product of distributions

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

#Contents#
* table of contents
{:toc}

## Idea

### Generally
 {#Generally}

The _product_ of two [[distributions]] is an operation that generalizes, when defined, the ordinary pointwise product of [[functions]], if we think of distributions as [[generalized functions]] (via [this prop.](non-singular+distribution#NonSingularDistributionsAreDenseInAllDistributions)).

However, as opposed to ordinary functions, the product of distributions -- if it is required to [[extension|extend]] the pointwise product of functions and to satisfy [[associativity]] and the [[product law]] for [[derivatives of distributions]] -- is not defined for arbitrary pairs of distributions, but only [[partial function|partially defined]] for those pairs whose [[Fourier transform of distributions|Fourier transforms]] satisfies a certain compatibility condition. Roughly: whenever the Fourier transform around any point of one factor does not decay exponentially in one direction of wave vectors, then the Fourier transform of the other factor has to decay exponentially in the _opposite_ direction of wave vectors.

This is known as _[[Hörmander's criterion]]_ on the _[[wave front set]]_ of the distributions ([Hörmander90](#Hoermander90)), and the study of distributions with attention to this condition is part of what is called _[[microlocal analysis]]_.

(If one is willing to consider changing the _ordinary_ pointwise product of functions, then is possible to define the product operation globally on all distributions, see at _[[Colombeau algebra]]_.)

The product of distributions with Hörmander's condition on their [[wave front set]] may be understood via [[Fourier transform of distributions|Fourier transform]], as follows.

Let $u,v \in  \mathcal{D}'(\mathbb{R}^1)$ be two [[distributions]], for simplicity of exposition taken on the [[real line]].

Since the product $u \cdot v$, is, if it exists, supposed to generalize the _pointwise_ product of smooth functions, it must be fixed locally: for every point $x \in \mathbb{R}$ there ought to be a [[compactly supported function|compactly supported]] [[smooth function]] ([[bump function]]) $b \in C^\infty_{cp}(\mathbb{R})$ with $f(x) = 1$ such that

$$
  b^2 u \cdot v = (b u) \cdot (b v)
  \,.
$$

But now $b v$ and $b u$ are both [[compactly supported distributions]], and these have the special property that their [[Fourier transform of distributions|Fourier transforms]] $\widehat{b v}$ and $\widehat{b u}$ are, in particular, [[smooth functions]] (by the [[Paley-Wiener-Schwartz theorem]]).

Moreover, the operation of [[Fourier transform]] intertwines pointwise products with [[convolution products]]. This means that if the product of distributions $u \cdot v$ exists, it must locally be given by the inverse Fourier transform of the [[convolution product]] of the Fourier transforms $\widehat {b u}$ and $\widehat b v$:

$$
  \widehat{ b^2 u \cdot v }(x)
  \;=\;
  \underset{\underset{k_{max} \to \infty}{\longrightarrow}}{\lim}
  \,
  \int_{- k_{max}}^{k_{max}} \widehat{(b u)}(k) \widehat{(b v)}(x - k) d k
  \,.
$$

(Notice that the converse of this formula holds as a fact: [this prop.](Fourier+transform#FourierTransformOfDistributionsInterchangesConvolutionOfDistributionsWithPointwiseProduct).)

This shows that the product of distributions exists once there is a [[bump function]] $b$ such that the [[integral]] on the right converges as $k_{max} \to \infty$.

Now the [[Paley-Wiener-Schwartz theorem]] says more, it says that the Fourier transforms $\widehat {b u}$ and $\widehat {b u}$ are polynomially bounded. On the other hand, the [[integral]] above is well defined if the [[integrand]] decreases at least quadratically with $k \to \infty$.
This means that for the convolution product to be well defined, either $\widehat {b u}$ has to polynomially decrease faster with $k \to \pm \infty$ than $\widehat {b v}$ grows in the _other_ direction, $k \to \mp \infty$ (due to the minus sign in the argument of the second factor in the [[convolution product]]), or the other way around.

Moreover, the degree of polynomial growth of the [[Fourier transform of distributions|Fourier transform]] increases by one with each [[derivative of distributions|derivative]]. Therefore if the [[product law]] for [[derivatives of distributions]] is to hold generally, we need that either $\widehat{b u}$ or $\widehat{b v}$ decays faster than _any_ polynomial in the opposite of the directions in which the respective other factor does not decay.

Here the set of directions of wave vectors in which the Fourier transform of a distribution localized around any point does not decay exponentially is called the _[[wave front set]]_ of a distribution. Hence the condition that the product of two distributions is well defined is that for each wave vector direction in the wave front set of one of the two distributions, the opposite direction must not be an element of the wave front set of the other distribution.

### In perturbative quantum field theory
 {#IdeaInPerturbativeQFT}

The issue of mutliplying distributions has prominently been perceived in [[perturbative quantum field theory]], where [[operator-valued distributions]] serve to give the [[algebra of observables]] such as the [[Wick algebra]] of the [[free fields]] or more generally the [[interacting field algebra]].

A popular impression is (or has been) that the failure of distributions to have a globally defined product is a failure of the mathematical formalism to support the structures needed to model [[perturbative quantum field theory]]. But in fact the opposite is true: Handling the product of distributions correctly via proper analysis of their [[wave front set]] and handling the [[point-extension of distributions]] properly via analysis of their [[scaling degree of a distribution|scaling degree]] leads to a mathematical rigorous construction and mathematically captures all the effects expected from the non-rigorous treatmeants, notably the [[renormalization]] freedom. This is the topic of [[causal perturbation theory]]/[[locally covariant perturbative quantum field theory]], see there for more.

## Definition
 {#Definition}

+-- {: .num_defn #ProductOfDistributions}
###### Definition
**(multiplication of distributions)

Let $u,v \in \mathcal{D}'(X)$ be two [[distributions]] such that the sum of their [[wave front sets]] $WF(u) + WF(v)$  does not intersect zero ([[Hörmander's criterion]]). Then their _product distribution_

$$
  u \cdot v \in \mathcal{D}'(X)
$$

is the [[pullback of a distribution|pullback]] (via [this prop.](pullback+of+a+distribution#PullbackOfDistributionsWhoseWaveFrontDoesNotIntersectNormalBundle)) of their [[tensor product of distributions|tensor product]] along the [[diagonal]] map $\Delta_X \;\colon\; X \to X \times X$:

$$
  u \cdot v
    \;\coloneqq\;
  \Delta_X^\ast( u \otimes v )
  \,.
$$

=--

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

Def. \ref{ProductOfDistributions} is indeed well defined.

=--

([Hörmander 90, theorem 8.2.10](#Hoermander90), [Melrose 03, prop. 4.11](#Melrose03))

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

We need to check that the [[pullback of distributions]] is well defined. By [this prop.](pullback+of+a+distribution#PullbackOfDistributionsWhoseWaveFrontDoesNotIntersectNormalBundle) this means to check that the [[wave front set]] of the $u \otimes v$ does not intersect the [[conormal bundle]] of the diagonal map

Now the conormal bundle of the diagonal map consists of those pairs of covectors whose sum vanishes:

$$
  N^\ast_{\Delta_X}
  =
  \left\{
    (x,(\xi, -\xi))
    \;\vert\;
    \xi \in T^\ast_x X
  \right\}
  \,.
$$

Moreover, by [this example](tensor+product+of+distributions#WaveFrontOfTensorProductDistribution) the wave front set of the tensor product distribution $u \otimes v$ is

$$
  WF(u \otimes v)
  \;\subset\;
  \left(
     WF(u) \times WF(v)
  \right)
    \;\cup\;
  \left(
    \left( supp(u) \times \{0\} \right)
    \times WF(v)
  \right)
    \;\cup\;
   \left(
     WF(u)
     \times
     \left(
       supp(v) \times \{0\}
     \right)
   \right)
  \,,
$$

Since any wave front set excludes the zero-section by definition, the second and the third summand in this union never intersect the above conormal bundle. The first summand intersects the above conormal bundle precisely if there is a covector in $WF(u)$ which is minus a covector contained in $WF(v)$. That this is not the case is precisely the assumption.

=--

## Properties

### General

+-- {: .num_prop #WaveFrontSetOfProductOfDistributionsInsideFiberProductOfFactorWaveFrontSets}
###### Proposition
**([[wave front set]] of [[product of distributions]] is inside [[fiber]]-wise [[sum]] of [[wave front sets]])**

Let $u,v \in \mathcal{D}'(X)$ be a [[pair]] of [[distributions]] satisfying [[Hörmander's criterion]], so that their product of distributions $u \cdot v$ (def. \ref{ProductOfDistributions}) exists by prop. \ref{ProductOfDistributionsViaDiagonalPullbackOfTensorProductIsWellDefined}. Then the [[wave front set]] of the product distribution is contained inside the fiberwise [[sum]] of the wave front set elements of the two factors:

$$
  WF(u \cdot v)
  \;\subset\;
  (WF(u) \cup (X \times \{0\}))
    +
  (WF(v) \cup (X \times \{0\}))
  \,.
$$

=--

([Hörmander 90, theorem 8.2.10](#Hoermander90))

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

By def. \ref{ProductOfDistributions} and prop. \ref{ProductOfDistributionsViaDiagonalPullbackOfTensorProductIsWellDefined} we have $u \cdot v = \Delta_X^\ast(u \otimes v)$. By [this example](tensor+product+of+distributions#WaveFrontOfTensorProductDistribution) the wave front set of the tensor product distribution $u \otimes v$ is

$$
  WF(u \otimes v)
  \;\subset\;
  \left(
     WF(u) \times WF(v)
  \right)
    \;\cup\;
  \left(
    \left( supp(u) \times \{0\} \right)
    \times WF(v)
  \right)
    \;\cup\;
   \left(
     WF(u)
     \times
     \left(
       supp(v) \times \{0\}
     \right)
   \right)
$$

and by [this prop. ](pullback+of+a+distribution#PullbackOfDistributionsWhoseWaveFrontDoesNotIntersectNormalBundle) we have

$$
  WF(\Delta_X^\ast (u \otimes v))
    \subset
  \Delta_X^\ast WF(u \otimes v)
  \coloneqq
  \left\{
    (x, k_1 + k_2 )
    \;\vert\;
    ((x,k_1),(x,k_2))
    \in
    WF(u \otimes v)
  \right\}
  \,.
$$

=--

More generally:

+-- {: .num_prop #PartialProductOfDistributionsOfSeveralVariables}
###### Proposition
**(partial product of [[distributions of several variables]])**

Let

$$
  K_1 \in \mathcal{D}'(X \times Y)
  \phantom{AAA}
  K_2 \in \mathcal{D}'(Y \times Z)
$$

be two [[distributions of two variables]]. For their [[product of distributions]] to be defined over $Y$, [[Hörmander's criterion]] on the [[pair]] of [[wave front sets]] $WF(K_1), WF(K_2)$  needs to hold for the [[wave front set|wave front]] [[wave vectors]] along $X$ and $Y$ taken to be zero.

If this is satisfied, then composition of [[integral kernels]] (if it exists)

$$
  (K_1 \circ K_2)(-,-)
  \;\coloneqq\;
  \underset{Y}{\int}
    K_1(-,y) K_2(y,-)
  dvol_Y(y)
  \;\in\;
  \mathcal{D}'(X \times Z)
$$

has [[wave front set]] constrained by

$$
  WF'(K_1 \circ K_2)
  \;\subset\;
  WF'(K_1) \circ WF'(K_2)
  \;\cup\;
  (X \times \{0\}) \times WF'(K_2)_Z
  \;\cup\;
  WF(K_1)_X \times (Z \times \{0\})
  \,,
$$

where on the left the composition symbol means composition of [[relations]] of [[wave vectors]] over points in $Y$. Explicitly this means that

$$
  \label{CompositionOfIntegralKernelsWaveFronConstraint}
  WF(K_1 \circ K_2)
  \;\subset\;
  \left\{
    (x,z, k_x, k_z)
    \;\vert\;
    \array{
      \left(
      (x,y,k_x,-k_y) \in WF(K_1)
      \,\,
      \text{and}
      \,\,
      (y,z,k_y, k_z) \in WF(K_2)
      \right)
      \\
      \text{or}
      \\
      \left(
        k_x = 0
        \,\text{and}\,
        (y,z,0,-k_z) \in WF(K_2)
      \right)
      \\
      \text{or}
      \\
      \left(
        k_z = 0
        \,\text{and}\,
        (x,y,k_x,0) \in WF(K_1)
      \right)
    }
  \right\}
$$

=--

([Hörmander 90, theorem 8.2.14](#Hoermander90))

$\,$

* for **continuity** see at _[[Hörmander topology]]_

### Non-existence of a global product
 {#NonExistenceOfAGlobalProduct}

The fact that there is no general extension of multiplication to distributions (without condition on the [[wave front set]]) is a famous [[no-go theorem]] of [[Laurent Schwartz]].

A quick way to see the problem is the following:

Let $H(x)$ be the [[Heaviside function]], we clearly have

$$
      H(x) = H^{n} (x)
$$

where the product on the right side is the product of classical functions. Applying differentiation and the product rule naivly results in a contradiction immediatly:
$$
   \delta(x) = n H^{n-1} (x) \delta(x)
$$

The inconsistency of this product is detected by the [[wave front sets]] as follows: The wave front set both of the [[Heaviside distribution]] as well as of the [[delta distribution]] on the line is $\{(0,k) \vert k \neq 0\}$ ([this example](wavefront+set#WaveFrontOfDeltaDistribution) and [this example](wavefront+set#WaveFrontSetOfHeavisideDistribution)). Therefore for both of these distributions mutliplication is only defined, according to def. \ref{ProductOfDistributions}, with a distribution $u$ for which there exists a [[bump function]] $b$ with $b(0) = 1$ such that $b \cdot u$ is again a bump function. This excludes the products of these distributions with themselves and with each other.

## Examples
 {#Examples}

+-- {: .num_defn #ProductOfADistributionWithANonSingularDistribution}
###### Definition
**(product of a [[distribution]] with a [[non-singular distributions]] is [[product of a distribution with a smooth function]])**

The [[wave front set]] of a [[non-singular distribution]] $u_f$ corresponding to a [[smooth function]] $f \in C^\infty(\mathbb{R}^n)$, is [[empty]] ([this prop.](Paley-Wiener-Schwartz+theorem#DecayPropertyOfFourierTransformOfCompactlySupportedFunctions)). Therefore the product of distributions (def. \ref{ProductOfDistributions}) of a [[non-singular distribution]] with any distribution $u$ is defined, and given by the [[product of distributions with smooth functions]]:

$$
  u_f \cdot u = f \cdot u = u(f\cdot (-))
  \,.
$$

=--

## Related concepts

* [[product of a distribution with a smooth function]]

* [[convolution product of distributions]]

* [[Hadamard distribution]]

* [[Hörmander topology]]

* [[microcausal functional]], [[Wick algebra]]

* [[operator-valued distribution]]

* [[Fourier transform of distributions]]

## References

* {#Hoermander71} [[Lars Hörmander]], _Fourier integral operators. I_.  Acta Mathematica 127, 79–183 (1971) ([Euclid](https://projecteuclid.org/euclid.acta/1485889699))

* {#Hoermander90} [[Lars Hörmander]], section 8.1 of _The analysis of linear partial differential operators_, vol. I, Springer 1983, 1990

* {#Melrose03} [[Richard Melrose]], chapter 1 of _Introduction to microlocal analysis_, 2003 ([pdf](http://www-math.mit.edu/~rbm/iml90.pdf))

* [[Michael Oberguggenberger]], _Products of distributions_, Journal für die reine und angewandte Mathematik (1986) Volume: 365, page 1-11 ([EuDML](https://eudml.org/doc/152801))

* [[Michael Oberguggenberger]], _Multiplication of Distributions and Applications to Partial Differential Equations_, Longman 1992

Slides with exposition of the role of multiplication of distributions in [[renormalization]] of [[Feynman diagrams]] via [[causal perturbation theory]]/[[perturbative AQFT]]:

* {#Brouder10} [[Christian Brouder]], _Multiplication of distributions_, 2010 ([[BrouderProductOfDistributions.pdf:file]])

[[!redirects products of distributions]]

[[!redirects multiplication of distributions]]
[[!redirects multiplications of distributions]]


product of distributions with a smooth function

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

#Contents#
* table of contents
{:toc}

## Definition

Let $n \in \mathbb{N}$ and consider $\mathbb{R}^n$ the [[Cartesian space]] of [[dimension]] $n$.

+-- {: .num_defn #ProductOfDistributionWithSmoothFunction}
###### Definition
**(product of a [[distribution]] with a [[smooth function]])**

For $u \in \mathcal{D}'(\mathbb{R}^n)$ a [[distribution]], and $f \in C^\infty(\mathbb{R}^n)$ a [[smooth function]], their _product_

$$
  f \cdot u \;\in\; C^\infty(\mathbb{R}^n)
$$

is the [[distribution]] given on a [[compactly supported function|compactly supported]] [[smooth function]] $g \in C^\infty_{cp}(\mathbb{R}^n) = \mathcal{D}(\mathbb{R}^n)$ by

$$
  (f\cdot u)(g) \;\coloneqq\; u\left( f \cdot g\right)
  \,,
$$

where on the right we have the application of $u$ regarded as a [[continuous linear functional]] $u \colon \mathcal{D}(\mathbb{R}^n) \to \mathbb{C}$ to the ordinary pointwise product of smooth functions $f \cdot g$.

=--

## Properties

+-- {: .num_defn #ProductOfADistributionWithANonSingularDistribution}
###### Definition
**([[product of distributions|product of a distribution]] with a [[non-singular distributions]] is product of distribution with a smooth function)**

The [[wave front set]] of a [[non-singular distribution]] $u_f$ corresponding to a [[smooth function]] $f \in C^\infty(\mathbb{R}^n)$, is [[empty]] ([this prop.](Paley-Wiener-Schwartz+theorem#DecayPropertyOfFourierTransformOfCompactlySupportedFunctions)). Therefore the product of distributions (def. \ref{ProductOfDistributions}) of a [[non-singular distribution]] with any distribution $u$ is defined, and given by the product of distributions with smooth functions according to def. \ref{ProductOfDistributionWithSmoothFunction}:

$$
  u_f \cdot u = f \cdot u = u(f\cdot (-))
  \,.
$$

=--

## Related concepts

* [[convolution product]]

* [[convolution product of distributions]]

* [[derivative of a distribution]]

## References

See also

* Wikipedia, _Distribution (mathematics) -- Multiplication by a smooth function_

[[!redirects products of distributions with a smooth function]]

[[!redirects product of a distribution with a smooth function]]
[[!redirects products of distributions with smooth functions]]

[[!redirects product of distributions with smooth functions]]
[[!redirects products of distributions with smooth functions]]

[[!redirects product of a distribution with smooth functions]]
[[!redirects products of a distribution with smooth functions]]


product of simplices

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Limits and colimits
+--{: .hide}
[[!include infinity-limits - contents]]
=--
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

This page describes aspects of the [[combinatorics]] of [[Cartesian products]] of [[simplicial sets]], mostly by describing (Prop. \ref{NonDegenerateSimplicesInProductOfSimplices}) the non-degenerate simplicices inside a [[Cartesian product]] $\Delta[p] \times \Delta[q]$ of basic [[n-simplices]] for $n = p,q$. These are enumerated by $(p,q)$-(un-)[[shuffles]] in a manner that for [[simplicial abelian groups]] is originally known from the [[Eilenberg-Zilber map]].

## Products of simplicial sets
 {#ProductsOfSimplicialSets}

\begin{proposition}\label{CartesianProductOfSimplicialSetsIsComponentwise}
**([[cartesian product]] of [[simplicial sets]])**\linebreak
For $K, L \,\in\,$ [[SimplicialSets]], their [[Cartesian product]],

$$
  K \times L
  \;\in\;
  SimplicialSets
$$

is the [[simplicial set]] whose $k$th component [[set]] is the [[Cartesian product]] of [[Sets]] of the components of the two factors

\[
  \label{ComponentSetsOfProductSimplicialSet}
  (K\times L)_k
  \;=\;
  K_k \times L_k
  \,,
\]

and whose face- and degeneracy maps are, similarly, the image under the Cartesian product-[[functor]] of the face- and degeneracy maps of the factors:

\[
  \label{StructureMapsOfProductSimpliciaSet}
  d^{K \times L}_i(x,y)
  \;=\;
  \big(
    d^K_i(x), \, d^L_i(y)
  \big)
  \,,
  \;\;\;\;\;
  s^{K \times L}_i(x,y)
  \;=\;
  \big(
    s^K_i(x), \, s^L_i(y)
  \big)
\]

\end{proposition}
\begin{proof}
Since [[SimplicialSets]] is a [[category of presheaves]], namely over the [[simplex category]], this is a special case of the general fact that [[limits of presheaves are computed objectwise]].

But it is also immediate to check that
(eq:StructureMapsOfProductSimpliciaSet) with (eq:ComponentSetsOfProductSimplicialSet)
satisfies the defining [[universal property]] of the [[Cartesian product]].
\end{proof}

\begin{remark}\label{DegenerateSimplicesInProductOfSimplicialSets}
**(degenerate simplices in product of simplicial sets)**
Prop. \ref{CartesianProductOfSimplicialSetsIsComponentwise}
means in particular that a [[simplex]]
of the form $\big(s_\alpha(x),\, s_\beta(y)\big) \,\in\, K \times L$  may be non-degenerate even though its two components are each degenerate (see the archetypical Example \ref{NonDegenerateSimplicesInSimplicialSquare} below).

Indeed, the proposition says that the degenerate simplices in $K \times L$ are precisely those such that their two simplex-components in $K$ and $L$, respectively, are in the image of the *same* degeneracy map $\alpha = \beta$.
\end{remark}

## Non-degenerate cells in a product of simplices
 {#NonDegenerateCellsInAProductOfSimplices}

Using the [above](#ProductsOfSimplicialSets) basic facts about products of simplicial sets, we have the following explicit characterization of the non-degenerated cells in products $\Delta[p] \times \Delta[q]$ of the basic [[simplices]] (the simplicial sets [[representable functor|represented]] by the objects in the [[simplex category]]).

### Characterization
 {#Characterization}

\begin{proposition}\label{NonDegenerateSimplicesInProductOfSimplices}
**(non-degenerate $(p+q)$-simplices in $\Delta[p] \times \Delta[q]$)**
\linebreak
For $p,q \,\in\, \mathbb{N}$,
the non-degenerate simplices in the [[Cartesian product]] (Prop. \ref{CartesianProductOfSimplicialSetsIsComponentwise})

$$
  \Delta[p] \times \Delta[q]
$$

of standard [[simplices]] in [[sSet]] correspond, under the [[Yoneda lemma]], to precisely those [[morphisms]] of [[simplicial sets]]

\[
  \label{GenericSimplexInProductOfSimplices}
  \Delta[p+q]
  \xrightarrow{\;\; \sigma \;\;}
  \Delta[p] \times \Delta[q]
\]

which satisfy the following equivalent conditions:

* as morphisms of [[posets]] they are [[strictly monotone]];

* as [[permutations]] of $(p+q)$ elements they are $(p,q)$-[[shuffles]];

* as morphisms of [[finitely generated object|finitely generated]] [[categories]] they take generating morphisms to generating morphisms;

\end{proposition}

(e.g. [Kerodon 2.5.7.2](#Kerodon): [00RH](https://kerodon.net/tag/00RH))

Here the first statement says that such morphisms may hence be represented by [[paths]]

* on a $(p+1)\times(q+1)$-lattice,

* from one corner to its opposite corner,

* consisting of $p+q$ unit steps,

* each either horizontally or vertically

and the second statement says that such paths are characterized by the $(p,q)$-[[shuffle]] $(\mu_1 \lt \cdots \lt \mu_p, \, \nu_1 \lt \cdots \lt \nu_q)$ of lists of step numbers $\mu_i$ where the path proceeds horizontally and step numbers $\nu_i$ where it proceeds vertically:

\begin{tikzcd}[
    sep=30pt
  ]
    &[-12pt]
    &
    \mathclap{
      \color{gray}
      \mu_1 = 1
    }
    &
    \mathclap{
      \color{gray}
      \mu_2 = 4
    }
    &
    \color{gray}
    \cdots
    \\[-20pt]
    &
    {(0,0)}
    \ar[r]
    \ar[d]
    \ar[
      r,-,
      color=blue,
      opacity=.25,
      line width=6pt
    ]
    &
    %
    {(1,0)}
    \ar[r]
    \ar[d]
    \ar[
      d,-,
      color=blue,
      opacity=.25,
      line width=6pt
    ]
    &
    {(2,0)}
    \ar[rr,dotted]
    \ar[d]
    &
    &
    {(p,0)}
    \ar[d]
    \\
    \color{gray}
    \nu_1 = 2
    &
    {(0,1)}
    \ar[r]
    \ar[d]
    &
    {(1,1)}
    \ar[r]
    \ar[d]
    \ar[
      d,-,
      color=blue,
      opacity=.25,
      line width=6pt
    ]
    &
    {(2,1)}
    \ar[rr, dotted]
    \ar[d]
    &
    &
    {(p,1)}
    \ar[d]
    \\
    \color{gray}
    \nu_2 = 3
    &
    {(0,2)}
    \ar[r]
    \ar[dd, dotted]
    &
    {(1,2)}
    \ar[r]
    \ar[dd, dotted]
    \ar[
      r,-,
      color=blue,
      opacity=.25,
      line width=6pt
    ]
    &
    {(2,2)}
    \ar[rr, dotted]
    \ar[dd, dotted]
    \ar[ddrr, dotted]
    \ar[
      ddrr,-,
      color=blue,
      opacity=.25,
      line width=6pt,
      dotted
    ]
    &
    &
    {(p,2)}
    \ar[dd, dotted]
    \\
    \color{gray}
    \vdots
    &
    &
    &
    &&
    \\
    &
    {(0,q)}
    \ar[r]
    &
    {(1,q)}
    \ar[r]
    &
    {(2,q)}
    \ar[rr, dotted]
    &
    &
    {(p,q)}
\end{tikzcd}

\begin{proof}
From Prop. \ref{CartesianProductOfSimplicialSetsIsComponentwise}
it is clear (Rem. \ref{DegenerateSimplicesInProductOfSimplicialSets}) that a
simplex $\sigma$ (eq:GenericSimplexInProductOfSimplices) is degenerate
precisely if, when regarded as a path as above, it contains a [[constant function|constant]] step, i.e. one which moves neither horizontally nor vertically. But then -- by degree reasons, since we are looking at paths of $p + q$ steps in a lattice of side length $p$ and $q$ -- it must be that the path proceeds by $p + q$ unit steps.
\end{proof}

\begin{remark}\label{NotationForSimplicesInAProductOfSimplices}
**(sequence- and shuffle-notation for simplices in a product of simplices)** \linebreak
Written as a [[pair]] of $(p+q)$-simplices, one in $\Delta[p]$ and one in $\Delta[q]$, the non-degenerate simplex (eq:GenericSimplexInProductOfSimplices) is hence a [[pair]] of [[monotone]] lists of [[natural numbers]]

\[
  \label{ListNotationForProductsOfSimplices}
  \left(
    \begin{array}{ccccccccc}
      0& \ldots & 0 & 1 & \ldots & 1 & 2 & \ldots & p
      \\
      0& \ldots & i & i & \ldots & j & j & \ldots & q
    \end{array}
  \right)
  \,,
\]

such that one row has a constant step precisely where the other has not.

Recording the step numbers where either of these lists is non-constant yields a *$(p,q)$-[[shuffle]]*:

\[
  \label{ShuffleNotationForProductsOfSimplices}
  \begin{array}{cc}
    \text{positions of horizontal steps}
    &
    \text{positions of vertical steps}
    \\
    \mu_1 \lt \mu_2 \lt \cdots \lt \mu_p
    &
    \nu_1 \lt \nu_2 \lt \cdots \lt \nu_p
  \end{array}
\]

namely a [[permutation]] of $(p+q)$ elements where each element is larger than its left neighbour, except possibly when going from the $p$th to the $p+1$st element.

\end{remark}

The shuffle-data yields a conveniently explicit re-construction of the product of simplices, as follows:

\begin{definition}
For $\mu = (\mu_1 \lt  \mu_2 \lt \cdots \lt \mu_p)$ a sequence of [[natural numbers]] in $\{1, \cdots,  p+q\}$, write

$$
  \sigma_\mu
  \;\colon\;
  X_q \to X_{p+q}
$$

for the [[composition|composite]]

$$
  [p+q]
    \overset{\;\;
      \sigma_{\mu_q - 1 }
    \;\;}{\longrightarrow}
  [p+q-1]
    \longrightarrow
    \cdots
    \longrightarrow
  [q + 2]
    \overset{\;\;
      \sigma_{ \mu_{2} - 1 }
    \;\;}{\longrightarrow}
   [q + 1]
   \overset{\;\;
     \sigma_{ \mu_1 - 1 }
   \;\;}{\longrightarrow}
   [q]
  \,,
$$

where $\sigma_i \,\colon\, [n+1] \to [n]$ denotes  the [[surjective map|surjective]] [[monotone map]] (the [[codegeneracy map]]) that repeats the index $i$.

\end{definition}

\begin{proposition}
The non-degenerate simplices in the [[Cartesian product]] (Prop. \ref{CartesianProductOfSimplicialSetsIsComponentwise})

$$
  \Delta[p] \times \Delta[q]
$$

of [[simplices]] in [[sSet]] are precisely those of the form

$$
  (\sigma_\nu, \sigma_\mu)
    \;\in\;
  Hom\big(
    \Delta[p + q]
    ,\,
    \Delta[p] \times \Delta[q]
  \big)
  \;=\;
  \big(
    \Delta[p] \times \Delta[q]
  \big)_{p+q}
$$

for $(\mu,\nu)$ a $(p,q)$-[[shuffle]].

\end{proposition}

\linebreak

### Examples
 {#Examples}

\begin{example}\label{CylindersOfSimplices}
**([[cylinders]] of [[simplices]])**

The non-degenerate (n+1)-simplices in the [[cylinder]] over the [[n-simplex]]
$$
  \begin{aligned}
           & \Delta[1]
    \\
    \times & \Delta[n]
  \end{aligned}
  \;\in\;
  sSet
$$

are, according to Prop. \ref{CartesianProductOfSimplicialSetsIsComponentwise}
 and in the notation of Remark \ref{NotationForSimplicesInAProductOfSimplices}, as follows:

$$
  \begin{array}{cc}
  \text{path} &\phantom{A}& (1,n)\text{-shuffle}
  \\
  \phantom{A}
  \\
  \left(
  \;\,
  \array{
    \mathllap{[}
    0 & 1 & 1 & 1 & 1 & \cdots & 1 & 1
    \mathrlap]
    \\
    \mathllap[
    0 & 0 & 1 & 2 & 3 & \cdots & n-1 & n
    \mathrlap]
  }
  \;\,
  \right)
  &&
  \Big(
    1
    ,
    \;
    2, 3, 4, 5, 6, \cdots, n
  \Big)
  \,,
  \\
  \phantom{A}
  \\
  \left(
  \;\,
  \array{
    \mathllap{[}
    0 & 0 & 1 & 1 & 1 & \cdots & 1 & 1
    \mathrlap]
    \\
    \mathllap[
    0 & 1 & 1 & 2 & 3 & \cdots & n-1 & n
    \mathrlap]
  }
  \;\,
  \right)
  &&
  \Big(
    2
    ,\;
    1, 3, 4, 5, 6, \cdots, n
  \Big)
  \\
  \phantom{A}
  \\
  \left(
  \;\,
  \array{
    \mathllap{[}
    0 & 0 & 0 & 1 & 1 & \cdots & 1 & 1
    \mathrlap]
    \\
    \mathllap[
    0 & 1 & 2 & 2 & 3 & \cdots & n-1 & n
    \mathrlap]
  }
  \;\,
  \right)
  &&
  \Big(
    3
    ,\;
    1, 2, 4, 5, 6,  \cdots, n
  \Big)
  \\
  \phantom{A}
  \\
  \left(
  \;\,
  \array{
    \mathllap{[}
    0 & 0 & 0 & 0 & 1 & \cdots & 1 & 1
    \mathrlap]
    \\
    \mathllap[
    0 & 1 & 2 & 3 & 3 & \cdots & n-1 & n
    \mathrlap]
  }
  \;\,
  \right)
  &&
  \Big(
    4
    ,\;
    1, 2, 3, 5, 6 \cdots, n
  \Big)
  \end{array}
$$

and so on.

\end{example}

\begin{example}
\label{NonDegenerateSimplicesInSimplicialSquare}
**(non-degenerate simplices in [[simplicial set|simplicial]] [[square]])**
\linebreak
The complete set of non-degenerate simplices in
$\Delta[1] \times \Delta[1]$
is, in specialization of Example \ref{CylindersOfSimplices},
according to Prop. \ref{CartesianProductOfSimplicialSetsIsComponentwise}
and in the notation of Remark \ref{NotationForSimplicesInAProductOfSimplices},
the following:

  \begin{tikzcd}
    \scalebox{.8}{$
      \left( {[0]} \atop {[0]} \right)
    $}
    \ar[
      rr,
      "{
        \left(
        {[0,0]}
        \atop
        {[0,1]}
        \right)
      }"{above, scale=.8}
    ]
    \ar[
      dd,
      "{
        \left(
        {[0,1]}
        \atop
        {[0,0]}
        \right)
      }"{left, scale=.8}
    ]
    \ar[
      ddrr,
      "{
        \left(
        { [0,1] }
        \atop
        { [0,1] }
        \right)
      }"{description, scale=.8}
    ]
    &&
    \scalebox{.8}{$
      \left( {[0]} \atop {[1]} \right)
    $}
    \ar[
      dd,
      "{
        \left(
        {[0,1]}
        \atop
        {[1,1]}
        \right)
      }"{right, scale=.8}
    ]
    \ar[
      ddll,
      phantom,
      "{
        \left(
        { [0,0,1] }
        \atop
        { [0,1,1] }
        \right)
      }"{pos=.15, scale=.8}
    ]
    \ar[
      ddll,
      phantom,
      "{
        \left(
        { [0,1,1] }
        \atop
        { [0,0,1] }
        \right)
      }"{pos=.85, scale=.8}
    ]
    \\
    \\
    \scalebox{.8}{$
      \left( {[1]} \atop {[0]} \right)
    $}
    \ar[
      rr,
      "{
        \left(
        {[1,1]}
        \atop
        {[0,1]}
        \right)
      }"{below, scale=.8}
    ]
    &&
    \scalebox{.8}{$
      \left( {[1]} \atop {[1]} \right)
    $}
  \end{tikzcd}

See also [Friedman 2008, Fig. 20](#Friedman08).
\end{example}

\begin{example}
  The non-degenerate simplices in
  $$
    \begin{aligned}
      & \Delta[2]
      \\
      \times & \Delta[1]
    \end{aligned}
  $$
  are, in specialization of Example \ref{CylindersOfSimplices},
  according to Prop. \ref{CartesianProductOfSimplicialSetsIsComponentwise}
  and in the notation of Remark \ref{NotationForSimplicesInAProductOfSimplices},
  the following three:

$$
  \begin{array}{cc}
  \text{ path }
  &&
  (2,1)\text{-shuffle}
  \\
  \phantom{A}
  \\
  \left(
  \;\,
  \array{
    \mathllap{[}
    0 & 1 & 2 & 2
    \mathrlap]
    \\
    \mathllap[
    0 & 0 & 0 & 1
    \mathrlap]
  }
  \;\,
  \right)
  &\phantom{AA}&
  \Big(
     1, 2 , \; 3
  \Big)
  \\
  \phantom{A}
  \\
  \left(
  \;\,
  \array{
    \mathllap{[}
    0 & 1 & 1 & 2
    \mathrlap]
    \\
    \mathllap[
    0 & 0 & 1 & 1
    \mathrlap]
  }
  \;\,
  \right)
  &&
  \Big(
     1, 3 , \; 2
  \Big)
  \\
  \phantom{A}
  \\
  \left(
  \;\,
  \array{
    \mathllap{[}
    0 & 0 & 1 & 2
    \mathrlap]
    \\
    \mathllap[
    0 & 1 & 1 & 1
    \mathrlap]
  }
  \;\,
  \right)
  &&
  \Big(
     2, 3 , \; 1
  \Big)
  \end{array}
$$

\end{example}

\begin{example}
  Some non-degenerate simplices in $\Delta[2] \times \Delta[2]$:

$$
  \begin{array}{cc}
    \text{path} &\phantom{AA}& (2,2)\text{-shuffle}
    \\
    \phantom{A}
    \\
  \left(
  \;\,
  \array{
    \mathllap{[}
    0 & 1 & 2 & 2 & 2
    \mathrlap]
    \\
    \mathllap[
    0 & 0 & 0 & 1 & 2
    \mathrlap]
  }
  \;\,
  \right)
  &&
  \Big(
    1, 2, \; 3, 4
  \Big)
  \\
  \phantom{AA}
  \\
  \left(
  \;\,
  \array{
    \mathllap{[}
    0 & 1 & 1 & 2 & 2
    \mathrlap]
    \\
    \mathllap[
    0 & 0 & 1 & 1 & 2
    \mathrlap]
  }
  \;\,
  \right)
  &&
  \Big(
    1, 3, \; 2, 4
  \Big)
  \\
  \phantom{AA}
  \\
  \left(
  \;\,
  \array{
    \mathllap{[}
    0 & 1 & 1 & 1 & 2
    \mathrlap]
    \\
    \mathllap[
    0 & 0 & 1 & 2 & 2
    \mathrlap]
  }
  \;\,
  \right)
  &&
  \Big(
    1, 4, \; 2, 3
  \Big)
  \\
  \phantom{AA}
  \\
  \left(
  \;\,
  \array{
    \mathllap{[}
    0 & 0 & 0 & 1 & 2
    \mathrlap]
    \\
    \mathllap[
    0 & 1 & 2 & 2 & 2
    \mathrlap]
  }
  \;\,
  \right)
  &&
  \Big(
    3, 4, \; 1, 2
  \Big)
  \end{array}
$$

\end{example}

\linebreak

## Related concepts

* [[Eilenberg-Zilber map]]

## References

The shuffle-formula is due (in terms of [[simplicial abelian groups]] of [[cochain on a simplicial set|chains on a simplicial set]], see at *[[Eilenberg-Zilber map]]*) to:

* [[Samuel Eilenberg]], [[Saunders MacLane]], (5.3) in: *On the groups $H(\Pi,n)$*, I, Ann. of Math. (2) 58, (1953), 55–106. ([jstor:1969820](https://www.jstor.org/stable/1969820))

following:

* [[Samuel Eilenberg]], [[Joseph A. Zilber]], American Journal of Mathematics Vol. 75, No. 1 (Jan., 1953), pp. 200-204 ([jstor:2372629](https://www.jstor.org/stable/2372629), [doi:10.2307/2372629](https://doi.org/10.2307/2372629))

The streamlined perspective of strictly monotonic morphisms $\Delta[p+q] \to \Delta[p] \times \Delta[q]$ is highlighted in

* {#Kerodon} [[Kerodon]], 2.5.7.2 *$(p,q)$-Shuffles* ([00RH](https://kerodon.net/tag/00RH))

Exposition:

* {#Friedman08} [[Greg Friedman]], §5 of: _An elementary illustrated introduction to simplicial sets_, Rocky Mountain J. Math. 42(2): 353-423 (2012) ([arXiv:0809.4221](http://arxiv.org/abs/0809.4221), [doi:10.1216/RMJ-2012-42-2-353](https://projecteuclid.org/journals/rocky-mountain-journal-of-mathematics/volume-42/issue-2/Survey-Article-An-elementary-illustrated-introduction-to-simplicial-sets/10.1216/RMJ-2012-42-2-353.full))

Textbook accounts:

* {#Friedman20} [[Greg Friedman]], §2.5.1 & §B.6 in: *Singular Intersection Homology*,  Cambridge University Press 2020 ([doi:10.1017/9781316584446](https://doi.org/10.1017/9781316584446), [pdf](http://faculty.tcu.edu/gfriedman/IHbook.pdf))

[[!redirects products of simplices]]

[[!redirects product of simplicial sets]]
[[!redirects products of simplicial sets]]

[[!redirects Cartesian product of simplices]]
[[!redirects Cartesian products of simplices]]


product theorem

* [[fundamental product theorem in topological K-theory]]

[[!redirects product theorems]]

[[!redirects fundamental product theorem]]
[[!redirects fundamental product theorem]]


product topological space

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

#Contents#
* table of contents
{:toc}

## Idea

Given two [[topological spaces]] $(X_1,\tau_1)$ and $(X_2, \tau_2)$, then the [[Cartesian product]] of their underlying sets $X_1 \times X_2$ is naturally equipped with a [[topological space|topology]] $\tau_{X_1 \times X_2}$ itself, generated from the [[base of a topology|base opens]] which are themselves Cartesian product $U_1 \times U_2 \subset X_1 \times X_2$, of [[open subsets]] of the original spaces: $U_i \subset X_i$. The resulting topological space

$$
  (X_1 \times X_2 , \tau_{X_1 \times X_2})
$$

is called the _product topological space_ of the two original spaces.

The [[formal duality|formally dual]] concept is that of [[disjoint union topological spaces]].

More generally, consider any index [[set]] $I$ and an $I$-indexed set $\{X_i, \tau_i\}_{i \in I}$ of [[topological spaces]]. Then the _product topology_ $\tau_{prod}$ or _[[Tychonoff topology]]_ on the [[Cartesian product]] $\underset{i \in I}{\prod} X_i$ of underlying sets is equivalently

1. the topology generated from the [[sub-base of a topology|sub-base]] given by products $\underset{i \in I}{\prod} U_i$ with $U_i \subset X_i$ open, but _all except one of the factors_ equal to the corresponding $X_i$,

   hence the topology whose open subsets are precisely those obtained as arbitrary unions of finite intersections of such subsets;

1. the topology generated from the [[base of a topology|base]] given by products $\underset{i \in I}{\prod} U_i$ with $U_i \subset X_i$ open, but _all except a finite number of factors_ equal to the corresponding $X_i$,

   hence the topology whose open subsets are precisely those obtained as arbitrary unions of such subsets.

This product topology is singled out by the fact that the resulting product topological space is the [[category theory|category theoretic]]  [[product]] of the original space in the [[category]] [[Top]] of topological spaces:

$$
  \left(
     \underset{i \in I}{\prod} X_i
     ,\;
     \tau_{prod}
  \right)
  \;\simeq\;
  \underset{i \in I}{\prod} (X_i, \tau_i)
  \,.
$$

This means that the product topology enjoys the [[universal property]] that for any topological space $(Y,\tau_Y)$ then sets of [[continuous functions]] $\{(Y, \tau_Y) \overset{\phi_i}{\to} (X_i, \tau_i)\}_{i \in I}$ into the factor spaces are in [[natural bijection]] with continuous functions $(\phi_i)_{i \in I} \colon (Y, \tau_Y) \to \left(\underset{i \in I}{\prod} X_i, \tau_{prod}\right)$ into the product topological space.

Beware that (among others) there is also the _[[box topology]]_ $\tau_{box}$ on the Cartesian product of underlying sets $\underset{i\in I}{\prod} X_i$, whose open subsets are the unions of those of the for $\underset{i \in I}{\prod} U_i$ with $U_i \subset X_i$ open and with _no_ further restriction on the factors.

For $I$ a [[finite set]], then these two topologies coincide, but for
$I$ not finite then the box topology is a strictly [[fine topology|finer topology]]

$$
  \tau_{prod} \subset \tau_{box}
$$

and hence in this case it does _not_ enjoy the [[universal property]] of the product topology above.

[[!include universal constructions of topological spaces -- table]]

## Definition

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

Let $\{(X_i, \tau_i)\}_{i \in I}$ be a [[set]] of [[topological spaces]]. Then their _product topological space_ has as underlying set the [[Cartesian product]] $\underset{i \in I}{\prod} X_i$ of the underlying sets, and has as topology $\tau_{prod} \subset P(\underset{i \in I}{\prod} X_i)$ the [[coarse topology|coarsest topology]] such that all the [[projection]] maps

$$
   p_i
  \;\colon\;
   \underset{i \in I}{\prod} X_i
   \longrightarrow
    X_i
$$

become [[continuous functions]] (called the _Tychonoff topology_).

This means equivalently that $\tau_{prod}$ is the topology generated from the [[sub-base of a topology|sub-base]]

$$
   \beta_{prod}
   \;\coloneqq\;
   \left\{
      p_i^{-1}(U_i) \subset \underset{i \in I}{\prod} X_i
      \;\vert\;
      i \in I, U_i \subset X_i \, \text{open}
   \right\}
   \,.
$$

=--

Notice that

$$
  p_i^{-1}(U_i)
  =
  U_i
  \times
  \left(\underset{j \in I \backslash \{i\}}{\prod} X_j\right)
$$

and that

$$
  p_i^{-1}(U_i)
  \cap
  p_j^{-1}(U_j)
  =
  U_i \times U_j
  \times
  \left(\underset{k \in I \backslash \{i,j\}}{\prod} X_k\right)
$$

etc.

## Examples

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

Let $X$ be any [[topological space]] and write $Disc(\{1,2\})$ for the [[discrete topological space]] on a set with two elements. Then there is a [[homeomorphism]]

$$
  X \times Disc(\{0,1\})
    \;\simeq\;
  X \sqcup X
$$

between the product space of $X$ with $Disc(\{1,2\})$ and the [[disjoint union space]] of $X$ with itself.

=--

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

By definition of [[disjoint union]] there is a [[bijection]] of underlying sets
$X \sqcup X \simeq X \times \{1,2\}$.

By unwinding the definitions

1. the open subsets of $X \times Disc(\{0,1\})$ are unions of those of the form $U \times S$, where $U \subset X$ is an open subset and $S \subset \{1,2\}$ is any subset

1. the open subsets of $X \sqcup X$ are of the form $U \sqcup V$ with $U,V \subset X$ open.

Under the above bijection the we have

$$
  U \sqcup V = \left(U \times \{1\}\right) \cup \left( V \times \{2\}\right)
  \,.
$$

Conversely, every union of elements of the form $U' \times \{1\}$, $V' \times \{2\}$ and $W \times \{1,2\} = W \times \{1\} \cup W \times \{2\}$ is of the form $U \times \{1\} \cup V \times \{2\}$.

This shows that the above bijection of underlying sets induces a bijection of the corresponding open subsets, hence is a homeomorphism.

=--

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

For $n \in \mathbb{N}$ consider the [[Cartesian space]] $\mathbb{R}^n$ with the [[metric topology]] induced from its [[Euclidean space|Euclidean metric]] structure. Then the product topological spaces satisfy

$$
  \mathbb{R}^{n_1}\times \mathbb{R}^{n_2}
  \simeq
  \mathbb{R}^{n_1 + n_2}
  \,.
$$

=--

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

Let $Disc(S)$ be a [[discrete topological space]] on a set with at least two elements. Then the infinite product space

$$
  \underset{n \in \mathbb{Z}}{\prod} Disc(S)
$$

is itself _not_ a discrete space.

=--

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

The open subsets of a discrete space include _all_ the subsets of the underlying set. Hence we need to see that there are subsets of the [[Cartesian product]] set   $\underset{n \in \mathbb{Z}}{\prod} Disc(S)$ which are not open subsets in the Tychonoff topology.

But by definition the open subsets in the Tychnoff topology are unions of products of open subsets of the factor spaces such that only a finite number of the factors is not the total space.

Since $S$ is assumed to contain at least two elements let $1 \in S$ be one of these. Then $\{1\} \subset S$ is a proper subset. Accordingly the product subset

$$
  \underset{n \in \mathbb{B}}{\prod} \{1\}
  \;\subset\;
  \underset{n\in \mathbb{N}}{\prod} Disc(S)
$$

is not open. For if it were, it would have to be the union of product subsets that contain the total set $S$ in at least one entry, which by construction it is not.

=--

+-- {: .num_example #CantorSpace}
###### Example
**([[Cantor space]])**

Write $Disc(\{0,1\})$ for the the [[discrete topological space]] with two points. Write $\underset{n \in \mathbb{N}}{\prod} Disc(\{0,2\})$ for the [[product topological space]] of a [[countable set]] of copies of this discrete space with itself (i.e. the corresponding [[Cartesian product]] of sets $\underset{n \in \mathbb{N}}{\prod} \{0,1\}$ equipped with the Tychonoff topology induced from the [[discrete topology]] of $\{0,1\}$).

Then consider the [[function]]

$$
  \array{
    \underset{n \in \mathbb{N}}{\prod}
      &\overset{\kappa}{\longrightarrow}&
    [0,1]
    \\
    (a_i)_{i \in \mathbb{N}}
    &\overset{\phantom{AAAA}}{\mapsto}&
    \underoverset{i = 0}{\infty}{\sum} \frac{2 a_i}{ 3^n}
  }
$$

which sends an element in the product space, hence a [[sequence]] of binary digits, to the value of the [[power series]] as shown on the right.

One checks that this is a [[continuous function]] (from the [[product topology]] to the [[Euclidean space|Euclidean]] [[metric topology]] on the [[closed interval]]). Moreover with its [[image]] $\kappa\left( \underset{n \in \mathbb{N}}{\prod} \{0,1\}\right) \subset [0,1]$ equipped with its [[subspace topology]], then this is a [[homeomorphism]] onto its image:

$$
  \underset{n \in \mathbb{N}}{\prod} Disc(\{0,1\})
    \overset{\phantom{AA}\simeq\phantom{AA}}{\longrightarrow}
  \kappa\left(
    \underset{n \in \mathbb{N}}{\prod} Disc(\{0,1\})
  \right)
    \overset{\phantom{AAAA}}{\hookrightarrow}
  [0,1]
  \,.
$$

This image is the _[[Cantor space]]_ as a subspace of the closed interval.

=--

###### Example

A Cartesian product of countably many copies of the real closed interval $[0,1]$ (as a topological space) is the [[Hilbert cube]].

## Properties

### Basic properties

+-- {: .num_prop #ProjectionsAreOpenMaps}
###### Proposition
**([[projections]] are [[open maps]])**

For $\{X_i\}_{i \in I}$ a set of topological spaces, the for each $j \in I$ the [[projection]] out of their product space into the $j$th component

$$
 p_j \;\colon\; \underset{i \in I}{\prod} X_i \longrightarrow X_j
$$

is an [[open map]].

=--

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

Since images preserve unions ([this prop.](interactions+of+images+and+pre-images+with+unions+and+intersections#PreservationOfUnionsAndIntersectionsOfSets)) it is sufficient to check that the image under $p_j$ of a [[base fo the topology|base open]] is still open. But base opens in the product topology by definition are, in particular, products of open subsets.

=--

### Universal property

The product topological space construction from def. \ref{ProductTopologicalSpace} is the [[limit]] in [[Top]] over the [[discrete category|discrete]] [[diagram]] consisting of the factor spaces, hence the [[category theory|category theoretic]] [[product]].

For **proof** see at [Top -- Universal constructions](Top#UniversalConstructions).

### The Tychonoff theorem

The [[Tychonoff theorem]] states that the product space of any set of [[compact topological spaces]] (with its Tychonoff topology) is itself compact.

### Relation to smash product of pointed spaces

+-- {: .num_prop #OnePointCompactificationAndSmashProduct}
###### Proposition
**([[one-point compactification intertwines Cartesian product with smash product]])

On the [[subcategory]] $Top_{LCHaus}$ of [[Top]] on the [[locally compact Hausdorff spaces]] with [[proper maps]] between them, the [[functor]] of [[one-point compactification]] (Prop. \ref{OnePointCompactificationFunctor})

$$
  (-)^{cpt}
  \;\colon\;
  Top_{LCHaus}
  \longrightarrow
  Top^{\ast/}
$$

sends [[Cartesian products]] ([[product topological spaces]]) to [[smash products]] of [[pointed topological spaces]], hence constitutes a [[strong monoidal functor]], in that there is a [[natural transformation|natural]] [[homeomorphism]]:

$$
  \big(
    X \times Y
  \big)^{cpt}
  \;\simeq\;
  X^{cpt} \wedge Y^{cpt}
  \,.
$$

=--

This is briefly mentioned in [Bredon 93, p. 199](one-point+compactification#Bredon93).
The argument is spelled out in: [MO:a/1645794/](https://math.stackexchange.com/a/1645794/58526), [Cutler 20, Prop. 1.6](one-point+compactification#Cutler20).

### Relation to singular (co)homology

The [[singular homology]] of product topological spaces is informed by

* [[Eilenberg-Zilber theorem]]

* [[Künneth theorem]]

## Related concepts

* [[product manifold]]

* [[tensor product of chain complexes]]

[[!include universal constructions of topological spaces -- table]]

## References

The _Tychonoff topology_ is named after [[A. N. Tychonoff]].

* [[Terence Tao]], _Notes on products of topological spaces_ ([pdf](http://www.math.ucla.edu/~tao/resource/general/121.1.00s/product.pdf))

* [[Florian Herzig]], _Product topology_ ([pdf](http://www.math.toronto.edu/~herzig/MAT327-lecturenotes08.pdf))

* {#Terilla14} [[John Terilla]],  _Notes on categories, the subspace topology and the product topology_ 2014 ([pdf](https://math.mit.edu/~jhirsh/terilla_subspace_quotient.pdf))

On [[product topological space|product spaces]] with [[Lindelöf topological spaces]]:

* [[Michael Barr]], [[John Kennison]], [[Robert Raphael]], *On productively  Lindelöf spaces*, Scient. Math. Japon. **65** (2007) 23–36 [[pdf](https://www.math.mcgill.ca/barr/papers/alster.pdf), [[BKR-ProductivelyLindeloef.pdf:file]]]

[[!redirects product topological spaces]]

[[!redirects topological product space]]
[[!redirects topological product spaces]]

[[!redirects product space]]
[[!redirects product spaces]]

[[!redirects Tychonoff product]]
[[!redirects Tychonoff products]]
[[!redirects Tychonov product]]
[[!redirects Tychonov products]]
[[!redirects Tykhonoff product]]
[[!redirects Tykhonoff products]]
[[!redirects Tykhonov product]]
[[!redirects Tykhonov products]]
[[!redirects Tichonoff product]]
[[!redirects Tichonoff products]]
[[!redirects Tichonov product]]
[[!redirects Tichonov products]]
[[!redirects Tikhonoff product]]
[[!redirects Tikhonoff products]]
[[!redirects Tikhonov product]]
[[!redirects Tikhonov products]]
[[!redirects Тиьонов product]]
[[!redirects Тиьонов products]]

[[!redirects Tychonoff topology]]
[[!redirects Tychonoff topologies]]
[[!redirects Tychonov topology]]
[[!redirects Tychonov topologies]]
[[!redirects Tykhonoff topology]]
[[!redirects Tykhonoff topologies]]
[[!redirects Tykhonov topology]]
[[!redirects Tykhonov topologies]]
[[!redirects Tichonoff topology]]
[[!redirects Tichonoff topologies]]
[[!redirects Tichonov topology]]
[[!redirects Tichonov topologies]]
[[!redirects Tikhonoff topology]]
[[!redirects Tikhonoff topologies]]
[[!redirects Tikhonov topology]]
[[!redirects Tikhonov topologies]]
[[!redirects Tihonov topology]]
[[!redirects Тиьонов topology]]
[[!redirects Тиьонов topologies]]

[[!redirects topological product]]
[[!redirects topological products]]

[[!redirects product topology]]
[[!redirects product topologies]]


product type

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

# Contents
* table of contents
{: toc}

## Idea

In [[type theory]] a _product type_ of two [[types]] $A$ and $B$ is the type whose [[terms]] are [[ordered pairs]] $(a,b)$ with $a\colon A$ and $b\colon B$.

In a [[model]] of the type theory in [[categorical semantics]], this is a [[product]]. In [[set theory]], it is a [[cartesian product]]. In [[dependent type theory]], it is a special case of a [[dependent sum]].

Note that a [[dependent product type]] is something different (a generalization of a [[function type]]).

## Overview
 {#Overview}

\linebreak

\begin{imagefromfile}
    "file_name": "ProductTypeInferenceRules-221230.jpg",
    "width": 650,
    "unit": "px",
    "margin": {
        "top": -40,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

> (table taken from [[schreiber:Topological Quantum Gates in Homotopy Type Theory|Myers et al.]])

## Definition

Like any type constructor in [[type theory]] (see at [[natural deduction]]), a product type is specified by rules saying when we can introduce it as a type, how to construct terms of that type, how to use or "eliminate" terms of that type, and how to compute when we combine the constructors with the eliminators.

### In natural deduction

We assume that our product types have typal [[conversion rules]], as those are the most general of the conversion rules. Both the propositional and judgmental conversion rules imply the typal conversion rules by the structural rules for propositional and judgmental equality respectively.

The formation and introduction rules are the same for both the positive and negative product types

Formation rules for product types:
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type}}{\Gamma \vdash A \times B \; \mathrm{type}}$$

Introduction rules for product types:
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma\vdash B \; \mathrm{type}}{\Gamma, x:A, y:B \vdash (x, y):A \times B}$$

#### As a positive type

* Elimination rules for product types:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma, z:A \times B \vdash C \; \mathrm{type} \quad \Gamma, x:A, y:B \vdash c:C[(x, y)/z]}{\Gamma, z:A \times B \vdash \mathrm{ind}_{A \times B}^C(c):C}$$

* Computation rules for product types:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma, z:A \times B \vdash C \; \mathrm{type} \quad \Gamma, x:A, y:B \vdash c:C[(x, y)/z]}{\Gamma, x:A, y:B \vdash \beta_{A \times B}^C(c):\mathrm{ind}_{A \times B}^C(c)[(x, y)/z] =_{C[(x, y)/z]} c}$$

* Uniqueness rules for product types:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma\vdash B \; \mathrm{type} \quad \Gamma, z:A \times B \vdash C \; \mathrm{type} \quad \Gamma, x:A, y:B \vdash c:C[(x, y)/z] \quad \Gamma, z:A \times B \vdash u:C \quad \Gamma, x:A, y:B \vdash i_{(-,-)}(u):u[(x, y)/z] =_{C[(x, y)/z]} c}{\Gamma, z:A \times B \vdash \eta_{A \times B}^C(c):u =_{C} \mathrm{ind}_{A \times B}^C(c)}$$

Given a [[pointed type|pointed]] [[h-proposition]] $(A, a)$, the positive product type results in $(a,-)$ being an [[equivalence of types]].

#### As a negative type

* Elimination rules for product types:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type}}{\Gamma, z:A \times B \vdash \pi_1(z):A} \qquad \frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type}}{\Gamma, z:A \times B \vdash \pi_2(z):B}$$

* Computation rules for product types:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma\vdash B \; \mathrm{type}}{\Gamma, x:A, y:B \vdash \beta_{\times 1}(x, y):\pi_1((x, y)) =_A x} \qquad \frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type}}{\Gamma, x:A, y:B \vdash \beta_{\times 2}(x, y):\pi_2((x, y)) =_{B} y}$$

* Uniqueness rules for product types:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type}}{\Gamma, z:A \times B \vdash \eta_{\times}(z):z =_{A \times B} (\pi_1(z), \pi_2(z))}$$

Given a [[pointed type|pointed]] [[h-proposition]] $(A, a)$, the negative product type results in $\pi_2(a,-)$ being a [[quasi-inverse function]] of $(a,-)$, rather than an [[equivalence of types]]. Thus, the positive definition is preferred to the negative definition. Alternatively, the negative product type should be defined as a [[coinductive type]] rather than an [[inductive type]].

#### As a special case of the dependent sum

In [[dependent type theory]] a product type $A \times B$ is the special case the [[dependent sum]] over $a : A$ for the special case that $B$ is regarded as an $A$-[[dependent type]] that actually happens to be $A$-independent. The rules are given as follows:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B \; \mathrm{type} \quad \Gamma, a:A, b:A \vdash B[a/x] \equiv B[b/x] \; \mathrm{type}}{\Gamma \vdash A \times B \; \mathrm{type}}$$

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B \; \mathrm{type} \quad \Gamma, a:A, b:A \vdash B[a/x] \equiv B[b/x] \; \mathrm{type}}{\Gamma \vdash A \times B \equiv \sum_{x:A} B\; \mathrm{type}}$$

#### As a special case of the dependent product

In [[dependent type theory]] given types $A$ and $B$, one could define a type family $C$ indexed by elements of the [[two-valued type]] $\mathbb{2}$ by $C(0) \coloneqq A$ and $C(1) \coloneqq B$. A product type $A \times B$ is a special case of the [[dependent product type]]

$$A \times B \coloneqq C(0) \times C(1) \coloneqq \prod_{x:\mathbb{2}} C(x)$$

and is given by the following rules:

Formation rules for product types:
$$\frac{\Gamma, x:\mathbb{2} \vdash C \; \mathrm{type}}{\Gamma \vdash C(0) \times C(1) \; \mathrm{type}}$$

Introduction rules for product types:
$$\frac{\Gamma, x:\mathbb{2} \vdash c:C}{\Gamma \vdash \lambda(x:\mathbb{2}).c(x):C(0) \times C(1)}$$

Elimination rules for product types:
$$\frac{\Gamma \vdash z:C(0) \times C(1) \quad \Gamma \vdash p:\mathbb{2}}{\Gamma \vdash z(p):C[p/x]}$$

Computation rules for product types:
$$\frac{\Gamma, x:\mathbb{2} \vdash c:C \quad \Gamma \vdash p:\mathbb{2}}{\Gamma \vdash \beta_\Pi(p):\lambda(x:\mathbb{2}).c(x)[p/x] =_{C[p/x]} c[p/x]}$$

Uniqueness rules for product types:
$$\frac{\Gamma \vdash z:C(0) \times C(1)}{\Gamma \vdash \eta_\Pi(z):z =_{C(0) \times C(1)} \lambda(x).z(x)}$$

#### Properties

Suppose $A$ and $B$ are types and $A \times B$ is the negative product type of $A$ and $B$. Then for all elements $c:A \times B$ and $c':A \times B$, there is a function
$$\mathrm{ap}_{(\pi_1, \pi_2)}(c, c'):(c =_{A \times B} c') \to \left((\pi_1(c) =_A \pi_1(c')) \times (\pi_2(c) =_B \pi_2(c'))\right)$$
There are functions $\mathrm{ap}_{\pi_1}(c, c'):(c =_{A \times B} c') \to (\pi_1(c) =_A \pi_1(c'))$ and $\mathrm{ap}_{\pi_2}(c, c'):(c =_{A \times B} c') \to (\pi_2(c) =_B \pi_2(c'))$ which are the [[function application to identities]] for the product projections $\pi_1$ and $\pi_2$, which means for all identities $p:c =_{A \times B} c'$, there are identities
$$\mathrm{ap}_{\pi_1}(c, c')(p):\pi_1(c) =_A \pi_1(c')$$
$$\mathrm{ap}_{\pi_2}(c, c')(p):\pi_2(c) =_A \pi_2(c')$$
Then, by the introduction rule for products, one could form the pair
$$(\mathrm{ap}_{\pi_1}(c, c')(p), \mathrm{ap}_{\pi_2}(c, c')(p))$$
with identites
$$\delta(c, c', p):\mathrm{ap}_{(\pi_1, \pi_2)}(c, c')(p) =_{(\pi_1(c) =_A \pi_1(c')) \times (\pi_2(c) =_B \pi_2(c'))} (\mathrm{ap}_{\pi_1}(c, c')(p), \mathrm{ap}_{\pi_2}(c, c')(p))$$
for all $p:c =_{A \times B} c'$.

Similarly, for all elements $a:A$, $a':A$, $b:B$, $b':B$, the [[binary function application to identities]] for $(-, -)$ defined in the [[introduction rule]] for product types is given by
$$\mathrm{apbinary}_{(-,-)}(a, a', b, b'):\left(a =_A a') \times (b =_B b')\right) \to ((a, b) =_{A \times B} (a', b')$$
inductively defined by the identities
$$\beta_{\mathrm{apbinary}_{(-,-)}}(a, b):\mathrm{apbinary}_{(-,-)}(a, a, b, b)(\mathrm{refl}_{A}(a), \mathrm{refl}_{B}(b)) =_{(a, b) =_{A \times B} (a, b)} \mathrm{refl}_{A \times B}((a, b)))$$

For all elements $c:A \times B$ and $c':A \times B$, the composition of functions
$$\mathrm{apbinary}_{(-,-)}(\pi_1(c), \pi_1(c'), \pi_2(c), \pi_2(c')) \circ \mathrm{ap}_{(\pi_1, \pi_2)}(c, c')$$
has domain $c =_{A \times B} c'$ and codomain $(\pi_1(c), \pi_2(c)) =_{A \times B} (\pi_1(c'), \pi_2(c'))$.

By the uniqueness rule for the negative product type, there are identities $\eta_{\times}(c):c =_{A \times B} (\pi_1(c), \pi_2(c))$ and $\eta_{\times}(c'):c' =_{A \times B} (\pi_1(c'), \pi_2(c'))$, and thus, for all identities $p:c =_{A \times B} c'$, there are identities

$$
  \array{& c & \overset{\eta_{\times}(c)}= & (\pi_1(c), \pi_2(c)) & \\
          p & \Vert & \rightarrow & \Vert & \mathrm{apbinary}_{(-,-)}(\pi_1(c), \pi_1(c'), \pi_2(c), \pi_2(c'))(\mathrm{ap}_{(\pi_1, \pi_2)}(c, c')(p))\\
          &c' & \underset{\eta_{\times}(c')}= & (\pi_1(c'), \pi_2(c')) & \\
}$$

For all elements $a:A$, $a':A$, and $b:B$, $b':B$, the composition of functions
$$\mathrm{ap}_{(\pi_1, \pi_2)}((a, b), (a', b')) \circ \mathrm{apbinary}_{(-,-)}(a, a', b, b')$$
has domain $(a =_A a') \times (b =_B b')$ and codomain $(\pi_1((a, b)) =_A \pi_1((a', b'))) \times (\pi_2((a, b)) =_B \pi_2((a', b')))$.

By the computation rules of the negative product type, there are identities $\beta_{\times 1}(a, b):\pi_1((a, b)) =_A a$, $\beta_{\times 2}(a, b):\pi_2((a, b)) =_A b$, $\beta_{\times 1}(a', b'):\pi_1((a', b')) =_A a'$, and $\beta_{\times 2}(a', b'):\pi_2((a', b')) =_A b'$, and for all pairs of identities $q:(a =_A a') \times (b =_B b')$, there are identities

$$
  \array{& \pi_1((a, b)) & \overset{\beta_{\times 1}(a, b)}= & a & \\
          \pi_1(\mathrm{ap}_{(\pi_1, \pi_2)}((a, b), (a', b'))(\mathrm{apbinary}_{(-,-)}(a, a', b, b')(q))) & \Vert & \leftarrow & \Vert & \pi_1(q) \\
          & \pi_1((a', b')) & \underset{\beta_{\times 1}(a', b')}= & a' & \\
}$$

$$
  \array{& \pi_2((a, b)) & \overset{\beta_{\times 2}(a, b)}= & b & \\
          \pi_2(\mathrm{ap}_{(\pi_1, \pi_2)}((a, b), (a', b'))(\mathrm{apbinary}_{(-,-)}(a, a', b, b')(q))) & \Vert & \leftarrow & \Vert & \pi_2(q) \\
          & \pi_2((a', b')) & \underset{\beta_{\times 2}(a', b')}= & b' & \\
}$$

Both of these square are in general not [[commutative squares]].

### In lambda-calculus

There are actually two ways to present product types, as a [[negative type]] or as a [[positive type]].  In both cases the [[type formation rule]]
is the following:

$$
  \frac{A\colon Type \qquad B\colon Type}{A\times B\colon Type}
$$

but the constructors and eliminators may be different.

#### As a negative type

When presented negatively, primacy is given to the [[term elimination rule|eliminators]].  We specify that there are two ways to eliminate a term of type $A\times B$: by projecting out the first component, or by projecting out the second.

$$ \frac{p \colon A\times B}{\pi_1 p\colon A} \qquad \frac{p\colon A\times B}{\pi_2 p\colon B}$$

This then determines the form of the [[term introduction rule|constructors]]: in order to construct a term of type $A\times B$, we have to specify what value that term should yield when all the eliminators are applied to it.  In other words, we have to specify a pair of elements, one of $A$ (to be the value of $\pi_1 p$) and one of $B$ (to be the value of $\pi_2 p$).

$$ \frac{a\colon A \qquad b\colon B}{(a,b)\colon A\times B} $$

Finally, we have [[computation rules]] which say that the relationship between the constructors and the eliminators is as we hoped.  We always have [[beta reduction]] rules

$$ \pi_1(a,b) \to_\beta a \qquad \pi_2(a,b) \to_\beta b $$

and we may or may not choose to have an [[eta reduction]] rule

$$ (\pi_1 p, \pi_2 p) \to_\eta p $$

#### As a positive type

When presented positively, primacy is given to the constructors.  We specify that the way to construct something of type $A\times B$ is to give something of type $A$ and something of type $B$:

$$ \frac{a\colon A \qquad b\colon B}{(a,b)\colon A\times B} $$

Of course, this is the same as the constructor obtained from the negative presentation.  However, the eliminator is different.  Now, in order to say how to *use* something of type $A\times B$, we have to specify how we should behave for all possible ways that it could have been constructed.  In other words, we have to say, *assuming* that $p$ were of the form $(a,b)$, what we want to do.  Thus we end up with the following rule:

$$ \frac{p\colon A\times B \qquad x\colon A, y\colon B \vdash c\colon C}{let (x,y) = p in c \;\colon C} $$

We need a term $c$ in the context of two variables of types $A$ and $B$, and the destructor or match "binds those variables" to the two components of $p$.  Note that the "ordered pair" $(x,y)$ in the destructor is just a part of the syntax; it is not an instance of the *constructor* ordered pair.  In [[dependent type theory]], this elimination rule must be generalized to allow the type $C$ to depend on $A\times B$.

Now we have [[beta reduction]] rule:

$$ let (x,y) = (a,b) \,in c \;\to_\beta\; c[a/x, b/y] $$

In other words, if we build an ordered pair and then break it apart, what we get is just the things we put into it.  (The notation $c[a/x, b/y]$ means to substitute $a$ for $x$ and $b$ for $y$ in the term $c$).

And (if we wish) the [[eta reduction]] rule, which is a little more subtle:

$$ let (x,y) = p in c[(x,y)/z] \;\to_\eta\; c[p/z] $$

This says that if we break something of type $A\times B$ into its components, but then we only use those two components by way of putting them back together into an ordered pair, then we might as well just not have broken it down in the first place.

Positively defined products are naturally expressed as [[inductive types]].  For instance, in [[Coq]] syntax we have

    Inductive prod (A B:Type) : Type :=
    | pair : A -> B -> prod A B.

(Coq then implements beta-reduction, but not eta-reduction.  However, eta-equivalence is provable with the internally defined [[identity type]], using the dependent eliminator mentioned above.)

Arguably, negatively defined products should be naturally expressed as [[coinductive type]]s, but this is not exactly the case for the presentation of coinductive types used in Coq.

#### Positive versus negative

In ordinary "nonlinear" type theory, the positive and negative product types are equivalent.  They manifestly have the same constructor, while we can define the eliminators in terms of each other as follows:

$$
\begin{aligned}
  \pi_1 p &\;\coloneqq\; let (x,y) = p in x\\
  \pi_2 p &\;\coloneqq\; let (x,y) = p in y\\
  let (x,y) = p in c &\;\coloneqq\; c[\pi_1 p / x, \pi_2 p / y]
\end{aligned}
$$

It is obvious that the $\beta$-reduction rules in the two cases correspond; see below for $\eta$-conversion.

In dependent type theory, in order to recover the *dependent* eliminator for the positive product type from the eliminators for the negative product type, we need the latter to satisfy the $\eta$-conversion rule so as to make the above definition well-typed.  It is sufficient to have the $\eta$-conversion up to [[propositional equality]], however, if we are willing to insert a substitution along such an equality in the definition of the dependent eliminator.  Conversely, the dependent eliminator for the positive product allows us to prove a propositional version of the negative $\eta$-conversion (without assuming the positive $\eta$-conversion).  See [propositional eta-conversions](/nlab/show/eta-conversion#Propositional).

Now from $\eta$-conversion for the negative product, we can also derive

$$
\begin{aligned}
  let (x,y) = p in c[(x,y)/z]
  &\;\coloneqq\; c[(\pi_1 p,\pi_2 p)/z]\\
  &\;\to_\eta\; c[p/z]
\end{aligned}
$$

so the defined positive product also satisfies its $\eta$-conversion, which will be definitional or propositional according to that of the negative product.

On the other hand, if the positive product has a definitional $\eta$-conversion, then for the defined negative product we have

$$
\begin{aligned}
  (\pi_1 p, \pi_2 p)
  &\;\coloneqq\; (let (x,y) = p in x, let (x,y) = p in y)\\
  &\;\leftarrow_\eta\; (let (x',y') = p in \;( let (x,y) = (x',y') in x , let (x,y) = (x',y') in y )\\
  &\;\to_\beta\; let (x',y') = p in (x',y')\\
  &\;\to_\eta\; p
\end{aligned}
$$

Note that this involves a beta-reduction step and also a "backwards" $\eta$-reduction step.  So from positive $\eta$ *reduction* we cannot derive negative $\eta$-reduction, only negative $\eta$-equivalence.  (However, the directionality of $\eta$-reduction is somewhat questionable anyway.)

In conclusion, we have:

* In non-dependent type theory, positive and negative products are equivalent, as are their definitional $\beta$-reduction rules.

* In dependent type theory with [[identity types]], improving the positive eliminator to a *dependent* eliminator is equivalent to asserting [propositional](/nlab/show/eta-conversion#Propositional) versions of either $\eta$-conversion rule.

* In any case, the two *definitional* $\eta$-conversion rules also correspond.

It is of importance to note that these translations require the [[contraction rule]] and the [[weakening rule]]; that is, they duplicate and discard terms.  In [[linear logic]] these rules are disallowed, and therefore the positive and negative products become different.  The positive product becomes "tensor" $A\otimes B$, and the negative product becomes "with" $A \& B$.

## Categorical interpretation

Under [[categorical semantics]], product types satisfying both beta and eta conversions correspond to [[products]] in a [[category]].  More precisely:

* categorical products may be used to interpret product types that validate both beta and eta rules, while

* the [[syntactic category]] of a type theory with product types has categorical products, as long as the type theory satisfies both beta and eta rules.

Of course, the categorical notion of product matches the *negative* definition of a product most directly.  In linear logic, therefore, the categorical product interprets "with" $\&$, while an additional [[monoidal category|monoidal structure]] interprets "tensor" $\otimes$.  On the other hand, in a representable [[cartesian multicategory]], the product has a "from the left" universal property which matches the *positive* definition.

There is also another interpretation in category theory of the product type $A$ and $B$, as the [[initial]] $A$-indexed family of [[parallel morphisms]] with [[source]] $B$, the object $A \times B$ with a family of morphisms

$$\mathrm{in}_A(x):B \to A \times B$$

such that for any other object $C$ with a family of morphisms $f(x):B \to C$, there exists a unique morphism $u_C:A \times B \to C$ such that

$$u_C \circ \mathrm{in}_A(x) = f(x)$$

This corresponds to the positive product types.

## Related concepts

* [[sum type]]

* [[dependent sum type]]

* [[dependent product type]]

## References

A textbook account in the context of [[programming languages]] is in section 11 of

* [[Robert Harper]], _[[Practical Foundations for Programming Languages]]_

On product types as [[inductive types]]:

* Kajetan Söhnen, §2.4.3 in : *Higher Inductive Types in Homotopy Type Theory*, Munich (2018) [[pdf](https://www.math.lmu.de/~petrakis/Soehnen.pdf), [[Soehnen-HigherInductiveTypes.pdf:file]]]

[[!redirects product type]]
[[!redirects product types]]

[[!redirects pair type]]
[[!redirects pair types]]


product-preserving functor

Let $F \colon C \to D$ be a [[functor]], and suppose a collection of objects $\{c_i\}$ in $C$ admits a [[product]], with projections

$$\pi_i \colon \prod_i c_i \to c_i.$$

We say $F$ [[preserved limit|preserves]] this product if the collection of maps

$$F(\pi_i): F(\prod_i c_i) \to F(c_i)$$

exhibits $F(\prod_i c_i)$ as a product of the collection of objects $F(c_i)$.

If $C$ has all (small) products, $F$ is _product-preserving_ if it preserves every product in $C$.

If $C$ does *not* have all small products, then one wants a more subtle condition; compare [[flat functor]] (which is about [[finite limits]] instead of products).

[[!redirects product-preserving functor]]
[[!redirects product-preserving functors]]


product-regular cardinal

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# Contents
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## Idea

The version of [[regular cardinal]] for [[products]]/[[Cartesian products]] instead of [[sums]]/[[unions]]/[[disjoint unions]].

## Definition

A cardinal $\kappa$ is **product-regular** if, for all families of sets $(X_i)_{i \in I}$, if $\vert I \vert \lt \kappa$ and for all elements $i \in I$ $\vert X_i \vert \lt \kappa$, then the indexed product

$$\left| \prod_{i \in I} X_i \right| \lt \kappa$$

## Properties

Every inaccessible cardinal is product-regular. Every [[uncountable]] product-regular cardinal is an [[inaccessible cardinal]], but the finite cardinals $0$ representing the [[empty set]] and the countable cardinal $\aleph_0$ representing the [[natural numbers]] are product-regular cardinals which are not inaccessible cardinals.

## See also

* [[regular cardinal]]

* [[inaccessible cardinal]]

## References

* [[Tom Leinster]], *Large Sets 9*, [web](https://golem.ph.utexas.edu/category/2021/07/large_sets_9.html)

[[!redirects product-regular cardinals]]


products in pQFT -- table

| product in [[perturbative QFT]] | $\,\,$ induces  |
|--|--|
| [[normal-ordered product]] | [[Wick algebra]] ([[free field]] [[quantum observables]])  |
| [[time-ordered product]] | [[S-matrix]] ([[scattering amplitudes]]) |
| [[retarded product]] | [[interacting quantum observables]] |


produoidal category

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# Produoidal category
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## Idea

A **produoidal category** is like a [[duoidal category]] in whose structure (namely, the two [[tensor product|tensor products]] and [[unit object|unit objects]]) we have replaced [[functors]] by [[profunctors]]. Alternatively, a produoidal category is a category with two [[promonoidal category|promonoidal structures]] which [[interchange law|interchange]] laxly.

## Definition

A **produoidal category** is a pair of [[pseudomonoid|pseudomonoids]] that interchange laxly in the [[monoidal bicategory]] [[Prof]].
This means that it is a [[category]] $\mathbb{C}$ together with

* a first [[promonoidal category|promonoidal structure]] $\mathbb{C}_{\otimes} \colon \mathbb{C} \times \mathbb{C} ⇸ \mathbb{C}$ and $\mathbb{C}_{I}\colon 1 ⇸ \mathbb{C}$;
* a second [[promonoidal category|promonoidal structure]] $\mathbb{C}_{\triangleleft} \colon \mathbb{C} \times \mathbb{C} ⇸ \mathbb{C}$ and $\mathbb{C}_{N}\colon 1 ⇸ \mathbb{C}$;
* the same laxators as a [[duoidal category]], including, for instance,
$$\begin{aligned}
& \int^{U,V}\mathbb{C}_{\otimes}(X;U,V) \times \mathbb{C}_{\triangleleft}(U;A,B) \times
\mathbb{C}_{\triangleleft}(V;C,D) \to \int^{P,Q}\mathbb{C}_{\triangleleft}(X;P,Q) \times \mathbb{C}_{\otimes}(P;A,C) \times
\mathbb{C}_{\otimes}(Q;B,D);
\end{aligned}$$
* the same coherence conditions as a [[duoidal category]].

## Examples

 * [[optic (in computer science)|Optics]] over a monoidal category form a produoidal category. [EHR23](#EarnshawHeffordRoman23)

## References

The first mention of produoidal categories as a duoidale seems to be:

 * [[Thomas Booker]], [[Ross Street]]. _Tannaka duality and convolution for duoidal categories_, ([link](https://arxiv.org/pdf/1111.5659.pdf)).

An explicit unpacking of the definition, along with examples including the category of [[optic (in computer science)|optics]] appears in

 * {#EarnshawHeffordRoman23} [[Matt Earnshaw]], [[James Hefford]], [[Mario Román]]. _The Produoidal Algebra of Process Decomposition_ ([link](https://arxiv.org/abs/2301.11867)).


Prof

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## Definition

$\mathbf{Prof}$ is the [[2-category]] of [[categories]], [[profunctors]], and [[natural transformations]].

Recall that a profunctor from $A$ to $B$ is a functor $B^{op}\times A\to Set$.  Composition of profunctors in $Prof$ is by the "tensor product of functors" [[coend]] construction: if $H\colon A\to B$ and $K\colon B\to C$, their composite is given as a functor $C^{op}\times A \to Set$ by
$$(c,a)\mapsto \int^{b\in B} H(b,a)\times K(c,b).$$
The identity on a category $A$ is its hom-functor $Hom_A(-,-)$.

$\mathbf{Prof}$ can also be obtained as the [[Kleisli 2-category]] of [[functoriality of categories of presheaves|PSh]].

## Properties

If profunctors are [[categorification|categorified]] [[binary relations]], then $Prof$ is a categorification of [[Rel]].

Note that as defined here, $Prof$ is a weak $2$-[[2-category|category]] or [[bicategory]].  A naturally defined equivalent [[strict 2-category]] has the same objects, but the morphisms $A\to B$ are [[cocontinuous functor]]s $P A \to P B$, where $P A$ is the [[presheaf category]] of $A$.  This is equivalent because a profunctor $A\to B$ can equivalently be regarded as a functor $A\to P B$, and $P A$ is the [[free cocompletion]] of $A$.  This equivalence is an instance of [[Lack's coherence theorem]].

Note that every functor $f\colon A\to B$ gives two [[representable functor|representable]] profunctors $B(f-,-)$ and $B(-,f-)$.  This defines two [[2-functors]] $Cat \to Prof$ that are  the identity on objects.  The relationship between [[Cat]] and $Prof$ encoded in this way makes them into an [[equipment]].

$Prof$ is a sort of classifying object for arbitrary functors; see [[displayed category]].

There are also [[enriched category|enriched]] and [[internal category|internal]] versions of $Prof$. These accordingly refine [[categories of enriched categories]].

$Prof$ is a [[locally cocomplete bicategory]].

## Related concepts

* [[Pos]]

* [[Set]]

  * [[Rel]], [[Span]], **Prof**

* [[Grpd]], [[∞Grpd]]

* [[Cat]],

  * [[Operad]]

  * [[VCat]]

* [[2Cat]]

* [[(∞,1)Cat]]

  * [[(∞,1)Operad]]

* [[(∞,n)Cat]]

## References

* [[Maren Justesen]], *Bikategorien af Profunktorer*, Aarhus 1968 ([pdf](https://ncatlab.org/nlab/files/Justesen_Profunktoren.pdf))

category: category


profinite completion of a group

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#Contents#
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## Definition

### Profinite completion of a group

The **profinite completion** $\hat{G}$ of a ([[discrete group|discrete]]) [[group]] $G$ is the [[limit]] (in the [[category]] of [[topological groups]]) over the [[diagram]] with [[objects]] all the [[finite group|finite]] [[quotient groups]] $G/N_{fin}$ where $N_{fin}$ is a [[normal subgroup]] of $G$ with [[finite number|finite]] [[index of a subgroup|index]], and [[morphisms]] induced from the [[lattice]] of [[subgroups]] of $G$.

Note that the profinite completion actually is a [[profinite group]], and there is a canonical homomorphism $G \to \hat{G}$.

More formally, we note that for any group $G$, the family of its normal finite index subgroups forms a [[filtered category|cofiltered category]] under inclusion. (Denote it by $\Omega_G$.)  The assignment of $G/N$ to $N$ gives a functor from  $\Omega_G^{op}$ to the category of finite groups. It is thus a [[profinite group]] in the sense given in that entry, i.e. a [[pro-object]] in the category of finite groups. This pro-object is the profinite completion of $G$.

The above topological version of this, with which we started, is obtained by means of the equivalence between the category of pro-(finite groups) and that of the groups internal to profinite spaces  that is by taking the limit in the category of topological groups of the diagram of (discrete) finite groups that the above construction gives one.

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

The inclusion [[functor]] $inc$ from the category, $FinGrps$, of finite groups, into that of groups does not have a [[left adjoint]].  It does have a [[pro-left adjoint]], that is to say, it induces a functor on [[pro-objects|procategories]]

$$
  pro\!-\!inc : pro\!-\! FinGrps\to pro\!-\Grps
$$

and that functor _does_ have a left adjoint. If we restrict that 'pro-adjoint' to the subcategory of $pro\!-\! FinGrps$ given by the 'constant' pro-objects, then the result is the pro-finite completion construction that is given above. Because of this, if we think of the natural functor $C\to pro\!-\! C$ to be an inclusion, i.e. think of an object as a pro-object indexed by the one arrow category, we can give a [[universal property]] for the pro-finite completion of a group $G$. This universal property gives a universal cone from $G$  to finite groups, and just encodes the obvious fact that any homomorphism from $G$ to a finite group factors through one of its finite quotient groups. If we write $\hat{G}$ for the pro-finite completion, the universal cone is a map $G\to \hat{G}$ in $pro\!-\! Grps$.

=--

### Pro-$\mathcal{C}$ completion of a group

Let $\mathcal{C}$ be any class of [[finite groups]] that is closed under the formation of [[subgroups]], homomorphic [[images]] and [[group extensions]].

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

A _pro-$\mathcal{C}$ group_ is an [[inverse limit]] of an inverse system of groups in the class $\mathcal{C}$ or alternatively a [[pro-object]] in the full subcategory $\mathcal{C}$ determined by the class $\mathcal{C}$.

=--

The subcategory of $pro\!-\!FinGrps$ consisting of the pro-$\mathcal{C}$ groups and the continuous homomorphisms between them will be denoted $pro\!-\!\mathcal{C}$.

This notation now has two definitions, but, as the corresponding categories are equivalent, this causes no problem.

The categories of the form $pro\!-\!\mathcal{C}$ form [[varieties]] in $prof-FinGrps$.  Recall that a [[variety]] in any algebraic context means a subcategory of 'algebras' closed under products, subobjects and quotients.  We note the condition on $\mathcal{C}$ implies the closure of $\mathcal{C}$ under finite products, so $\mathcal{C}$ is what is called a _pseudovariety_.  The category $pro\!-\!FinGrps$ is [[monadic]] over the category of spaces. This means that free objects exist in all the $pro\!-\!\mathcal{C}$.  A good reference for this is Gildenhuys and Kennison, (1971), see below.

## Examples

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

Consider the profinite completion of the
[[fundamental group]] of an complex [[projective variety]] $X$. Since $X$ has an underlying [[topological space]], its [[fundamental group]] of loops $\pi_1^{top}(X)$ can be defined in the usual way. But one can also define the [[algebraic fundamental group]] $\pi_1^{alg}(X)$. This is a profinite group, which is isomorphic to the profinite completion of $\pi_1^{top}(X)$.

=--

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

The [[profinite completion of the integers]] is

$$
  \widehat {\mathbb{Z}} \coloneqq \underset{\leftarrow}{\lim}_n \mathbb{Z}/n\mathbb{Z}
  \,.
$$

This is [[isomorphism|isomorphic]] to the [[product]] of the [[p-adic integers]] for all $p$

$$
  \widehat{\mathbb{Z}} \simeq \underset{p}{\prod} \mathbb{Z}_p
  \,.
$$

For more on this see at _[[p-adic integers]]_, at _[[adele]]_ and _[[idele]]_.

=--

## Related concepts

### Profinite completion of spaces

* [[profinite completion of a space|Profinite completion of a space]].

(Beware there are two possible interpretations of this term. One is handled in the section above, being profinite completion of the homotopy type of a space.  The entry linked to here treats another more  purely topological concept.)

### Equational or monadic completions

Profinite completion of groups is a special case of a general process that 'completes' a category together with a 'forgetful functor' to some 'base' category, replacing it by a category which is equational/monadic over the base.

*  D. Gildenhuys and J. Kennison, _Equational completions, model induced
triples and pro-objects_, J. Pure Applied Alg., 1(4), (1971), 317–346.

### Profinite rings

[[compact Hausdorff rings are profinite]]

### Pro-finite completions of homotopy types.

Artin and Mazur in their lecture note on étale homotopy introduced a process of profinite completion, generalising that for groups in as much as the profinite completion of an Eilenberg-Mac Lane space having $G$ as fundamental group has the profinite completion of $G$ as _its_ fundamental group. (WARNING: This needs a bit more detail to make it true! so this part of the entry needs more work.)

## References

*  L. Ribes and P. Zalesskii, 2000, _Profinite groups_ , volume 40 of _Ergebnisse der Mathematik und ihrer Grenzgebiete_. 3. Folge , Springer-Verlag, Berlin.

* J. Dixon, M. du Sautoy, A. Mann and D. Segal, 1999, _Analytic pro-p groups_, volume 61 of Cambridge Studies in Advanced Mathematics , Cambridge Univ. Press.

[[!redirects profinite completion]]


profinite completion of a space

> (Beware there are two possible interpretations of this term. One is handled in the entry on [[profinite completion of a group]], being profinite completion of the [[homotopy type]] of a space. The entry  here treats another more purely [[topology|topological]] concept.)

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##Idea

The _profinite completion_ functor (on [[topological spaces]]) is the [[left adjoint]] to the inclusion of the category of  [[profinite spaces]] into that of all [[topological spaces]]. It is particularly useful when applied to [[discrete topological spaces]] (i.e. really: [[sets]]!).

## Definition:

Let $X$ be a [[topological space]].  A *profinite completion* of $X$ is a [[profinite space]], $\hat{X}$, together with a [[continuous map]], $\eta_X : X \to \hat{X}$, such that, if given any profinite space, $Y$, and a continuous map, $g : X \to Y$, there is a unique continuous map $\psi : \hat{X}\to Y$ with $\psi \eta_X = g$

## Related concepts

* [[localic reflection]]

## References

* Abolfazl Tarizadeh, _On the category of profinite spaces as a reflective subcategory_ ([arXiv:1207.5963](http://arxiv.org/abs/1207.5963))

[[!redirects profinite reflection]]


profinite completion of the integers

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## Definition

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###### Definition

The [[profinite completion of a group|profinite completion]] of the [[integers]] is the [[inverse limit]]

$$
  \widehat{\mathbb{Z}}
    \coloneqq
  \underset{\leftarrow}{\lim}_{n \in \mathbb{N}} (\mathbb{Z}/n\mathbb{Z})
$$

of all the [[cyclic groups]] over their canonical [[filtered diagram]].

=--

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

This is [[isomorphism|isomorphic]] to the [[product]] of the [[p-adic integers]] for all [[prime numbers]] $p$

$$
  \widehat{\mathbb{Z}} \simeq \underset{p\; prime}{\prod} \mathbb{Z}_p
  \,.
$$

=--

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

From this perspective the concept of the [[ring of adeles]] is natural, see there for more.

=--

## Properties

### Pontryagin duality

Under [[Pontryagin duality]], $\hat \mathbb{Z}$ maps to $\mathbb{Q}/\mathbb{Z}$, see at _[[Pontryagin duality for torsion abelian groups]]_.

$$
  \array{
     &\mathbb{Z}[p^{-1}]/\mathbb{Z}
     &\hookrightarrow&
     \mathbb{Q}/\mathbb{Z}
     &\hookrightarrow&
     \mathbb{R}/\mathbb{Z}
     \\
     {}^{\mathllap{hom(-,\mathbb{R}/\mathbb{Z})}}\downarrow
     \\
     &\mathbb{Z}_p
     &\leftarrow&
     \hat \mathbb{Z}
     &\leftarrow&
     \mathbb{Z}
  }
$$

## Related concepts

* the [[modular group]]/[[general linear group]] with profinite integer coefficients $GL_2(\widehat{\mathbb{Z}})$ appears as the symmetry group in [[modular equivariant elliptic cohomology]].

## References

* {#Lenstra} [[Hendrik Lenstra]], example 2.2 in _Profinite groups_ ([pdf](http://websites.math.leidenuniv.nl/algebra/Lenstra-Profinite.pdf))

* [[Groupprops]], _[Profinite completion of the integers](http://groupprops.subwiki.org/wiki/Profinite_completion_of_the_integers)_

[[!redirects profinite integers]]


profinite group

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# Profinite groups
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## Definition

### Categorical form

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###### Definition

A **profinite group** is a [[pro-object]] in the [[category]] of [[finite group|finite groups]] (thus it might more precisely be called a pro-(finite group)).

=--

Equivalently:

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

A _profinite group_ is an [[internalization|internal]] [[group]] object in the [[category]] of [[profinite sets]].

=--

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

This means that a profinite group is a [[filtered category|cofiltered]] [[diagram]] of [[finite groups]], which is thought of as a "formal limit" but the limit is not actually computed.  In most cases, the limit would not actually exist in the category of finite groups, and while it would exist in the category of all groups, it would be "wrong" category-theoretically: maps between profinite groups are not the same as maps between their honest limits in [[Grp]].

However, because of [[Stone duality]], it turns out that maps between profinite groups _are_ the same as maps between their honest limits in the category of [[topological group|topological groups]], where the finite groups are given the discrete topology.  Thus, the category of profinite groups can alternately be defined as the category of topological groups that are filtered inverse limits of finite groups.  Moreover, the topological groups that arise in this way can be characterized as those which are [[Hausdorff space|Hausdorff]], [[compact space|compact]], and [[totally disconnected space|totally disconnected]], giving a more elementary definition. In other words, their underlying topological spaces are [[profinite space|profinite]].

=--

### Inverse limit form

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

A   __profinite group__ is an [[inverse limit]] of a system of finite groups.

The finite groups are considered as compact [[discrete groups|discrete]] [[topological groups]] and so the inverse limit, as a closed subspace of the compact space that is the product of all those finite groups has the _inverse limit topology_, hence is, as is said above, a compact Hausdorff, totally disconnected group.

=--

## Examples

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

A [[finite group]] is profinite.

=--

Historically a motivating example was:

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

The absolute [[Galois group]] of a [[number field]] is profinite.

=--

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

For a [[prime number]] $l$ the (additive) group of [[p-adic integer|l-adic integers]] is profinite in that it is the [[inverse limit]] $\mathbb{Z}_l=lim\; \mathbb{Z}/l^n\mathbb{Z}$.

=--

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

In [[SGA1]], Grothendieck defined the [[algebraic fundamental group]] of a [[scheme]] as a profinite group.  (This is linked with his work on [[pro-representable functor|pro-representable functors]].)

=--

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

Any group has a [[profinite completion of a group|profinite completion]], given by taking the projective limit of the group's quotients by its finite index normal subgroups.

A finite index subgroup of a profinite group is not necessarily open. Here is a standard way to obtain examples of such. Let $G$ be a finite group, and let $G^{\mathcal{U}}$ be its [[ultrapower]] with respect to some [[ultrafilter]] $\mathcal{U}$ on $\mathbb{N}$. Since the cardinality and group structure of the _finite_ group $G$ is first-order expressible, the [[Los ultraproduct theorem]] tells us $G^{\mathcal{U}} \simeq G.$ There is a canonical quotient map

$$
\prod_{\mathbb{N}} G \overset{\pi_{\mathcal{U}}}{\twoheadrightarrow} G^{\mathcal{U}} \simeq G
$$

whose kernel $K \overset{\mathrm{df}}{=} \operatorname{ker}(\pi_{\mathcal{U}})$ has finite index in the profinite group $\prod_{\mathbb{N}} G$ because its surjective image has finite index also. Since every ultrafilter contains the cofinite filter, $K$ meets every basic open of the product topology on $\prod_{\mathbb{N}} G$ and so is dense. Since no proper open subset of $\prod_{\mathbb{N}} G$ is dense, $K$ is not open.

A profinite group is called **strongly complete** if it is isomorphic to its own profinite completion. Since a profinite group $G$ is isomorphic to the projective limit of its quotients by open finite index normal subgroups, a profinite group is strongly complete if and only if its open finite index normal subgroups are cofinal among its finite index normal subgroups, if and only if all of its finite index subgroups are open (see e.g. [this MO answer](https://mathoverflow.net/a/122475)).

=--

## Properties

* The category of profinite groups has nice 'exactness' properties. The [[projective limit]] of a system of profinite groups is an [[exact functor]], unlike its behaviour on groups themselves. To extend this behaviour beyond (pro)finite groups sometimes pro-[[localic groups]] have been used; see [[progroup]].

* [[profinite completion of a group|Profinite completions]] have been extended from groups to homotopy types for the analysis of finitary properties of the homotopy type. Various constructions in algebraic geometry lead  naturally to [[profinite homotopy type|profinite homotopy types]].

* Subclasses of profinite groups are extensively studied.  For instance, if $p$ is a prime number, a pro-$p$ group is a pro-object in the category of $p$-groups.

* Pro-p analytic groups have been introduced as an analogue of Lie groups, with certain rings of formal power series replacing differentiable functions.

## Analogues of the group algebra construction

If $G$ is a profinite or pro-p group, the best replacement for the group algebra of $G$  in this context will be  a [[pseudocompact ring|pseudocompact algebra]].  This is the completed group algebra defined as the inverse limit of the ordinary group algebras $k[G/U]$ as $U$ varies through the open normal subgroups of $G$. Here the coefficient ring $k$ will be chosen itself to be a [[pseudocompact ring]]. As finite rings are pseudocompact, one of the most appropriate choices will be a $k = \mathbb{Z}_p$, the field of $p$ elements; (see the book by Dixon et al, below).

## Related concepts

* [[finite group]], [[finitely generated group]]

* [[pro-object]]

* [[profinite completion of a group]]

* [[profinite set]]

* [[profinite space]]

* [[pro-etale site]]

* [[compact Hausdorff rings are profinite]]

## References

Introductory lectures include

* [[Hendrik Lenstra]], _Profinite groups_ ([pdf](http://websites.math.leidenuniv.nl/algebra/Lenstra-Profinite.pdf))
 {#Lenstra}

A survey is given in

* Alexander Lubowsky, _Book review_, BAMS 38(4), pp. 475-479, ([pdf](https://www.ams.org/journals/bull/2001-38-04/S0273-0979-01-00914-4/))

A fairly recent textbook is

* L. Ribes and P. Zalesskii, 2000, _Profinite groups_, volume 40 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, Springer-Verlag,
Berlin.

For the connections with, amongst other things, Galois theory from a categorical viewpoint:

* [[Francis Borceux]], [[George Janelidze]], _[[Galois Theories]]_, Cambridge Studies in Advanced Mathematics __72__, Cambridge University Press, 2001.

For the corresponding 'analytic theory' see:

* J. Dixon, M. du Sautoy, A. Mann and D. Segal, _Analytic pro-p groups_, volume 61 of Cambridge Studies in Advanced Mathematics,
Cambridge Univ. Press 1999.

A standard starting point for the study of the homological properties of the completed group algebra of a profinite group is

* A. Brumer, _Pseudocompact algebras, profinite groups and class formations_,  J. Algebra __4__ (1966) 442-470, [MR202790](http://www.ams.org/mathscinet-getitem?mr=202790), doi

[[!redirects profinite group]]
[[!redirects profinite groups]]
[[!redirects pro-finite group]]
[[!redirects pro-finite groups]]


profinite homotopy type

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## Idea

In [[algebraic topology]], profinite homotopy types are frequently formed by [[profinite completion]] of a homotopy type in order to generate more 'accessible' information. The theory has applications both in [[algebraic geometry]] and [[algebraic topology]].

One origin of the theory can be found in [[Grothendieck's Galois theory]], in which he defined an [[algebraic fundamental group]] of a scheme using its _finite_ 'covering spaces'.  These correspond to the finite field extensions in the [[Grothendieck's Galois theory| classical case of fields]], and from that perspective one can ask what the higher profinite homotopy $n$-types of a scheme should classify.

In the 1960s Artin and Mazur constructed a functor which associates to
each locally noetherian scheme $X$ its [[etale homotopy type|étale homotopy type]], $X_{et}$ , an object of
$pro\Ho(SSets)$, the [[pro-object|pro-category]] of the homotopy category $Ho(SSets)$ of simplicial sets. They observed that this did not correspond to a homotopy category of a [[model category]] on a category of pro simplicial sets. (This observation was the start of the search for a suitable [[pro-homotopy theory]].)

Friedlander gave a rigidified version of the Artin-Mazur homotopy type, which he called the [[etale topological type|étale topological type]] of the scheme. This was used by Quillen in the proof of the Adams conjecture, a result purely in Algebraic Topology.

* Dennis Sullivan introduced profinite completions into topology in his work:
D. Sullivan, _Genetics of homotopy theory and the Adams conjecture_,
Annals of Math., 100, (1974), 1–79. The Adams conjecture, which is a statement about purely topological phenomena, was then proved by Quillen using these ideas.

* At about the same time, Bousfield and Kan, studied profinite completions amongst a wealth of other stuff in
A. Bousfield and D. Kan, 1972, _Homotopy limits, Completions and localizations_, volume 304 of Lecture Notes in Maths , Springer-Verlag.

* In the 1990s, Morel and Voevodsky defined a neat framework
for the use of topological methods in algebraic geometry. They embedded the
category of smooth schemes of 'finite type over a field $k$ into a larger category
of '$k$-spaces', which carries the structure of a closed model category. The study of these $k$-spaces is linked to [[etale homotopy theory|étale homotopy theory]], see Schmidt, _On the étale homotopy type of
Morel-Voevodsky spaces_, and [[Dan Isaksen]], _Etale realization on the $A^1$-homotopy theory of schemes_. Adv. in Math. 184, 37--63 (2004).

* A well motivated approach to profinite homotopy theory has been published by [[Gereon Quick]], in order to give a good [[profinite completion]]  construction for homotopy types;  (see  G. Quick, _Profinite homotopy theory_, Documenta Mathematica, 13,
(2008), 585–612, (Archiv [0803.4082](http://arxiv.org/abs/0803.4082)) and _Some remarks on profinite completion of spaces_ , Arxiv
preprint [arXiv:1104.4659](http://arxiv.org/pdf/1104.4659.pdf). The published version of the model category structure does not give enough generators of the fibrations, but this is corrected in a later article.

[[!redirects profinite homotopy theory]]