state monad

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### Context
#### Type theory
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[[!include type theory - contents]]
=--
#### Modalities, Closure and Reflection
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[[!include modalities - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

The _state monad_ is the name for the [[monad in computer science]] which is used to implement the functionality of read/write on a global "mutable state" (a global variable) in the context of [[functional programming languages]].

A [[functional program]] with

1. input of ([[data type|data-]])[[type]] $X$,

1. output of [[type]] $Y$

1. and "mutable state" of [[type]] $W$

   (e.g. the state of a "random access memory" device, cf. [Yates (2019), p. 26 & Fig. 1.10](#Yates19))

{#IsClearly} is clearly a [[function]] ([[morphism]]) of [[function type]] between the [[product types]] with $W$ (also known as a *[[Mealy machine]]*, see [there](Mealy+machine#MealyMachinesAsEffectfulMaps) and cf. [Oliveira & Miraldo 2016, p. 462](#OliveiraMiraldo16)):

$$
  prog
    \;\colon\;
  X \times W
    \longrightarrow
  Y \times W
  \,.
$$

Now, under the [hom-isomorphism](adjoint+functor#InTermsOfHomIsomorphism) of the ([[Cartesian product]] $\dashv$ [[internal hom]])-[[adjoint functor|adjunction]], this is equivalently given by its [[adjunct]], which is a function into the [[function type]] $[-,-]$ ([[internal hom]]):

$$
  X \longrightarrow [W, W \times Y ]
 \,.
$$

Here the operation $[W, W\times (-)]$ is the [[monad]] on the type system which is induced by the above [[adjunction]]; and this latter function is naturally regarded as a morphism in the [[Kleisli category]] of this monad.

{#RAM} In the context of [[monad (in computer science)|monads in computer science]], this is called the _state monad_ for mutable states of type $W$, such as for a data type $W$ of  Random-Access-Memory:

## Definition
 {#Definition}

Concretely, with the [[underlying]] functor being

$$
  W State \;\colon\; D \,\mapsto\, \big[W,\, W \times D \big]
$$

the join operation of the $W$-state monad is given by [[evaluation]] on the intermediate $W$-variable, hence the bind operation is given as follows:

\begin{imagefromfile}
    "file_name": "StateMonadBinding-230828.jpg",
    "width": 750,
    "unit": "px",
    "margin": {
        "top": -30,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

To see more in detail how this encodes access to a global variable $w \colon W$ in the sense of [[programming languages]] consider the following operations which isolate the processes of reading from and writing to that variable, respectively:

$$
  \array{
    read &\colon& W State(W)
    \\
    read &\equiv& w \mapsto (w,w)
  }
  \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;
  \array{
    write &\colon& W \to W State(\ast)
    \\
    write &\equiv& w \mapsto (w' \mapsto w)
  }
$$

With these, for instance the procedure
$$
  \array{
    inc &\colon& \mathbb{N}State(\ast)
    \\
    inc &\equiv& n \mapsto n+1
  }
$$
which first reads-in the state of a global natural number variable and then overwrites it by its increment may be constructed as shown (on the right in [do-notation](monad+in+computer+science#DoNotation)) here:

\begin{imagefromfile}
    "file_name": "StatefulNumberIncrementInDONotation-230828.jpg",
    "width": 560,
    "unit": "px",
    "margin": {
        "top": -30,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

(This is essentially the example from [Benton, Hughes & Moggi 2002 p. 68 & 71](#BentonHughesMoggi02))

## Properties

### Realization in dependent type theory

In a [[locally Cartesian closed category]]/[[dependent type theory]] $\mathbf{H}$, then to every type $W$ is associated its [[base change]] [[adjoint triple]]

$$
  \mathbf{H}_{/W}
   \stackrel{\overset{\sum_W}{\longrightarrow}}{\stackrel{\overset{W^\ast}{\longleftarrow}}{\underset{\prod_W}{\longrightarrow}}}
  \mathbf{H}
  \,.
$$

In terms of this the state monad is the composite

$$
  State = \prod_W W^\ast \sum_W W^\ast
$$

of [[context extension]] followed by [[dependent sum]], followed by [[context extension]], followed by [[dependent product]].

Here $\prod_W W^\ast = [W,-]$ is called the _[[function monad]]_ or _[[reader monad]]_ and $\sum_W W^\ast = W \times (-)$ is the _[[coreader comonad]]_.

## Related concepts

* [[Mealy machine]]

* [[maybe monad]]

* [[continuation monad]]

* [[function monad]] (reader monad)

* [[coreader comonad]]

* [[writer monad]]

* [[QRAM]]

* [[store comonad]]

## References

Original discussion of the state/side-effect monad as a [[monad in computer science]]:

* {#Moggi91} [[Eugenio Moggi]], Exp. 1.1 in: *Notions of computation and monads*, Information and Computation, **93** 1 (1991) [doi:10.1016/0890-5401(91)90052-4, [pdf](http://www.disi.unige.it/person/MoggiE/ftp/ic91.pdf)]

* [[Philip Wadler]], Section 2.3 in: *Monads for functional programming*, in M. Broy (eds.) *Program Design Calculi* NATO ASI Series, **118** Springer (1992) [[doi;10.1007/978-3-662-02880-3_8](https://doi.org/10.1007/978-3-662-02880-3_8), [pdf](https://homepages.inf.ed.ac.uk/wadler/papers/marktoberdorf/baastad.pdf)]

* {#BentonHughesMoggi02} [[Nick Benton]], [[John Hughes]], [[Eugenio Moggi]], Ex. 42 in: *Monads and Effects*, in: *Applied Semantics*, Lecture Notes in Computer Science **2395**, Springer (2002) 42-122 [[doi:10.1007/3-540-45699-6_2](https://doi.org/10.1007/3-540-45699-6_2)]

* {#PlotkinPower02} [[Gordon Plotkin]], [[John Power]], Section 3 of: *Notions of computation determine monads* In: M. Nielsen, U. Engberg, (eds.) FOSSACS 2002. LNCS, Lecture Notes in Computer Science **2303**,  Springer (2002) 342-356 [[doi:10.1007/3-540-45931-6_24p](https://doi.org/10.1007/3-540-45931-6_24)]

Exposition:

* {#Milewski19} [[Bartosz Milewski]], §21.2.5 in: *Category Theory for Programmers*, Blurb (2019) [[pdf](https://github.com/hmemcpy/milewski-ctfp-pdf/releases/download/v1.3.0/category-theory-for-programmers.pdf), [github](https://github.com/hmemcpy/milewski-ctfp-pdf), [webpage](https://bartoszmilewski.com/2014/10/28/category-theory-for-programmers-the-preface/), [ISBN:9780464243878](https://www.blurb.com/b/9621951-category-theory-for-programmers-new-edition-hardco)]

On the [[module of a monad|modules]] ("algebras") of the state monad:

* [[François Métayer]], *State monads and their algebras* [[arXiv:math/0407251](https://arxiv.org/abs/math/0407251)]

Concrete applications via implementation in [[Haskell]]:

* {#Yates19} Ryan Yates, *Improving Haskell Transactional Memory* (2019) [[hdl:1802/35367](https://hdl.handle.net/1802/35367), [pdf](file:///C:/Users/us13/Downloads/Yates_rochester_0188E_11946.pdf)]

Lecture notes:

* [[Tarmo Uustalu]], p.24 of: *Monads and Interaction Lecture 1* [[pdf](https://cs.ioc.ee/~tarmo/mgs21/mgs1.pdf), [[Uustalu-Monads1.pdf:file]]], lecture notes for [MGS 2021](https://staffwww.dcs.shef.ac.uk/people/G.Struth/mgs21.html) (2021):

In the generality of the *local* state monad:

* [Plotkin & Power (2002), Sec. 4](#PlotkinPower02)

* [[Paul-André Melliès]], *Local States in String Diagrams*, In: G. Dowek (eds.) *Rewriting and Typed Lambda Calculi* RTA TLCA 2014, Lecture Notes in Computer Science **8560**, Springer (2014) [[doi:10.1007/978-3-319-08918-8_23](https://doi.org/10.1007/978-3-319-08918-8_23)]

Explicit understanding of [[Kleisli morphisms]] for the state moand as [[Mealy machines]]:

* {#OliveiraMiraldo16} José Nuno Oliveira, Victor Cacciari Miraldo, p. 462 in: *A practical approach to state-based system calculi*, Journal of Logical and Algebraic Methods in Programming **85** 4 (2016) 449-474 [[doi:10.1016/j.jlamp.2015.11.007](https://doi.org/10.1016/j.jlamp.2015.11.007)]

and (almost):

* {#PeroneKarachalias23} Marco Perone, Georgios Karachalias, *Composable Representable Executable Machines* [[arXiv:2307.09090](https://arxiv.org/abs/2307.09090)]

[[!redirects state monads]]

[[!redirects local state monad]]
[[!redirects local state monads]]

[[!redirects random access memory]]
[[!redirects random access memories]]

[[!redirects RAM]]
[[!redirects RAMs]]


state on a star-algebra

> This entry describes a concrete formalization of the general notion of [[state]] in the context of [[quantum probability theory]] and [[algebraic quantum field theory]] and [[operator algebra]]. For other conceptualizations of [[states]] see there.

***

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### Context
#### Measure and probability theory
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[[!include measure theory - contents]]
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#### Functional analysis
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[[!include functional analysis - contents]]
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#### Algeraic Quantum Field Theory
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[[!include AQFT and operator algebra contents]]
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# Contents
* table of contents
{: toc}

## Idea
 {#Idea}

The concept of _state on a star-algebra_ is the formalization of the general idea of *[[states]]* from the point of view of  [[quantum probability theory]] and [[AQFT|algebraic quantum theory]].

In order to motivate the definition from more traditional formulations in  [[physics]], recall that there a _[[state]]_ $\langle - \rangle$ is the information that allows to  assign to each [[observable]] $A$ the [[expectation value]] $\langle A\rangle$ that this observable has when the [[physical system]] is assumed to be in that state.

Often this is formalized in the [[Schrödinger picture]] where a [[Hilbert space|Hilbert]] [[space of states]] $\mathcal{H}$ is taken as primary, and the [[observables]] are [[representation|represented]] as suitable [[linear operators]] $A$ on $\mathcal{H}$. Then for $\psi \in \mathcal{H}$ a state ([[pure state]]) the [[expectation value]] of $A$ in this state is the [[inner product]] $\langle \psi \vert A \vert \psi \rangle \coloneqq (\psi, A \psi)$. This defines a [[linear function]]

$$
  \langle \psi \vert - \vert \psi \rangle
  \;\colon\;
  \mathcal{A}
  \longrightarrow
  \mathbb{C}
$$

on the [[algebra of observables]] $\mathcal{A}$, satisfying some extra properties.

Conversely, in the [[Heisenberg picture]] one may take the "abstract" [[associative algebra|algebra]] [[algebra of quantum observables|of observables]] as primary (i.e. not necessarily manifested as an [[operator algebra]]), and declare that a state is any linear functional

$$
  \langle - \rangle
  \;\colon\;
  \mathcal{A}
  \longrightarrow
  \mathbb{C}
$$

which is _positive_ in that $\langle A^\ast A\rangle \geq 0$ and _normalized_ in that $\langle 1\rangle = 1$.   Under suitable conditions a [[Hilbert space|Hilbert]] [[space of states]] may be (re-)constructed from this *a posteriori* via the _[[GNS construction]]_.

Traditionally this definition is considered for [[algebras of observables]] which are [[C*-algebras]] (as usually required for [[non-perturbative quantum field theory]], see e.g. [Fredenhagen (2003) Sec. 2](#Fredenhagen03), [Landsman (2017) Def. 2.4](#Landsman17)), but the definition makes sense generally for plain [[star-algebras]] ([Meyer (1995), I.1.1](#Meyer95)), such as for instance for the [[formal power series algebras]] that appear in [[perturbative quantum field theory]] (e.g. [Bordemann-Waldmann (1996) Def. 1](#BordemannWaldmann96), [Fredenhagen & Rejzner (2012) Def. 2.4](#FredenhagenRejzner12), [Khavkine & Moretti (2015) Def. 6](#KhavkineMoretti15), [Dütsch (2018) Def. 2.11](#Duetsch18)).

{#AsPositiveLinearFunctionals} Of course, the notion of states depends only on the $\ast$-algebra's [[underlying]] [[partial order|partially ordered]] [[complex vector space]] (see [here](C-star-algebra#PartialOrderAndPositiveElements)) and hence makes sense in the generality of any [[partial order|partially ordered]] [[vector space]], in which case one refers to them also as *positive linear functionals* or similar (e.g. [Murphy (1990) §3.3](#Murphy90)).

The perspective that states are normalized positive linear functionals on the algebra of observables is implicit in traditional [[perturbative quantum field theory]], where it is encoded in the [[2-point function]] corresponding to a [[vacuum state]] or more generally a [[quasi-free quantum state]] (the _[[Hadamard propagator]]_).  The perspective is made explicit in [[algebraic quantum field theory]] (see e.g. [Fredenhagen 03, section 2](#Fredenhagen03)) and for [[star-algebras]] of observables that are not necessarily [[C*-algebras]] in [[perturbative algebraic quantum field theory]] (e.g. [Bordemann-Waldmann 96](#BordemannWaldmann96), [Fredenhagen-Rejzner 12, def. 2.4](#FredenhagenRejzner12), [Khavkine-Moretti 15, def. 6](#KhavkineMoretti15), [Dütsch 18, def. 2.11](#Duetsch18)).

## Definition

### For unital star algebras
 {#DefinitionForUnitalStarAlgebras}

+-- {: .num_defn #StateOnAStarAlgebra}
###### Definition
**([[state]] on a [[unital algebra|unital]] [[star algebra]])**

Let $\mathcal{A}$ be a [[unital]] [[star-algebra]] over the [[complex numbers]] $\mathbb{C}$. A _state_ on $\mathcal{A}$ is a [[linear function]]

$$
  \rho \;\colon\; \mathcal{A} \longrightarrow \mathbb{C}
$$

such that

1. {#PositivityCondition} (positivity) for all $A \in \mathcal{A}$ the value of $\rho$ on the product $A^\ast A$ is

   1. [[real part|real]] $\,\rho(A^\ast A) \in \mathbb{R} \hookrightarrow \mathbb{C}$

   1. as such [[non-negative real number|non-negative]]:

   \[
     \label{Positivity}
     \rho(A^\ast A) \geq 0
   \]

1. (normalization)

   \[
     \label{Normalization}
     \rho(\mathbf{1}) = 1
   \]

   for $\mathbf{1} \in \mathcal{A}$ the [[unit]] in the algebra.

=--

(e.g. [Meyer 95, I.1.1](#Meyer95), [Bordemann-Waldmann 96](#BordemannWaldmann96), [Fredenhagen-Rejzner 12, def. 2.4](#FredenhagenRejzner12), [Khavkine-Moretti 15, def. 6](#KhavkineMoretti15), [Landsman 2017, Def. 2.4](#Landsman17))

+-- {: .num_remark}
###### Remark
**([[probability theory|probability theoretic]] interpretation of [[state on a star-algebra]])**

A [[star algebra]] $\mathcal{A}$ equipped with a [[state on a star-algebra|state]] is also called a _[[quantum probability space]]_, at least when $\mathcal{A}$ is in fact a [[von Neumann algebra]].

=--

+-- {: .num_remark #StatesFormAConvexSet}
###### Remark
**([[state on a star-algebra|states]] form a [[convex set]])**

For $\mathcal{A}$ a unital [[star-algebra]], the [[set]] of states on $\mathcal{A}$ according to def. \ref{StateOnAStarAlgebra} is naturally a [[convex set]]: For $\rho_1, \rho_2 \colon \mathcal{A} \to \mathbb{C}$ two states then for every $p \in [0,1] \subset \mathbb{R}$ also the [[linear combination]]

$$
  \array{
    \mathcal{A}
     &\overset{p \rho_1 + (1-p) \rho_2}{\longrightarrow}&
    \mathbb{C}
    \\
    A &\mapsto& p \rho_1(A) + (1-p) \rho_2(A)
  }
$$

is a state.

=--

+-- {: .num_defn #PureStateOnAStarAlgebra}
###### Definition
**([[pure state]])**

A state $\rho \colon \mathcal{A} \to \mathbb{C}$ on a unital [[star-algebra]] (def. \ref{StateOnAStarAlgebra}) is called a _[[pure state]]_ if it is extremal in the [[convex set]] of all states (remark \ref{StatesFormAConvexSet}) in that an identification

$$
  \rho = p \rho_1 + (1-p) \rho_2
$$

for $p \in (0,1)$ implies that $\rho_1 = \rho_2$ (hence $= \rho$).

=--

### For $C^\ast$-Algebras

The following discusses states specifically on [[C*-algebras]].

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

An element $A$ of an (abstract) $C^*$-algebra is called **[[positive operator|positive]]** if it is [[self-adjoint operator|self-adjoint]] and its [[spectrum of an operator|spectrum]] is contained in $[0, \infinity)$. We write $A \ge 0$ and say that the set of all [[positive operators]] is the positive cone (of a given $C^*$-algebra).
=--

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

This definition is motivated by the [[Hilbert space]] situation, where an operator $A \in \mathcal{B} (\mathcal{H})$ is called [[positive operator|positive]] if for every vector $x \in \mathcal{H}$ the inequality $ \langle x, A x \rangle \ge 0$ holds. If the abstract $C^*$-algebra of the definition above is represented on a Hilbert space, then we see that by [[functional calculus]] we can define a self adjoint operator $B$ by $B \coloneqq f(A)$ with $f(t) := t^{1/2}$ and get  $ \langle x, A x \rangle = \langle B x, B x \rangle \ge 0$. This shows that the positive elements of the abstract algebra, if represented on a Hilbert space, become positive operators as defined here in the Hilbert space setting.

=--

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

A [[linear functional]] $\rho$ on an $C^*$-algebra is **positive** if $A \ge 0$ implies that  $\rho(A) \ge 0$.

A **state** of a unital $C^*$-algebra is [[linear functional]] $\rho$ such that $\rho$ is positive and $\rho(1) = 1$.

=--

Though the mathematical notion of state is already close to what physicists have in mind, they usually restrict the set of states further and consider normal states only. We let $\mathcal{R}$ be an $C^*$-algebra and $\pi$ an representation of $\mathcal{R}$ on a Hilbert space $\mathcal{H}$.

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

A **normal state** $\rho$ is a state that satisfies one of the following equivalent conditions:

* $\rho$ is weak-operator continuous on the unit ball of $\pi(\mathcal{R})$.

* $\rho$ is strong-operator continuous on the unit ball $\pi(\mathcal{R})$.

* $\rho$ is ultra-weak continuous.

* There is an operator $A$ of [[trace class]] of $\mathcal{H}$ with $tr(A) = 1$ such that $\rho(\pi(R)) = tr(A \pi(R))$ for all $R \in \mathcal{R}$.

=--

This appears as [KadisonRingrose, def. 7.1.11, theorem 7.1.12](#KadisonRingrose)

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

This list is not complete, there are more commonly used equivalent characterizations of normal states.

The last one is most frequently used by physicists, in that context the operator $A$ is also called a [[density matrix]] or density operator.

=--

Sometimes the observables of a system are described by an abstract $C^*$-algebra, in this case an important notion is the folium:

+-- {: .num_defn}
###### Definition
The **folium** of a representation $\pi$ of an $C^*$-algebra  $\mathcal{R}$ on a Hilbert space is the set of normal states of $\pi(\mathcal{R})$.
=--

+-- {: .num_defn}
###### Definition
A state $\rho$ of a representation is called a **vector state** if there is a $x \in \mathcal{H}$ such that $\rho(\pi(R)) = \langle \pi(R)x, x \rangle$ for all $R \in \mathcal{R}$.
=--

+-- {: .num_theorem}
###### Theorem
Normal states are vector states if $\mathcal{R}$ is a [[von Neumann algebra]] with a separating vector. More precisely: Let $\mathcal{R}$ be a von Neumman algebra acting on a Hilbert space $\mathcal{H}$, let $\rho$ be a normal state of $\mathcal{R}$ and $x \in \mathcal{H}$ be a separating vector for $\mathcal{R}$, then there is a $y \in \mathcal{H}$ such that $\rho(R) = \langle Ry, y \rangle$ for all $R \in \mathcal{R}$.
=--

This appears as [KadisonRingrose, theorem 7.2.3](#KadisonRingrose).

The set of states of an $C^*$-algebra is sometimes called the **[[space of quantum states|state space]]**.

The state space is non-empty (define a state on the subalgebra $\mathbb{C} 1$ and extend it to the whole $C^*$-algebra via the [[Hahn-Banach theorem]]), convex and weak$^*$-compact, so it has extreme points. By the [[Krein-Milman theorem]] (see Wikipedia: [Krein-Milman theorem] (http://en.wikipedia.org/wiki/Krein%E2%80%93Milman_theorem)) it is the weak$^*$-closure of its extreme points.

+-- {: .num_defn}
###### Definition
A **pure state** is a state that is an extreme point of the state space.
=--

The term "pure" originates from the notion of [[entanglement]], a pure state is not a mixture of two distinct other states.

## Examples
 {#Examples}

\begin{example}\label{ClassicalProbabilityMeasureAsStateOnMeasurableFunctions}
**(classical [[probability measure]] as state on [[measurable functions]])**
\linebreak
For $\Omega$ a [[locally compact topological space|locally compact]] [[Hausdorff space]] equipped with a compatible structure of a classical [[probability space]], hence a [[measure space]] which normalized total measure $\int_\Omega d\mu = 1$, let $\mathcal{A} \coloneqq C_0(\Omega)$ be the algebra of [[continuous function]] with values in the [[complex numbers]] and [[vanishing at infinity]], regarded as a [[star algebra]] by pointwise [[complex conjugation]]. Then forming the [[expectation value]] with respect to $\mu$ defines a [[state on a star-algebra|state]] (def. \ref{StateOnAStarAlgebra}):

$$
  \array{
    C_0(\Omega)
      &\overset{\langle - \rangle_\mu}{\longrightarrow}&
    \mathbb{C}
    \\
    A &\mapsto& \int_\Omega A \, d\mu
  }
$$

\end{example}

(e.g. [Landsman 2017, p. 16-17](#Landsman17))

+-- {: .num_example #ElementsOfHilbertSpaceAsPureStates}
###### Example
**(elements of a [[Hilbert space]] as [[pure states]] on [[bounded operators]])**

Let $\mathcal{H}$ be a [[complex numbers|complex]] [[separable Hilbert space|separable]] [[Hilbert space]] with [[inner product]] $\langle -,-\rangle$ and let $\mathcal{A} \coloneqq \mathcal{B}(\mathcal{H})$ be the algebra of [[bounded operators]], regarded as a [[star algebra]] under forming [[adjoint operators]]. Then for every element $\psi \in \mathcal{H}$ of unit [[norm]] $\langle \psi,\psi\rangle = 1$ there is the [[state on a star-algebra|state]] (def. \ref{StateOnAStarAlgebra}) given by

$$
  \array{
    \mathcal{B}(\mathcal{H})
      &\overset{\langle -\rangle_\psi}{\longrightarrow}&
     \mathbb{C}
     \\
     A &\mapsto& \langle \psi \vert\, A \, \vert \psi \rangle &\coloneqq& \langle \psi, A \psi \rangle
  }
$$

These are [[pure states]] (def. \ref{PureStateOnAStarAlgebra}).

More general states in this case are given by [[density matrices]].

=--

\begin{example}
\label{StatesOnGroupAlgebrasAreUnitaryRepresentations}
**(states on group algebras are unitary representations)**
\linebreak
  Let $G$ be a [[discrete group]] and $\mathbb{C}[G]$ its [[group algebra]] regarded as a complex [[star-algebra]] under the combined operation of [[inverse element|inversion]] of group elements and [[complex conjugation]] of the [[coefficients]].

Then each state $\rho \,\colon\,\mathbb{C}[G] \longrightarrow \mathbb{C}$ (Def. \ref{StateOnAStarAlgebra}) arises from a [[unitary representation]] $\widehat{(\text{-})} \,\colon\, \longrightarrow U(\mathscr{H})$ on a [[Hilbert space]] $(\mathscr{H}, \langle -\vert-\rangle)$ with a [[cyclic vector]] $\psi \in \mathscr{H}$ as
$$
  \rho(g)
  \;=\;
  \big\langle
    \psi
   \big\vert
    \widehat{g} \cdot \psi
  \big\rangle
  \,.
$$

(With suitable adjustments, this statement generalizes from discrete to [[topological groups]].)
\end{example}

This is due to [Gelfand & Raikov (1943) (2)](#GelfandRaikov43), [Naimark (1956) §30 Thm. 1](#Naimark56) (reviewed at [eom](https://encyclopediaofmath.org/wiki/Positive-definite_function)) further generalization in [Saworotnow (1970)](#Saworotnow70), [(1972)](Saworotnow72), textbook account in [Dixmier (1977) Thm. 13.4.5 (ii)](#Dixmier77).

## Properties

### Closure properties

\begin{example}\label{Mixtures}
**(mixtures, convex combinations)**

For $k \in \mathbb{N}_+$, let

* $\big( \rho_i \colon \mathcal{A} \to \mathbb{C}  \big)_{i = 1}^k$

  be a $k$-[[tuple]] of states;

* $\big( p_i \in \mathbb{R}_{\geq 0} \big)_{i = 1}^k$,
  $\underoverset{i = 1}{k}{\sum} p_i \;=\; 1$

  be a [[probability distribution]] on the [[finite set]] $\{1, \cdots, k\}$

then the [[convex combination]]

$$
  \underoverset{i = 1}{k}{\sum} p_i \cdot \rho_i
  \;\;\;
  \in
  \;
  \mathcal{A}^\ast
$$

is another state on the star-algebra $\mathcal{A}$.

\end{example}

\begin{prop}\label{OperatorStateCorrespondence}
**(operator-state correspondence)**

For $\rho \;\colon\; \mathcal{A} \to \mathbb{C}$ a state, with a non-null observable $O \in \mathcal{A}$, $\rho(O^\ast O) \neq 0$, then also

\[
  \label{OperatorImageOfState}
  \rho_O
  \;\colon\;
  A
    \;\mapsto\;
  \tfrac{1}{ \rho(O^\ast O) }
  \cdot
  \rho\big( O^\ast \cdot A \cdot O \big)
\]

is a state.

\end{prop}
\begin{proof}

To check positivity (eq:Positivity), we compute for any $A \in \mathcal{A}$ as follows:

$$
  \begin{aligned}
    \rho_O
    \big(
      A^\ast
      \cdot
      A
    \big)
    & =
    \tfrac{1}{\rho(O^\ast O)}
    \cdot
    \rho
    \big(
      O^\ast
      \cdot
      (
        A^\ast
        \cdot
        A
      )
      \cdot
      O
    \big)
    \\
    & =
    \tfrac{1}{\rho(O^\ast O)}
    \cdot
    \rho
    \big(
      (
        A \cdot O
      )^\ast
      \cdot
      (
        A
        \cdot
        O
      )
    \big)
    \\
    & \geq 0
    \,,
  \end{aligned}
$$

where the first step is the definition (eq:OperatorImageOfState) the second step uses the [[anti-homomorphisms]]-property of the star-involution, and the last step follows by the assumed positivivity of $\rho$.

To check normalization (eq:Normalization), we observe that:

$$
  \begin{aligned}
    \rho_O(\mathbf{1})
    & =
    \tfrac{1}{\rho(O^\ast O)}
    \cdot
    \rho
    \big(
      O^\ast \cdot \mathbf{1} \cdot O
    \big)
    \\
    & =
    \tfrac{1}{\rho(O^\ast O)}
    \cdot
    \rho
    \big(
      O^\ast \cdot O
    \big)
    \\
    & = 1
    \,.
  \end{aligned}
$$

\end{proof}

### Fell's theorem

See at *[[Fell's theorem]]*.

### Gleason's theorem

See at *[[Gleason's theorem]]*.

## Related concepts

* [[GNS construction]]

[[!include states and observables -- content]]

## References

### Exposition

* {#Meyer95} [[Paul-André Meyer]], Section I.1.1 in: *Quantum Probability for Probabilists*,  Lecture Notes in Mathematics **1538**, Springer 1995 ([doi:10.1007/BFb0084701](https://link.springer.com/book/10.1007/BFb0084701))

* {#Gleason09} [[Jonathan Gleason]], *The $C^*$-algebraic formalism of quantum mechanics* (2009) [[[Gleason09.pdf:file]], [[GleasonAlgebraic.pdf:file]]]

* {#Gleason11} [[Jonathan Gleason]], *From Classical to Quantum: The $F^\ast$-Algebraic Approach*, contribution to *[VIGRE REU 2011](http://www.math.uchicago.edu/~may/VIGRE/VIGREREU2011.html)*, Chicago (2011) [[pdf](https://www.math.uchicago.edu/~may/VIGRE/VIGRE2011/REUPapers/Gleason.pdf), [[GleasonFAlgebraic.pdf:file]]]

Textbook account:

* {#BratteliRobinson79} [[Ola Bratteli]], [[Derek W. Robinson]], §2.3 in: *Operator Algebras and Quantum Statistical Mechanics* -- vol 1: *$C^\ast$- and $W^\ast$-Algebras. Symmetry Groups. Decomposition of States.*, Springer (1979, 1987, 2002) [[doi:10.1007/978-3-662-02520-8](https://doi.org/10.1007/978-3-662-02520-8)]

* [[Nikolay Bogolyubov]], A. A. Logunov, A. I. Oksak, I. T. Todorov, G. G. Gould, p. 234 in: *Algebra of Observables and State Space* [[doi:10.1007/978-94-009-0491-0_6](https://doi.org/10.1007/978-94-009-0491-0_6)],  Chapter in: *General principles of quantum field theory*, Mathematical Physics and Applied Mathematics **10**, Kluwer (1990) [[doi:10.1007/978-94-009-0491-0](https://doi.org/10.1007/978-94-009-0491-0)]

Discussion under the name "positive linear functionals":

* {#Steward76} [[James D. Steward]], *Positive definite functions and generalizations, an historical survey*, The Rocky Mountain Journal of Mathematics **6** 3 (1976) 409-434 [[jstor:44236118](https://www.jstor.org/stable/44236118)]

* {#Dixmier77} [[Jacques Dixmier]], §2 of: *$C^\ast$-algebras*, North Holland (1977)

* {#Murphy90} [[Gerard Murphy]], §3.3 in: _$C^\ast$-algebras and Operator Theory_, Academic Press (1990) [[doi:10.1016/C2009-0-22289-6](https://doi.org/10.1016/C2009-0-22289-6)]

* Wikipedia, *[Positive linear functional](https://en.wikipedia.org/wiki/Positive_linear_functional)*

* [[eom]], *[Positive-definite function](https://encyclopediaofmath.org/wiki/Positive-definite_function)*

With an eye towards [[density matrices]] and their [[entropy]]:

* Paolo Facchi, Giovanni Gramegna, Arturo Konderak, around (6) in: *Entropy of quantum states* ([arXiv:2104.12611](https://arxiv.org/abs/2104.12611), [spire:1860877](https://inspirehep.net/literature/1860877))

With an eye towards [[Gelfand-Tsetlin algebra|Gelfand-Tsetlin algebras]] and in the generality of [[conditional expectation values]]:

* [[Anatoly Vershik]], Def. 1 in: *Gel'fand-Tsetlin algebras, expectations, inverse limits, Fourier analysis*, pp. 619 in: [[Pavel Etingof]], [[Vladimir S. Retakh]], [[Isadore Singer]] (eds.) *The Unity of Mathematics* In Honor of the Ninetieth Birthday of [[I. M. Gelfand]], ([arXiv:math/0503140](https://arxiv.org/abs/math/0503140), [ISBN:978-0-8176-4467-3](https://www.springer.com/gp/book/9780817640767))

### Original articles

* {#HaagKastler64} [[Rudolf Haag]], [[Daniel Kastler]], equation (3) in: *An algebraic approach to quantum field theory*, Journal of Mathematical Physics, **5** (1964) 848-861 [[doi:10.1063/1.1704187](https://doi.org/10.1063/1.1704187), [spire:9124](https://inspirehep.net/literature/9124)]

* {#KadisonRingrose} Richard Kadison, John Ringrose, _Fundamentals of the theory of operator algebras_, AMS (1991)

* {#BordemannWaldmann96} [[Martin Bordemann]], [[Stefan Waldmann]], _Formal GNS Construction and States in Deformation Quantization_, Commun. Math. Phys. (1998) 195: 549.  ([arXiv:q-alg/9607019](https://arxiv.org/abs/q-alg/9607019), [doi:10.1007/s002200050402](https://doi.org/10.1007/s002200050402))

* {#Fredenhagen03} [[Klaus Fredenhagen]], section 2 of _Algebraische Quantenfeldtheorie_, lecture notes, 2003 ([[FredenhagenAQFT2003.pdf:file]])

* {#HalvorsonMueger06} [[Hans Halvorson]], [[Michael Müger]], def. 1.11 in _Algebraic Quantum Field Theory_ ([arXiv:math-ph/0602036](https://arxiv.org/abs/math-ph/0602036))

* {#FredenhagenRejzner12} [[Klaus Fredenhagen]], [[Katarzyna Rejzner]], definition 2.4 in _Perturbative algebraic quantum field theory_, In _Mathematical Aspects of Quantum Field Theories_, Springer 2016 ([arXiv:1208.1428](https://arxiv.org/abs/1208.1428))

* {#KhavkineMoretti15}  [[Igor Khavkine]], [[Valter Moretti]], _Algebraic QFT in Curved Spacetime and quasifree Hadamard states: an introduction_, Chapter 5 in [[Romeo Brunetti]] et al. (eds.) _Advances in Algebraic Quantum Field Theory_, Springer, 2015 ([arXiv:1412.5945](https://arxiv.org/abs/1412.5945))

* {#Rejzner16} [[Katarzyna Rejzner]], section 2.1.2 of _Perturbative Algebraic Quantum Field Theory_, Mathematical Physics Studies, Springer 2016 ([web](https://link.springer.com/book/10.1007%2F978-3-319-25901-7))

* {#FredenhagenLindner13} [[Klaus Fredenhagen]], Falk Lindner, _Construction of KMS States in Perturbative QFT and Renormalized Hamiltonian Dynamics_, Communications in Mathematical Physics Volume 332, Issue 3, pp 895-932, 2014-12-01 ([arXiv:1306.6519](https://arxiv.org/abs/1306.6519))

* {#Landsman17} [[Klaas Landsman]], around def. 2.4 in _Foundations of quantum theory -- From classical concepts to Operator algebras_, Springer Open 2017 ([doi:10.1007/978-3-319-51777-3](https://link.springer.com/book/10.1007/978-3-319-51777-3), [pdf](https://link.springer.com/content/pdf/10.1007%2F978-3-319-51777-3.pdf))

* {#Duetsch18} [[Michael Dütsch]], section 2.5 of _[[From classical field theory to perturbative quantum field theory]]_, 2018

* Nicolò Drago, [[Valter Moretti]], _The notion of observable and the moment problem for $\ast$-algebras and their GNS representations_ ([doi:1903.07496](https://arxiv.org/abs/1903.07496), [spire:1725528](http://inspirehep.net/record/1725528))

For more references see at _[[operator algebra]]_.

### The case of group algebras

The characterization of states on [[group algebras]]:

* {#GelfandRaikov43} И. М. Гельфанд, Д. А. Райков, *Неприводимые унитарные представления локально бикомпактных групп*, Матем. сб., 13(55):2–3 (1943)  301–316 [[mathnet pdf](https://www.mathnet.ru/links/728d630fdf2220b1ea8bdb2a5f6a5488/sm6181.pdf)]

  \linebreak

  [[Israel Gelfand]], [[Dmitri Raikov]], *Irreducible unitary representations of locally bicompact groups*, Recueil Mathématique. N.S., 13(55) 2–3 (1943) 301–316 [[mathnet:eng/sm6181](https://www.mathnet.ru/eng/sm6181)]

* {#Naimark56} [[Mark A. Naimark]], *Normed Rings*, Moscow (1956)

* {#Saworotnow70} Parfeny P. Saworotnow, *Representation of a topological group on a Hilbert module*, Duke Math. J. **37** 1  (1970) 145-150 [[doi:10.1215/S0012-7094-70-03720-8](https://projecteuclid.org/journals/duke-mathematical-journal/volume-37/issue-1/Representation-of-a-topological-group-on-a-Hilbert-module/10.1215/S0012-7094-70-03720-8.short)]

* {#Saworotnow72} Parfeny P. Saworotnow, *Generalized Positive Linear Functionals on a Banach Algebra with an Involution*, Proceedings of the AMS, **31** 1 (1972) 299-304  [[jstor:2038564](https://www.jstor.org/stable/2038564)]

* [Steward (1976) §4](#Steward76)

* [Dixmier (1977) Thm. 13.4.6](#Dixmier77)

* [[eom]], *[Positive-definite function](https://encyclopediaofmath.org/wiki/Positive-definite_function)*

* Wikipedia, *[Positive-definite function on a group](https://en.wikipedia.org/wiki/Positive-definite_function_on_a_group)*

[[!redirects states on a star-algebra]]

[[!redirects states on star-algebras]]

[[!redirects state on a star algebra]]
[[!redirects states on a star algebra]]
[[!redirects states on star algebras]]

[[!redirects state on a C-star algebra]]
[[!redirects states on a C-star algebra]]
[[!redirects state on a C* algebra]]
[[!redirects states on a C* algebra]]

[[!redirects state on an operator algebra]]
[[!redirects state of an operator algebra]]
[[!redirects states of an operator algebra]]
[[!redirects states of operator algebras]]
[[!redirects state on an operator algebra]]
[[!redirects states on an operator algebra]]
[[!redirects states on operator algebras]]
[[!redirects state in AQFT]]
[[!redirects states in AQFT]]
[[!redirects state in operator algebra]]
[[!redirects states in operator algebra]]
[[!redirects state in AQFT and operator algebra]]
[[!redirects states in AQFT and operator algebra]]
[[!redirects state of an algebra]]
[[!redirects states of an algebra]]
[[!redirects states of algebras]]
[[!redirects state on an algebra]]
[[!redirects states on an algebra]]
[[!redirects states on algebras]]

[[!redirects normal state]]
[[!redirects normal states]]

[[!redirects positive-definite function]]
[[!redirects positive-definite functions]]

[[!redirects positive linear functional]]
[[!redirects positive linear functionals]]


state-field correspondence

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#Contents#
* table of contents
{:toc}

## Idea

> The following needs clarification.

A field operator $\phi$ in a [[quantum field theory]] with a distinguished [[vacuum]] vector $|0\rangle$ defines its incoming state as the $|\phi_{in}\rangle \coloneqq U(0,-\infty)\phi |0\rangle$ i.e. as the limit when time goes to infinity of the state $\phi|0\rangle$, here $U(t_1,t_2)$ is the evolution operator from $t_1$ to $t_2$ (which may be written as $U(t_2-t_1)$ when the Hamiltonian is time-independent), which is by definition the inverse of $U(t_2,t_1)$ for $t_2\gt t_1$. The assignment $\phi\mapsto |\phi_{in}\rangle$ is a [[bijection]] for [[conformal field theory|conformal field theories]].

## Related concepts

* [[state-operator correspondence]]

## References

Discussion in [[conformal field theory]]:

* [[Paul Ginsparg]], Section 3.4 in: *Applied Conformal Field Theory*, lectures at: *[Fields, strings, critical phenomena, Les Houche Summer School 1988](https://inis.iaea.org/search/search.aspx?orig_q=RN:21047524)*([arXiv:hep-th/9108028](https://arxiv.org/abs/hep-th/9108028))

* [[Martin Schottenloher]], Rem. 9. 11 in: *Foundations of Two-Dimensional Conformal Quantum Field Theory* ([pdf](https://www.mathematik.uni-muenchen.de/~schotten/LNP-cft-pdf/09_978-3-540-68625-5_Ch09_23-08-08.pdf)), chapter 9 in: *A Mathematical Introduction to Conformal Field Theory*, Lecture Notes in Physics, Springer 2008 ([doi:10.1007/978-3-540-68628-6](https://link.springer.com/book/10.1007/978-3-540-68628-6), [web](https://www.mathematik.uni-muenchen.de/~schotten/LNP-cft-pdf/))


states and observables -- content

**[[quantum probability theory]] -- [[observables]] and [[states]]**

* **[[states]]**

  * [[classical state]]

  * [[quantum state]]

    * [[space of states (in geometric quantization)]]

    * [[state on a star-algebra]], [[quasi-state]]

    * [[qbit]], [[Bell state]]

      [[dimer]], [[tensor network state]]

    * [[quantum state preparation]]

    * [[probability amplitude]], [[quantum fluctuation]]

    * [[pure state]]

      * [[wave function]]

      * [[bra-ket]]

      * [[Bell state]]

    * [[quantum superposition]], [[quantum interference]]

    * [[quantum entanglement]]

    * [[quantum measurement]]

      [[wave function collapse]]

      [[Born rule]]

      [[deferred measurement principle]]

      [[quantum reader monad]]

      [[measurement problem]]

    * [[superselection sector]]

    * [[mixed state]], [[density matrix]]

      [[entanglement entropy]]

      [[holographic entanglement entropy]]

    * [[coherent quantum state]]

  * [[ground state]], [[excited state]]

  * [[quasi-free state]]

  * [[Fock space]], [[second quantization]]

  * [[vacuum]], [[vacuum state]]

    * [[Hadamard state]]

    * [[vacuum diagram]]

    * [[vacuum expectation value]], [[vacuum amplitude]], [[vacuum fluctuation]]

    * [[vacuum energy]]

    * [[vacuum polarization]]

    * [[interacting vacuum]]

    * [[thermal vacuum]], [[KMS state]]

    * [[vacuum stability]]

      * [[false vacuum]], [[tachyon]], [[Coleman-De Luccia instanton]]

    * [[theta vacuum]]

    * [[perturbative string theory vacuum]]

      * [[non-geometric string theory vacuum]]

      * [[landscape of string theory vacua]]

  * [[entangled state]]

  * [[tensor network state]]

    * [[matrix product state]]

    * [[tree tensor network state]]

* **[[observables]]**

  * [[quantum observable]], [[beable]]

  * [[algebra of observables]], [[star-algebra]]

  * [[Bohr topos]]

  * [[quantum operator (in geometric quantization)]]

  * [[quantum operation]], [[quantum effect]], [[effect algebra]]

  * in [[quantum field theory]]

    * [[local observable]]

    * [[polynomial observable]]

      * [[linear observable]]

        * [[field observable]]

    * [[regular observable]]

    * [[microcausal observable]]

    * [[normal-ordered product]], [[time-ordered products]], [[retarded product]]

    * [[Wick algebra]]

    * [[scattering amplitude]]

    * [[interacting field algebra of observables]], [[Bogoliubov's formula]]

* [[GNS construction]]

* [[theorems]]

  * [[order-theoretic structure in quantum mechanics]]

    * [[Gleason's theorem]]

    * [[Alfsen-Shultz theorem]]

    * [[Harding-Döring-Hamhalter theorem]]

    * [[Kochen-Specker theorem]]

    * [[Nuiten's lemma]]

  * [[Wigner's theorem]]

  * [[no-cloning theorem]]

  * [[Bell's theorem]]


static spacetime

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=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A [[spacetime]] is called _static_ if it is [[stationary spacetime|stationary]] (admits a [[Killing vector field]] which is [[timelike]]) and in addition irrotational (...).

## Properties

### Ehlers transformation

An _[[Ehlers transformation]]_ is a transformation that [[action|acts]] on the parameters of [[static spacetimes]] solving the [[Einstein field equations]] and generates other solutions of the field equations that are in general [[stationary spacetimes]].

## References

See also

* Wikipedia, _[Static spacetime](https://en.wikipedia.org/wiki/Static_spacetime)_

[[!redirects static spacetimes]]


stationary spacetime

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=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A [[spacetime]] is called _stationary_ if it admits a [[Killing vector field]] which is [[timelike]].

If the spacetime in addition is irrotational (...) it is called a _[[static spacetime]]_.

## Properties

### Ehlers transformation

An _[[Ehlers transformation]]_ is a transformation that [[action|acts]] on the parameters of [[static spacetimes]] solving the [[Einstein field equations]] and generates other solutions of the field equations that are in general [[stationary spacetimes]].

## References

See also

* Wikipedia, _[Stationary spacetime](https://en.wikipedia.org/wiki/Stationary_spacetime)_

[[!redirects stationary spacetimes]]


statistic

# Statistic-s
* table of contents
{: toc}

## Idea

A _statistic_ is a function of a sample where the function itself is independent of the sample's distribution.

Warning: Although the plural is 'statistics', the entry [[statistics]] is about the wider subject and not a collection of statistic-s.

## Literature

* wikipedia: [statistic](http://en.wikipedia.org/wiki/Statistic), [U-statistic](http://en.wikipedia.org/wiki/U-statistic), [statistical theory](http://en.wikipedia.org/wiki/Statistical_theory)

* Leah Jager, Jon A. Wellner, _On the "Poisson boundaries" of the family of weighted Kolmogorov statistics_, Anirban DasGupta, ed., Lecture Notes--Monograph Series __45__,
A Festschrift for Herman Rubin (Beachwood, Ohio, USA: Institute of Mathematical Statistics, 2004), 319-331, [MR2126907](http://www.ams.org/mathscinet-getitem?mr=2126907) [euclid](http://projecteuclid.org/euclid.lnms/1196285400)  [doi](http://dx.doi.org/10.1214/lnms/1196285400)

[[!redirects statistic]]


statistical ensemble

## References

See also

* Wikipedia, _[Statistical ensemble](https://en.wikipedia.org/wiki/Statistical_ensemble_%28mathematical_physics%29)_

[[!redirects statistical ensembles]]

[[!redirects ensemble]]
[[!redirects ensembles]]


statistical inference

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\tableofcontents

## Idea

By *statistical inference* broadly one refers to deducing from partial [[data analysis]] information of [[statistics|statistical]] nature, such as about the [[probability distributions]] that may give rise to the observed data.

## Related concepts

* [[statistical significance]]

## References

Textbook accounts:

* George Casella, Roger L. Berger, *Statistical Inference*, Duxbury (2002) [[pdf](https://mybiostats.files.wordpress.com/2015/03/casella-berger.pdf)]

See also:

* Wikipedia, *[Statistical inference](https://en.wikipedia.org/wiki/Statistical_inference)*


statistical learning theory

#Contents#
* table of contents
{:toc}

## Idea

A framework for [[machine learning]].

## References

Textbook accounts:

* [[Vladimir Vapnik]], _Statistical Learning Theory_, Wiley (1998) [[ISBN:978-0-471-03003-4](https://www.wiley.com/en-us/Statistical+Learning+Theory-p-9780471030034)]

* Trevor Hastie, Robert Tibshirani, Jerome Friedman, *The Elements of Statistical Learning: Data Mining, Inference, and Prediction*  [[webpage](https://web.stanford.edu/~hastie/ElemStatLearn/), [pdf](https://web.stanford.edu/~hastie/ElemStatLearn/printings/ESLII_print12_toc.pdf), [doi:10.1007/978-0-387-84858-7](https://link.springer.com/book/10.1007/978-0-387-84858-7)]

See also:

* Wikipedia, *[Statistical learning theory](https://en.wikipedia.org/wiki/Statistical_learning_theory)*

* [[David Corfield]], [[Bernhard Schölkopf]], [[Vladimir Vapnik]], _Falsificationism and statistical learning theory: Comparing the Popper and Vapnik-Chervonenkis dimensions_, Journal for General Philosophy of Science **40** 1 (2009) 51-58 [[doi:10.1007/s10838-009-9091-3](https://doi.org/10.1007/s10838-009-9091-3)]

[[!redirects statistical learning theories]]


statistical mechanics

+-- {: .rightHandSide}
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#### Measure and probability theory
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__Statistical mechanics__ is the theoretical study of systems with a large number of degrees of freedom, and in particular statistical features of ensembles of large systems. An ensemble is a theoretical term denoting a collection of similar large systems under consideration, where statistical features of the ensemble are considered. Alternatively, an ensemble may be a probability distribution of similar physical systems. Statistical mechanics wants to extract a small number of relevant "macroscopical" features of a physical system determined as the average over an ensemble. One may consider ensembles with fixed features, such as the number of particles or the total energy, and in which hypothetically an ergodic hypothesis is true: after sufficient passage of time the system will evolve in a system which is arbitrary close to any given member of the ensemble.

## Related entries

* [[thermodynamics]], [[ergodic theory]]

* [[Brownian motion]]

* [[Liouville's equation]]

* [[Langevin equation]]

* [[stochastic process]]

* [[dissipation]], [[fluctuation-dissipation theorem]]

* [[random matrix theory]]

* [[Euclidean field theory]]

* [[KMS state]]

category: physics

[[!redirects statistical physics]]


statistical significance

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#Contents#
* table of contents
{:toc}

## Idea

The _statistical significance_ of a [[measurement]] outcome $x$ of some [[variable]] $X$ is a measure of how unlikely it is that this outcome would be observed if $X$ were a [[random variable]] under given default assumptions, the latter often called the "standard model" or the "[[null hypothesis]]" and denoted $H_0$. Hence

1. a low statistical significance of the observed value $x$ _suggests_ that the default assumptions, hence the "standard model" or the "null hypothesis", is correct, whence $x$ is then also called a _[[null result]]_;

1. a high statistical significance of the observed value $x$ _suggests_ that the default assumptions, hence the "standard model" or the "null hypothesis", might be inaccurate and hence "to be rejected", or, rather, to be replaced by an improved assumption/model/hypothesis $H_1$ under which the observation $x$ becomes implied with higher probability.

More precisely, in general statistical significance is given as the [[probability]] $P(X \geq x\vert H_0)$ that a [[random variable]] $X$ with values in the [[real numbers]] takes a value in the [[interval]] $w$ of values equal or greater the observed value $x$. This probability is known as the _[[p-value]]_. The lower this probability, the higher the statistical significance of observing $x$.

> graphics grabbed from [Sinervo 02](#Sinervo02)

In the special, but, by the [[central limit theorem]], generic situation that the [[probability distribution]] $p(X\vert H_0)$ of $X$ under hypothesis $H_0$ is a [[normal distribution]],
it follows that the [[probability]] $P(X \geq x\vert H_0)$ is a [[monotone decreasing function]] of $x$, so that the value of $x$ itself is a measure of its statistical significance -- the larger $x$, the larger its statistical significance, in this case. It is natural then to express the magnitude $x \in \mathbb{R}$ in [[physical unit|units]] of [[standard deviations]] $\sigma$ of the given [[normal distribution]] $p(X\vert H_0)$. This expression of statistical significances in terms of [[standard deviations]] is common in [[particle physics]] (e.g. [Sinervo 02](#Sinervo02)), see [below](#ParticlePhysics).

As with much of [[statistics]], the concept of statistical significance is mostly used at the interface between [[theory (physics)|theory]] and [[experiment]], hence is part of the process of _[[coordination]]_, and as such subject to subtleties of real world activity that are not manifest in its clean mathematical definition. Often policy making and/or financial decisions depend on estimating and interpreting statistical significances. This has led to some lively debate about their use and misuse, see [below](#PossibleMisuse) for more. As a general rule of life, for a mathematical result to work well in applications, you need to understand what it says and what it does not say. There has also been considerable debate between [[Bayesianism|Bayesian]] and frequentist statisticians as to cogency of the use of statistical significance, see [below](#Bayes).

## Origin of significance threshold

The threshold $p\lt 0.05$ seems to date back to ([Fisher 1926](#Fisher26))

>… it is convenient to draw the line at about the level at which we can say: “Either there is something in the treatment, or a coincidence has occurred such as does not occur more than once in twenty trials.”…

>If one in twenty does not seem high enough odds, we may, if we prefer it, draw the line at one in fifty (the 2 per cent point), or one in a hundred (the 1 per cent point). Personally, the writer prefers to set a low standard of significance at the 5 per cent point, and ignore entirely all results which fail to reach this level. A scientific fact should be regarded as experimentally established only if a properly designed experiment rarely fails to give this level of significance.

This last sentence—implicitly about experimental replication—is often conveniently forgotten by researchers with only a rough working knowledge statistics, whereby a _single_ $p$-value under $0.05$ is seen as a discovery, rather than merely evidence.

## In particle physics
 {#ParticlePhysics}

In [[particle physics]], it has become customary to require statistical significance levels of $5 \sigma$ (5 [[standard deviations]]) in order to claim that a given observation is a real effect (e.g. [Sinervo 02, Section 5.3](#Sinervo02), [Dorigo 15, Section 3](#Dorigo15)). This corresponds to a probability ([[p-value]]) of about

$$
  5 \sigma
  \;\leftrightarrow\;
  p
  \;=\;
  2.8 \cdot 10^{-7}
  \;=\;
  0.00000028
$$

that the observation is a random fluctuation under the null hypothesis.

Notice that this is rather more stringent than the p-value of $p = 0.05$, corresponding to $\lt 2 \sigma$, which has been much used and much criticized in other areas of science, see [below](#PossibleMisuse).

Of course the $5 \sigma$-criterion for detectoin is a convention as any other. It has been argued in [Lyons 13b](#Lyons13b), [Dorigo 15](#Dorigo15) that different detection-thresholds should be used for different experiments. These suggestions are subjective and not generally agreed on either:

\begin{center}

\end{center}

> table taken from [Lyons 13b, p. 4](#Lyons13b)

\begin{center}

\end{center}

> table taken from [Dorigo 15, p. 16](#Dorigo15)

Regarding effects of current interest, notice that

1. the significance $4 \sigma$ for anomalies in the [[anomalous magnetic moment]] $(g_\mu-2)$ of the [[muon]] required by [Lyons 13b, p. 4](#Lyons13b) has been reached by summer 2018, see [there](anomalous+magnetic+moment#Anomalies);

1. the significance $3 \sigma$ for [[flavour anomalies]] in [[B meson]] decays required both by [Lyons 13b, p. 4](#Lyons13b) and by [Dorigo 15, p. 16](#Dorigo15) has also been much exceeded by summer 2018, see [there](https://ncatlab.org/nlab/show/flavour+anomaly#Idea).

## In other sciences

In pharmacology, for instance, hypothesis testing at some significance level is used in drug trials. Where those taking drug $A$ fare on average better than those on drug $B$, one might calculate the likelihood that such an effect would be found by chance given that the two drugs were in fact equally effective.

## Possible misuse
 {#PossibleMisuse}

There have been a number of criticisms of the uses to which $p$-values have been put in scientific practice (e.g., [ZilMcCl 08](#ZilMcCl08), [GSRCPGA 16](#GSRCPGA16)). The American Statistical Association has published a statement on $p$-values ([WassLaz 16](#WassLaz16)), claiming that

> the widespread use of 'statistical significance' (generally interpreted as '$p \leq 0.05$') as a license for making a claim of a scientific finding (or implied truth) leads to considerable distortion of the scientific process.

## Frequentist versus Bayesian statistics
 {#Bayes}

On a more fundamental level, [[Bayesianism|Bayesian]] statisticians have taken issue with the frequentist practice of hypothesis testing (see, e.g., [Jaynes 03, sec 9.1.1](#Jaynes03), [D'Agostini 03](#DAgostini03)), arguing that the value that should be sought is the probability of a proposed hypothesis conditional on the data, rather than the probability of the data under some null hypothesis. The frequentist concept of probability cannot allow for such a thing as a hypothesis having a probability.

A useful comparison of the frequentist and Bayesian approaches as employed in particle physics, and a call for their reconciliation, is in ([Lyons 13a](#Lyons13a)):

> for physics analyses at the CERN’s LHC, the aim is, at least for determining parameters and setting upper limits in searches for various new phenomena, to use both approaches; similar answers would strengthen confidence in the results, while differences suggest the need to understand them in terms of the somewhat different questions that the two approaches are asking. It thus seems that the old war between the two methodologies is subsiding, and that they can hopefully live together in fruitful cooperation.

## Related concepts

* [[statistical inference]]

## References

One early discussion is in

* {#Fisher26}  RA Fisher, _The Arrangement of Field Experiments_, Journal of the Ministry of Agriculture of Great Britain **33** (1926) pp 503–513.

Review includes

* {#Sinervo02} Pekka K. Sinervo, _Signal Significance in Particle Physics_, in Proceedings of _Advanced Statistical Techniques in Particle Physics_  Durham, UK, March 18-22, 2002 ([arXiv:hep-ex/0208005](https://arxiv.org/abs/hep-ex/0208005), [spire:601052](http://inspirehep.net/record/601052))

* {#Jaynes03} [[Edwin Jaynes]], _Probability theory: The logic of science_, Cambridge University Press, 2003.

* {#Lyons13a} [[Louis Lyons]], _Bayes and Frequentism: a Particle Physicist's perspective_,  Journal of Contemporary Physics Volume 54, 2013 - Issue 1
 ([arXiv:1301.1273](https://arxiv.org/abs/1301.1273), [doi:10.1080/00107514.2012.756312](https://doi.org/10.1080/00107514.2012.756312))

* {#Lyons13b} [[Louis Lyons]], _Discovering the Significance of 5 sigma_ ([arXiv:1310.1284](https://arxiv.org/abs/1310.1284))

* {#Dorigo15} [[Tommaso Dorigo]], _Extraordinary claims: the $0.000029\%$ solution_, EPJ Web of Conferences 95, 02003 (2015) ([doi:10.1051/epjconf/20159502003](https://doi.org/10.1051/epjconf/20159502003))

* {#Barlow19} Roger John Barlow, _Practical Statistics for Particle Physics_ ([arXiv:1905.12362](https://arxiv.org/abs/1905.12362))

See also

* Wikipedia, _[Statistical significance](https://en.wikipedia.org/wiki/Statistical_significance)_

* Leland Wilkinson and Task Force on Statistical Inference, Statistical methods in psychology journals: guidelines and explanations, American Psychologist 54 (1999), 594–604, [pdf](http://www.apa.org/pubs/journals/releases/amp-54-8-594.pdf)

Cautionary remarks on misuse of the concept include the following

* {#ZilMcCl08} Stephen Ziliak and Deirdre McCloskey, _The Cult of Statistical Significance_, Michegan University Press, 2008, [book](https://www.press.umich.edu/186351/cult_of_statistical_significance).

* {#DAgostini03} Giulio D'Agostini, _Bayesian reasoning in data analysis: A critical introduction_, World Scientific Publishing, 2003.

* $n$Cafe: [Fetishizing p-Values](http://golem.ph.utexas.edu/category/2010/09/fetishizing_pvalues.html) by Tom Leinster

* {#GSRCPGA16} Greenland, Senn, Rothman, Carlin, Poole, Goodman, Altman, _Statistical tests, P values, confidence intervals, and power: a guide to misinterpretations_, Eur J Epidemiol. 2016 Apr;31(4):337-50. ([doi:10.1007/s10654-016-0149-3](https://www.ncbi.nlm.nih.gov/pubmed/27209009))

* {#WassLaz16} Ronald Wasserstein, Nicole Lazar, _The ASA's Statement on $p$-Values: Context, Process, and Purpose_, The American Statistician 70(2), 2016, pp. 129-133 ([doi:10.1080/00031305.2016.1154108](https://doi.org/10.1080/00031305.2016.1154108))

[[!redirects statistical significances]]

[[!redirects p-value]]
[[!redirects p-values]]


statistics

# Statistics
* table of contents
{: toc}

This entry (apart from a few remarks) will be mainly about general statistics and mathematical statistics. For more about physical applications in [[statistical mechanics]] and probabilistic interpretation of [[quantum mechanics]], see there. There is also a technical notion of [[statistic]] (singular).

## Idea

Statistics studies the [[analysis]] of collections of
random or sample data, and the probabilistic
likelihood of various inferences
on the basis of these data, as well as the mathematical regularities in large [[ensemble]]s of occurrences of such data.

In physics, statistics also pertains to the behaviour of
large ensembles of particles. For identical particles,
this is the subject of [[particle statistics]] and for
general systems the subject of [[statistical mechanics]].

Mathematical statistics is based on [[probability theory]].
Most of the standard formalism uses [[measure theory]] as used in probability. Statistical mechanics in addition heavily uses [[ergodic theory]].

## Definitions

A __statistical model__ is a [[measurable function]] $P : \Theta \to \Delta(X)$, where $X$ and $\Theta$ are [[measurable spaces]] and $\Delta(X)$ is the [[simplex]] of [[probability measures]] over $X$.

## Related entries

* [[probability]], [[probability theory]]

* [[statistical inference]]

* [[statistical significance]]

* [[stochastic process]],

* [[statistical mechanics]]

* [[time series]],

* [[Giry monad]],

* [[Bayesian reasoning]]

* [[conditional expectation]]

* [[information geometry]]

* [[Bayesian interpretation of quantum mechanics]],

## Literature

* Michael Schmitt, _Statistics for theorists_,  quick 3-lecture intro for theoretical physicists at TASI 2020 (each around 1 and half hours) lec1 [mp4](https://physicslearning.colorado.edu/tasi/tasi_2020/videos/Schmitt%20Lecture%202/Schmitt%20Lecture%202.mp4) slides [pdf](https://physicslearning.colorado.edu/tasi/tasi_2020/lectureNotes/Schmitt%20Lecture%201.pdf), lec2 [mp4](https://physicslearning.colorado.edu/tasi/tasi_2020/videos/Schmitt%20Lecture%202/Schmitt%20Lecture%202.mp4) [pdf](https://physicslearning.colorado.edu/tasi/tasi_2020/lectureNotes/Schmitt%20Lecture%202.pdf), lec3 [mp4](https://physicslearning.colorado.edu/tasi/tasi_2020/videos/Schmitt%20Lecture%203/Schmitt%20Lecture%203.mp4) [pdf](https://physicslearning.colorado.edu/tasi/tasi_2020/lectureNotes/Schmitt%20lecture%203.pdf)

* wikipedia: [statistics](http://en.wikipedia.org/wiki/Statistics), [statistic](http://en.wikipedia.org/wiki/Statistic) (singular!), [list of statistical packages](http://en.wikipedia.org/wiki/List_of_statistical_packages),
[descriptive statistics](http://en.wikipedia.org/wiki/Descriptive_statistics), [inferential statistics](http://en.wikipedia.org/wiki/Inferential_statistics), [data mining](http://en.wikipedia.org/wiki/Data_mining), [statistical theory](http://en.wikipedia.org/wiki/Statistical_theory), [estimation theory](http://en.wikipedia.org/wiki/Estimation_theory), [statistical significance](http://en.wikipedia.org/wiki/Statistical_significance), [regression analysis](http://en.wikipedia.org/wiki/Regression_analysis), [time series analysis](http://en.wikipedia.org/wiki/Time_series_analysis), [misuse of statistics](http://en.wikipedia.org/wiki/Misuse_of_statistics), [applied statistics](http://en.wikipedia.org/wiki/Applied_statistics), [Bayesian inference](https://en.wikipedia.org/wiki/Bayesian_inference), [multivariate statistics](https://en.wikipedia.org/wiki/Multivariate_statistics), [maximal likelihood estimation](https://en.wikipedia.org/wiki/Maximum_likelihood_estimation), [Fisher information](https://en.wikipedia.org/wiki/Fisher_information)

* [[Peter McCullagh]], _What is a statistical model?_,      Ann. Statist. __30__:5 (2002), 1225-1310 [euclid](http://projecteuclid.org/euclid.aos/1035844977) [MR1936320](http://www.ams.org/mathscinet-getitem?mr=1936320) [doi](http://dx.doi.org/10.1214/aos/1035844977) -- on applying category theory to describe statistical models.

* Lior Pachter, Bernd Sturmfels, _Tropical geometry of statistical models_, PNAS 101 no. 46,  16132–16137, [doi](htTP://dx.doi.org/10.1073/pnas.0406010101)

category: probability, physics


Stavros Garoufalidis

* [webpage](http://people.mpim-bonn.mpg.de/stavros)

## Selected writings

On the [[universal Vassiliev invariant]] of the [[unknot]] ([[Wheels theorem]]):

* [[Dror Bar-Natan]], [[Stavros Garoufalidis]], [[Lev Rozansky]], [[Dylan Thurston]], _Wheels, wheeling, and the Kontsevich integral of  the unknot_ ([q-alg/9703025](http://arxiv.org/abs/q-alg/9703025))

On [[analytically continued Chern-Simons theory]]:

* {#Garoufalidis07} [[Stavros Garoufalidis]], _Chern-Simons theory, analytic continuation and arithmetic_ ([arXiv:0711.1716](http://arxiv.org/abs/0711.1716))

On the [[colored Jones polynomial]]:

* [[Stavros Garoufalidis]], Martin Loebl, _A non-commutative formula for the colored Jones function_ ([arXiv:math/0411505](https://arxiv.org/abs/math/0411505))

Relation of the [[colored Jones polynomial]] to [[Lie algebra weight systems]] on [[chord diagrams]]:

* [[Dror Bar-Natan]], [[Stavros Garoufalidis]], _On the Melvin–Morton–Rozansky conjecture_, Invent math (1996) 125: 103 ([doi:10.1007/s002220050070](https://doi.org/10.1007/s002220050070), [[BarNatanGaroufaldis96.pdf:file]])

On the [[complex volume]] of [[hyperbolic 3-manifolds]] and the [[volume conjecture]]:

* {#GaroufalidisThurstonZickert11} [[Stavros Garoufalidis]], [[Dylan Thurston]], [[Christian Zickert]], _The complex volume of $SL(n,\mathbb{C})$-representations of 3-manifolds_ ([arXiv:1111.2828](http://arxiv.org/abs/1111.2828), [Euclid](http://projecteuclid.org/euclid.dmj/1259332507))


Steady State

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Cosmology
+-- {: .hide}
[[!include cosmology -- contents]]
=--
#### Gravity
+--{: .hide}
[[!include gravity contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

The _Steady State theory_ is a [[model (in theoretical physics)|model]] in [[cosmology]]. Originally proposed as a [[phenomenology|phenomenologically viable]] model, it was later ruled out by observation, which instead prefers the [[Big Bang]] [[FRW model]].

## Related concepts

* [[cosmic inflation]]

## References

* Wikipedia, _[Steady State theory](http://en.wikipedia.org/wiki/Steady_State_theory)_

[[!redirects Steady State]]
[[!redirects Steady State model of cosmology]]
[[!redirects steady state model of cosmology]]
[[!redirects Steady State cosmology]]
[[!redirects steady state cosmology]]
[[!redirects Steady State model]]
[[!redirects steady state model]]
[[!redirects Steady State theory]]
[[!redirects steady state theory]]


Steenrod algebra

> under construction

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

#Contents#
* table of contents
{:toc}

## Idea

For $p$ a [[prime number]], the _Steenrod algebra_ $\mathcal{A}$ is the [[associative algebra]] over the [[prime field]] $\mathbb{F}_p$ of [[cohomology operations]] on [[ordinary cohomology]] with [[coefficients]] in $\mathbb{F}_p$. For $p = 2$ this is the $\mathbb{F}_2$-algebra generated by the [[Steenrod square]] operations. (Often the case $p = 2$ is understood by default.)

For any prime $p$, the mod-$p$ Steenrod algebra furthermore has the structure of a [[Hopf algebra]] over $\mathbb{F}_p$, non-commutative but co-commutative. Under forming [[linear duals]] this gives the _dual Steenrod algebra_, traditionally denoted $\mathcal{A}_\ast$ instead of $\mathcal{A}^\ast$. This is a [[Hopf algebra]] over $\mathbb{F}_p$ that is commutative, but non-co-commutative.

The dual $\mathbb{F}_p$ Steenrod algebra is a special case of a [[commutative Hopf algebroid]] structure canonically induced on the self [[generalized homology]] $E_\bullet(E)$ of any [[ring spectrum]] $E$ for which $E_\bullet \to E_\bullet(E)$ is a [[flat morphisms]]. Therefore in this general case one sometimes speaks of "dual $E$-Steenrod algebras".

The [[Ext]]-groups between [[comodules]] for these [[commutative Hopf algebroids]] $E_\bullet(E)$ prominently appear on the second page of the $E$-[[Adams spectral sequence]], see there for more.

## Definition

### For ordinary cohomology

(...) (e.g. [Mosher-Tangora 68, section 6](#MosherTangora68), [Lurie 07](#Lurie07)) (...)

### For generalized cohomology

The Steenrod algebra and its standard properties, such as the [Adem relations](#OnAdemRelations), follow abstractly from the [[Cotor]] groups of [[comodules]] over any [[commutative Hopf algebroid]].

This is due to ([May 70, 11.8](#May70)). A review is in ([Ravenel, appendix 1, theorem A1.5.2](#Ravenel)).

In particular for $E$ a suitable [[E-infinity ring]], its self-[[generalized homology]] $E_\bullet(E)$ forms a (graded-)[[commutative Hopf algebroid]] over $E_\bullet$.

See at _[Hopf algebroid structure -- For generalized cohomology](#HopfAlgebroidstructureForGeneralized)_ below. For more see at _[[Adams spectral sequence]] -- [The first page](Adams+spectral+sequence#FirstPageAndHopfAlgebroid)_.

## Properties

### Presentation by generators and relations

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

The _Serre-Cartan basis_ is the [[subset]] of elements of the Steenrod algebra on those of the form

$$
  Sq^{i_1} \circ Sq^{i_2} \circ \cdots \circ Sq^{i_n}
$$

where $i_k \in \mathbb{N}$ subject to the relation

$$
  i_k \geq 2 i_{k+1}.
$$

=--

This is indeed a [[linear basis]] for the $\mathbb{F}_2$-[[vector space]] underlying the Steenrod algebra.

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

The [[Steenrod square]] operations satisfy the following relation, for all
for all $0 \lt i \lt 2 j$:

$$
  Sq^i \circ Sq^j
   =
  \sum_{k = 0}^{[i/2]}
  \left(
    \array{
      j - k - 1
      \\
      i - 2k
    }
  \right)_{mod 2}
  Sq^{i + j -k} \circ Sq^k
$$

=--

These are called the **Ádem relations** ([Ádem 52](#Adem52)).

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

The Ádam relations precisely generate the [[ideal]] of relations among the Serre-Cartan basis elements, def. \ref{AdemRelations}, in the Steenrod algebra.

=--

More generally, for other [[prime numbers]]:

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

Let $p$ be a [[prime number]]. Write $\mathbb{F}_p$ for the corresponding [[prime field]].

The **mod $p$-Steenrod algebra** $\mathcal{A}_{\mathbb{F}_p}$ is the graded co-commutative [[Hopf algebra]] over $\mathbb{F}_p$ which is

* for $p = 2$ generated by elements denoted $Sq^n$ for $n \in \mathbb{N}$, $n \geq 1$;

* for $p \gt 2$ generated by elements denoted $\beta$ and $P^n$ for $n \in \mathbb{N}$, $n \geq 1$

(called the **Serre-Cartan basis elements**)  whose product is subject to the following relations (called the **Ádem relations**):

**for $p = 2$**:

for $0 \lt h \lt 2k$  the

$$
  Sq^h Sq^k
  \;=\;
  \underoverset{i = 0}{[h/2]}{\sum}
  \left(
    \array{
      k -i - 1
      \\
      h - 2i
    }
  \right)
  Sq^{h + k -i} Sq^i
  \,,
$$

**for $p \gt 2$**:

for $0 \lt h \lt p k$ then

$$
  P^h P^k
  \;=\;
  \underoverset{i = 0}{[h/p]}{\sum}
  (-1)^{h+i}
  \left(
    \array{
      (p-1)(k-i) - 1
      \\
      h - p i
    }
  \right)
  P^{h +k - i}P^i
$$

and if $0 \lt h \le p k$ then

$$
  \begin{aligned}
    P^h \beta P^k
    & =\;
    \underoverset{[h/p]}{i = 0}{\sum}
    (-1)^{h+i}
    \left(
      \array{
        (p-1)(k-i)
        \\
        h - p i
      }
    \right)
    \beta P^{h+k-i}P^i
    \\
    & +
    \underoverset{[(h-1)/p]}{i = 0}{\sum}
    (-1)^{h+i-1}
    \left(
      \array{
        (p-1)(k-i) - 1
        \\
        h - p i - 1
      }
    \right)
    P^{h+k-i} \beta P^i
  \end{aligned}
$$

and whose coproduct $\Psi$ is subject to the following relations:

**for $p = 2$**:

$$
  \Psi(Sq^n)
  \;=\;
  \underoverset{k = 0}{n}{\sum} Sq^k \otimes Sq^{n-k}
$$

**for $p \gt 2$**:

$$
  \Psi(P^n)
  \;=\;
  \underoverset{n}{k = 0}{\sum}
  P^k \otimes P^{n-k}
$$

and

$$
  \Psi(\beta) = \beta \otimes 1 + 1 \otimes \beta
  \,.
$$

=--

e.g. ([Kochmann 96, p. 52](#Kochmann96))

### Dual Steenrod algebra

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

The $\mathbb{F}_p$-[[linear dual]] of the mod $p$-Steenrod algebra (def. \ref{SteenrodAlgebraForHFp}) is itself naturally a graded [[commutative Hopf algebroid|commutative Hopf algebra]] (with coproduct the linear dual of the original product, and vice versa), called the **dual Steenrod algebra** $\mathbb{A}_{\mathbb{F}_p}^\ast$.

=--

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

There is an isomorphism

$$
  \mathcal{A}^\ast_{\mathbb{F}_p}
  \simeq
  H_\bullet( H \mathbb{F}_p, \mathbb{F}_p )
  =
  \pi_\bullet( H \mathbb{F}_p \wedge H \mathbb{F}_p )
  \,.
$$

=--

(e.g. [Rognes 12, remark 7.24](#Rognes12))

#### Milnor's theorem
 {#MilnorTheorem}

We now give the generators-and-relations description of the dual Steenrod algebra $\mathcal{A}^\ast_{\mathbb{F}_p}$ from def. \ref{DualSteenrodAlgebraForHPf}, in terms of linear duals of the generators for $\mathcal{A}_{\mathbb{F}_p}$ itself, according to def. \ref{SteenrodAlgebraForHFp}.

In the following, we use for $p = 2$ the notation

$$
  P^n \coloneqq Sq^{2n}
$$

$$
  \beta \coloneqq Sq^1
  \,.
$$

This serves to unify the expressions  for $p = 2$ and for $p \gt 2$ in the following. Notice that for all $p$

* $P^n$ has even degree $deg(P^n) = 2n(p-1)$;

* $\beta$ has odd degree $deg(\beta) = 1$.

+-- {: .num_theorem #MilnorTheoremOnDualSteenrodAlgebra}
###### Theorem
**([[Milnor's theorem]])**

The dual mod $p$-Steenrod algebra $\mathcal{A}^\ast_{\mathbb{F}_p}$ (def. \ref{DualSteenrodAlgebraForHPf}) is, as an [[associative algebra]], the free [[graded commutative algebra]]

$$
  \mathcal{A}^\ast_{\mathbb{F}_p}
    \simeq
  Sym_{\mathbb{F}_p}(\xi_1, \xi_2, \cdots, \;\tau_0, \tau_1, \cdots)
$$

on generators:

* $\xi_n$ being the linear dual to $P^{p^{n-1}} P^{p^{n-2}} \cdots P^p P^1$, has degree $2p^i - 2$;

* $\tau_n$ being linear dual to $P^{p^{n-1}} P^{p^{n-2}} \cdots P^p P^1\beta$, has degree $2p^i - 1$.

Moreover, the coproduct on $\mathcal{A}^\ast_{\mathbb{F}_p}$ is given by

$$
  \Psi(\xi_n)
  =
  \underoverset{k = 0}{n}{\sum} \xi_{n-k}^{p^k} \otimes \xi_k
$$

and

$$
  \Psi(\tau_n)
  =
  \tau_n \otimes 1
  +
  \underoverset{k=0}{n}{\sum}
  \xi_{n-k}^{p^k}\otimes \tau_k
  \,,
$$

where we set $\xi_0 \coloneqq 1$.

=--

This is due to ([Milnor 58](#Milnor58)). See for instance ([Kochmann 96, theorem 2.5.1](#Kochmann96))

### Hopf algebroid structure
 {#HopfAlgebraStructure}

#### For ordinary cohomology

The Steenrod algebra for mod $p$ coefficients is a [[Hopf algebra]] over $\mathbb{F}_p$ which is graded commutative and non-co-commutative.

This is due to ([Milnor 58](#Milnor58)). A review is in [Ravenel, ch. 3, section 1](#Ravenel).

#### For generalized cohomology
 {#HopfAlgebroidstructureForGeneralized}

More generally:

+-- {: .num_lemma #SelfHomologyIsModuleOverCohomologyRing}
###### Lemma

Let $R$ be an [[E-∞ ring]] and let $A$ an [[E-∞ algebra]] over $R$.
The self-[[generalized homology]] $A^R_\bullet(A)$
is naturally a [[module]] over the [[cohomology ring]] $A_\bullet$
via applying the [[homotopy groups]] $\infty$-functor $\pi_\bullet$
to the canonical inclusion

$$
  A \stackrel{\simeq}{\rightarrow}
  A \underset{R}{\wedge} R
  \stackrel{}{\rightarrow}
  A \underset{R}{\wedge} A
  \,.
$$

=--

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

Let $R$ be an [[E-∞ ring]] and let $A$ an [[E-∞ algebra]] over $R$.
If the the $A_\bullet$-[[module]] $A^R_\bullet(A)$ of
lemma \ref{SelfHomologyIsModuleOverCohomologyRing} is a [[flat module]],
then

1. $(A_\bullet, A_\bullet(A))$ is a [[commutative Hopf algebroid]] over $R_\bullet$;

1. $A^R_\bullet(X)$ is a left $A^R_\bullet(A)$-module for every $R$-[[∞-module]] $X$.

=--

This is due to ([Baker-Lazarev 01](#BakerLazarev01)), further discussed  in ([Baker-Jeanneret 02](#BakerJeanneret02)) (there expressed in terms of the presentation by [[highly structured ring spectra]]).
A review is also in ([Ravenel, chapter 2, prop. 2.2.8](#Ravenel)).

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

Using the arguments of ([Adams 74](#Adams74), [Ravenel 86](#Ravenel86)).

The [[flat module|flatness]] condition implies that there is an [[equivalence]]

$$
  \pi_\bullet\left(
    A \underset{R}{\wedge} A \underset{R}{\wedge} X
  \right)
  \stackrel{\simeq}{\longrightarrow}
  A_\bullet^R(A) \underset{A_\bullet}{\otimes} A_\bullet^R(X)
  \,.
$$

Combining this with the map in lemma \ref{SelfHomologyIsModuleOverCohomologyRing}
yields the [[coaction]]

$$
  A_\bullet^R(X)
  \longrightarrow
  \pi_\bullet\left(
    A \underset{R}{\wedge} A \underset{R}{\wedge} X
  \right)
  \stackrel{\simeq}{\longrightarrow}
  A_\bullet^R(A) \underset{A_\bullet}{\otimes} A_\bullet^R(X)
  \,.
$$

=--

These (dual) $E$-Steenrod algebra Hopf algebroids have also been called "[[brave new algebra|brave new Hopf algebroids]]" ([Baker](#Baker),  [Baker-Jeanneret 02](#BakerJeanneret02))

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

For $E = H \mathbb{F}_2$ the [[Eilenberg-MacLane spectrum]],
this reproduces the [[Hopf algebra]] structure on the dual
ordinary Steenrod algebra as above.

=--

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

For $E = $ [[MU]] then by [[Quillen's theorem on MU]] $\pi_\bullet(MU) \simeq L$ is the [[Lazard ring]] and the Hopf algebroid $(\pi_\bullet(MU), MU_\bullet(MU))$ is described by the [[Landweber-Novikov theorem]].

=--

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

For $E = $ [[BP]] the analog is the content of the [[Adams-Quillen theorem]]. The [[Landweber exact functor theorem]] was proven using the $B P_\bullet(B P)$-Hopf algebroid.

=--

### Relation to Adams spectral sequence

#### For ordinary cohomology

By construction, the total [[cohomology]] $H^\bullet(X,\mathbb{Z}_2)$ of a [[topological space]] $X$, naturally is a [[module]] over the Steenrod algebra:

a cohomology element is represented by a [[cocycle]] which is a map $X \longrightarrow B^n \mathbb{Z}_2$ and the action of a [[Steenrod square]] on this is just by [[composition]].

In this form the Steenrod algebra appears in the second page of the
[[Adams spectral sequence]] which computes $[\Sigma^\infty X, \Sigma^\infty Y]$ for [[topological spaces]] $X$ and $Y$: that second page is given by the [[Ext]]-groups

$$
  E_2 = Ext_A(H^\bullet(X, \mathbb{Z}_2), H^\bullet(Y,\mathbb{Z}_2))
$$

computed in the [[category of modules|category of A-modules]] for $A$ the Steenrod algebra.

#### For generalized cohomology

More generally, For $R$ an [[E-infinity ring]] such that its dual $R$-[[Steenrod algebra]] in the form of the self-[[homology]] $R_\bullet(R)$ is a (graded-)[[commutative Hopf algebroid]] over $R_\bullet = \pi_\bullet(R)$ (see at [Steenrod algebra -- Hopf algebroid structure](Steenrod+algebra#HopfAlgebraStructure)), then the $E^2$-term of the $E$-Adams spectral sequence is an [[Ext]] of $E_\bullet(E)$-[[comodules]]

$$
  E^2 \simeq Ext_{R_\bullet(R)}(R_\bullet, R_\bullet(X))
  \,.
$$

See the [references below](#ReferencesHopfAlgebroid).

## Related concepts

* [[cohomology operation]]

* [[power operation]]

* [[Pi-algebra]]

* [[Lambda-algebra]]

* [[Adams resolution]], [[Adams spectral sequence]], [[Adams-Novikov spectral sequence]]

* [[Brown-Gitler spectrum]]

* [[generic representation theory]]

## References

### General

Original articles include

* {#Adem52} [[José Ádem]], _The iteration of the Steenrod squares in algebraic topology_ , Proceedings of the National Academy of Sciences of the United States of America 38: 720–726 (1952)

* {#Milnor58} [[John Milnor]], _The Steenrod algebra and its dual_, Ann. of Math. 67 (1958), 150–171.

* {#SteenrodEpstein62} [[Norman Steenrod]], [[David Epstein]], _Cohomology operations_, Ann. of Math. Studies, no. 50, Princeton University Press, Princeton, N. J., 1962.

* {#Adams74} [[John Frank Adams]], part III, section 12 of _[[Stable homotopy and generalised homology]]_, University of Chicago Press (1974).

Comprehensive discussion of the ordinary Steenrod algebra, with proof is the Adem relations includes

* {#MosherTangora68} [[Robert Mosher]], [[Martin Tangora]], _Cohomology Operations and Application in Homotopy Theory_, Harper and Row (1968) ([pdf](www.maths.ed.ac.uk/~aar/papers/moshtang.pdf))

* {#Lurie07} [[Jacob Lurie]], 18.917 _[Topics in Algebraic Topology: The Sullivan Conjecture, Fall 2007](http://ocw.mit.edu/courses/mathematics/18-917-topics-in-algebraic-topology-the-sullivan-conjecture-fall-2007)_. (MIT OpenCourseWare: Massachusetts Institute of Technology), _[Lecture notes](http://ocw.mit.edu/courses/mathematics/18-917-topics-in-algebraic-topology-the-sullivan-conjecture-fall-2007/lecture-notes/)_

  Lecture 2 _Steenrod operations_ ([pdf](http://ocw.mit.edu/courses/mathematics/18-917-topics-in-algebraic-topology-the-sullivan-conjecture-fall-2007/lecture-notes/lecture2.pdf))

  Lecture 3 _Basic properties of Steenrod operations_ ([pdf](http://ocw.mit.edu/courses/mathematics/18-917-topics-in-algebraic-topology-the-sullivan-conjecture-fall-2007/lecture-notes/lecture3.pdf))

  Lecture 4 _The Adem relations_ ([pdf](http://ocw.mit.edu/courses/mathematics/18-917-topics-in-algebraic-topology-the-sullivan-conjecture-fall-2007/lecture-notes/lecture4.pdf))

  Lecture 5 _The Adem relations (cont.)_ ([pdf](http://ocw.mit.edu/courses/mathematics/18-917-topics-in-algebraic-topology-the-sullivan-conjecture-fall-2007/lecture-notes/lecture5.pdf))

The general algebraic approach was laid out in

* {#May70} [[Peter May]], _A general algebraic approach to Steenrod operations_, in _The Steenrod Algebra and its Applications_ (Proc. Conf. to Celebrate N. E. Steenrod's Sixtieth Birthday, Battelle Memorial Inst., Columbus, Ohio, 1970), Lecture Notes in Mathematics, Vol. 168, Springer, Berlin, 1970, pp. 153–231. ([pdf](http://www.math.uchicago.edu/~may/BOOKS/Algebraic.pdf))

* [[Robert Bruner]], [[Peter May]], J. E. McClure, M. Steinberger,
_$H_\infty$ ring spectra and their applications_, Lecture Notes in Math., Springer-Verlag, Berlin,

reviewed in ([Ravenel 86, A1.5](#Ravenel)).

Further textbook accounts include

* Harvey Margolis, _Spectra and the Steenrod algebra_, 1983 North-Holland

* {#Kochmann96} [[Stanley Kochmann]], section 2.5 of _[[Bordism, Stable Homotopy and Adams Spectral Sequences]]_, AMS 1996

* {#Schwede12} [[Stefan Schwede]], chapter II section 10 of _[[Symmetric spectra]]_, 2012 ([pdf](http://www.math.uni-bonn.de/~schwede/SymSpec-v3.pdf))

Lecture notes include

* {#Rognes12} [[John Rognes]], section 3 of _The Adams spectral sequence_, 2012 ([pdf](http://folk.uio.no/rognes/papers/notes.050612.pdf))

Review:

* {#Ravenel} [[Doug Ravenel]], _[[Complex cobordism and stable homotopy groups of spheres]]_ Academic Press 1986

  > (appendix 1, section 5 reviews the abstract algebraic definition).

* [[Agnès Beaudry]], [[Jonathan A. Campbell]], §3 in: *A Guide for Computing Stable Homotopy Groups*, in *Topology and Quantum Theory in Interaction*, Contemporary Mathematics **718**, Amer. Math. Soc. (2018)  [[arXiv:1801.07530](https://arxiv.org/abs/1801.07530), [ams:conm/718](https://bookstore.ams.org/view?ProductCode=CONM/718)]

  > (with an eye towards [[functorial quantum field theory]])

* Cary Malkievich, _The Steenrod Algebra_ ([pdf](http://math.stanford.edu/~carym/steenrod.pdf))

On the dual Steenrod algebra:

* J. Palmieri, _Some quotient Hopf algebras of the dual Steenrod algebra_ ([pdf](http://www.math.washington.edu/~palmieri/Papers/quotient.pdf))

For a commented list of furhter references see also

* _[Steenrod algebra learning seminar](http://www.math.uwo.ca/~sisaacso/steenrod-seminar/steenrod.xhtml)_

See also

* [online Adem relation calculator](http://math.berkeley.edu/~aaron/adem/)

### $E_\bullet$-Hopf algebroid structure
 {#ReferencesHopfAlgebroid}

The [[commutative Hopf algebroid]] structure on the dual $E$-Steenrod algebra $E_\bullet(E)$ and
its relation to the $E^2$-term in the
[[Adams spectral sequence]] is discussed in

* {#Baker} [[Andrew Baker]], _Brave new Hopf algebroids_, [pdf](http://www.maths.gla.ac.uk/~ajb/dvi-ps/brave-ha.pdf)

* {#BakerLazarev01} [[Andrew Baker]], [[Andrey Lazarev]], _On the Adams Spectral Sequence for R-modules_, Algebr. Geom. Topol. 1 (2001) 173-199 ([arXiv:math/0105079](http://arxiv.org/abs/math/0105079))

* {#BakerJeanneret02} [[Andrew Baker]] and Alain Jeanneret, _Brave new Hopf algebroids and extensions of $MU$-algebras_, Homology Homotopy Appl. Volume 4, Number 1 (2002), 163-173. ([Euclid](http://projecteuclid.org/euclid.hha/1139840059))

* {#Hovey03} [[Mark Hovey]], _Homotopy theory of comodules over a Hopf algebroid_ ([arXiv:math/0301229](http://arxiv.org/abs/math/0301229))

[[!redirects Steenrod algebras]]

[[!redirects Adem relation]]
[[!redirects Ádem relation]]
[[!redirects Adem relations]]
[[!redirects Ádem relations]]

[[!redirects dual Steenrod algebra]]
[[!redirects dual Steenrod algebras]]

[[!redirects E-Steenrod algebra]]
[[!redirects E-Steenrod algebras]]
[[!redirects dual E-Steenrod algebra]]
[[!redirects dual E-Steenrod algebras]]


Steenrod approximation theorem

[[!redirects Steenrod-Wockel approximation theorem]]

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Differential geometry
+--{: .hide}
[[!include synthetic differential geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The _Steenrod-approximation theorem_ states mild conditions under which an extension of a [[smooth function]] on a [[closed subset]] by a [[continuous function]] may itself be improved to an extension by a smooth function.

This is a smooth enhancement of the [[Tietze extension theorem]].

## Statement

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

Let $X$ be a finite [[dimension]]al [[connected]] [[smooth manifold]] [[manifold with corners|with corners]]. Let $\pi : E \to X$ be a locally trivial [[Diff|smooth]]  [[bundle]] with a [[locally convex]] [[manifold]] $N$ as typical [[fiber]] and $\sigma : X \to E$ a [[continuous function|continuous]] [[section]].

If $L \subset X$ is a [[closed subset]] and $U \subset X$ is an [[open subset]] such that $\sigma$ is [[smooth function|smooth]] in a [[neighbourhood]] of $L \setminus U$, then:

1. for each [[open neighbourhood]] $O$ of $\sigma(X)$ in $E$ there exists a [[section]] $\tau : X \to O$

   * which is [[smooth function|smooth]] in a neighbourhood of $L$;

   * and which equals $\sigma$ on $X \setminus U$;

1. there exists a [[homotopy]] $F : [0,1] \times X \to O$ between $\sigma$ and $\tau$ such that

   * each $F(t,-)$ is a section of $\pi$;

   * for $(t,x) \in [0,1] \times (X \setminus U)$ we have $F(t,x) = \sigma(x) = \tau(x)$.

=--

See ([Wockel](#Wockel))

$$
  \array{
    && O &\stackrel{id}{\to} & O &  \stackrel{id}{\to}& O
    \\
    & {}^{\mathllap{smooth}}\nearrow
    &
    {}_{\mathllap{\sigma|_{X \setminus U}}}\uparrow
     = \uparrow_{\mathrlap{\tau|_{X \setminus U}}}
    &&
   \uparrow^{\mathrlap{\sigma}}
    & \swArrow_F&
    \uparrow^{\exists \tau}
    & \nwarrow^{\mathrlap{smooth}}
    \\
    L \setminus U &\hookrightarrow & X \setminus U
    &\hookrightarrow& X
    &\stackrel{id}{\to}&
   X
   &\stackrel{}{\hookleftarrow}& L
  }
$$

## Examples

### Smoothing of delayed homotopies {#SmoothingOfDelayedHomotopies}

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

Let $f,g : Z \to Y$ be two [[smooth function]]s between [[smooth manifold]]s. Let  $\eta : Z \times [0,1] \to Y$ be a continuous [[delayed homotopy]] between them, constant in a [[neighbourhood]] $Z \times ([0,\epsilon) \coprod (1-\epsilon,1])$.

Then there exists also smooth homotopy between $f$ and $g$ which is itself continuously homotopic to $\eta$.

=--

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

To apply the [generalized Steenrod theorem](#GeneralizedSteenrodTheorem) with the notation as stated there, make the following identifications

* set $X \coloneqq Z \times [0,1]$;

* set $N = Y$;

* let $E = Z \times [0,1] \times Y$ be the trivial $Z$-bundle over $X$

  (so that sections of $E$ are equivalently functions $Z \times[0,1] \to Y$)

* let $(\sigma : X \to E) := (\eta : Z \times [0,1] \to Y)$ be the given continuous homotopy;

* set $L \coloneqq Z \times [0,1]$;

* let $U \coloneqq Z \times (0,1)$.

Then because by assumption $\eta$ is a continuous [[delayed homotopy]] between [[smooth function]]s, it follows that $\sigma$ is smooth in a neighbourhood $Z \times ([0,\epsilon) \coprod (1-\epsilon,1])$ of $L$.

So the theorem applies and provides a smooth  homotopy

$$
  \tau \colon [0,1] \times Z \to Y
$$

which moroever is itself (continuously) homotopic to $\eta$ via some continuous $F : [0,1] \times [0,1] \times Z \to Y$.

=--

## Related theorems

* the [[Hadamard lemma]]

* [[Borel's theorem]]

* the [[Tietze extension theorem]]

* the [[Whitney extension theorem]]

* [[embedding of smooth manifolds into formal duals of R-algebras]]

* [[smooth Serre-Swan theorem]]

* [[derivations of smooth functions are vector fields]]

## References

* [[Norman Steenrod]], Section 6.7 of: _The topology of fibre bundles_, Princeton Mathematical Series **14**, Princeton Univ. Press (1951) [[jstor:j.ctt1bpm9t5](https://www.jstor.org/stable/j.ctt1bpm9t5)]

* [[Morris Hirsch]], Chapter 2.2 (Thm. 2.5) in: _Differential topology_, Springer Graduate Texts in Mathematics **33** (1976) [[doi:10.1007/978-1-4684-9449-5](https://link.springer.com/book/10.1007/978-1-4684-9449-5), [gBooks](http://books.google.com/books/about/?id=iSvnvOodWl8C)]

* {#Wockel} [[Christoph Wockel]], _A Generalisation of Steenrod's Approximation Theorem_, Archivum mathematicum **45** 2 (2009) [[arXiv:0610252](http://arxiv.org/abs/math/0610252)]


Steenrod approximation theorem > history

See _[[Steenrod approximation theorem]]_.


Steenrod homology

## Idea

Steenrod Homology of a [[metric space|metric]] (or [[uniform space|uniform]], or [[Hausdorff space|Hausdorff]], or ...) space measures separations/holes by asking which _discrete_ or _approximate_ simplices can be filled-in to arbitrary fineness.

### Notation ###

The dimension-$p$ simplices of a simplicial complex $K$ will be denoted $K_p$, and we shall write $A \ll X$ to mean that $A$ is a compact subspace of $X$.

## The Definition

Given a metric space $(X,d)$, define the category $Reg (X,d)$, "of regular complexes", with objects pairs $(K,f)$ where

  * $ K $ is a locally-finite [simplicial complex]
  * $ f : K_0 \to X $ satisfies $ limsup_{e:K_1} d( \partial_0 e, \partial_1 e) = 0$

and with morphisms from $ (K,f)$ to $(L,g)$ being the inclusions of subcomplexes $ h : K \to L $ making $g$ an extension of $f$:
\[ Reg_d((K,f),(L,g)) = \{ h : hom(K,L) | f = g \circ h \} \]
Note that $Reg_d$ has finite pushouts.

The condition that $K$ be locally-finite means that arbitrary formal sums $\sum g_x x$ of $p$-simplices $x\in K_p$ have a well-defined boundary in the usual way:
\[ \partial \sum g_x x = \sum (-1)^j g_x \partial_j x \]
mentions a given simplex only finitely-many times.  Taking $g_x\in G$, for any abelian group $G$, this gives a functor $C_* : Reg(X,d) \to Ch $, valued in chain complexes.

**Definition** : *The Steenrod Homology $H_p(X,d)$ is the homology of the chain complex $\colim_{K,f} C_*$ in degree $p+1$.*

In pieces, this means that a Steenrod $p$-cycle may be represented by a triple $(K,f,\varphi)$ of a locally finite complex $K$, a $K_1$-regular map $K_0 \to X$, and a formal sum $\varphi$ of simplices of dimension $p+1$ of $K$ with $\partial \varphi = 0$.  The group operations may as well be represented in terms of the chains $\varphi$ for a single $K$ and $f$.

### Historical note ###

What we have just called $H_p$, Steenrod himself notated $H^{p+1}$.  It was early days yet.

## Pairing with Alexander-Spanier Cohomology

A compactly-supported Alexander-Spanier Cocycle is a (compactly-supported) cochain $\chi : X^{p+1} \to G$ whose derivation $d \chi$ vanishes on tuples that are *small enough*.  For such a cochain, the sums
\[ \varphi \frown \chi = \sum_{x :K_{p+1}} n_x (d\chi)(x) = \sum_x n_x \chi(\partial x)\]
are finite sums, and vanish for both coboundaries $\chi = d\omega$ and for boundaries $x = \partial y$ (because $d d = \partial \partial = 0$.  Thus the pairings descend to
\[ H_p( X,d ;\mathbb{Z}) \otimes H_c^p( X , G ) \to G \]
and all the reasonable variations you might consider.

## Continuities

### Colimits

Given a set $\{ X_i \}$ of metric spaces, their disjoint union can be given a metric in which separate summands are all distance $1$ from eachother.  This metric has
$ Reg( * \sqcup \coprod X_i) = colim Reg(* \sqcup X_i) $ and therefore also
\[ H_p(* \sqcup \coprod X_i) \simeq \bigoplus H_p( * \sqcup X_i ) \]

While $limsup d(\partial_0 x,\partial_1 x) = 0$ is a *topological* condition on *compact* metric spaces, it definitely is not so on noncompact spaces.  For this reason, one frequently considers the compactly-supported (compactly-generated?) homology as well,
\[ {}_c H_p(X) = \colim_{A \ll X } H_p(A) . \]

### Limits

Let $\dots \to X_{i+1} \to X_i \to \dots $ be a tower of *compact* metric spaces; in generous versions of Set-Theory, its limit is again a *compact* metric space.  In contrast to $colim$, one can reasonably say $Reg(lim_i X_i) = lim_i Reg(X_i)$; and moreover, since the morphisms of $Reg(X_i)$ are *inclusions of subcomplexes*, it follows that
\[ \colim_{Reg(lim X)} C_* \simeq lim_i \colim_{Reg(X_i)} C_* \]
the usual abstract nonsense relating to the [[Milnor sequence]], then gives short exact sequences
\[ 0 \to lim_i^1 H_{p+1}(X_i) \to H_p (lim X) \to lim_i H_p( X_i ) \to 0 \]

## Related entries

* [[Steenrod-Sitnikov homology]]

## References

* N. E. [[Steenrod]], Regular Cycles of Compact Metric Spaces,  Annals of Mathematics, Second Series, Vol. 41, No. 4 (Oct., 1940), pp. 833-851 [JSTOR](https://www.jstor.org/stable/1968863)

*  [[John Milnor]], _On axiomatic homology theory_, Pacific J. Math. Volume 12, Number 1 (1962), 337-341 ([Euclid](http://projecteuclid.org/euclid.pjm/1103036730))