euclid.msjm/1389985812))

### Black brane conical singularities

Discussion of [[supergravity]] [[black brane]]-solutions at conical singularities ([[cone branes]]) includes the following (see also at _[[far-horizon geometry]]_)

* [[Bobby Acharya]], [[Jose Figueroa-O'Farrill]], [[Chris Hull]], B. Spence, _Branes at conical singularities and holography_, Adv. Theor.Math. Phys.2:1249-1286, 1999 ([arXiv:hep-th/9808014](https://arxiv.org/abs/hep-th/9808014))

* {#FigOFar98} [[Jose Figueroa-O'Farrill]], _Near-horizon geometries of supersymmetric branes_, talk at [SUSY 98](http://inspirehep.net/record/971430/) ([arXiv:hep-th/9807149](https://arxiv.org/abs/hep-th/9807149), [talk slides](http://www.maths.ed.ac.uk/~jmf/CV/Seminars/SUSY98.pdf))

* {#AtiyahWitten01} [[Michael Atiyah]], [[Edward Witten]], _$M$-Theory dynamics on a manifold of $G_2$-holonomy_, Adv. Theor. Math. Phys. 6 (2001) ([arXiv:hep-th/0107177](http://arxiv.org/abs/hep-th/0107177))

A suggestion that there is an analogue of [[AdS-CFT duality]] for conical bulk spacetimes and their conical singularities:

* Rong-Xin Miao, _Codimension-$n$ Holography for the Cones_ ([arXiv:2101.10031](https://arxiv.org/abs/2101.10031))

[[!include G2-conifolds -- references]]

[[!redirects conical singularity]]
[[!redirects conical singularities]]

[[!redirects conifold singularity]]

[[!redirects metric cone]]
[[!redirects metric cones]]


cone morphism > history

> see at *[[cone]]* the section *[cones over a diagram](cone#ConesOverADiagram)*


cone type

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### Context
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## Idea

The cone type is an axiomatization of the [[cone]] in the context of homotopy type theory.

## Definition

A cone type on a type $A$ is a [[higher inductive type]] $Cone(A)$ inductively generated by

* A term $e:Cone(A)$

* A function $i: A \to Cone(A)$

* A term

$$p: \prod_{a:A} e = i(a)$$

In [[Coq]] pseudocode, the cone is given by

    Inductive Cone (A : Type) : Type
      | vertex : Cone A
      | base :  A -> Cone A
      | edge : A -> Id Cone A vertex base(a)

It can equivalently be defined as the (homotopy) [[pushout]] of $A$ and the unit type $1$ under $A$. Similarly, it is the [[cofiber]] of the [[identity function]] of $A$. This definition makes it clear that the cone type is always a [[contractible type]]. As a result, cone types could be though of as a way of constructing free contractible types for any type $A$.

The cone type of $A$ can also be defined as the [[localization of a type|localization of $A$]] at the [[empty type]], $\mathrm{Cone}(A) \coloneqq L_\emptyset(A)$, and thus it is the [[(-2)-truncation]] of $A$.

### Inference rules

Formation rules for cone types:

$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash \mathrm{cone}(A)}$$

Introduction rules for cone types:

$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash *:\mathrm{cone}(A)}$$

$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash i:A \to \mathrm{cone}(A)}$$

$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash \mathcal{g}:\prod_{x:A} * =_{\mathrm{cone}(A)} i(x)}$$

Elimination rules for cone types:

$$\frac{\begin{array}{l}
\Gamma \vdash A \; \mathrm{type} \quad \Gamma, y:\mathrm{cone}(A) \vdash C(y) \; \mathrm{type} \\
\Gamma \vdash c_*:C(*) \quad \Gamma \vdash c_i:\prod_{x:A} C(i(x)) \\
\Gamma \vdash c_\mathcal{g}:\prod_{x:A} c_* =_{y:\mathrm{cone}(A).C(y)}^{\mathcal{g}(x)} c_i(x)
\end{array}}{\Gamma \vdash \mathrm{ind}_{\mathrm{cone}(A)}(c_*, c_i, c_\mathcal{g}):\prod_{y:\mathrm{cone}(A)} C(y)}$$

Computation rules for cone types:

* Judgmental computation rules

$$\frac{\begin{array}{l}
\Gamma \vdash A \; \mathrm{type} \quad \Gamma, y:\mathrm{cone}(A) \vdash C(y) \; \mathrm{type} \\
\Gamma \vdash c_*:C(*) \quad \Gamma \vdash c_i:\prod_{x:A} C(i(x)) \\
\Gamma \vdash c_\mathcal{g}:\prod_{x:A} c_* =_{y:\mathrm{cone}(A).C(y)}^{\mathcal{g}(x)} c_i(x)
\end{array}}{\Gamma \vdash \mathrm{ind}_{\mathrm{cone}(A)}(c_*, c_i, c_\mathcal{g})(*) \equiv c_*:C(*)}$$

$$\frac{\begin{array}{l}
\Gamma \vdash A \; \mathrm{type} \quad \Gamma, y:\mathrm{cone}(A) \vdash C(y) \; \mathrm{type} \\
\Gamma \vdash c_*:C(*) \quad \Gamma \vdash c_i:\prod_{x:A} C(i(x)) \\
\Gamma \vdash c_\mathcal{g}:\prod_{x:A} c_* =_{y:\mathrm{cone}(A).C(y)}^{\mathcal{g}(x)} c_i(x)
\end{array}}{\Gamma, x:A \vdash \mathrm{ind}_{\mathrm{cone}(A)}(c_*, c_i, c_\mathcal{g})(i(x)) \equiv c_i(x):C(i(x))}$$

$$\frac{\begin{array}{l}
\Gamma \vdash A \; \mathrm{type} \quad \Gamma, y:\mathrm{cone}(A) \vdash C(y) \; \mathrm{type} \\
\Gamma \vdash c_*:C(*) \quad \Gamma \vdash c_i:\prod_{x:A} C(i(x)) \\
\Gamma \vdash c_\mathcal{g}:\prod_{x:A}
\end{array}}{\Gamma, x:A \vdash \mathrm{apd}_{y:\mathrm{cone}(A).C(y)}(\mathrm{ind}_{\mathrm{cone}(A)}(c_*, c_i, c_\mathcal{g}), *, i(x), \mathcal{g}(x)) \equiv c_\mathcal{g}(x):c_* =_{y:\mathrm{cone}(A).C(y)}^{\mathcal{g}(x)} c_i(x)}$$

* Propositional computation rules

$$\frac{\begin{array}{l}
\Gamma \vdash A \; \mathrm{type} \quad \Gamma, y:\mathrm{cone}(A) \vdash C(y) \; \mathrm{type} \\
\Gamma \vdash c_*:C(*) \quad \Gamma \vdash c_i:\prod_{x:A} C(i(x)) \\
\Gamma \vdash c_\mathcal{g}:\prod_{x:A} c_* =_{y:\mathrm{cone}(A).C(y)}^{\mathcal{g}(x)} c_i(x)
\end{array}}{\Gamma \vdash \beta_{\mathrm{cone}(A)}^{*}(c_*, c_i, c_\mathcal{g}):\mathrm{ind}_{\mathrm{cone}(A)}(c_*, c_i, c_\mathcal{g})(*) =_{C(*)} c_*}$$

$$\frac{\begin{array}{l}
\Gamma \vdash A \; \mathrm{type} \quad \Gamma, y:\mathrm{cone}(A) \vdash C(y) \; \mathrm{type} \\
\Gamma \vdash c_*:C(*) \quad \Gamma \vdash c_i:\prod_{x:A} C(i(x)) \\
\Gamma \vdash c_\mathcal{g}:\prod_{x:A} c_* =_{y:\mathrm{cone}(A).C(y)}^{\mathcal{g}(x)} c_i(x)
\end{array}}{\Gamma \vdash \beta_{\mathrm{cone}(A)}^{i}(c_*, c_i, c_\mathcal{g}):\prod_{x:A} \mathrm{ind}_{\mathrm{cone}(A)}(c_*, c_i, c_\mathcal{g})(i(x)) =_{C(i(x))} c_i(x)}$$

$$\frac{\begin{array}{l}
\Gamma \vdash A \; \mathrm{type} \quad \Gamma, y:\mathrm{cone}(A) \vdash C(y) \; \mathrm{type} \\
\Gamma \vdash c_*:C(*) \quad \Gamma \vdash c_i:\prod_{x:A} C(i(x)) \\
\Gamma \vdash c_\mathcal{g}:\prod_{x:A} c_* =_{y:\mathrm{cone}(A).C(y)}^{\mathcal{g}(x)} c_i(x)
\end{array}}{\Gamma \vdash \beta_{\mathrm{cone}(A)}^{\mathcal{g}}(c_*, c_i, c_\mathcal{g}):\prod_{x:A} \mathrm{apd}_{y:\mathrm{cone}(A).C(y)}(\mathrm{ind}_{\mathrm{cone}(A)}(c_*, c_i, c_\mathcal{g}), *, i(x), \mathcal{g}(x)) =_{y:\mathrm{cone}(A).C(y)}^{\mathcal{g}(x)} c_\mathcal{g}(x)}$$

Uniqueness rules for cone types:

$$\frac{\begin{array}{l}
\Gamma \vdash A \; \mathrm{type} \quad \Gamma, y:\mathrm{cone}(A) \vdash C(y) \; \mathrm{type} \\ \Gamma \vdash c:\prod_{y:\mathrm{cone}(A)} C(y) \quad \Gamma \vdash p:\mathrm{cone}(A) \\
\Gamma \vdash l:\prod_{x:A} p =_{\mathrm{cone}(A)} i(x)
\end{array}}{\Gamma \vdash \eta_{\mathrm{cone}(A)}(c, p, l):\mathrm{ind}_{\mathrm{cone}(A)}(c(p), \lambda x:A.c(i(x)), \lambda x:A.\mathrm{apd}_{y:\mathrm{cone}(A).C(y)}(c, p, i(x), l(x)))(p) =_{C(p)} c(p)}$$

## Properties

### Propositionally constant functions

In [[dependent type theory]], a [[propositionally constant function]] from type $A$ to $B$ at an element $b:B$ is a function $f:A \to B$ with a dependent function $p:\prod_{x:A} f(x) =_B b$.

Given a propositionally constant function at $b:B$, $f:A \to B$, with $p:\prod_{x:A} f(x) =_B b$, by the recursion principle of the cone type, one has

$$\mathrm{rec}_{\mathrm{Cone}(A)}^B(b, f, p):\mathrm{Cone}(A) \to B$$

such that

$$\mathrm{rec}_{\mathrm{Cone}(A)}^B(b, f, p)(\mathrm{vertex}) \equiv b:B$$

$$\mathrm{rec}_{\mathrm{Cone}(A)}^B(b, f, p)(\mathrm{base}(x)) \equiv f(x):B$$

$$\mathrm{ap}_{\mathrm{rec}_{\mathrm{Cone}(A)}^B(b, f, p)}(\mathrm{edge}(x)) \equiv p:f(x) =_B b$$

In particular, by the judgmental [[congruence rule]] for the introduction rule of [[function types]], the standard constant function $\lambda x:A.b:A \to B$ for $b:B$ with $\beta_{A, B}^{x:A.b}:\prod_{x:A} (\lambda x:A.b)(x) =_B b$ from the typal [[computation rules]] for [[function types]] factors through $\mathrm{Cone}(A)$ by

$$\lambda x:A.b \equiv \mathrm{rec}_{\mathrm{Cone}(A)}^B(b, \lambda x:A.b, \beta_{A, B}^{x:A.b}) \circ \mathrm{base}$$

where for [[strict function types]] we have $\beta_{A, B}^{x:A.b} \equiv \lambda x:A.\mathrm{refl}_B(b)$.

Propositionally constant functions can also be regarded directly as functions $\mathrm{Cone}(A) \to B$, similar to how paths in $A$ can be regarded as functions $\mathbb{I} \to A$ from the the [[interval type]] rather than the application of said function along the path generator of the interval type.

Since the [[cone type]] is a [[contractible type]], it is equivalent to the [[unit type]], with equivalences $e:\mathrm{Cone}(A) \simeq \mathbb{1}$, and every constant function $(b, f, p)$ factors the unit type through $e \circ \mathrm{base}:A \to \mathbb{1}$ and $\mathrm{rec}_{\mathrm{Cone}(A)}^B(b, f, p) \circ e^{-1}:\mathbb{1} \to B$.

## Examples

* The [[unit type]] $1$ is the cone type of an empty type $0$.

* The [[interval type]] $I$ is the cone type of the unit type $1$

* A [[simplex type]] $\Delta_n$ is the cone type of the simplex type $\Delta_{n-1}$. $\Delta_1 = I$ and $\Delta_0 = 1$.

## See also

* [[cone]]

* [[constant function]]

* [[empty type]], [[localization of a type]]

* [[cone type]], [[bracket type]], [[set truncation]]

[[!redirects cone type]]
[[!redirects cone types]]

[[!redirects weak cone type]]
[[!redirects weak cone types]]
[[!redirects strict cone type]]
[[!redirects strict cone types]]


configuration category

## Idea

[…]

## References

[[Pedro Boavida de Brito]] and [[Michael S. Weiss]]:

* _Spaces of smooth embeddings and configuration categories_, [arXiv](https://arxiv.org/abs/1502.01640).

* _Configuration categories and homotopy automorphisms_, [arXiv](https://arxiv.org/abs/1503.00501).

* _The configuration category of a product_, [arXiv](https://arxiv.org/abs/1701.06987).

* _The configuration category of a covering space_, [arXiv](https://arxiv.org/abs/2312.17631).

* _Presentations of configuration categories_, [arXiv](https://arxiv.org/abs/2312.17632).

* _The torus trick for configuration categories_, [arXiv](https://arxiv.org/abs/2401.00799).


configuration scheme

Configuration schemes are a generalization of [[schemes]] which are obtained via formal gluing of the usual schemes along closed embeddings. In fact, the categories of [[quasicoherent sheaves]] are glued. Standard [[cohomology theory]] and the study of [[coherent sheaves]] extend to them. Configuration schemes were thought of to provide a nonclassical type of resolution of [[singularity|singularities]], and they can be a part of a categorical version of resolution of singularities.

Valery A. Lunts, _Coherent sheaves on configuration schemes_, [math.AG/0012061](http://arxiv.org/abs/math/0012061)

Valery A. Lunts, _Categorical resolution of singularities_, [math.AG/0905.4566](http://arxiv.org/abs/0905.4566)


configuration space

A _configuration space_ is, naturally, a [[space]] of configurations of some system. The term appears as a technical term both in [[mathematics]] and in [[physics]]. While the idea is essentially the same in both cases, the emphasis of examples and perspectives differs. Notably in mathematics "the configuration space of a [[topological space]] $X$" is usually taken by default to refer very specifically to the space of pairwise distinct points $X$, also called _[[Fadell's configuration space]]_.

Therefore see either of:

* [[configuration space (mathematics)]]

* [[configuration space (physics)]]

[[!redirects configuration spaces]]

category: disambiguation


configuration space (physics)

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## Idea

The _configuration space_ of a [[physical system]] is the [[space]] of all possible configurations of the system.

For example the configuration space of $n \in \mathbb{N}$ point particles that  roam around in some [[manifold]] $X$ (e.g. thought of as physical [[space]]) is the $n$-fold [[Cartesian product]] $X^n$. If the particles are constrained not to be at coincident points, then the configuration space is the corresponding [[subspace]] of $X^n$. This happens to be the kind of  [[configuration space (mathematics)|configuration space as typically considered in mathematics]] see at _[[Fadell's configuration space]]_.

In [[field theory]] over [[Lorentzian manifolds|Lorentzian]] [[spacetime]] one is typically interested not in the configurations of fields "at a given time" (say on a fixed [[Cauchy surface]]) but in configuratios of fields on (an [[open subset]] of) [[spacetime]]. This is then called the _[[space of trajectories]]_ or _[[space of histories]]_ of the system.

## Examples

* in [[gravity]]: [[Wheeler superspace]]

## Related concepts

* [[configuration space (mathematics)]]

* [[space of field histories]]

* [[phase space]]

[[!redirects space of field configurations]]
[[!redirects spaces of field configurations]]