Pontrjagin dual

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

#Contents#
* table of contents
{:toc}

## Definition

\begin{definition}\label{PontrjaginDualGroup}
Let $A$ be a [[abelian group|commutative]] ([[Hausdorff space|Hausdorff]]) [[topological group]]. A (continuous) *[[group character]]* of $A$ is any continuous homomorphism $\chi: A\to S^1$ to the [[circle group]]. The __Pontrjagin dual group__

$$
  \widehat{A}
  \;\coloneqq\;
  TopGrps
  \big(
    A
    ,\,
    S^1
  \big)
$$
is the [[abelian group]] of all these [[group characters]] of $A$, equipped with pointwise multiplication (that is multiplication induced by multiplication in the [[circle group]], the multiplication of norm-$1$ complex numbers in $S^1\subset\mathbb{C}$) and with the [[topological space|topology]] of [[uniform convergence]] on each [[compact space|compact]] $K\subset A$ (this is equivalent to the [[compact-open topology]]).
\end{definition}

## Examples

\begin{example}
The Pontrjagin dual of the additive group of [[integers]] $\mathbb{Z}$ is the [[circle group]] $S^1$, and conversely, $\mathbb{Z}$ is the Pontrjagin dual of $S^1$. This pairing of dual topological groups, given by $(n,z) \mapsto z^n$, is related to the subject of [[Fourier transforms]].
\end{example}

\begin{example}
  More generally, the Pontrjagin dual of $\mathbb{Z}^n$ is the [[n-torus]] $(S^1)^n$. In [[solid state physics]] this example appears in the guise of the *[[Brillouin torus]]*.
\end{example}

In general, the dual of a [[discrete space|discrete]] group is a [[compact space|compact]] group and conversely. In particular, therefore, the dual of a [[finite group]] is again finite.

\begin{example}
The [[finite group|finite]] [[cyclic groups]] are Pontrjagin self-dual: $\widehat{\mathbb{Z}/n} \,\simeq\, \mathbb{Z}/n$.
\end{example}

\begin{prop}\label{PontrajginDualOfCompactGroupAsSecondGroupCohomologyGroup}
  **(Pontrjagin dual of compact group as second group cohomology group)**
  \linebreak
  If $G$ is a [[finite group]] (more generally: a [[compact Lie group]]) then its Pontrjagin dual is equivalently its [[cohomology group]] in degree-2 [[group cohomology]] (more generally: refined [[Lie group cohomology]]) with [[integer]] [[coefficients]]:
  $$
    \widehat{G}
    \;\simeq\;
    H^2_{grp}
    \big(
      G
      ;\,
      \mathbb{Z}
    \big)
    \;=\;
    H^2
    \big(
      B G
      ;\,
      \mathbb{Z}
    \big)
    \,.
  $$
\end{prop}
\begin{proof}
  The key point is that, by assumption on $G$, we have
  \[
    \label{RealGroupCohomologyOfCompactGroupVanishes}
    H^{\bullet \geq 1}_{grp}
    \big(
      G;\, \mathbb{R}
    \big)
    \;=\;
    0
    \,.
  \]

Using this, the conclusion is obtained as follows:
The defining [[short exact sequence]] of [[groups]]
$$
  \mathbb{Z} \xhookrightarrow{\;} \mathbb{R} \twoheadrightarrow S^1
$$
extends to a [[homotopy fiber sequence]] of [[2-groups]] (and further of [[n-groups]])
$$
  \mathbb{Z}
    \xrightarrow{\;\;}
  \mathbb{R}
    \xrightarrow{\;\;}
  S^1
    \xrightarrow{\;\;}
  B \mathbb{Z}
    \xrightarrow{\;\;}
  B \mathbb{R}
    \xrightarrow{\;\;}
  \cdots
  \,.
$$

This induces a [[long exact sequence]] of [[cohomology groups]] induced from the [[long exact sequence of homotopy groups]] of the image of this fiber sequence under the [[derived hom-space]] $\mathbf{H}(B G,-) \coloneqq Maps(B G;\, -)$ (of $\mathbf{H} =$ [[∞Grpd]]):
$$
  \array{
  \cdots
  &\to&
  \pi_1
  \left(
    \mathbf{H}(B G, \, B^2 \mathbb{R})
  \right)
  &\xrightarrow{\;\;}&
  \pi_1
  \left(
    \mathbf{H}(B G, \, B^2 S^1)
  \right)
  &\xrightarrow{\;\;}&
  \pi_0
  \left(
    \mathbf{H}(B G, \, B^2 \mathbb{Z})
  \right)
  &\xrightarrow{\;\;}&
  \pi_0
  \left(
    \mathbf{H}(B G, \, B^2 \mathbb{R})
  \right)
  &\to&
  \cdots
  \\
  &&
  =
  &&
  =
  &&
  =
  &&
  =
  \\
  \cdots
  &\to&
  \underset{
    = 0
  }{
  \underbrace{
  H^1
  \big(
    B G
    \;,
    \mathbb{R}
  \big)
  }
  }
  &\xrightarrow{\;\;}&
  H^1
  \big(
    B G
    \;,
    S^1
  \big)
  &\xrightarrow{\;\simeq\;}&
  H^2
  \big(
    B G
    \;,
    \mathbb{Z}
  \big)
  &\xrightarrow{\;\;}&
  \underset{
    = 0
  }{
  \underbrace{
  H^2
  \big(
    B G
    \;,
    \mathbb{R}
  \big)
  }
  }
  &\to&
  \cdots
  \,.
  }
$$
Using the assumption (eq:RealGroupCohomologyOfCompactGroupVanishes) under the braces, this implies the middle isomorphism, as shown.

Now the claim follows by re-expressing $H^1(B G;\, S^1)$ as follows:

$$
  \begin{aligned}
  H^2(B G;\, \mathbb{Z})
  \;\simeq\;
  H^1(B G;\, S^1)
  &
  \;\simeq\;
  \pi_0 \mathbf{H}\big( B G, \, B S^1 \big)
  \\
  &
  \;\simeq\;
  \pi_0 Groupoids\big( G \rightrightarrows \ast, \, S^1 \rightrightarrows \ast \big)
  \\
  & \;\simeq\;
  \pi_0
  \Big(
    Groups\big(G,S^1\big) \sslash S^1
  \Big)
  \\
  & \;\simeq\;
  \pi_0
  \Big(
    Groups\big(G,\,S^1\big) \times B S^1
  \Big)
  \\
  & \;\simeq\;
  Groups(G,S^1)
  \;\simeq\;
  \widehat G
  \,,
  \end{aligned}
$$
where the third line expresses the [[functor groupoid]] of [[functors]] and [[natural transformations]] between [[delooping groupoids]], while the last step uses that the [[circle group]], being [[abelian group|abelian]], has [[trivial action|trivial]] [[conjugation action]] on the [[hom-set]] of [[group homomorphisms]]. (For $G$ a [[compact Lie group]] the analogous argument applies to the [[delooping]]/[[quotient stacks]] $\mathbf{B}G$ in $\mathbf{H} = $[[Smooth∞Grpd]].)
\end{proof}

\begin{example}\label{EquivariantFundamentalGroupOf3TwistOfKTheory}
**(equivariant fundamental group of 3-twists of equivariant K-theory)**
\linebreak
  For $G$ a [[finite group]], the [[fundamental group]] $\pi_1(-)$ of the $G$-[[fixed locus]] $(-)^G$ of the base space $\mathcal{B} PU(\mathcal{H})$ of the [[universal equivariant PU(H)-bundle|universal equivariant $PU(\mathbb{H})$-bundle]] ([[classifying space|classifying]] 3-[[twisted cohomology|twists]] in [[twisted equivariant K-theory]]) is
$$
  \pi_1
  \Big(
     \big(
       \mathcal{B} PU(\mathcal{H})
     \big)^G
  \Big)
  \;\simeq\;
  Grps(G, S^1)
  \,=\,
  \widehat G
$$
(in any [[connected component]] of a "stable map" $G \to PU(\mathcal{H})$, that is) and hence is the Pontrjagin dual group (Def. \ref{PontrjaginDualGroup}) when $G$ is [[abelian group|abelian]].
\end{example}
By [BEJU 2014, Thm. 1.10](universal+equivariant+PU-bundle#BEJU14), see [this Prop.](universal+equivariant+PU-bundle#EquivariantHomotopyGroupsOfBaseSpace).

\begin{example}
The Pontrjagin dual $\hat{\mathbb{R}}$ of the additive group of [[real numbers]] is isomorphic again to $\mathbb{R}$ itself, with the pairing given by $(x,p) \mapsto \mathrm{e}^{\mathrm{i} x p}$. More generally, $\widehat{\mathbb{R}^n} \,=\, \mathbb{R}^n$.
\end{example}

## Properties

### Pontrjagin duality theorem

\begin{theorem}
For every [[abelian group|abelian]] [[Hausdorff space|Hausdorff]] [[locally compact topological group]] $A$, the natural function $A \mapsto \widehat{\widehat{A}}$ from $A$ into the Pontrjagin dual of the Pontrjagin dual of $A$, assigning to every $g\in A$ the continuous character $f_g$ given by $f_g(\chi)=\chi(g)$, is an [[isomorphism]] of topological groups (that is, a group isomorphism that is also a [[homeomorphism]]).
\end{theorem}

Thus, the [[functor]]

$$LocCompAb^{op} \to LocCompAb: G \to \widehat{G}$$

is an [[equivalence of categories]], in fact an [[adjoint equivalence]] whose [[unit of an adjunction|unit]] is

$$A \to \widehat{\widehat{A}}: g \mapsto f_g$$

and whose [[counit of an adjunction|counit]] (the same arrow read in the opposite category) are [[isomorphisms]]. This contravariant self-equivalence restricts to equivalences

$$Ab^{op} \to CompAb$$

$$CompAb^{op} \to Ab$$

where [[Ab]] is the category of ([[discrete topological space|discrete topological]]) groups and $CompAb$ is the category of abelian [[Hausdorff space|Hausdorff]] [[compact topological groups]], each
embedded in $LocCompAb$ in the evident way.

The [[Fourier transform]] on abelian [[locally compact groups]] is formulated in terms of Pontrjagin duals (see below).

### Basic properties of dual groups
 {#BasicPropertiesOfDualGroups}

There are many properties of Hausdorff abelian [[locally compact groups]] that implies properties of their Pontrjagin duals.  For example:

* If $A$ is [[finite group|finite]], then $\widehat{A}$  is finite.

* If $A$ is [[compact topological group|compact]], then $\widehat{A}$ is [[discrete group|discrete]].

  (see also at *[[nearby homomorphisms from compact Lie groups are conjugate]]*)

* If $A$ is [[discrete group|discrete]], then $\widehat{A}$  is [[compact topological group|compact]].

* If $A$ is [[torsion subgroup|torsion]]-free and [[discrete group|discrete]], then $\widehat{A}$ is [[connected topological space|connected]] and [[compact topological group|compact]].

* If $A$ is an [[abelian group|abelian]] [[torsion group]], then $\widehat A$ is an abelian [[profinite group]] (for more see at _[[Pontryagin duality for torsion abelian groups]]_)

* If $A$ is [[connected topological space|connected]] and [[compact topological group|compact]], then $\widehat{A}$ is [[torsion subgroup|torsion]]-free and [[discrete group|discrete]].

* If $A$ is a [[Lie group]], then $\widehat{A}$ is compactly generated in that there is a [[compact topological space|compact]] [[neighborhood]] of the [[neutral element]] that generates $\widehat{A}$ as a group.

* If $A$ is compactly generated, then $\widehat{A}$ is a Lie group.

* If $A$ is [[second countable topological space|second countable]], then $\widehat{A}$ is second countable.

* If $A$ is [[separable topological space|separable]], then $\widehat{A}$ is [[metrizable topological space|metrizable]].

For a discussion of these facts see [Morris 77](#Morris77), [Armacost 81](#Armacost81), exposition in:

* Variations on Pontryagin duality, ([nCafe](http://golem.ph.utexas.edu/category/2008/11/variations_on_pontryagin_duali.html))

Another statement of this type is presented in [Mackey 1948](#Mackey48):

* $A$ is connected if and only if $\widehat{A}$ is a product of a discrete torsion-free group with a finite dimensional real vector space.

## Applications

Pontrjagin duality underlies the abstract framework of [[Fourier analysis]] on locally compact Hausdorff abelian groups $A$: by Fourier duality on $A$, there is a Hilbert space isomorphism ([[Fourier transform]])

$$\mathcal{F}_A: L^2(A, d\mu) \to L^2(\hat{A}, d\hat{\mu})\,,$$

where $d\mu$ is a suitable choice of [[Haar measure]] on $A$, and $d\hat{\mu}$ is a suitable choice of Haar measure on the dual group. Fourier duality is compatible with Pontrjagin duality in the sense that if $\hat{\hat{A}}$ is identified with $A$, then $\mathcal{F}_{\hat{A}}$ is the inverse of $\mathcal{F}_A$.

## Related concepts

* [[Pontryagin duality for torsion abelian groups]]

* [[locally compact topological group]]

* [[Fourier transform]]

* [[character group]]

* [[dual object]]

* [[Cartier duality]], [[Poincaré line bundle]]

## References

### General

The original article:

* {#Pontrjagin34} [[Lev Pontrjagin]], *Theory of topological commutative groups*, Annals of Mathematics Second Series, Vol. 35, No. 2 (Apr., 1934), pp. 361-388 ([doi:10.2307/1968438](https://doi.org/10.2307/1968438))

  Russian translation: Uspekhi Mat. Nauk, 1936, no. 2, 177–195 ([mathnet:umn8882](http://mi.mathnet.ru/eng/umn8882))

Gentle exposition:

* Partha Sarathi Chakraborty, *Pontrjagin duality for finite groups* ([[Chakraborty_PontrjaginDuality.pdf:file]])

Textbook accounts:

* {#Morris77} [[Sidney A. Morris]], _Pontryagin Duality and the Structure of Locally Compact Abelian Groups_, London Math. Soc. Lecture Notes 29, Cambridge U. Press, 1977 ([doi:10.1017/CBO9780511600722](https://doi.org/10.1017/CBO9780511600722))

* {#Armacost81} David L. Armacost, *The Structure of Locally Compact Abelian Groups*, Dekker, New York, 1981.

See also

* Wikipedia, *[Pontryagin duality](https://en.wikipedia.org/wiki/Pontryagin_duality)*

Also:

* {#Mackey48} [[George Mackey]], *The Laplace Transform For Locally Compact Abelian Groups*, Proceedings of the National Academy of Sciences of the United States of America Vol. 34, No. 4 (Apr. 15, 1948), pp. 156-162 ([jstor:87968](https://www.jstor.org/stable/87968))

* [[Michael Barr]], *On duality of topological abelian groups* ([pdf](http://www.math.mcgill.ca/barr/ftp/pdffiles/abgp.pdf), [[Barr_PontrjaginDuality.pdf:file]])

  which provides a perhaps better context for Pontryagin duality than the category of locally compact Hausdorff abelian groups (also known as 'LCA groups').  Barr explains:

  > Did you know that there is a [[star-autonomous category|*-autonomous category]] of topological abelian groups that includes all the LCA groups and whose duality extends that of Pontrjagin?  The groups are characterized by the property that among all topological groups on the same underlying abelian group and with the same set of continuous homomorphisms to the circle, these have the finest topology.  It is not obvious that such a finest exists, but it does and that is the key.

### For higher groups

Discussion of [[categorification|categorified]] Pontrjagin duality for [[2-groups]] and application to [[topological T-duality]]:

* {#BSST07} [[Ulrich Bunke]], [[Thomas Schick]], [[Markus Spitzweck]], [[Andreas Thom]], _Duality for topological abelian group stacks and T-duality_, Proceedings of the ICM 2006 satellite conference, Valladolid, Spain, 2006. Zürich: EMS. Series of Congress Reports, 227-347 (2008) ([arXiv:math/0701428](http://arxiv.org/abs/math/0701428))

* [[Calder Daenzer]], _A groupoid approach to noncommutative T-duality_, PhD, 2007 ([pdf](http://www.math.upenn.edu/grad/dissertations/DaenzerThesis.pdf))

[[!redirects Pontrjagin duals]]

[[!redirects Pontryagin dual]]
[[!redirects Pontryagin duals]]

[[!redirects Pontrjagin dual group]]
[[!redirects Pontrjagin dual groups]]

[[!redirects Pontryagin dual group]]
[[!redirects Pontryagin dual groups]]

[[!redirects Pontrjagin duality]]
[[!redirects Pontrjagin dualities]]

[[!redirects Pontryagin duality]]
[[!redirects Pontryagin dualities]]

[[!redirects Pontriagin dual]]
[[!redirects Pontriagin duals]]

[[!redirects Pontriagin duality]]
[[!redirects Pontriagin dualities]]

[[!redirects Понтрягин dual]]
[[!redirects Понтрягин duals]]

[[!redirects Понтрягин duality]]
[[!redirects Понтрягин dualities]]


Pontrjagin ring

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

#Contents#
* table of contents
{:toc}

## Idea

The operations on an [[H-space]] $X$ (such as a [[topological group]] or a  [[loop space]]) equip its [[homology]] with the [[mathematical structure|structure]] of  [[ring]]. At least for [[ordinary homology]] this is known as the _[[Pontrjagin ring]]_ $H_*(X)$ of $X$.

## Properties

### Relation to Whitehead product
 {#RelationToWhiteheadProduct}

Under the [[Hurewicz homomorphism]], the [[commutator]] of the Pontrjagin product on homology is the [[Whitehead product]] on [[homotopy groups]] of a [[based loop space]].

This is due to [Samelson (1953)](#Samelson53) and for higher Whitehead brackets due to [Arkowitz (1971)](#Arkowitz71).

In fact, in [[characteristic zero]] the [[Pontrjagin ring]] structure on [[connected topological space|connected]] [[based loop spaces]] $\Omega X$ is identified via the [[Hurewicz homomorphism]] with the [[universal enveloping algebra]] (see [there](universal+enveloping+algebra#ForSuperLieAlgebras)) of the [[Whitehead bracket super Lie algebra]] of $X$ [[Milnor & Moore (1965) Appendix](#MilnorMoore65); [Whitehead (1978) Thm. X.7.10](#Whitehead78); [Félix, Halperin & Thomas 2000, Thm. 16.13](#FélixHalperinThomas00)]. Moreover, in this case the [[underlying]] [[ordinary cohomology]] ([[universal coefficient theorem|hence]] the [[ordinary homology]]) [[vector space]] may be read off from any [[Sullivan model]] of $X$ (by the proposition [here](Sullivan+model+of+loop+space#SullivanModelForBasedLoopSpace)).

For the following examples we use these notational conventions:

* $\mathbb{K}$ denotes a [[field]] of [[characteristic zero]],

* a subscript on a generator denotes its degree,

* $\mathbb{K}\langle\cdots \rangle$ denotes the [[graded vector space|graded]] $\mathbb{K}$-[[vector space]] [[linear span|spanned]] by the [[linear basis]] of generators listed inside the angular brackets;

* $\mathbb{K}[\cdots]$ denotes the [[underlying]] [[vector space]] of the free [[graded-commutative algebra]] on the set of generators listed inside the square brackets;

* $T(-)$ denotes the graded [[tensor algebra]] on a given [[graded vector space]],

* the [[semifree dgc-algebra]] on a given set $\{\cdots\}$ of generators subject to differential relations we denote, with slight abuse of notation, by

  $$
    \mathbb{K}[\cdots]\big/
    \left(
      \begin{array}{c}
        \mathrm{d}(-) = \ldots
        \\
        \vdots
      \end{array}
    \right)
  $$

\begin{example}
\label{RationalPontrAlgOfLoopsOfSpheresAndCPn}
**(rational Pontrjagin algebra of loops of spheres and $\mathbb{C}P^n$s)**
\linebreak
The [[Sullivan model of spheres|Sullivan model of]] the [[2-sphere]] is
$$
  CE(\mathfrak{l} S^2)
  \;\simeq\;
  \mathbb{Q}\big[
    \omega_2, \omega_3
  \big]\big/
  \left(
  \begin{array}{l}
    \mathrm{d} \omega_2 \,=\, 0
    \\
    \mathrm{d} \omega_3
      \,=\,
    \tfrac{1}{2} \omega_2 \wedge \omega_2
  \end{array}
  \right)
  \,.
$$
From this it follows (since [[the co-binary Sullivan differential is the dual Whitehead product]])
that the binary [[Whitehead brackets|Whitehead]] [[super Lie brackets]] of $S^2$ are:
$$
  \begin{array}{l}
    [v_1, v_1] = v_2
    \\
    [v_2, v_2] = 0
    \\
    [v_1, v_2] = 0
  \end{array}
$$

The rational Pontrjagin ring of $\Omega S^2$ is the [[universal enveloping algebra]] of this [[super Lie algebra]], hence:
$$
  H_\bullet\big(
    \Omega S^2
    ;\,
    \mathbb{K}
  \big)
  \;\simeq\;
  T\big(
    \mathbb{K}\langle v_1, v_2\rangle
  \big)
  \big/
  \left(
    \begin{array}{l}
      2 v_1^2 - v_2
      \\
      v_1 v_2 - v_2 v_1
    \end{array}
  \right)
  \mathrlap{\,.}
$$
Therefore the [[underlying]] [[graded vector space|graded]] $\mathbb{K}$-[[vector space]] of the Pontrjagin ring of $\Omega S^2$ is
$\mathbb{K}[v_1, v_2]$ but the product of the $v_1$ is deformed from $v_1 \cdot v_1 = 0$ to $v_1 \cdot v_1 = \tfrac{1}{2}v_2$.

Similarly, the rational Pontrjagin algebra of the loop space of the [[4-sphere]] is
$$
  H_\bullet\big(
    \Omega S^4
    ;\,
    \mathbb{K}
  \big)
  \;\simeq\;
  T\big( \mathbb{K}\langle v_3, v_6\rangle \big)
  \big/
  \left(
    \begin{array}{l}
      2 v_3^2 - v_6
      \\
      v_3 v_6 - v_6 v_3
    \end{array}
  \right)
$$
whose [[underlying]] $\mathbb{K}$-[[vector space]] is $\mathbb{K}[v_3, v_6]$ but with the product of the $v_3$ deformed from $v_3 \cdot v_3 = 0$ to $v_3 \cdot v_3 = \tfrac{1}{2}v_6$.

On the other hand, the differential of the [[Sullivan model of complex projective space|Sullivan model of]] [[complex projective space]] $\mathbb{C}P^n$ for $n \geq 2$ has vanishing co-binary part, so that
$$
  H_\bullet\big(
    \Omega \mathbb{C}P^{n \geq 2}
    ;\,
    \mathbb{K}
  \big)
  \;\simeq\;
  T\big( \mathbb{K}\langle v_2, v_{2n}\rangle \big)
  \big/
  \left(
    \begin{array}{l}
      2 v_1^2
      \\
      v_1 v_{2n} - v_{2n} v_1
    \end{array}
  \right)
  \;\simeq\;
  \mathbb{K}[v_1, v_{2n}]
$$
is just a plain graded-commutative algebra.

For $n = \infty$ (ie. for the [[classifying space]] $\mathbb{C}P^\infty \simeq B U(1)$) this becomes
$$
  H_\bullet\big(
    \Omega \mathbb{C}P^{\infty}
    ;\,
    \mathbb{K}
  \big)
  \;\simeq\;
  T\big( \mathbb{K}\langle v_2\rangle \big)
  \big/
  \big(
      2 v_1^2
  \big)
  \;\simeq\;
  \mathbb{K}[v_1]
  \mathrlap{\,,}
$$
reflecting the fact that $\Omega \, \mathbb{C}P^\infty \,\simeq\, \Omega B U(1)\,\simeq\, S^1$.
\end{example}

### Homological group completion

The homological version of
the _[[group completion theorem]]_ relates the [[Pontrjagin ring]] of a [[topological monoid]] $A$ to that of its [[group completion]] $\Omega B A$.

### Relation to quantum cohomology
 {#RelationToQuantumCohomology}

The homology Pontrjagin product of certain [[compact Lie groups]] is identified with the [[quantum cohomology]] of corresponding [[flag varieties]] (see references [below](#ReferencesOnQuantumCohomologyAsPontrjaginRings)).

## Related concepts

* [[homology of loop spaces]]

## References

### General

The concept and the terminology "Pontryagin-multiplication" is due to

* [[Raoul Bott]], [[Hans Samelson]], *On the Pontryagin product in spaces of paths*, Commentarii Mathematici Helvetici **27** (1953) 320–337 [[doi:10.1007/BF02564566](https://doi.org/10.1007/BF02564566)]

who name it in honor of the analogous product operation on the homology of [[compact Lie groups]] due to:

* [[Lev Pontrjagin]], *Homologies in compact Lie groups*, Rec. Math. [Mat. Sbornik] N. S. **6** (**48**) 3 (1939) 389–422 [[mathnet:5835](http://m.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=sm&paperid=5835&option_lang=eng)]

see also:

* [[Hans Samelson]], §3 in: *Beiträge Zur Topologie der Gruppen-Mannigfaltigkeiten*, Annals of Mathematics, Second Series, **42** 5 (1941) 1091-1137 [[jsotr:1970463](https://www.jstor.org/stable/1970463)]

Proof that the [[commutator]] of the Pontrjagin product is the [[Whitehead product]], under the [[Hurewicz homomorphism]]:

* {#Samelson53} [[Hans Samelson]], *A Connection Between the Whitehead and the Pontryagin Product*, American Journal of Mathematics **75** 4 (1953) 744–752  [[doi:10.2307/2372549](https://doi.org/10.2307/2372549)]

and in [[characteristic zero]]:

* {#MilnorMoore65} [[John Milnor]], [[John Moore]], Appendix (pp. 262) of: _On the structure of Hopf algebras_, Annals of Math. __81__ (1965) 211-264 [[doi:10.2307/1970615](https://doi.org/10.2307/1970615), [pdf](http://www.uio.no/studier/emner/matnat/math/MAT9580/v12/undervisningsmateriale/milnor-moore-ann-math-1965.pdf)]

* {#FélixHalperinThomas00} [[Yves Félix]], [[Stephen Halperin]], [[Jean-Claude Thomas]], Thm. 16.13 in: _Rational Homotopy Theory_, Graduate Texts in Mathematics **205** Springer (2000) [[doi:10.1007/978-1-4613-0105-9](https://link.springer.com/book/10.1007/978-1-4613-0105-9)]

and slightly beyond

* {#Halperin92} [[Stephen Halperin]], *Universal enveloping algebras and loop space homology*, Journal of Pure and Applied Algebra **83** 3 (1992) 237-282 \[doi:10.1016/0022-4049(92)90046-I\]

* [[Jonathan A. Scott]], _Algebraic Structure in the Loop Space Homology Bockstein Spectral Sequence_,  Transactions of the American Mathematical Society **354** 8 (2002) 3075-3084 [[jstor:3073034](https://www.jstor.org/stable/3073034)]

and for higher Whitehead brackets:

* {#Arkowitz71} [[Martin Arkowitz]], *Whitehead Products as Images of Pontrjagin Products*, Transactions of the American Mathematical Society, **158** 2 (1971) 453-463 [[doi:10.2307/1995917](https://doi.org/10.2307/1995917)]

and full lifting of the theorem [Milnor & Moore, 1965 (Appendix)](#MilnorMoore65), equipping the rational Pontrjagin algebra with [[A-infinity algebra|$A_\infty$-algebra]] structure and identifying it with the universal envelope of the [[Whitehead L-infinity algebra]]:

* [[José M. Moreno-Fernández]], Thm. 4.1 in: *The Milnor-Moore theorem for $L_\infty$-algebras in rational homotopy theory*, Mathematische Zeitschrift **300** (2022) 2147–2165  [[arXiv:1904.12530](https://arxiv.org/abs/1904.12530), [doi:10.1007/s00209-021-02838-z](https://doi.org/10.1007/s00209-021-02838-z)]

Refinement to algebra structure on the [[singular chain complex]] (Adams-Hilton model):

* [[John F. Adams]], [[Peter J. Hilton]], *On the chain algebra of a loop space*, Commentarii Mathematici Helvetici **30** (1956) 305–330 [[doi:10.1007/BF02564350](https://doi.org/10.1007/BF02564350)]

reviewed and further developed in:

* [[Kathryn Hess]], [[Paul-Eugène Parent]], [[Jonathan Scott]], [[Andrew Tonks]], *A canonical enriched Adams-Hilton model for simplicial sets*, Advances in Mathematics **207** 2 (2006) 847-875 [[doi:10.1016/j.aim.2006.01.013](https://doi.org/10.1016/j.aim.2006.01.013), [arXiv:math/0408216](https://arxiv.org/abs/math/0408216)]

More on the Pontrjagin rings of the [[classical Lie groups]]:

* [[Ichiro Yokota]], *On the homology of classical Lie groups*, J. Inst. Polytech. Osaka City Univ. Ser. A **8** 2 (1957) 93-120 [[euclid:ojm/1353054814](https://projecteuclid.org/journals/osaka-journal-of-mathematics/volume-8/issue-2/On-the-homology-of-classical-Lie-groups/ojm/1353054814.full)]

reviewed in:

* [[Inna Zakharevich]], pp. 35 of: *Stable homotopy*, Section 11 of: *[K-theory and characteristic classes](https://pi.math.cornell.edu/~zakh/6530/)*, lecture notes (2017) [[pdf](https://pi.math.cornell.edu/~zakh/6530/sec11.pdf), [[Zakharevich-StableHomotopy.pdf:file]]]

Further early discussion:

* [[Raoul Bott]], *The space of loops on a Lie group*, Michigan Math. J. **5** 1 (1958) 35-61 [[doi:10.1307/mmj/1028998010](https://projecteuclid.org/journals/michigan-mathematical-journal/volume-5/issue-1/The-space-of-loops-on-a-Lie-group/10.1307/mmj/1028998010.full)]

  > (for loop spaces of [[Lie groups]])

* [[William Browder]], *Homology and Homotopy of H-Spaces*, Proceedings of the National Academy of Sciences of the United States of America **46** 4 (1960) 543-545 [[jstor:70867](https://www.jstor.org/stable/70867)]

* [[William Browder]], p. 36 of: *Torsion in H-Spaces*, Annals of Mathematics, Second Series **74** 1 (1961) 24-51  [[jstor:1970305](https://www.jstor.org/stable/1970305)]

* [[William Browder]], *Homology Rings of Groups*, American Journal of Mathematics **90** 1 (1968) [[jstor:2373440](https://www.jstor.org/stable/2373440)]

In the context of [[string topology]]:

* [[Moira Chas]], [[Dennis Sullivan]], §3 in: *String Topology* [[arXiv:math/9911159](https://arxiv.org/abs/math/9911159)]

* [[Ralph L. Cohen]], [[John D. S. Jones]], Jun Yan, pp. 15 of: *The loop homology algebra of spheres and projective spaces*, in *Categorical Decomposition Techniques in Algebraic Topology*, Progress in Mathematics **215**, Birkhäuser (2003) [[arXiv:math/0210353](https://arxiv.org/abs/math/0210353), [doi:10.1007/978-3-0348-7863-0_5](https://doi.org/10.1007/978-3-0348-7863-0_5)]

* Gwénaël Massuyeau, [[Vladimir Turaev]], *Brackets in the Pontryagin algebras of manifolds*, Mém. Soc. Math. France **154** (2017) [[arXiv:1308.5131](https://arxiv.org/abs/1308.5131)]

Textbook accounts:

* {#Whitehead78} [[George W. Whitehead]], p. 98 in: *Elements of Homotopy Theory*, Springer (1978) [[doi:10.1007/978-1-4612-6318-0](https://link.springer.com/book/10.1007/978-1-4612-6318-0)]

* {#Hatcher02} [[Allen Hatcher]], §3.C, pp. 287 in: *Algebraic Topology*, Cambridge University Press (2002) [[ISBN:9780521795401](https://www.cambridge.org/gb/academic/subjects/mathematics/geometry-and-topology/algebraic-topology-1?format=PB&isbn=9780521795401), [webpage](https://pi.math.cornell.edu/~hatcher/AT/ATpage.html)]

Lecture notes

* [[Tyrone Cutler]], §3 in: *The Bott-Samelson Theorem* (2020) [[pdf](https://www.math.uni-bielefeld.de/~tcutler/pdf/Week%2011%20-%20The%20Bott-Samelson%20Theorem.pdf)]

See also

* Wikipedia, _[Pontryagin product](https://en.wikipedia.org/wiki/Pontryagin_product)_

Relating the Pontrjagin algebra on [[loop groups]] of [[compact Lie groups]] to their [[Langlands dual groups]]:

* [[Zhiwei Yun]], [[Xinwen Zhu]], *Integral homology of loop groups via Langlands dual groups*, Represent. Theory **15** (2011) 347-369 [[arXiv:0909.5487](https://arxiv.org/abs/0909.5487), [doi:10.1090/S1088-4165-2011-00399-X](https://doi.org/10.1090/S1088-4165-2011-00399-X)]

Computation of the Pontryagin products for (loop spaces of) [[flag manifolds]]:

* Jelena Grbic, Svjetlana Terzic, *The integral Pontrjagin homology of the based loop space on a flag manifold*, Osaka J. Math. **47** (2010) 439-460 [[arXiv:math/0702113](https://arxiv.org/abs/math/0702113)

[[!include quantum cohomology as Pontrjagin rings -- references]]

[[!redirects Pontrjagin rings]]

[[!redirects Pontryagin ring]]
[[!redirects Pontryagin rings]]
[[!redirects Pontrjagin ring]]
[[!redirects Pontrjagin rings]]

[[!redirects Pontryagin product]]
[[!redirects Pontryagin products]]
[[!redirects Pontrjagin product]]
[[!redirects Pontrjagin products]]

[[!redirects Pontryagin multiplication]]
[[!redirects Pontryagin multiplications]]

[[!redirects Pontryagin-multiplication]]
[[!redirects Pontryagin-multiplications]]

[[!redirects Pontryagin algebra]]
[[!redirects Pontryagin algebras]]
[[!redirects Pontrjagin algebra]]
[[!redirects Pontrjagin algebras]]

[[!redirects Peterson isomorphism]]
[[!redirects Peterson isomorphisms]]

[[!redirects Adams-Hilton model]]
[[!redirects Adams-Hilton models]]


Pontrjagin-Thom collapse map

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Manifolds and cobordisms
+--{: .hide}
[[!include manifolds and cobordisms - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea
 {#Idea}

Given an [[embedding of topological spaces|embedding]] of [[smooth manifolds]] $i \colon X \hookrightarrow Y$ of [[codimension]] $n$, the _Thom collapse map_ ([Thom 54](#Thom54)) is the [[continuous function]] from $Y$ to the [[n-sphere]] which assigns **asymptotic normal distance** from the [[submanifold]], measured

1. in [[direction vector|direction]] [[orthogonality|perpendicular]] to the submanifold, with respect to a [[normal framing]];

1. asymptotically, regarding all points outside a [[tubular neighbourhood]] as being [[one-point compactification|at infinity]].

> graphics grabbed from [SS 19](Cohomotopy+charge#SatiSchreiber19)

For maximal codimension $n$, hence for 0-dimensional [[submanifolds]], hence for [[configuration space of points|configurations of points]], this is alternatively known as the "electric field map" ([Salvatore 01](cohomotopy+charge#Salvatore01) following [Segal 73, Section 1](cohomotopy+charge#Segal73), see also [Knudsen 18, p. 49](cohomotopy+charge#Knudsen18)) or the "scanning map" ([Kallel 98](cohomotopy+charge#Kallel98)).

The [[homotopy class]] of the Thom collpase map may be regarded as the _[[Cohomotopy charge]]_ of the submanifolds, as measured in $n$-[[Cohomotopy]]-[[generalized cohomology|cohomology theory]].

The PT collapse is a useful approximation to the would-be [[left inverse]] of the [[embedding of topological spaces]]

As such, it is is used to define pushforward of [[cohomology]]-classes along $i$ ("[[Umkehr maps]]"). It also appears as the key step in [[Thom's theorem]].

## Definition

### Component definition in topological spaces
 {#ComponentDefinitionInTopologicalSpaces}

All [[topological spaces]] in the following are taken to be [[compact space|compact]].

Consider $X$ and $Y$ two [[manifolds]] and

$$
  i \colon X \hookrightarrow Y
$$

an [[embedding]].

Write

* $N_i X \coloneqq i^* T Y/ T X $ for the [[normal bundle]];

* $Th(N_i X)$ for the [[Thom space]] of the normal bundle;

* $f \colon N_i X \longrightarrow Y$ for any choice of [[tubular neighbourhood]] of $i$.

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

The **collapse map** (or the _Pontrjagin-Thom construction_) associated to $i$ and the choice of tubular neighbourhood $f$ is

$$
  c_i
    \colon
  Y
    \to
  Y/(Y - f(N_i X))
    \stackrel{\simeq}{\to}
  Th(N_i X)
  \,,
$$

where the first morphism is the [[projection]] onto the [[quotient]] and the second is the canonical [[homeomorphism]] to the [[Thom space]] of the [[normal bundle]].

=--

+-- {: .num_defn #RefinedCollapseMap}
###### Remark

Since in the construction of remark \ref{CollapseMap} every point of $N_i X$ is associated to a particular point of $X$, the collapse map lifts to a map

$$
  Y
    \longrightarrow
  X_+ \wedge Th(N_i X)
$$

from $Y$ to the [[smash product]] of the [[Thom space]] (canonically regarded as a [[pointed topological space]]) and the topological space $X$ with a base point adjoined.

=--

(e.g. [Rudyak 98, p. 317](#Rudyak98))

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

Of particular interest is the case where $Y$ in the above is a [[Cartesian space]] $\mathbb{R}^{dim X + k}$ or rather its [[one-point compactification]], the [[n-sphere|sphere]] $S^{dim X + k}$. By the [[Whitney embedding theorem]], for every $n \in \mathbb{N}$ there exists an $k \in \mathbb{N}$ such that every [[manifold]] $X$ of [[dimension]] $n$ has an [[embedding]] $X \hookrightarrow \mathbb{R}^{n+k} \to S^{n+k}$. In this case the collapse map of def. \ref{CollapseMap} has the form

$$
  S^{n+k} \longrightarrow Th(N_i X)
  \,.
$$

Composing this further with the canonical map $N_i X \longrightarrow E O(k) \underset{O(k)}{\times} \mathbb{R}^{k}$ to the universal vector bundle of rank $k$ yields a map

$$
  S^{n+k} \longrightarrow M O(k)
$$

from to the $k$th space in the [[Thom spectrum]] $M O$. This hence defines an element in the [[homotopy group of a spectrum|homotopy group]] $\pi_{k}(M O)$ of the [[Thom spectrum]]. [[Thom's theorem]] says that all elements in the homotopy groups of $M O$ arise this way, and that they retain precisely the information of the [[cobordism]] [[equivalence class]] of manifolds $X$.

In this case the refined Thom collapse map of def. \ref{RefinedCollapseMap} is of the form

$$
  S^{n+k}
    \longrightarrow
  X_+ \wedge Th(N_i X)
  \,.
$$

=--

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

The refined map in example \ref{ForEmbeddingsIntoAnNSphere} lifts to a morphism of [[spectra]]

$$
  \mathbb{S} \longrightarrow \Sigma_+^\infty X \wedge \Sigma^{-n-k} Th(N_i X)
$$

where $\mathbb{S}$ denotes the [[sphere spectrum]] and $Th(N_i X)$ now the [[Thom spectrum]] of the normal bundle.

This morphism is the [[unit of an adjunction]] which exhibits the [[suspension spectrum]] $\Sigma_+^\infty X$ as a [[dualizable object]] in the [[stable homotopy category]], with [[dual object]] $\Sigma^{-n-k} Th(N_i X)$.  See at _[[Atiyah duality]]_ and at _[[n-duality]]_.

=--

Equivalently, one may proceed as follows. For a framed manifold i.e. a manifold $M^n$ with a chosen trivialization of the normal bundle $N_i (M^n)$ in some $\mathbf{R}^{n+r}$ one has $T N_i(M^n)\cong \Sigma^r(M^n_+)$ where $M^n_+$ is the union of $M^n$ with a disjoint base point. Identify a sphere $S^{n+r}$ with a one-point compactification  $\mathbf{R}^{n+r}\cup \{\infty\}$. Then the Pontrjagin-Thom construction is the map $S^{n+r}\to Th(N_i X)$ obtained by collapsing the complement of the interior of the unit disc bundle $D(N_i M^n)$ to the point corresponding to $S(N_i M^n)$ and by mapping each point of $D(N_i M^n)$ to itself. Thus to a framed manifold $M^n$ one associates the composition

$$
S^{n+r}\to Th(N_i X)\cong \Sigma^r M^n_+\to S^r
$$

and its homotopy class defines an element in $\pi_{n+r}(S^r)$.

### Abstract definition in terms of duality
 {#AbstractDefinitionInTermsOfDuality}

The following is a more abstract description of Pontryagin-Thom collapse in the [[stable homotopy theory]] of [[sphere spectrum]]-[[(∞,1)-module bundles]].

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

Write

$$
  D \coloneqq (-)^\vee\circ \Sigma^\infty_+ \coloneqq L_{whe} Top \to \mathbb{S}Mod
$$

for the [[Spanier-Whitehead duality]] map which sends a [[topological space]] first to its [[suspension spectrum]] and then that to its [[dual object]] in the [[(∞,1)-category of spectra]].

=--

([ABG 11, def 10.3](#ABG11)).

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

For $X$ a [[compact manifold]], let $X \to \mathbb{R}^n$ be an [[embedding]] and write $S^n \to X^{\nu_n}$ for the classical [[Pontryagin-Thom collapse map]] for this situation, and write

$$
  \mathbb{S} \to X^{-T X}
$$

for the corresponding [[looping]] map from the [[sphere spectrum]] to the [[Thom spectrum]] of the negative [[tangent bundle]] of $X$. Then [[Atiyah duality]] produces an [[equivalence]]

$$
  X^{- T X} \simeq D X
$$

which identifies the [[Thom spectrum]] with the [[dual object]] of $\Sigma^\infty_+ X$ in $\mathbb{S} Mod$ and this constitutes a [[commuting diagram]]

$$
  \array{
    && X^{- T X}
    \\
    & \nearrow & \downarrow^{\mathrlap{\simeq}}
    \\
    \mathbb{S}
    &\underset{D(X \to \ast)}{\to}&
    D X
  }
$$

identifying the classical [[Pontryagin-Thom collapse map]] with the abstract [[dual morphism]] construction of prop. \ref{SpanierDualityOperation}.

More generally, for $W \hookrightarrow X$ an [[embedding]] of [[manifolds]], then [[Atiyah duality]] identifies the [[Pontryagin-Thom collapse maps]]

$$
  \mathbb{S} \to X^{-T X} \to W^{- T W}
$$

with the abstract [[dual morphisms]]

$$
  \mathbb{S} \to D X \to D W
  \,.
$$

=--

([ABG 11, prop. 10.5](#ABG11)).

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

Given now $E \in CRing_\infty$ an [[E-∞ ring]], then the [[dual morphism]] $\mathbb{S} \to D X$ induces under [[smash product]] a similar Pontryagin-Thom collapse map, but now not in [[sphere spectrum]]-[[(∞,1)-modules]] but in $E$-[[(∞,1)-modules]].

$$
  E \to D X \otimes_{\mathbb{S}} E
  \,.
$$

The image of this under the $E$-[[generalized cohomology theory|cohomology]] functor produces

$$
  [D X \otimes_{\mathbb{S}} E, E] \to E
  \,.
$$

If now one has a [[Thom isomorphism]] ($E$-[[orientation in generalized cohomology|orientation]]) $ [D X \otimes_{\mathbb{S}} E, E] \simeq [X,E]$ that identifies the cohomology of the dual object with the original cohomology, then together with produces the [[Umkehr map]]

$$
  [X,E] \simeq [D X \otimes_{\mathbb{S}} E, E] \to E
$$

that pushes the $E$-cohomology of $X$ to the $E$-cohomology of the point. Analogously if instead of the terminal map $X \to \ast$ we start with a more general map $X \to Y$.

More generally a [[Thom isomorphism]] may not exists, but $[D X \otimes_{\mathbb{S}} E, E]$ may still be equivalent to a [[twisted cohomology]]-variant $[X,E]_{\chi}$ of $[X,E]$, namely to $[\Gamma_X(\chi),E]$, where $\chi \colon \Pi(X) \to E Line \hookrightarrow E Mod$ is an ([[flat (∞,1)-bundle|flat]]) $E$-[[(∞,1)-module bundle]] on $X$ and and $\Gamma \simeq \underset{\to}{\lim}$ is the [[(∞,1)-colimit]] (the [[generalized Thom spectrum]] construction). In this case the above yields a _[[twisted Umkehr map]]_.

=--

([ABG 10, 9.1](#ABG10))

## Properties

### General

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

For given $i$ all collapse maps for different choices of [[tubular neighbourhood]] $f$ are [[homotopy|homotopic]].

=--

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

By the fact that the space of [[tubular neighbourhood]]s (see there for details) is [[contractible]].

=--

### Relation between cohomotopy and cobordism

For $X$ a [[closed manifold|closed]] [[smooth manifold]] of [[dimension]] $D$, the [[Pontryagin-Thom construction]] (e.g. [Kosinski 93, IX.5](#Kosinski93)) identifies the [[set]]

$$
  SubMfd_{/bord}^{d}(X)
$$

of [[cobordism classes]] of  [[closed manifold|closed]] and [[normally framed submanifolds]] $\Sigma \overset{\iota}{\hookrightarrow} X$ of [[dimension]] $d$ inside $X$ with the [[cohomotopy]] $\pi^{D-d}(X)$ of $X$ in degree $D- d$

$$
  SubMfd_{/bord}^{d}(X)
    \underoverset{\simeq}{PT}{\longrightarrow}
  \pi^{D-d}(X)
  \,.
$$

(e.g. [Kosinski 93, IX Theorem (5.5)](#Kosinski93))

In particular, by this [[bijection]] the canonical [[group]] [[structure]] on [[cobordism groups]] in sufficiently high [[codimension]] (essentially given by [[disjoint union]] of [[submanifolds]]) this way induces a group structure on the cohomotopy sets in sufficiently high degree.

## Related concepts

* [[Thom's theorem]]

* [[Thom isomorphism]]

[[!include generalized fiber integration synonyms - table]]

## References

[[!include Pontryagin-Thom construction -- references]]

See also:

* [[Ralph Cohen]], John Klein, _Umkehr Maps_ ([arXiv:0711.0540](http://arxiv.org/abs/0711.0540))

* [[Victor Snaith]], _Stable homotopy around the arf-Kervaire invariant_, Birkhauser 2009

The general abstract formulation in [[stable homotopy theory]] is in sketched in section 9 of

* {#ABG10} [[Matthew Ando]], [[Andrew Blumberg]], [[David Gepner]], _Twists of K-theory and TMF_, in Robert S. Doran, Greg Friedman, [[Jonathan Rosenberg]], _Superstrings, Geometry, Topology, and $C^*$-algebras_, Proceedings of Symposia in Pure Mathematics [vol 81](http://www.ams.org/bookstore-getitem/item=PSPUM-81), American Mathematical Society ([arXiv:1002.3004](http://arxiv.org/abs/1002.3004))

and is in section 10 of

* {#ABG11} [[Matthew Ando]], [[Andrew Blumberg]], [[David Gepner]], _Parametrized spectra, multiplicative Thom spectra, and the twisted Umkehr map_ ([arXiv:1112.2203](http://arxiv.org/abs/1112.2203))

with an emphases on [[parameterized spectra]].

### Stratified versions

Instead of relating (structured) cobordisms with homs into Thom spectra, one may also relate [[stratified space|stratified]] cobordisms with homs into [[cell complex|cell complexes]]. A version of this relation is spelled out in *Section VII.4* of:

* {#BRS} Sandro Buonchristiano, Colin Rourke, and Brian Sanderson. _A geometric approach to homology theory_. Vol. 18. Cambridge University Press, 1976.

The stratified Pontrjagin-Thom construction may also be considered in higher categorical terms, where it relates functors between [[geometric computad|geometric computads]] to [[manifold diagram|manifold diagrams]]. This is discussed at:

* {#DornCategoricalTP} [[Christoph Dorn]], _The categorical Pontryagin-Thom construction_. [link](https://cxdorn.github.io/tcptc/)

[[!redirects Pontrjagin-Thom construction]]
[[!redirects Pontryagin-Thom construction]]

[[!redirects Pontryagin-Thom collaps map]]
[[!redirects Pontrjagin-Thom collaps map]]
[[!redirects Pontrjagin-Thom collapse map]]

[[!redirects Thom collapse map]]
[[!redirects Thom collapse maps]]
[[!redirects Thom collaps map]]
[[!redirects Thom collaps maps]]

[[!redirects Pontrjagin-Thom collapse]]
[[!redirects Pontryagin-Thom collapse]]
[[!redirects Thom collapse]]

[[!redirects Pontryagin-Thom collapse map]]

[[!redirects Pontryagin-Thom collapse maps]]
[[!redirects Pontrjagin-Thom collapse maps]]

[[!redirects Pontrjagin-Thom collapse construction]]
[[!redirects Pontrjagin-Thom collapse constructions]]
[[!redirects Pontryagin-Thom collapse construction]]
[[!redirects Pontryagin-Thom collapse constructions]]


Pontryagin class

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

#Contents#
* table of contents
{:toc}

## Idea

The _Pontryagin classes_ are [[characteristic class]]es on the [[classifying space]] $B O(n)$ of the [[orthogonal group]]
and, by pullback, on the base of any bundle with structural group the
[[orthogonal group]]. The latter is where they were originally defined.

The analogs for the [[unitary group]] are the [[Chern class]]es.

## Definition

The universal Pontryagin [[characteristic classes]] $P_k$ on the [[classifying space]] $B O(n)$ are, up to a sign, the pullbacks of the [[Chern classes]] $c_{2k}$ along the [[complexification]] inclusion

$$
  B O(n) \to B U(n)
  \,.
$$

## Properties

### As generating universal characteristic classes

The torsion-free quotient of the [[cohomology ring]] $H^\bullet(B SO(2n+1), \mathbb{Z})$
is the [[polynomial ring]] on  all Pontryagin classes
$\{P_i\}_{i = 1}^n$. The torsion is generated by [[Bockstein homomorphism | Bockstein]] images of $H^\bullet(BSO(2n+1);\mathbb{F}_2)$, which is generated by the [[Stiefel-Whitney classes]].

The torsion-free quotient of the [[cohomology ring]] $H^\bullet(B SO(2n), \mathbb{Z})$
is the [[quotient]] of the [[polynomial ring]] on  Pontryagin classes
$P_i$ and the [[Euler class]] $\chi$ by the relation $\chi^2 = P_n$; again the torsion is generated by Bocksteins of monomials in the Stiefel–Whitney classes.

### Further relation to Chern classes
 {#FurtherRelationToChernClasses}

Under the other canonical map

$$
  j \;\colon\; B U(n) \to BO(2n)
$$

one has

$$
  j^\ast(P_k) = \sum_{a + b = 2 k} (-1)^{a+k} c_a c_b
$$

and

$$
  j^\ast(\chi) = c_n
  \,.
$$

### Splitting principle and Chern roots
 {#SplittingPrincipleAndChernRoots}

Under the inclusion

$$
  i \;\colon\; U(1)^n \hookrightarrow U(n) \to O(2n)
$$

of the [[maximal torus]] one has that

$$
  (B i)^\ast(P_k) = \sigma_k(x_1, \cdots, x_n)^2
$$

and

$$
  (B i)^\ast(\chi) = \sigma_n(x_1, \cdots, x_n)
$$

where the $x_i \in H^\bullet(B U(1)^n, \mathbb{Z})$ are the "[[Chern roots]]".

See at _[Chern class - Properties -- Splitting principle and Chern roots](Chern%20class#SplittingPrinciple)_ and at _[splitting principle - Examples - Real vector bundles](splitting+principle#RealVectorBundles)_ for more.

[[!include Chern- and Pontrjagin forms -- section]]

## Trivializations and structures

The [[twisted differential c-structures]] corresponding to Pontryagin class include

* [[twisted differential string structure]] for the first fractional Pontryagin class $\frac{p_1}{2} : B Spin \to B^3 U(1)$;

* [[twisted differential fivebrane structure]] for the second fractional Pontryagin class $\frac{1}{6}p_2 : B String \to B^7 U(1)$.

## Related concepts

* [[Pontrjagin character]]

* [[p1-structure]], [[string structure]], [[fivebrane structure]]

* [[shifted C-field flux quantization]]

* [[Stiefel-Whitney class]], [[Wu class]], [[one-loop anomaly polynomial I8]]

* [[Chern class]]

* [[Stiefel-Whitney class]]

* [[Chern character]]

* [[orthogonal calculus]]

* [[quaternionic oriented cohomology theory]]

## References

### General

The original definition is in

* [[Л. С. Понтрягин]], _Характеристические циклы многообразий_, ДАН, XXXV, № 2 (1942), 35–39.
English translation: _Characteristic Cycles of Manifolds_.  L. S. Pontryagin Selected Works.  Volume 1.  Selected Research Papers.  Edited by R. V. Gamkrelidze.  CRC Press, 1986.  283–287.  [doi](https://doi.org/10.1201/9780367813758).

* [[Л. С. Понтрягин]], _Характеристические циклы дифференцируемых многообразий_, Матем. сб., 21(63):2 (1947), 233–284.  [MathNet.Ru PDF](http://mi.mathnet.ru/msb6237 ).
English translation by A. A. Brown: [[Lev Pontrjagin]], _Characteristic cycles on differentiable manifolds_, Mat. Sbornik N. S. 21(63) (1947), 233-284; A.M.S. Translation 32 (1950).  [PDF](https://www.maths.ed.ac.uk/~v1ranick/papers/pont4.pdf).
English translation by P. S. V. Naidu:
_Characteristic Cycles of Differentiable Manifolds_.  L. S. Pontryagin Selected Works.  Volume 1.  Selected Research Papers.  Edited by R. V. Gamkrelidze.  CRC Press, 1986.  375–433.  [doi](https://doi.org/10.1201/9780367813758).

Early accounts:

* {#Hirzebruch56} [[Friedrich Hirzebruch]], Chapter 1, Section 4 of: _Neue topologische Methoden in der Algebraischen Geometrie_, Ergebnisse der Mathematik und Ihrer Grenzgebiete. 1. Folge, Springer 1956 ([doi:10.1007/978-3-662-41083-7](https://www.springer.com/de/book/9783662406052))

Classical textbook references are

* {#KobayashiNomizu63} [[Shoshichi Kobayashi]], [[Katsumi Nomizu]], Section  XII.4 in: _Foundations of Differential Geometry, Volume 1_, Wiley 1963 ([web](https://www.zuj.edu.jo/download/foundations-of-differential-geometry-vol-1-kobayashi-nomizu-pdf/), [ISBN:9780471157335](https://www.wiley.com/en-us/Foundations+of+Differential+Geometry%2C+Volume+1-p-9780471157335), [Wikipedia](https://en.wikipedia.org/wiki/Foundations_of_Differential_Geometry))

* [[John Milnor]], [[Jim Stasheff]], _Characteristic classes_, Princeton Univ. Press 1974 ([ISBN:9780691081229](https://press.princeton.edu/books/paperback/9780691081229/characteristic-classes-am-76-volume-76), [doi:10.1515/9781400881826](https://doi.org/10.1515/9781400881826), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/milnstas.pdf))

* [[Werner Greub]], [[Stephen Halperin]], [[Ray Vanstone]], _[[Connections, Curvature, and Cohomology]]_ Academic Press (1973)

With an eye towards [[mathematical physics]]:

* [[Mikio Nakahara]], Section 11.4.1 of: *[[Geometry, Topology and Physics]]*, IOP 2003 ([doi:10.1201/9781315275826](https://doi.org/10.1201/9781315275826), pdf)

* {#RudolphSchmidt} [[Gerd Rudolph]], [[Matthias Schmidt]], around Def. 4.2.19 in: _Differential Geometry and Mathematical Physics: Part II. Fibre Bundles, Topology and Gauge Fields_, Theoretical and Mathematical Physics series, Springer 2017 ([doi:10.1007/978-94-024-0959-8](https://link.springer.com/book/10.1007/978-94-024-0959-8))

See also

* Paul Bressler, _The first Pontryagin class_, [math.AT/0509563](http://arxiv.org/abs/math/0509563)

* Ivan Panin, Charles Walter, _Quaternionic Grassmannians and Pontryagin classes in algebraic geometry_, [arxiv/1011.0649](http://arxiv.org/abs/1011.0649)

A brief introduction is in chapter 23, section 7

* [[Peter May]], _A concise course in algebraic topology_ ([pdf](http://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf))

### Relation to gravitational instantons

First Pontrjagin class as counting [[gravitational instanton number]]:

* {#EguchiFreund67} [[Tohru Eguchi]], [[Peter Freund]], _Quantum Gravity and World Topology_, Phys. Rev. Lett. 37, 1251 (1967) ([doi:10.1103/PhysRevLett.37.1251](https://doi.org/10.1103/PhysRevLett.37.1251))

* {#BelavinBurlankov76} [[Alexander Belavin]], D. Burlankov, _The renormalisable theory of gravitation and the Einstein equations_, Physics Letters A Volume 58, Issue 1, 26 July 1976, Pages 7-8 (doi:10.1016/0375-9601(76)90530-2)

* Serdar Nergiz, Cihan Saclioglum, equation (27) in: _A Quasiperiodic Gibbons--Hawking Metric and Spacetime Foam_, Phys. Rev. D 53, 2240 (1996) ([arXiv:hep-th/9505141](https://arxiv.org/abs/hep-th/9505141))

[[!redirects Pontryagin classes]]

[[!redirects Pontrjagin class]]
[[!redirects Pontrjagin classes]]

[[!redirects first Pontryagin class]]
[[!redirects second Pontryagin class]]
[[!redirects first Pontrjagin class]]
[[!redirects second Pontrjagin class]]

[[!redirects first fractional Pontryagin class]]
[[!redirects first fractional Pontrjagin class]]

[[!redirects second fractional Pontryagin class]]
[[!redirects second fractional Pontrjagin class]]

[[!redirects fractional first Pontryagin class]]
[[!redirects fractional second Pontryagin class]]


Pontryagin duality > history

> see *[[Pontrjagin dual]]*


Pontryagin duality for torsion abelian groups

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

## Statement

[[Pontryagin duality]] restricts to an [[equivalence of categories]] between [[abelian group|abelian]] [[profinite groups]] and (the [[opposite category|opposite]] of) [[abelian group|abelian]] [[torsion groups]]

Notice that for $G$ a [[profinite group]] then $hom(G,\mathbb{R}/\mathbb{Z})\simeq hom(G,\mathbb{Q}/\mathbb{Z})$ and so one usually writes this equivalence as

$$
  hom(-,\mathbb{Q}/\mathbb{Z})
  \;\colon\;
  ProFinAb
  \stackrel{\simeq}{\longrightarrow}
  TorsionAb^{op}
$$

## Examples

* $\mathbb{Q}/\mathbb{Z}$ is dual to the [[profinite completion of the integers]] $\hat \mathbb{Z}$. And so the canonical map $\mathbb{Z} \longrightarrow \hat \mathbb{Z}$ is dual to $\mathbb{Q}/\mathbb{Z} \longrightarrow \mathbb{R}/\mathbb{Z}$.

* The abelian group underlying the [[p-adic integers]] $\mathbb{Z}_p$ is Pontryagin dual to the [[Prüfer p-group]] $\mathbb{Z}[p^{-1}]/\mathbb{Z}$.

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

## References

* [[Jean-Pierre Serre]], section 1.1. of _Galois cohomology_

* Ramakrishnan, Valenza, _Fourier analysis on number fields_

* [[Clark Barwick]], _Exercises on locally compact abelian groups: An invitation to harmonic analysis_  ([pdf](https://www.maths.ed.ac.uk/~cbarwick/papers/suppprobsLCA.pdf))


Pontryagin's theorem

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

#Contents#
* table of contents
{:toc}

## Idea

The _[[Pontryagin theorem]]_ ([Pontryagin 38a](#Pontryagin38a), [50](#Pontryagin50), [55 II.6](#Pontryagin55)) identifies, for a [[closed manifold|closed]] [[smooth manifold]] $M^d$

* the [[cobordism classes]] of [[normally framed submanifolds|normally framed]] [[closed manifold|closed]] [[submanifolds]] of [[codimension]] $n$  in $M^d$

with

* the $n$-[[Cohomotopy]] of $M^d$ -- the [[homotopy classes]] of [[maps]] from $M^d$ into the [[n-sphere]];

via

\begin{imagefromfile}
    "file_name": "PontryaginConstruction.jpg",
    "float": "right",
    "width": 540,
    "unit": "px",
    "margin": {
        "top": -40,
        "bottom": 20,
        "right": 0,
        "left": 10
    },
    "caption": "From [SS 2021](https://ncatlab.org/schreiber/show/M-Theory+as+Mf-Theory)"
\end{imagefromfile}

* the _[[Cohomotopy charge map]]_, often known as the _[[Pontryagin-Thom collapse construction]]_, which sends a [[normally framed submanifold]] $\Sigma^{d-n} \subset M^d$ to the [[continuous function]] taking values in the [[one-point compactification]] $\big( \mathbb{R}^n \big)^{cpt} \simeq S^n$ that assigns to points in $M^d$ sufficiently close to $\Sigma$ their "directed distance" from $\Sigma$, namely their [[normal vector]], and regards all other points as being "[[vanishing at infinity|at infinity]]":

\begin{xymatrix@C=20pt}
  {\phantom{AAA}}
  \overset{
    \mathclap{
    \raisebox{3pt}{
      \tiny
      \color{blue}
      \bf
      \begin{tabular}{c}
        unstable
        \\
        framed Cobordism
      \end{tabular}
    }
    }
  }{
  \mathrm{Cob}^{n}_{\mathrm{Fr}}
    \big(
      M^d
    \big)
  }
  \ar@{}[r]
  |-{:=}
  &
  \mathrm{NFramedSubMflds}_{d-n}
  \big(
    M^d
  \big)_{\!\!\big/\mathrm{brdsm}}
  \ar@<+10pt>[rr]
    ^-{
      \mbox{
        \tiny
        \color{olive}
        \bf
        \begin{tabular}{c}
          asymptotic directed distance
        \end{tabular}
      }
    }
  \ar@{}[rr]
    |-{ \simeq }
  \ar@<-10pt>@{<-}[rr]
    _-{
      \mbox{
        \tiny
        \color{olive}
        \bf
        regular pre-image of $0$
      }
    }
  &
  {\phantom{AAAAA}}
  &
  \mathrm{Maps}
  \big(
    M^d
    \,,\,
    S^n
  \big)_{\!\!\big/ \mathrm{hmpty}}
  \ar@{}[r]
    |-{ =: }
  &
  \overset{
    \mathclap{
    \raisebox{3pt}{
      \tiny
      \color{blue}
      \bf
      \begin{tabular}{c}
        unstable
        \\
        Cohomotopy
      \end{tabular}
    }
    }
  }{
    \pi^n
    \big(
      M^d
    \big)
  }
\end{xymatrix}

In this form, with the assumption that $M^d$ is [[closed manifold|closed]], hence [[compact topological space|compact]], the statement appears for instance in [Kosinski 93, Sec. IX Prop. 5.5](#Kosinski93).

More generally, if the [[smooth manifold]] $M^d$ is not assumed to be [[compact topological space|compact]], essentially the same [[Pontryagin-Thom construction]] still gives an identification of the [[cobordism classes]] of its [[normally framed submanifolds]] with the _[reduced Cohomotopy](cohomotopy#ReducedCohomotopy)_ of its [[one-point compactification]]:

\begin{xymatrix@C=20pt}
  \overset{
    \mathclap{
    \raisebox{3pt}{
      \tiny
      \color{blue}
      \bf
      \begin{tabular}{c}
        unstable
        \\
        framed Cobordism
      \end{tabular}
    }
    }
  }{
  \mathrm{Cob}^{n}_{\mathrm{Fr}}
    \big(
      M^d
    \big)
  }
  \ar@{}[r]
  |-{:=}
  &
  \mathrm{NFramedSubMflds}_{d-n}
  \big(
    M^d
  \big)_{\!\!\big/\mathrm{brdsm}}
  \ar@<+10pt>[rr]
    ^-{
      \mbox{
        \tiny
        \color{olive}
        \bf
        \begin{tabular}{c}
          asymptotic directed distance
        \end{tabular}
      }
    }
  \ar@{}[rr]
    |-{ \simeq }
  \ar@<-10pt>@{<-}[rr]
    _-{
      \mbox{
        \tiny
        \color{olive}
        \bf
        regular pre-image of $0$
      }
    }
  &
  {\phantom{AAAAA}}
  &
  \pi_0
  \mathrm{Maps}^{\ast/\!}
  \Big(
    \big(
      M^d
    \big)^{\mathrm{cpt}}
    \,,\,
    S^n
  \Big)
  \ar@{}[r]
    |-{ =: }
  &
  \overset{
    \mathclap{
    \raisebox{3pt}{
      \tiny
      \color{blue}
      \bf
      \begin{tabular}{c}
        unstable
        \\
        reduced Cohomotopy
      \end{tabular}
    }
    }
  }{
    {\widetilde \pi}{}^n
    \Big(
      \big(
        M^d
      \big)^{\mathrm{cpt}}
    \Big)
  }
\end{xymatrix}

This form of Pontryagin's theorem seems to be [[folklore]] (e.g. [here](https://www.tdx.cat/bitstream/handle/10803/129511/FCM_THESIS.pdf#page=8)). It is made fully explicit in [Csépai 20, p. 12-13](#Csepai20).

An analogous statement, identifying [[cobordism classes]] of [[normally oriented submanifolds]] with [[homotopy classes]] of maps into the [[universal vector bundle|universal]] [[special orthogonal group|special orthogonal]] [[Thom space]] $M SO(n)$, is _[[Thom's theorem]]_ ([Thom 54](#Thom54)):

\begin{xymatrix@C=20pt}
  {\phantom{AA}}
  \overset{
    \mathclap{
    \raisebox{3pt}{
      \tiny
      \color{blue}
      \bf
      \begin{tabular}{c}
        unstable
        \\
        oriented Cobordism
      \end{tabular}
    }
    }
  }{
  \mathrm{Cob}^{n}_{\mathrm{SO}}
    \big(
      M^d
    \big)
  }
  \ar@{}[r]
  |-{:=}
  &
  \mathrm{NOrientedSubMflds}_{d-n}
  \big(
    M^d
  \big)_{\!\!\big/\mathrm{brdsm}}
  \ar@<+10pt>[rr]
    ^-{
      \mbox{
        \tiny
        \color{olive}
        \bf
        \begin{tabular}{c}
          asymptotic directed distance
        \end{tabular}
      }
    }
  \ar@{}[rr]
    |-{ \simeq }
  \ar@<-10pt>@{<-}[rr]
    _-{
      \mbox{
        \tiny
        \color{olive}
        \bf
        regular pre-image of $B \mathrm{SO}(n)$
      }
    }
  &
  {\phantom{AAAAA}}
  &
  \pi_0
  \mathrm{Maps}^{\ast/\!}
  \Big(
    \big(
      M^d
    \big)^{\mathrm{cpt}}
    \,,\,
    M \mathrm{SO}(n)
  \Big)
  {\phantom{AAAAAAAA}}
\end{xymatrix}

(Now the notion of asymptotic directed distance depends on the [[normal bundle|normal tangent spaces]], along $\Sigma$, which themselves vary now in the [[Grassmannian]] $Gr_n$, hence in the [[classifying space]] $B SO(n) \subset M SO(n)$.)

Both statements, Pontryagin's and Thom's, as well as their joint generalization to other [[tangential structures]] (besides framing and orientation structure) and notably their [[stabilization]] to [[Whitehead-generalized cohomology theory|Whitehead-generalized]] [[Cobordism cohomology theory]], have all come to be widely known as _the [[Pontryagin-Thom construction]]_, or similar, a term commonly used also for rather more involved cases, such as in [[MUFr]]-theory. This type of construction constitutes the basis of modern [[cobordism theory]] and its application in [[stable homotopy theory]].

\linebreak

## Related concepts

* [[equivariant Pontrjagin theorem]], [[twisted Pontrjagin theorem]]

* [[Hopf degree theorem]], [[equivariant Hopf degree theorem]]

* [[Poincaré–Hopf theorem]]

## References

[[!include Pontryagin-Thom construction -- references]]

[[!include cohomotopy in topological data analysis -- references]]

[[!redirects Pontryagin isomorphism]]
[[!redirects Pontryagin's isomorphism]]

[[!redirects Pontrjagin isomorphism]]
[[!redirects Pontrjagin's isomorphism]]

[[!redirects Pontryagin theorem]]
[[!redirects Pontrjagin theorem]]

[[!redirects Pontryagin's theorem]]
[[!redirects Pontrjagin's theorem]]

[[!redirects Pontryagin construction]]
[[!redirects Pontryagin's construction]]
[[!redirects Pontryagin constructions]]
[[!redirects Pontryagin's constructions]]

[[!redirects Pontrjagin construction]]
[[!redirects Pontrjagin's construction]]
[[!redirects Pontrjagin constructions]]
[[!redirects Pontrjagin's constructions]]


Pontryagin-Thom construction -- references

### Pontrjagin-Thom construction
 {#ReferencesPontrjaginThomConstruction}

#### Pontrjagin's construction
 {#ReferencesPontrjaginConstruction}

##### General

The _[[Pontryagin theorem]]_, i.e. the unstable and [[normally framed submanifold|framed]] version of the [[Pontrjagin-Thom construction]], identifying [[cobordism classes]] of [[normally framed submanifolds]] with their [[Cohomotopy charge]] in unstable [[Karol Borsuk|Borsuk]]-[[Edwin Spanier|Spanier]] [[Cohomotopy sets]], is due to:

* {#Pontryagin38a} [[Lev Pontrjagin]], _[[Classification of continuous maps of a complex into a sphere]]_, _Communication I_, Doklady Akademii Nauk SSSR 19(3) (1938), 147-149

* {#Pontryagin50} [[Lev Pontryagin]], _[[Homotopy classification of mappings of an (n+2)-dimensional sphere on an n-dimensional one]]_, Doklady Akad. Nauk SSSR (N.S.) 19 (1950), 957–959 ([pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/pont3.pdf))

(both available in English translation in [Gamkrelidze 86](Revaz+Gamkrelidze#Gamkrelidze86)),

as presented more comprehensively in:

* {#Pontryagin55} [[Lev Pontrjagin]], _[[Smooth manifolds and their applications in Homotopy theory]]_, Trudy Mat. Inst. im Steklov, No 45, Izdat. Akad. Nauk. USSR, Moscow, 1955 (AMS Translation Series 2, Vol. 11, 1959) ([doi:10.1142/9789812772107_0001](https://www.worldscientific.com/doi/abs/10.1142/9789812772107_0001), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/pont001.pdf))

The [[Pontrjagin theorem]] must have been known to Pontrjagin at least by 1936, when he announced the computation of the [[second stable homotopy group of spheres|second stem of homotopy groups of spheres]]:

* {#Pontrjagin36} [[Lev Pontrjagin]], _Sur les transformations des sphères en sphères_ ([[PontrjaginSurLesTransformationDesSpheres.pdf:file]]) in: _Comptes Rendus du Congrès International des Mathématiques -- Oslo 1936_ ([pdf](https://www.mathunion.org/fileadmin/ICM/Proceedings/ICM1936.2/ICM1936.2.ocr.pdf))

Review:

* {#FreedUhlenbeck91} [[Daniel Freed]], [[Karen Uhlenbeck]], Appendix B of: _Instantons and Four-Manifolds_, Mathematical Sciences Research Institute Publications, Springer 1991 ([doi:10.1007/978-1-4613-9703-8](https://link.springer.com/book/10.1007/978-1-4613-9703-8))

* {#Kosinski93} [[Antoni Kosinski]], chapter IX of: _Differential manifolds_, Academic Press (1993) \[pdf, [ISBN:978-0-12-421850-5](https://www.sciencedirect.com/bookseries/pure-and-applied-mathematics/vol/138/suppl/C)\]

* {#Milnor97} [[John Milnor]], Chapter 7 of: _Topology from the differentiable viewpoint_, Princeton University Press, 1997. ([ISBN:9780691048338](https://press.princeton.edu/books/paperback/9780691048338/topology-from-the-differentiable-viewpoint), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/milnortop.pdf))

* [[Mladen Bestvina]] (notes by [[Adam Keenan]]), Chapter 16 in: _Differentiable Topology and Geometry_, 2002 ([[BestvinaKeenanDifferentialTopology.pdf:file]])

* [[Michel Kervaire]], _La méthode de Pontryagin pour la classification des applications sur une sphère_, in: E. Vesentini (ed.), _Topologia Differenziale_, CIME Summer Schools, vol. 26, Springer 2011 ([doi:10.1007/978-3-642-10988-1_3](https://doi.org/10.1007/978-3-642-10988-1_3))

* [[Rustam Sadykov]], Section 1 of: _Elements of Surgery Theory_, 2013 ([pdf](https://www.math.ksu.edu/~sadykov/Lecture%20Notes/Surgery%20Theory.pdf), [[SadykovSurgeryTheory.pdf:file]])

* {#Csepai20} [[András Csépai]], _Stable Pontryagin-Thom construction for proper maps_,  Period Math Hung 80, 259–268 (2020) ([arXiv:1905.07734](https://arxiv.org/abs/1905.07734), [doi:10.1007/s10998-020-00327-0](https://doi.org/10.1007/s10998-020-00327-0))

Discussion of the early history:

* [Kosinski 93, Section IX.9](#Kosinski93)

##### Twisted/equivariant generalizations
 {#TwistedEquivariantPontrjaginConstruction}

The (fairly straightforward) generalization of the [[Pontrjagin theorem]] to the [[twisted Pontrjagin theorem]], identifying [[twisted Cohomotopy]] with [[cobordism classes]] of [[normally twisted-framed submanifolds]], is made explicit in:

* {#Cruickshank03} [[James Cruickshank]], Lemma 5.2 using Sec. 5.1 in: _Twisted homotopy theory and the geometric equivariant 1-stem_, Topology and its Applications Volume 129, Issue 3, 1 April 2003, Pages 251-271 (doi:10.1016/S0166-8641(02)00183-9)

A general [[equivariant Pontrjagin theorem]] -- relating [[equivariant Cohomotopy]] to normal equivariant framed submanifolds -- remains elusive, but on [[free action|free]] [[G-manifolds]] it is again straightforward (and reduces to the [[twisted Pontrjagin theorem]] on the [[quotient space]]), made explicit in:

* {#Cruickshank99} [[James Cruickshank]], Thm. 5.0.6, Cor. 6.0.13 in: _Twisted  Cobordism and its Relationship to Equivariant Homotopy Theory_, 1999 ([pdf](http://www.collectionscanada.gc.ca/obj/s4/f2/dsk1/tape9/PQDD_0030/NQ46823.pdf), [[Cruickshank99.pdf:file]])

##### In negative codimension
 {#InNegativeCodimension}

In [[negative number|negative]] [[codimension]], the [[Cohomotopy charge map]] from the [[Pontrjagin theorem]] gives the [May-Segal theorem](configuration+space+of+points#LoopSpacesOfSuspensions), now identifying [[Cohomotopy]] _[[cocycle spaces]]_ with [[configuration spaces of points]]:

* {#May72} [[Peter May]], _The geometry of iterated loop spaces_, Springer 1972 ([pdf](https://www.math.uchicago.edu/~may/BOOKS/geom_iter.pdf))

* {#Segal73} [[Graeme Segal]], _Configuration-spaces and iterated loop-spaces_, Invent. Math. __21__ (1973), 213–221. MR 0331377 ([pdf](http://dodo.pdmi.ras.ru/~topology/books/segal.pdf))
c
Generalization of these constructions and results is due to

* {#McDuff75} [[Dusa McDuff]], _Configuration spaces of positive and negative particles_, Topology Volume 14, Issue 1, March 1975, Pages 91-107 (doi:10.1016/0040-9383(75)90038-5)

* {#Boedigheimer87} [[Carl-Friedrich Bödigheimer]], _Stable splittings of mapping spaces_, Algebraic topology. Springer 1987. 174-187 ([pdf](http://www.math.uni-bonn.de/~cfb/PUBLICATIONS/stable-splittings-of-mapping-spaces.pdf), [[BoedigheimerStableSplittings87.pdf:file]])

#### Thom's construction

[[Thom's theorem]] i.e. the unstable and _oriented_ version of the [[Pontrjagin-Thom construction]], identifying [[cobordism classes]] of [[normally oriented submanifolds]] with [[homotopy classes]] of [[maps]] to the [[universal vector bundle|universal]] [[special orthogonal group|special orthogonal]] [[Thom space]] $M SO(n)$, is due to:

* {#Thom54} [[René Thom]], _[[Quelques propriétés globales des variétés différentiables]]_, Comment. Math. Helv. 28, (1954). 17-86 ([doi:10.1007/BF02566923](https://doi.org/10.1007/BF02566923), [dml:139072](https://eudml.org/doc/139072), [digiz:GDZPPN002056259](http://www.digizeitschriften.de/dms/img/?PID=GDZPPN002056259), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/thomcob.pdf))

Textbook accounts:

* {#Stong68} [[Robert Stong]], _Notes on Cobordism theory_, 1968 ([toc pdf](http://pi.math.virginia.edu/StongConf/Stongbookcontents.pdf), [publisher page](http://press.princeton.edu/titles/6465.html))

#### Lashof's construction

The joint generalization of [Pontryagin 38a](#Pontryagin38a), [55](#Pontryagin55) (framing structure) and [Thom 54](#Thom54) (orientation structure) to any family of [[tangential structures]] ("[[(B,f)-structure]]") is first made explicit in

* {#Lashof63} [[Richard Lashof]], _Poincaré duality and cobordism_, Trans. AMS 109 (1963), 257-277 ([doi:10.1090/S0002-9947-1963-0156357-4](https://doi.org/10.1090/S0002-9947-1963-0156357-4))

and the general statement that has come to be known as the _[[Pontryagin-Thom isomorphism]]_ (identifying the stable [[cobordism classes]] of normally [[tangential structure|(B,f)-structured]] [[submanifolds]] with [[homotopy classes]] of [[maps]] to the [[Thom spectrum]] [[Mf]]) is really due to [Lashof 63, Theorem C](#Lashof63).

Textbook accounts:

* [[Theodor Bröcker]], [[Tammo tom Dieck]], Satz 3.1 & 4.9 in: *Kobordismentheorie*, Lecture Notes in Mathematics **178**, Springer (1970) [[ISBN:9783540053415](https://link.springer.com/book/9783540053415)]

* {#Kochman96} [[Stanley Kochman]], section 1.5 of: _[[Bordism, Stable Homotopy and Adams Spectral Sequences]]_, AMS 1996

* {#Rudyak98} [[Yuli Rudyak]], _On Thom spectra, Orientability and Cobordism_, Springer 1998 ([pdf](http://www.maths.ed.ac.uk/~aar/papers/rudyakthom.pdf))

Lecture notes:

* [[John Francis]], _Topology of manifolds_ course notes (2010) ([web](http://math.northwestern.edu/~jnkf/classes/mflds/)), Lecture 3: _Thom's theorem_ ([pdf](http://math.northwestern.edu/~jnkf/classes/mflds/3thom.pdf)),  Lecture 4 _Transversality_ (notes by I. Bobkova) ([pdf](http://math.northwestern.edu/~jnkf/classes/mflds/4transversality.pdf))

* {#Malkiewich11} [[Cary Malkiewich]], Section 3 of: _Unoriented cobordism and $M O$_, 2011 ([pdf](http://math.uiuc.edu/~cmalkiew/cobordism.pdf))

* Tom Weston, Part I of _An introduction to cobordism theory_ ([pdf](http://people.math.umass.edu/~weston/oldpapers/cobord.pdf))

See also:

* [[Manifold Atlas]], _[The Pontrjagin-Thom isomorphism](http://www.map.mpim-bonn.mpg.de/B-Bordism#The_Pontrjagin-Thom_isomorphism)_


Porter Williams

* [webpage](http://philosophy.columbia.edu/directories/student/porter-williams)

## related $n$Lab entries

* [[philosophy of physics]]

* [[naturalness]]

category: people


Pos

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

#Contents#
* table of contents
{:toc}

## Definition

$Pos$ (or $Poset$ or $Posets$) is the [[category]] whose [[objects]] are [[posets]] and whose [[morphisms]] are monotone (weakly increasing) maps.

Since posets can be identified with [[(0,1)-categories]], $Pos$ can be identified with the full subcategory of [[Cat]] spanned by these; thus it might also be called $(0,1)Cat$.

## Properties

The [[hom-sets]] of $Pos$ themselves have the structure of posets, because $Pos$ is a [[cartesian closed category]]. This means $Pos$ is a [[2-poset]] (aka $(1,2)$-category) or [[locally posetal 2-category]].  If [[Set]] is the primordial example of a [[category]] and [[Cat]] is the primordial example of a [[2-category]], then $Pos$ is the primordial example of a $2$-poset.

The category $Pos$ is a [[locally presentable category]]. This implies it is [[complete category|complete]] and [[cocomplete category|cocomplete]]. This can also be seen by viewing $Pos$ as a [[reflective subcategory]] of [[PreOrd]], which is [[topological functor|topological]] over $Set$ and therefore cocomplete.

We may illustrate the latter point of view by showing how to construct coequalizers in $Pos$. Concretely, to form the coequalizer of a pair of poset maps $f, g: X \rightrightarrows Y$, one first takes their coequalizer $q: Y \to Z$ in $Set$, and endows $Z$ with the smallest reflexive transitive relation that makes $q$ order-preserving. This makes $Z$ a [[preorder]]; it is the coequalizer of $f, g$ in $PreOrd$. Then one passes to the poset [[reflective subcategory|reflection]] of this preorder (i.e., identifying $z$ and $z'$ in $Z$ if $z \leq z'$ and $z' \leq z$) to form the coequalizer in $Pos$.

## Related categories

* [[Set]]

* [[Grpd]], [[∞Grpd]]

* [[Cat]], [[(∞,1)Cat]]

* [[(∞,n)Cat]]

[[!redirects Poset]]
[[!redirects Posets]]
[[!redirects category of posets]]
[[!redirects (0,1)Cat]]
[[!redirects (0,1)-Cat]]

category: category


Poset > history

< [[Pos]]


poset > history

< [[poset]]
[[!redirects poset -- history]]


poset of commutative subalgebras

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### $(0,1)$-Category theory
+--{: .hide}
[[!include (0,1)-category theory - contents]]
=--
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
#### Operator algebra
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

### General

For $A$ a an [[associative algebra]], not necessarily [[commutative algebra|commutative]], its collection $ComSub(A)$ of commutative subalgebras $B \hookrightarrow A$ is naturally a [[poset]] under inclusion of subalgebras.

### As a site for noncommutative geometry

Various authors have proposed ([Butterfield-Hamilton-Isham](#ButterfieldHamiltonIsham), [Döring-Isham](#DoeringIsham),
[Heunen-Landsmann-Spitters](#HeunenLandsmanSpitters)) that for the case that $A$ is a [[C-star algebra]] the [[noncommutative geometry]] of the [[Isbell duality|formal dual]] [[space]] $\Sigma(A)$ of $A$ may be understood as a commutative [[geometry]] [[internalization|internal]] to a [[sheaf topos]] $\mathcal{T}_A$ over $ComSub(A)$ or its [[opposite category|opposite]] $ComSub(A)^{op}$. An advantage of the latter is that $\Sigma$ becomes a [[compact locale|compact]] [[regular locale]].

### As a site for noncommutative phase spaces

Specifically, consider the case that the algebra $A = B(\mathcal{H})$ is that of [[bounded operator]]s on a [[Hilbert space]]. This is interpreted as an [[algebra of quantum observables]] and the commutative subalgebras are then "[[classical contexts]]".

Applying [[Bohrification]] to this situation (see there for more discussion), one finds that the [[locale]] $\Sigma(A)$ internal to $\mathcal{T}_A$ behaves like the noncommutative [[phase space]] of a system of [[quantum mechanics]], which however internally looks like an ordinary commutative geometry. Various statements about [[operator algebra]] then have geometric analogs in $\mathcal{T}_A$.

Notably the [[Kochen-Specker theorem]] says that $\Sigma(B(\mathcal{H}))$, while nontrivial, has no [[point]]s/no [[global element]]s. (This topos-theoretic geometric reformulation of the Kochen-Specker theorem had been the original motivation for considering $ComSub(A)$ in the first place in [ButterfieldIsham](#ButterfieldIsham)).

Moreover, inside $\mathcal{T}_A$ the [[quantum mechanics|quantum mechanical]] kinematics encoded by $B(\mathcal{H})$ looks like [[classical mechanics]] kinematics internal to $\mathcal{T}_A$ ([HeunenLandsmannSpitters](#HeunenLandsmanSpitters), following [DöringIsham](#DoeringIsham)):

1. the [[open subset]]s of $\Sigma(A)$ are identified with the quantum [[state]]s on $A$. Their collection forms the [[Heyting algebra]] of quantum logic.

1. [[observable]]s are morphisms of internal [[locale]]s $\Sigma(A) \to IR$, where $IR$ is the [[interval domain]].

The assignment to a noncommutative algebra $A$ of a [[locale]] $\underline{\Sigma}_A$ internal to $\mathcal{T}_A$ has been called [[Bohrification]], in honor of [[Nils Bohr]] whose heuristic writings about the nature of [[quantum mechanics]] as being probed by classical (= commutative) context one may argue is being formalized by this construction.

## Properties

### General

+-- {: .num_lemma}
###### Observation

The poset of commutative subalgebras $C(A)$ is always an (unbounded) meet-[[semilattice]]. If $A$ itself is commutative then it is a bounded meet semilattice, with $A$ itself being the top element.

=--

### Relation to Jordan algebras
 {#RelationToJordanAlgebras}

For $A$ an [[associative algebra]] write $A_J$ for its corresponding [[Jordan algebra]], where the commutative product $\circ : A_J \otimes A_J \to A_J$ is the symmetrization of the product in $A$: $a \circ b = \frac{1}{2}(a b + b a)$.

+-- {: .num_lemma}
###### Observation

There exist von Neumann algebras $A$, $B$ such that there exists a Jordan algebra isomorphism $A_J \to B_J$ but not an algebra isomorphism $A \to B$.

=--

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

By

* [[Alain Connes]], _A factor not anti-isomorphic to itself_, Annals of Mathematics, 101 (1962), no. 3, 536–554. ([JSTOR](http://www.jstor.org/stable/1970940))

there is a [[von Neumann algebra factor]] $A$ with no algebra isomorphism to its opposite algebra $A^{op}$. But clearly $A_J \simeq (A^{op})_J$.

=--

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

Let $A, B$ be [[von Neumann algebra]]s without a type $I_2$-[[von Neumann algebra factor]]-summand and let $ComSub(A)$, $ComSub(B)$ be their posets of [[commutative C-star-algebra|commutative]] sub-von Neumann algebras.

Then every [[isomorphism]] $ComSub(A) \to ComSub(B)$ of [[poset]]s comes from a unique [[Jordan algebra]] isomorphism $A_J \to B_J$.

=--

This is the theorem in ([Harding-Döring](#HardingDoering)).

There is a generalization of this theorem to more general [[C-star algebras]] in ([Hamhalter](#Hamhalter)).

For more on this see at _[[Harding-Döring-Hamhalter theorem]]_.

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

This is related to the [[Alfsen-Shultz theorem]], which says that two $C^*$-algebras have the
same [[state on an operator algebra|states]] precisely if they are Jordan-isomorphic.

=--

### The presheaf topos over $ComSub(A)^{op}$

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

For $A$ a [[C-star algebra]], write $ComSub(A)$ for its [[poset]] of sub-$C^*$-algebras. Write

$$
  \mathcal{T}_A := [ComSub(A),Set]
$$

for the [[presheaf topos]] on $ComSub(A)^{op}$. This is alse called the _[[Bohr topos]]_.

=--

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

This opposite order on commutative subalgebras may be seen as the information order from [[Kripke semantics]]: a larger subalgebra contains more information. In this light the presheaf topos on $ComSub(A)$, as used by ([Döring-Isham 07](#DoeringIsham)) and co-workers, may be seen as the co-Kripke model. This model is also referred to as the _coarse-graining semantics_ of quantum mechanics. See also at _[[spectral presheaf]]_.

=--

+-- {: .num_lemma}
###### Observation

The topos $\mathcal{T}_A$ is a [[localic topos]].

=--

Because $ComSub(A)$ is a [[posite]].

### The locale $\Sigma(A)$

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

The [[presheaf]]

$$
  (\mathbb{A} : B \mapsto U(B)) \;\; \in \mathcal{T}_A
  \,,
$$

where $U(B)$ is the underlying [[set]] of the commutative subalgebra $B$, is canonically a commutative $C^*$-algebra [[internalization|internal]] to $\mathcal{T}_A$.

=--

This is ([HeunenLandsmanSpitters, theorem 5](#HeunenLandsmanSpitters)).

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

By the [[constructive Gelfand duality theorem]] there is uniquely a [[locale]] $\Sigma(A)$ internal to $\mathcal{T}_A$ such that $\mathbb{A}$ is the internal commutative $C^*$-algebra of functions on $\Sigma(A)$.

=--

This observation is amplified in ([HeunenLandsmanSpitters](#HeunenLandsmanSpitters)).

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

If $A = \mathcal{B}(H)$ is the algebra of [[bounded operator]]s on a [[Hilbert space]] $H$ of [[dimension]] $\gt 2$, then then [[Kochen-Specker theorem]] implies that $\Sigma(A)$ has no points/no [[global element]].

=--

This is ([HeunenLandsmanSpitters, theorem 6](#HeunenLandsmanSpitters)), following ([Butterfield-Hamilton-Isham](#ButterfieldHamiltonIsham)).

## References

### General

The proposal that the the noncommutative geometry of $A$ is fruitfully studied via the commutative geometry over $ComSub(A)$ goes back to

* {#ButterfieldHamiltonIsham} [[Jeremy Butterfield]],  John Hamilton, [[Chris Isham]], _A topos perspective on the Kochen-Specker theorem_

  _I. quantum states as generalized valuations_ International Journal of Theoretical Physics,
37(11):2669–2733, 1998.

  _II. conceptual aspects and classical analogues_ International Journal of Theoretical Physics,
38(3):827–859, 1999

  _III. Von Neumann algebras as the base category_ International Journal of
Theoretical Physics, 39(6):1413–1436, 2000.

The proposal that the non-commutativity of the [[phase space]] in [[quantum mechanics]] is fruitfully understood in the light of this has been amplified in a series of articles

* {#DoeringIsham} [[Andreas Döring]], [[Chris Isham]], _A topos foundation for theories of physics_ Journal of Mathematical Physics (2008)

The presheaf topos on $ComSub(A)^{op}$ ([[Bohr topos]]) and its internal localic Gelfand dual to $A$ is discussed in

* {#HeunenLandsmanSpitters} [[Chris Heunen]], [[Klaas Landsman]], [[Bas Spitters]], _A topos for algebraic quantum theory_ ([arXiv:0709.4364](http://arxiv.org/abs/0709.4364))

See also [[higher category theory and physics]].

### Relation to Jordan algebras
 {#ReferencesRelationToJordanAlgebras}

The [relation to Jordan algebras](#RelationToJordanAlgebras) of $ComSub(A)$ is discussed in

* {#HardingDoering} [[John Harding]], [[Andreas Döring]], _Abelian subalgebras and the Jordan structure of a von Neumann algebra_ ([arXiv:1009.4945](http://arxiv.org/abs/1009.4945))

for $A$ a [[von Neumann algebra]] and more generally for $A$ a [[C*-algebra]] in

* {#Hamhalter} [[Jan Hamhalter]], _Isomorphisms of ordered structures of abelian $C^\ast$-subalgebras of $C^\ast$-algebras_, J. Math. Anal. Appl. 383 (2011) 391–399, doi:[10.1016/j.jmaa.2011.05.035](https://dx.doi.org/10.1016/j.jmaa.2011.05.035)

See at _[[Harding-Döring-Hamhalter theorem]]_.

[[!redirects semilattice of commutative subalgebras]]


poset of subobjects

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
#### $(0,1)$-Category theory
+--{: .hide}
[[!include (0,1)-category theory - contents]]
=--
#### Topos Theory
+-- {: .hide}
[[!include topos theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Given any [[object]] $X$ in any [[category]] $C$, the [[subobjects]] of $X$ form a [[poset]] (though in general it may be a [[large category|large]] one, i.e. a partially ordered [[proper class]]; cf. [[well-powered category]]). This is called, naturally enough, the __poset of subobjects__ of $X$, or the __subobject poset__ of $X$.

Sometimes it is the __poset of [[regular subobjects]]__ that really matters (although these are the same in any ([[pretopos|pre]])[[topos]]).

## Properties

If $C$ is [[finitely complete category|finitely complete]], then the subobjects form a meet-[[semilattice]], so we may speak of the __semilattice of subobjects__.

In any [[coherent category]] (such as a [[pretopos]]), the subobjects form a distributive [[lattice]], so we may speak of the __lattice of subobjects__.

In any [[Heyting category]] (such as a [[topos]]), the subobjects of $X$ form a [[Heyting algebra]], so we may speak of the __algebra of subobjects__.

The reader can probably think of other variations on this theme.

If $f : X \to Y$ is a morphism that has pullbacks along monomorphisms, then pullback along $f$ induces a poset morphism $f^* : Sub(Y) \to Sub(X)$, called the **[[preimage]]** or **[[inverse image]]**. This is functorial in the sense that if $g : Y \to Z$ also has this property, then $f^* \circ g^* = (g \circ f)^*$.

If $C$ has pullbacks of monomorphisms, $Sub$ is often used to denote the contravariant functor $C^{op} \to Poset$ whose action on morphisms is $Sub(f) = f^*$.

## Relation to the preorder of monomorphisms

The *poset* of [[subobjects]] $Sub(X)$ is the posetal reflection of the [[preorder]] $Mono(X)$ of [[monomorphisms]] into $X$.

If one opts for the alternative[^1] definition that subobjects _are_ monomorphisms into the object (not isomorphism classes thereof), then the reflection quotient map $Mono(X) \to Sub(X)$ is an [[equivalence]].

[^1]: Discussions of this can be found in [A.1.3](#Johnstone) of Johnstone's Elephant, and also [this MO discussion](#MO).

## Subobject poset functor and relation to hyperdoctrines

* Every [[category]] with pullbacks of monomorphisms has a contravariant functor $\mathrm{Sub}:C^\op \to Pos$ to the category of [[posets]] called the **subobject poset functor**, making it into a [[hyperdoctrine]].

* Every [[finitely complete category]] has a subobject poset functor $\mathrm{Sub}:C^\op \to SemiLat$ to the category of [[semilattices]].

* Every [[regular category]] $C$ is a [[regular hyperdoctrine]] induced by the subobject poset functor, where the subobject poset functor has codomain the [[subcategory]] of [[infimum]]-[[semilattices]] $\mathrm{Sub}:C^\op \to InfSemiLat$.

* Every [[coherent category]] $C$ is a [[coherent hyperdoctrine]] induced by the subobject poset functor, where the subobject poset functor has codomain the [[subcategory]] of [[distributive lattices]] $\mathrm{Sub}:C^\op \to DistLat$.

* Every [[Heyting category]] $C$ is a [[first-order hyperdoctrine]] induced by the subobject poset functor, where the subobject poset functor has codomain the [[subcategory]] of [[Heyting algebras]] $\mathrm{Sub}:C^\op \to HeytAlg$.

* Every [[Boolean category]] $C$ is a [[Boolean hyperdoctrine]] induced by the subobject poset functor, where the subobject poset functor has codomain the [[subcategory]] of [[Boolean algebras]] $\mathrm{Sub}:C^\op \to BoolAlg$.

## Related concepts

* [[lattice of subgroups]]

* [[well-powered category]]

* [[(n+1,1)-category of n-truncated objects]]

## References

* {#Johnstone}  [[Elephant]]

* {#MO} Martin Brandenburg, _Concise definition of subobjects_ ([mathoverflow:184196](https://mathoverflow.net/q/184196))

[[!redirects groupoid of subobjects]]

[[!redirects category of subobjects]]
[[!redirects categories of subobjects]]
[[!redirects subobject category]]
[[!redirects subobject categoriess]]

[[!redirects preorder of subobjects]]
[[!redirects preorders of subobjects]]
[[!redirects subobject preorder]]
[[!redirects subobject preorders]]

[[!redirects proset of subobjects]]
[[!redirects prosets of subobjects]]
[[!redirects subobject proset]]
[[!redirects subobject prosets]]

[[!redirects poset of subobjects]]
[[!redirects posets of subobjects]]
[[!redirects subobject poset]]
[[!redirects subobject posets]]

[[!redirects semilattice of subobjects]]
[[!redirects semilattices of subobjects]]
[[!redirects subobject semilattice]]
[[!redirects subobject semilattices]]

[[!redirects lattice of subobjects]]
[[!redirects lattices of subobjects]]
[[!redirects subobject lattice]]
[[!redirects subobject lattices]]

[[!redirects algebra of subobjects]]
[[!redirects algebras of subobjects]]
[[!redirects subobject algebra]]
[[!redirects subobject algebras]]


poset-stratified space

# Spaces stratified by a poset

* table of contents
{:toc}

## Idea

A **poset-stratified space** is a particular way to define a [[stratified space]] that is convenient for some purposes.

## Definition

A **poset-stratified space** is a continuous map $X\to P$, where $X$ is an arbitrary topological space and $P$ is a $T_0$ [[Alexandroff space]], i.e. a [[poset]] with the Alexandroff topology.

The category of stratified spaces is a full subcategory of the [[arrow category]] of [[Top]].  Note that Alexandroff topologies embed [[Pos]] fully-faithfully in [[Top]], so a map of stratified spaces consists of a continuous map $X\to Y$ and a poset map $P\to Q$ making a square commute.

## Examples

* The *cone* on a stratified space $X\to P$ is the quotient $X\times [0,1) / X\times \{0\}$, stratified by the poset $P^\lhd$ that adjoins a new bottom element to $P$.

* The standard stratification of the $n$-simplex $\Delta^n$ is obtained by regarding it as the $(n+1)$-fold cone of $\emptyset$.  It is stratified by the poset $[n] = \{0,\dots,n\}$.  Note that every stratified map $\Delta^n \to \Delta^m$ has an underlying poset map $[n]\to [m]$; it is a theorem the resulting map $Strat(\Delta^n,\Delta^m) \to Pos([n],[m])$ is a homotopy equivalence.  Thus, we have an embedding of the [[simplex category]] into $Strat$.

* Any manifold $M$ can be equipped with the trivial stratification over the terminal poset.  As a special case, the *extended simplices* $\Delta^n_e = \{ \vec{t} \in \mathbb{R}^{n+1} : \sum t_i = 1\}$ assemble into a cosimplicial object $\Delta^\bullet_e \in cStrat$.

## Conically smooth atlases and the $\infty$-category $Strat$

There is a notion of a *conically smooth atlas* on a stratified space.  (The definition is quite involved, inducting on a number of parameters simultaneously; see ([Ayala--Francis--Tanaka, section 3](#AFT)).)  The category of conically smooth stratified spaces (with conically smooth maps among them) is often again simply called $Strat$.

One can endow this new category $Strat$ with an enrichment in Kan complexes via the extended simplices, by defining $\mathrm{map}_{Strat}(X,Y)_\bullet = \mathrm{hom}_{Strat}(X \times \Delta^\bullet_e,Y)$.  This presents the $\infty$-categorical localization $\mathcal{S}trat$ of $Strat$ at the _stratified homotopy equivalences_ (see ([Ayala--Francis--Rozenblyum, Theorem 2.4.5](#AFRshh))).

## Exit path $\infty$-categories

For a stratified space $X$, its **exit path $\infty$-category** is a [[simplicial space]] (in the $\infty$-categorical sense) defined by

$$Exit(X)_p = \hom_{\mathcal{S}trat}(\Delta^p,X)$$

using the above embedding of $\Delta$ into $Strat$.  It is proven in [Ayala-Francis-Rozenblyum](#AFRshh) that this is indeed a [[complete Segal space]].  The main result of that paper is the [[stratified homotopy hypothesis]], which is the assertion that this construction defines a fully-faithful embedding $Exit : \mathcal{S}trat \hookrightarrow Cat_\infty$, and that certain sheaves on  $\mathcal{S}$trat, known as **striation** sheaves, are equivalent to $(\infty, 1)$-categories.

## Related concepts

* [[stratified category]]

## References

* [[Jacob Lurie]], [[Higher Algebra]], Appendix A.5

* {#AFT} [[David Ayala]], [[John Francis]], [[Hiro Lee Tanaka]], *Local structures on stratified spaces*, [arXiv](http://arxiv.org/abs/1409.0501)

* [[David Ayala]], [[John Francis]], [[Nick Rozenblyum]], *Factorization homology from higher categories*, [arXiv](http://arxiv.org/abs/1504.04007)

* {#AFRshh} [[David Ayala]], [[John Francis]], [[Nick Rozenblyum]], *A stratified homotopy hypothesis*, [arXiv](http://arxiv.org/abs/1502.01713)

* [[Peter J. Haine]], _On the homotopy theory of stratified spaces_ ([arXiv:1811.01119](https://arxiv.org/abs/1811.01119))

whilst an earlier paper on exit paths is

* [[David Treumann]], *[Exit paths and constructible stacks](https://www2.bc.edu/david-treumann/epacs.pdf)*, Compositio Math.145 (2009), 1504–1532

[[!redirects poset-stratified spaces]]


poset-valued set

# Poset-valued sets

* table of contents
{: toc}

## Idea

Let $P$ be a [[poset]] and $F:Rel\to Rel$ an endofunctor of [[Rel]].  A **$P_F$-valued set** is a set $A$ equipped with a function $F A\to P$.

The category of $P_F$-valued sets inherits a lot of structure from $P$: if $P$ is [[closed monoidal category|closed]] [[symmetric monoidal category|symmetric monoidal]], [[star-autonomous]], etc., and $F$ is well-behaved, then so is the category of $P_F$-valued sets.  This provides a general method for constructing models of variants of [[linear logic]].

## Category of Poset-Valued Sets

As above, let $F:Rel\to Rel$ be a [[functor]].  A $P_F$-valued set $A$ consists of

1. An underlying set $A_0$.
2. A valuation $\alpha_A : F A_0 \to P$

A morphism $f : A \to B$ of $P_F$-valued sets consists of

1. An underlying *relation* $f_0 : A_0 ⇸ B_0$
2. Such that if $x F(f_0) y$ then $\alpha_A(x) \leq_{P} \alpha_B(y)$

This gives a category of $P_F$-valued sets using relational composition.
We can further get a [[double category]] of $P_F$-valued sets by noticing that the category structure above is the horizontal category of a [[comma double category]], see there for more detail.

## Examples

If $P=\mathbf{3} = (0\le 1\le 2)$, with the star-autonomous structure where $1$ is both the unit object and the dualizing object, and $F(A) = A\times A$, then the category of $P_F$-valued sets includes the category of [[coherence spaces]] as a full subcategory, preserving the closed monoidal structure as well as products and coproducts.

## References

* [[Andrea Schalk]], [[Valeria de Paiva]], *Poset-valued sets or how to build models for linear logics*, In Theoretical Computer Science, Volume 315, Issue 1, 2004, Pages 83-107, [DOI](https://doi.org/10.1016/j.tcs.2003.11.014).

[[!redirects poset-valued sets]]


posetal reflection

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### $(0,1)$-Category theory
+--{: .hide}
[[!include (0,1)-category theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The *posetal reflection* of a [[preorder]] is the [[poset]] obtained by enforcing antisymmetry by quotienting out [[isomorphisms]].

## Construction

Let $(A, \leq)$ be a [[preorder]]. Define the [[equivalence relation]]:
\[
a \simeq b \quad\text{iff}\quad a \leq b \ \text{and}\ b \leq a.
\]
The corresponding posetal reflection $A'$ of $A$ is the preorder on the [[quotient]] $A/\simeq$ given by
\[
[a] \leq' [b] \quad\text{iff}\quad a \leq b
\]
It's easy to show both $\simeq$ and $\leq'$ are well-defined.

Abstractly, the correspondence $(A, \leq) \mapsto (A', \leq')$ is [[functor|functorial]] from preorders to posets, and in fact exhibits posets as a [[reflective subcategory]] of preorders (hence the name).