Polyakov gauge-string duality -- section

## Polyakov gauge/string duality
 {#PolyakovGaugeStringDualitySection}

Key ideas underlying what is now known as *[[AdS-CFT correspondence|holographic]] [[duality in string theory|duality in]] [[string theory]]* and specifically as *[[AdS-QCD correspondence|holographic QCD]]* (see also at *[[holographic light front QCD]]*) were preconceived by [[Alexander Polyakov]] (cf. historical reminiscences in [Polyakov (2008)](#Polyakov08)) under the name *[[gauge/string duality]]*, in efforts to understand [[confinement|confined]] [[QCD]] (the [[mass gap problem]]) by regarding [[color charge|color]]-[[flux tubes]] ([[Wilson lines]]) between [[quarks]] as dynamical [[strings]].

The logic here proceeds in the following steps (cf. [Polyakov (2007), §1](#Polyakov07) and see the commentary [below](#NoticeTheDecisiveInsight)):

\begin{imagefromfile}
    "file_name": "HadronicStringInPionScattering.jpg",
    "float": "right",
    "width": 400,
    "unit": "px",
    "margin": {
        "top": -0,
        "bottom": 10,
        "right": 0,
        "left": 15
    },
    "caption": "From [Veneziano 2012](string+theory#Veneziano12), Fig 2.9"
\end{imagefromfile}

1. **flux tubes confine as dynamical strings**

   The starting point is the [[hypothesis]] that the strong coupling of particles (such as [[quarks]]) by a (non-abelian) [[gauge field]] (such as the [[strong nuclear force]]) is embodied by the formation of "[[flux tubes]]" ("[[Wilson lines]]") between pairs of such particles, which in themselves behave like [[strings]] with a given [[tension]].

   $[$[Kogut & Susskind (1974)](#KogutSusskind74), [(1975)](#KogutSusskind75); [Wilson (1974)](#Wilson74); [Polyakov (1979)](#Polyakov79), [(1980)](#Polyakov80), [(1987)](#Polyakov80); [Makeenko & Migdal (1981)](#MakeenkoMigdal81), following [Nambu (1970)](Nambu-Goto+action#Nambu70), [Gotō (1971)](Nambu-Goto+action#Gotō71)$]$

   Under this assumption it would be:

   1. the [[flux tube]]/[[string]]'s [[tension]] which keeps the particles at theirs endpoints [[confinement|confined]],

   1. the excitation of these [[flux tubes]]/[[strings]] which follow [[Regge trajectories]] (such as of [[hadrons]]);

   1. the [[scattering]] of these [[flux tubes]]/[[strings]] which explains the observed [[Veneziano amplitudes]],

   which are the main qualitative features to be explained.

1. **quantum flux tubes probe effective higher dimensions**

   But if so, famous quantum effects make such [[flux tubes]]/[[strings]] behave like propagating in an effective/emergent higher-dimensional spacetime:

   * either because one regards the string as *a priori* propagating in non-critical [[spacetime]]-[[dimension]] $4$, in which case the [[dilaton field]]/[[Liouville field]] $\phi$ appears to cancel the [[quantum anomaly]] and acts like a warped 5th dimension:

     this was the point of view published in [Polyakov (1981)](#Polyakov81) but later claimed, in [Polyakov (2008)](#Polyakov08), to have originated already "by the end of '77";

   * or because one regards the string as propagating in critical dimension (10 for the [[superstring]]) right away,

     which is the point of view adopted later in [[AdS-CFT duality]], starting with [Maldacena (1997)](AdS-CFT+correspondence#Maldacena97a)

   with only the endpoints of the [[flux tube]]/[[string]] constrained to lie in the original lower dimensional spacetime

   $[$[Polyakov (1998)](#Polyakov98), [(1999)](#Polyakov99)$]$

   which now appears (in modern language that Polyakov did not originally use) as a "[[brane]]" inside a higher dimensional [[bulk]] spacetime.

   > Notice that in this picture the observable physics that we set out to describe takes place *on the brane* ([[underlying]] which is typically flat [[Minkowski spacetime]]!) at the [[asymptotic boundary|asymptotic]] [[boundary]] of a higher-dimensional [[bulk]] spacetime, while the (potentially large) extra dimensions of a possibly $\sim$ [[AdS]]-bulk remain primarily unobservable. In fact, in Polyakov's original picture the extra 5th dimension is not so much a spacetime dimension but a parameter for the *thickness* of the [[flux tube]], which becomes non-vanishing due to quantum effects $[$cf. [Polyakov (2008), p. 3](#Polyakov08)$]$.

1. **large/small $N$ confined gauge theory is holographic string theory/M-theory**

   Thus the description of strongly coupled matter via [[flux tubes]]/[[strings]] now reveals a [[holography|holographic]] situation where strongly-coupled quantum fields on [[intersecting brane models|intersecting branes]] are equivalently described by a theory of [[quantum gravity]] mediated by [[strings]] propagating in a higher dimensional [[bulk]] [[spacetime]].

   In relation to gauge string duality this is due to [Gubser, Klebanov & Polyakov (1998)](#GubserKlebanovPolyakov98), which is now understood as part of *[[AdS/CFT duality]]*, but it is actually meant to be more general, cf. [Polyakov & Rychkov (2000)](#PolyakovRychkov00).

> While this dual bulk string theory is itself strongly-coupled unless the "number of coincident branes" is humongous (the "[[large-N limit]]") and thus unrealistic after all, the difference is that recognizing the [[branes]] as physical objects reveals a web of concrete hints as to the string's strongly-coupled ([[non-perturbative quantum field theory|non-perturbative]]) completion, going under the working title *[[M-theory]]*, cf. at *[AdS-CFT -- Small $N$ corrections](AdS-CFT+correspondence#SmallNCorrections)*.

> In summary, the plausible approach of understanding strongly-coupled quantum gauge theories by regarding their [[flux tubes]] as dynamical [[strings]] seems to recast the [Millennium Problem](mass+gap#ReferencesMassGapProblem) of understanding strongly-coupled matter into the problem of formulating [[M-theory]]: Given [[M-theory]], it ought to be possible to find [[intersecting brane models]] of single (or a small number of [[coincident branes|coincident]]) [[M-branes]] (such as the *[Witten-Sakai-Sugimoto model](AdS-QCD+correspondence#WittenSakaiSugimotoModel)* [[M5-brane]] system) on whose [[worldvolume]] the desired strongly-coupled field theory is realized (such as [[QCD]]).

{#NoticeTheDecisiveInsight} Notice the decisive early insight of [[Alexander Polyakov]] here: While the idea that [[strings]] somehow describe [[hadron|hadronic]] [[bound states]] was the very origin of [[string theory]] in the early 1970s ("dual resonance models"), the mainstream *abandoned* this perspective in the later 1970s when the critical dimension and the full spectrum of the string became fully understood (cf. *[[Goddard-Thorn no-ghost theorem]]*) and declared that *instead* string should be understood as a [[GUT|grand unified]] [[theory of everything]] including [[quantum gravity]] (see e.g. the historical review of [Veneziano (2012), esp. pp. 30-31](string+theory#Veneziano12) which still clings to this perspective). From here it was only through the long detour of first discovering, inside this grander theory: [[D-branes]] (and [[M5-branes]]) in the 1990s, then their [[near-horizon geometry|near-horizon]] [[AdS-CFT duality]] just before the 2000s and then another decade of exploring [[intersecting D-brane models]] that the community in the 2010s came back full circle to Polyakov's holographic perspective on [[QCD]], now dubbed *[[holographic QCD]]*, in which [[strings]] are [[flux tubes]] that propagate not (alone) in the observable 4 spacetime dimensions but in a primarily unobservable (meanwhile known as *[[Randall-Sundrum model|Randall-Sundrum]]*-like) higher-dimensional [[bulk]] spacetime -- a holographic description of reality that [Polyakov (1999)](#Polyakov99) referred to as the *wall of the cave*, in allusion to [[Plato]] (cf. also [Polyakov (2008), p. 6](#Polyakov08)).

Our whereabouts in this remarkable picture are still often misunderstood today: If string theory is a theory of nature, then, it seems, we see the wall but not the cave: we live on a $\sim$[[Minkowski spacetime|Minkowskian]] [[intersecting D-brane model|brane intersection]] at the ([[asymptotic boundary|asympotic]]) [[boundary]] of a primarily unobserved $\sim$[[anti de Sitter spacetime|anti de Sitter]] [[bulk]] -- which may better be thought of not as physical space but as a configuration space of quantum flux.


polycategory

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#### Category theory
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## Idea

A **polycategory** is like a [[category]] or a [[multicategory]], but where both the [[domain]] and the [[codomain]] of a [[morphism]] can be [[finite set|finite]] [[lists]] of [[objects]] rather than just single objects (much like for [[properads]], but see
[below](#RelationToProperads)). Like multicategories, polycategories have both symmetric and non-symmetric variants.

Just as a symmetric multicategory with one object is also called an [[operad]], and so a general symmetric multicategory is sometimes called a "colored operad", a symmetric polycategory with one object is also called a **dioperad** and a general symmetric polycategory is sometimes called a "colored dioperad".

## Properties

### Relation to properads and PROPs
 {#RelationToProperads}

(Coloured) [[properads]] and [[PROPs]] are both similar to polycategories: what distinguishes polycategories from both properads and PROPs is that in a polycategory, we can only compose along one object at once.  That is, we have a composition operation

$$\circ_D \colon Hom(A,B;C,D,E) \times Hom(F,D,G; H) \to Hom(F,A,B,G; C,H,E)$$

but not an operation such as

$$\circ_{B,C} \colon Hom(A; B,C) \times Hom(B,C; D) \to Hom(A,D).$$

In properads and PROPs, we allow composition along multiple objects at once. This is analogous to the definition of (coloured) [[operads]] in terms of partial composition or simultaneous composition. However, though both kinds of operads are equivalent, there exist coloured properads that are not equivalent to polycategories. For instance, if we have polymorphisms $f : A \to B, B$ and $g : B, B \to C$, we can form the composite $g \circ f : A \to C$ in a properad, but not a polycategory.

PROPs generalise properads further by, in addition to having the multiple composition operator of properads, also having a tensoring operator (given by the action of $\otimes$ on morphisms) that allows for composition along _zero_ objects.

### Internal logic

Polycategories provide a natural [[categorical semantics]] for [[classical logic|classical]] [[linear logic]].

### Representability

A polycategory that is "representable on both sides", meaning informally that morphisms in $Hom(A_1,\dots,A_n;B_1,\dots,B_m)$ correspond to morphisms $A_1 \otimes \cdots\otimes A_n \to B_1 \invamp \cdots \invamp B_m$, is a [[linearly distributive category]].  A formal definition can be found in [(Cockett-Seely)](#CockettSeely97).

### Internal structures

Some categorical structures that are normally defined in a monoidal category can instead be defined in a polycategory, including [[Frobenius algebras]] and [[dual objects]].  Dual objects, in particular, are one approach to [[star-polycategories]]; they are the form of "negation" that makes a linearly distributive category into a [[star-autonomous category]].

### Symmetry

The relation between symmetric and non-symmetric (a.k.a. planar) polycategories is more subtle than that between symmetric and non-symmetric multicategories.  To obtain a truly "planar" notion of polycategory, it is not sufficient to omit the symmetric group actions; in addition one must place restrictions on the composition operations allowed so that no "wires cross".  (All the composition operations in a symmetric polycategory can then be recovered from planar composites together with symmetric group actions.)

One concrete consequence of this is that unlike for multicategories, it is not possible to freely generate a symmetric polycategory from a planar one by simply adding the symmetric group actions.  That is, any planar multicategory gives rise to a symmetric multicategory by duplicating each $n$-ary morphism into $n!$ new morphisms with every possible re-ordering of the objects in the domain, but an attempt to do something similar for a planar polycategory fails because not all the putative composites of such morphisms can be expressed in terms of composition in the original planar polycategory.  See for instance Example 1.3 of [Koslowski](#Koslowski).

## Generalizations

Just as [[multicategories]] are a special case of [[generalized multicategories]], which can be defined relative to any suitable [[monad]], polycategories are a special case of [[generalized polycategories]], which can be defined relative to any suitable [[pseudo-distributive law]].

## Related concepts

* [[dioperad]]
* [[PROP]]
* [[properad]]
* [[multicategory]]
* [[operad]]
* [[LNL Polycategory]]

## References

* M.E. Szabo, *Polycategories*, Comm. Algebra 3 (1975) 663-689.  [DOI](http://www.tandfonline.com/doi/abs/10.1080/00927877508822067)

* {#CockettSeely97} [[Robin Cockett]], [[Robert Seely]],  _Weakly Distributive Categories_, _Journal of Pure and Applied Algebra_, 114(1997)2, pp 133-173 ([ps.gz](http://www.math.mcgill.ca/rags/linear/wdc.ps.gz))

* {#Koslowski} Juergen Koslowski, *A monadic approach to polycategories*, [TAC](http://www.tac.mta.ca/tac/volumes/14/7/14-07abs.html) Vol. 14, 2005, No. 7, pp 125-156.

* [[Richard Garner]], *Polycategories via pseudo-distributive laws*, [arXiv](http://arxiv.org/abs/math/0606735)

* Wee Liang Gan, _Koszul duality for dioperads_, [arxiv](https://arxiv.org/abs/math/0201074)

[[!redirects polycategories]]
[[!redirects dioperad]]
[[!redirects dioperads]]
[[!redirects colored dioperad]]
[[!redirects colored dioperads]]
[[!redirects coloured dioperad]]
[[!redirects coloured dioperads]]


polydisc

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## Idea

### General

In the context of [[rigid analytic geometry|rigid]] [[analytic geometry]], a **polydisc** is a [[product]] of [[discs]]: the [[analytic space]] which is [[Isbell duality|formally dual]] to the [[Tate algebra]] $T_n$ (for an $n$-dimensional polydisk).

This is a basic [[analytic space]]. It is the analog in [[analytic geometry]] of the [[affine space]] $\mathbb{A}^n$ in [[algebraic geometry]].

Those [[analytic spaces]] which are subspaces of polydiscs are called _[[affinoids]]_.

### In complex analytic geometry
 {#InComplexAnalyticGeometry}

Specifically in [[complex analytic geometry]] a polydisc is a sub-[[complex manifold]] of a [[product]] of [[complex planes]]

$$
  D \hookrightarrow \mathbb{C}^n
$$

of the form

$$
  D = D_{\delta_1, \cdots, \delta_n}
  \coloneqq
  \left\{
    (z_1, \cdots, z_n) \in \mathbb{C}^n \;\big|\; {\vert z_i\vert \lt \delta_i}
  \right\}
$$

for $\delta_i \in (0,\infty]$.

e.g. ([Maddock, p.6](#Maddock))

Every [[complex analytic manifold]] is [[covering|locally]] [[isomorphism|isomorphic]] to a complex polydisc, in that it may be covered by [[open subsets]] which are [[biholomorphism|biholomorphic]] to complex polydiscs. e.g. ([Maddock, p. 7](#Maddock)).

In fact ([Fornæss-Stout 77a, lemma II.1](#FornaessStout77a)) states that every [[connected topological space|connected]] and [[second countable topological space|second countable]] [[complex manifold]] may already be [[cover|covered]] by [[finite number|finitely]] many [[open subsets]] [[biholomorphism|biholomorphic]] to a polydisc.

See also at _[good covers by Stein manifolds](Stein+manifold#GoodCoversBySteinManifolds)_.

## Related concepts

* [[Stein manifold]]

## References

### General

* Leonard Lipshitz, Zachary Robinson, _Rings of separated power series_ ([[LipshitzRobinson.pdf:file]])

### In complex analytic geometry

Original articles on [[coverings]] of [[complex manifolds]] by complex polydiscs include

* {#FornaessStout77a} [[John Fornæss]], [[Edgar Stout]], _Spreading Polydiscs on Complex Manifolds_ American Journal of Mathematics Vol. 99, No. 5 (Oct., 1977), pp. 933-960 ([JSTOR](http://www.jstor.org/stable/2373992))

* [[John Fornæss]], [[Edgar Stout]], _Polydiscs in complex manifolds_, Mathematische Annalen 1977, Volume 227, Issue 2, pp 145-153

Introductory lecture notes in the context of the [[Dolbeault theorem]] include

* {#Maddock} Zachary Maddock, _Dolbeault cohomology_ ([pdf](http://www.math.columbia.edu/~maddockz/notes/dolbeault.pdf))

[[!redirects polydisk]]

[[!redirects polydiscs]]
[[!redirects polydisks]]


polyfold

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## Idea

The notion of _polyfolds_, due to [[Helmut Hofer]], is a notion of [[generalized smooth spaces]]. The bulk of the theory is a generalized approach to [[Fredholm operator]] [[analysis]] in geometrical situations; which can be considered as an analytic framework designed to systematically resolve the issue of [[transversal map|transversality]]. It gives a unified framework for [[symplectic field theory]], [[Floer homology]], and some constructions in [[contact geometry]] etc.

## References

## Reviews

* [[Katrin Wehrheim]]'s [quick introduction](http://math.mit.edu/~katrin/slides/slides_polyfolds.pdf).

* Joel W. Fish, [[Helmut Hofer]], _Lectures on Polyfolds and Symplectic Field Theory_ ([arXiv:1808.07147](https://arxiv.org/abs/1808.07147))

## Original articles

* [[Helmut Hofer]],  _A general Fredholm theory and applications_ ([arXiv:math/0509366](http://arxiv.org/abs/math/0509366))

* [[Helmut Hofer]], Kris Wysocki, Eduard Zehnder,

  * _A general Fredholm theory I: A splicing-based differential geometry_ ([arXiv/0612604](http://arxiv.org/abs/math/0612604))

  * _A general Fredholm theory II: implicit function theorems_ ([arXiv:0705.1310](http://arxiv.org/abs/0705.1310))

  * _A general Fredholm theory III: Fredholm functors and polyfolds_ ([arXiv:0810.0736](http://arxiv.org/abs/0810.0736))

* [[Helmut Hofer]], Kris Wysocki, Eduard Zehnder, _Integration theory for zero sets of polyfold Fredholm sections_ ([arXiv/0711.0781](http://arxiv.org/abs/0711.0781))

* [[Helmut Hofer]], _Polyfolds and a general Fredholm theory_ ([arXiv/0809.3753](http://arxiv.org/abs/0809.3753))

* [[Helmut Hofer]], _Polyfolds and Fredholm Theory_,  Oxford University Press (2017) ([doi:10.1093/oso/9780198784913.003.0004](http://dx.doi.org/10.1093/oso/9780198784913.003.0004))

We thank Eugene Lerman for some of the information here.

[[!redirects polyfolds]]


polygon

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## Idea

A __polygonal line__ is the union of segments $\overline{A_0 A_1}$, $\overline{A_1 A_2}, \ldots, \overline{A_{n-1} A_n}$ where
$n\in\mathbf{N}$ and $(A_0,A_1,\ldots, A_n)$ is an $(n+1)$-tuple of consecutively distinct points in an affine plane or in a $k$-dimensional [[affine space]]. A polygonal line is called
__closed__ if $A_0 = A_n$ (regarding that consecutive points are distinct, this implies that $n\geq 2$). A closed polygonal line is __non-self-intersecting__ if for any $k \lt l$, $0\leq k\lt l \leq n-1$, $\overline{A_k A_{k+1}}\cap \overline{A_l A_{l+1}}= \emptyset$ except in the case $(k,l) = (0,n-1)$ when the intersection is $\{A_0\}$. A __simple closed (planar) polygonal line__ is a closed non-self-intersecting line which is in its entirety contained in some affine 2-plane.

By [[Jordan's curve theorem]], a [[complement]] of any simple closed polygonal line within its 2-plane has two components, one bounded and one unbounded.

A __polygon__ is a set in an affine $k$-space ($k\geq 2$) which can be represented as a simple closed planar polygonal line union, being the bounded component of its complement within its 2-plane. The bounded and unbounded components of the complement of the corresponding polygonal line in its 2-plane are called the interior and the exterior of the polygon. For fixed $n$ we also say $n$-gon, and the points $A_1,\ldots, A_n$ are called the vertices of the polygon, the segments $\overline{A_1 A_2}, \ldots, \overline{A_{n-1} A_n}, \overline{A_n A_1}$ are called the __edges__ or the sides of the polygon.

Often one restricts further and also requires that no 2 consecutive edges of a polygon are on the same line (equivalently, no 3 consecutive vertices are on the same line).

A diagonal of an $n$-gon is a segment from $A_k$ to $A_l$ when $|k-l|\gt 1$.

In other geometries instead of segments we may consider segments on geodesical lines, so we can talk about $n$-gones in spherical and non-Euclidean geometries.

For some related ideas see [[polyhedron]].

(...)

## Discussion

A 3-gon is also called a [[triangle]] and it may be alternatively defined as the smallest [[convex set]] in the [[Euclidean plane]] containing 3 non-colinear points. This definition does not rely on the Jordan's curve theorem (for which anyway we just need a more elementary case of simple closed polygonal lines).  Sometimes, by a polygon one means its boundary, that is a simple closed polygonal line, then no need to consider the interior.
$n$-gons for $n\geq 4$ an $n$-gon may be convex or nonconvex.
If it is convex there is seemingly no need for the considerations of (the polygonal case of) Jordan's curve theorem for considering the interior as we can take a convex hull. But how to make a simple criterium without consideration of interior when an $n$-gon is convex ?  In nonconvex/general case, one may resort to subdivision (of would-be interior) into triangles, what is simple in examples, but intricate in full generality.

## Examples

* [[convex polygons]]

  * ([[bigon]])

  * [[triangle]]

  * [[square]]

  * [[pentagon]]

  * [[hexagon]]

* [[regular polygons]]

...

## Related pages

* [[tesselation]]

* [[scissors congruence]]

* [[dihedral group]]

* [[dihedron]]

## References

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

* Jeff Erickson, _Jordan polygon theorem_, from notes for a combinatorial topology course in 2009 ([pdf](https://jeffe.cs.illinois.edu/teaching/comptop/2009/notes/jordan-polygon-theorem.pdf))

* M. Shimrat, _Algorithm 112: Position of point relative to polygon_, Commun. ACM 5(8):434, 1962 [doi](https://doi.org/10.1145/368637.368653); R. Hacker, _Certification of Algorithm 112: Position of point relative to polygon_, Commun. ACM 5(12):606, 1962

[[!redirects polygons]]


polygraph > history

See *[[computad]]*.


polyhedron

> This article is about polyhedra in [[algebraic topology]].  For polyhedra in convex [[geometry]], see the article [[polytope]].

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## Definition

\begin{definition}
A [[topological space]] $X$ is a **polyhedron** if it is [[homeomorphic]] to the [[geometric realization]] of a [[finite set|finite]] [[simplicial complex]] and hence has a [[triangulation]].
\end{definition}

## Properties

\begin{remark}
The study of polyhedra has been one of the sources for methods and applications in [[algebraic topology]], and it is often useful to go back to see the motivations and applications in the older sources.  For instance,  [[shape]] morphisms between polyhedral spaces are just [[homotopy classes]] of [[continuous maps]] so the Cech invariants of polyhedra coincide with their ordinary 'standard' invariants.
\end{remark}

## Related concepts

* [[simplicial complex]]

* [[triangulation]]

* [[geometric realization]]

* [[polygon]]

  * [[pentagon]]

* [[Lefschetz fixed point theorem]]

## Literature

Among textbook accounts:

* {#Spanier66} [[Edwin Spanier]], _Algebraic topology_, Springer 1966 ([doi:10.1007/978-1-4684-9322-1](https://link.springer.com/book/10.1007/978-1-4684-9322-1))

[[!redirects polyhedra]]
[[!redirects polyhedral space]]


polylogarithm

Polylogairthm $Li_n(x)$ generalization of [[dilogarithm]]. There are also multiple polylogarithms of many variables, where the index $n$ is replaced by a partition of number $n$.

* wikipedia [polylogarithm](http://en.wikipedia.org/wiki/Polylogarithm)
* L. Lewin, _Polylogarithms and Associated Functions_, North-Holland, Amsterdam, 1981 [MR618278](http://www.ams.org/mathscinet-getitem?mr=618278)
* Don Zagier, _Polylogarithms, Dedekind zeta functions, and the algebraic K-theory of fields_, in Arithmetic Algebraic Geometry, Progr. Math. __89__, Birkhäuser, Boston, 1990, 391–430 [MR1085270](http://www.ams.org/mathscinet-getitem?mr=1085270)
* A.B. Goncharov, _Polylogarithms, regulators and Arakelov motivic complexes _, JAMS [math.AG/0207036](http://arxiv.org/abs/math/0207036); _A simple construction of Grassmannian polylogarithms__, [arxiv/0908.2238](http://arxiv.org/abs/0908.2238); _Multiple polylogarithms, cyclotomy and modular complexes_, Math Res. Letters 5, (1998) 497-516, [arxiv/1105.2076](http://arxiv.org/abs/1105.2076)

* Herbert Gangl, Alexander B. Goncharov, Andrey Levin, _Multiple logarithms, algebraic cycles and trees_, in "Frontiers in Number Theory, Physics and Geometry", vol. 2 (Cartier, Julia, Moussa, Vanhove, eds.) [math.AG/0504552](http://arxiv.org/abs/math/0504552); _Multiple polylogarithms, polygons, trees and algebraic cycles_,
[arXiv:math.NT/0508066](http://arxiv.org/abs/math/0508066)

Applications to [[renormalization]] in quantum field theory and in relation to motives is in

* Alexander B. Goncharov, Marcus Spradlin, C. Vergu, Anastasia Volovich, _Classical polylogarithms for amplitudes and [[Wilson loop]]s_, Phys.Rev.Lett.105:151605,2010, [arxiv/1006.5703](http://arxiv.org/abs/1006.5703)

[[!redirects polylogarithm]]


polymonad

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\tableofcontents

## Idea

Unlike the [[extension system|bind operation]] of a [[monad]] which is of type
$$
  (\forall\alpha)
  (\forall\beta)
  \big(
    m\alpha \to (\alpha\to m\beta) \to m\beta
  \big)
$$
a *polymonadic bind* involves three abstract types:
$$
  (\forall\alpha)
  (\forall\beta)
  \big(
    m_1\alpha\to(\alpha\to m_2\beta)\to m_3\beta
  \big)
  \,.
$$

This notion subsumes monads in the sense that every set of monads and monad morphisms can be encoded using polymonads while obeying the polymonad laws. Moreover, it encompasses the [[monad (in computer science)|monad-like programming patterns]].

(Note that, from a category theoretic perspective, it is more natural to change the order of the arguments to match the form of [[Kleisli extension]], so that the operations are of the form
$$
  (\forall\alpha)
  (\forall\beta)
  \big(
    (\alpha\to m_2\beta)\to (m_1\alpha\to m_3\beta)
  \big)
  \,.
$$
.)

## Definition

A polymonad is the data of a collection $\mathcal{M}$ of unary type constructors cointaining an element $\mathrm{Id}$, and a bind set $\Sigma_{\mathcal{M}}$ containing an element $b_{\mathrm{Id},\mathrm{Id},\mathrm{Id}}$ (reverse apply).

It must obey the following polymonad laws:

* functorial law;
* left identity;
* right identity;
* associativiy;
* paired morphisms law;
* composition closure law.

The first four laws are chosen to generalise monads, and the last two are chosen so that if certain binds can be uniquely defined in terms of other ones, then these must be present in $\Sigma$.

## Examples

* [[Hoare monad]]

* [[effect monad]]

## References

* Nataliya Guts, [[Michael Hicks]], Nikhil Swamy, Daan Leijen, [[Gavin Bierman]], _Polymonads_, Extended version of POPL'13 submission [[pdf](https://www.cs.umd.edu/~mwh/papers/polymonadsTR.pdf), [[GHSLB-Polymonads.pdf:file]]]

* [[Michael Hicks]], [[Gavin Bierman]], Nataliya Guts, Daan Leijen, Nikhil Swamy, *Polymonadic Programming*, EPTCS **153** 79-99 (2014) [[arXiv:1406.2060](https://arxiv.org/abs/1406.2060), [doi:10.4204/EPTCS.153.7](https://doi.org/10.4204/EPTCS.153.7)]

* Jan Bracker, Henrik Nilsson, _Polymonad programming in Haskell_, Proceedings of the 27th Symposium on the Implementation and Application of Functional Programming, (2015) 1-12 [[doi:10.1145/2897336.2897340](https://dl.acm.org/doi/10.1145/2897336.2897340)]

[[!redirects polymonad]]
[[!redirects polymonads]]


polymorphism

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

> This page is about the concept of polymorphism in computer science. For the concept in [[type theory]] see [[universe polymorphism]]. For the generalisation of morphisms introduced by [[Shinichi Mochizuki]], see [[poly-morphism]].

# Polymorphism
* table of contents
{: toc}

## Idea

In [[computer science]], *polymorphism* refers to situations either where the same name is used to refer to more than one [[function]], or where the same function is used at more than one [[type]]. One usually distinguishes these two kinds of polymorphism as  *ad hoc polymorphism* and *parametric polymorphism*, respectively.

## Ad hoc polymorphism

In **ad hoc polymorphism**, one simply defines multiple functions with the same name and different [[types]], relying on the compiler (or, in some cases, the run-time system) to determine the correct function to call based on the types of its arguments and return value.  This is also called **overloading**.  For instance, using a mathematical notation, one might define functions

$$ add : \mathbb{N} \times \mathbb{N} \to \mathbb{N}$$
$$ add : \mathbb{R} \times \mathbb{R} \to \mathbb{R}$$

and then when $add(3,2)$ is invoked, the compiler knows to call the first function since $3$ and $2$ are [[natural numbers]], whereas when $add(4.2,\pi)$ is invoked it calls the second function since $4.2$ and $\pi$ are [[real numbers]].

Note that there is nothing which stipulates that the *behavior* of a class of ad-hocly polymorphic functions with the same name should be at all similar.  Nothing prevents us from defining $ add : \mathbb{N} \times \mathbb{N} \to \mathbb{N}$ to add its arguments but $ add : \mathbb{R} \times \mathbb{R} \to \mathbb{R}$ to subtract its arguments.  Of course, it is good programming practice to make overloaded functions similar in their behavior.

In the example above, there might even be a [[coercion]] function $c : \mathbb{N} \to \mathbb{R}$, to be invoked whenever a natural number appears where the compiler expects a real number, giving a [[commutative diagram]]
$$ \array {
   \mathbb{N} \times \mathbb{N}     & \overset{add}\to  & \mathbb{N} \\
   \mathllap{c \times c} \downarrow &                   & \downarrow \mathrlap{c} \\
   \mathbb{R} \times \mathbb{R}     & \underset{add}\to & \mathbb{R}
} $$
But things don\'t always work out this way.

## Parametric polymorphism

In **parametric polymorphism**, one writes code to define a function *once*, which contains a "type variable" that can be instantiated at many different types to produce different functions.  For instance, we can define a function

$$ first : A\times A \to A $$

where $A$ is a type variable (or *parameter*), by

$$ first(x,y) \coloneqq x. $$

Now the compiler automatically instantiates a copy of this function, with identical code, for any type at which it is called.  Thus we can behave as if we had functions

$$ first : \mathbb{N} \times \mathbb{N} \to \mathbb{N}$$
$$ first : \mathbb{R} \times \mathbb{R} \to \mathbb{R}$$

and so on, for any types we wish.  In contrast to *ad hoc* polymorphism, in this case we do have a guarantee that all these same-named functions are doing "the same thing", because they are all instantiated by the same original polymorphic code.

In a [[dependent type theory|dependently typed programming language]] with a [[type of types]], such as [[Coq]] or [[Agda]], a parametrically polymorphic family of functions can simply be considered to be a single dependently typed function whose first argument is a type.  Thus our function above would be typed as

$$ first : \prod_{A:Type} A\times A \to A$$

However, parametric polymorphism makes sense and is very useful even in languages with less rich type systems, such as [[Haskell]] and [[Standard ML]].

### Theorems for Free

In type systems with parametric polymorphism, such as [[System F]], there is a family of proofs for every polymorphic type signature which each give an algebraic law for any function with that signature. For example, the type signature $A \times B \to A$ has the [[free theorem]]

$$ \alpha \circ f = f \circ (\alpha \times \beta) $$

for all functions $f$ with that signature ([Wadler 89](#Wadler89), figure 1).

## Related concepts

* [[logical relations]]

## References

The distinction between ad hoc and parametric polymorphism is originally due to [[Christopher Strachey]]:

* [[Christopher Strachey]], _Fundamental concepts in programming languages_, Lecture Notes from August 1967 (International Summer School in Computer Programming, Copenhagen), later published in Higher-Order and Symbolic Computation, 13, 11-49, 2000. ([pdf](https://www.itu.dk/courses/BPRD/E2009/fundamental-1967.pdf))

Other classic papers include:

* [[John C. Reynolds]], _Towards a Theory of Type Structure_, Lecture Notes in Computer Science 19, 408-425, 1974. ([pdf](http://repository.cmu.edu/cgi/viewcontent.cgi?article=2289&context=compsci))

* [[Robin Milner]], _A Theory of Type Polymorphism in Programming_, Journal of Computer and System Sciences 17, 348-375, 1978. ([pdf](https://courses.engr.illinois.edu/cs421/sp2013/project/milner-polymorphism.pdf))

* [[John C. Reynolds]], _Types, Abstraction, and Parametric Polymorphism_, Information Processing 83, 1983. ([pdf](http://www.cse.chalmers.se/edu/year/2010/course/DAT140_Types/Reynolds_typesabpara.pdf))

Theorems for free were explained in:

* {#Wadler89} [[Philip Wadler]], _Theorems for free!_, in Proceedings of the fourth international conference on Functional programming languages and computer architecture (FPCA 1989). 1989. ([pdf](https://www2.cs.sfu.ca/CourseCentral/831/burton/Notes/July14/free.pdf), [doi:10.1145/99370.99404](https://doi.org/10.1145/99370.99404))

A recent paper extending Reynolds' notion of relational parametricity to dependent types:

* [[Robert Atkey]], [[Neil Ghani]] and Patricia Johann, _A Relationally Parametric Model of Dependent Type Theory_, In Proceedings of the 41st ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages (POPL 2014). 2014. ([web](http://bentnib.org/dtt-parametricity.html))

[[!redirects polymorphism]]
[[!redirects ad hoc polymorphism]]
[[!redirects parametric polymorphism]]
[[!redirects polymorphic]]
[[!redirects parametricity]]


polynomial

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

#Contents#
* table of contents
{:toc}

## Definitions and meanings

Let $R$ be a [[commutative ring]]. A _polynomial_ with coefficients in $R$ is an element of a **polynomial ring** over $R$. A polynomial ring over $R$ consists of a [[set]] $X$ whose elements are called "[[variables]]" or "indeterminates", and a [[function]] $X \to R[X]$ to (the underlying set of) a commutative $R$-algebra that is universal among such functions, so that $R[X]$ is the [[free object|free]] commutative $R$-algebra generated by $X$; a polynomial is then an element of the underlying set of $R[X]$.

+-- {: .num_remark}
###### Remark
Much like "[[vector]]", the term "polynomial" in this sense may seem slightly deprecated from the viewpoint of modern mathematics. We no longer think of a [[vector space]] as consisting of things called "vectors" (i.e. we don't assign an objective meaning to "vectors"); it's the other way around, where we introduce a type of structure called a vector space and then, relative to a given vector space context, declare that "vector" just means an element therein. Similarly, the term "polynomial" in the sense above has seemingly been subordinated to the structural concept of polynomial ring. (In Linderholm's *Mathematics Made Difficult*, page 152, there is an amusing passage where someone points at the expression $a_0 + a_1 X + a_2 X^2 + \ldots + a_n X^n$ and asks "Well, how about it? Is it a polynomial, or isn't it?" and the respondent says, "Yeah, sure, I guess. Looks like one. Yeah, sure, that's a polynomial all right" -- an answer which is not wrong exactly, but not quite right either, since it fails to recognize that there is something questionable about the question.)
=--

+-- {: .num_remark}
###### Remark
On further reflection, however, we might more objectively identify the concept of "polynomial" (let us say a polynomial in $n$ variables) with a definable $n$-[[signature (in logic)|ary]] [[signature (in logic)|operation]] in the [[theory]] of commutative $R$-algebras. From a categorical perspective, if $U: CommAlg_R \to Set$ is the forgetful functor, a definable $n$-ary operation means a [[natural transformation]] $U^n \to U$. The connection is that the functor $U^n$ is [[representable functor|representable]], being represented by the [[free object]] $F[n] = R[x_1, \ldots, x_n]$, so that a natural transformation $U^n \to U$ is canonically identified with a transformation $\hom(R[x_1, \ldots, x_n], -) \to U$ or to an element of $U R[x_1, \ldots, x_n]$, by the [[Yoneda lemma]]. In pursuit of this objective meaning (which is essentially due to [[Lawvere]]), we find that the "variable" $x_i$ stands for the $i^{th}$ projection map $U^n \to U$, and that the meaning of (let's say) $x_1 x_2 \in R[x_1, x_2, x_3]$ is that it is the definable operation whose instantiation at any commutative $R$-algebra $A$ is the function $A^3 \to A$ taking $(a, b, c)$ to $a b$.

Of course there are traditional standard expressions that people usually have in mind when they speak of "a polynomial" as such. But leaving it at that, where polynomials are merely identified with certain types of expressions (as by the characters in Linderholm's book), ignores the deeper objective meaning of definable operations which of course is the actual point of it all.
=--

+-- {: .num_remark}
###### Remark
Finally, sometimes "polynomial" is construed to mean a *[[polynomial function]]*. This is actually just a particular instantiation of a definable operation. The default meaning is that, if we are working for instance with a polynomial ring in one variable $R[x]$, then we have a composite
$$U R[x] \cong \hom(U, U) \stackrel{ev_R}{\to} \hom(U R, U R)$$
where the second map sends a natural transformation to its component at $R$ as an $R$-algebra. Put differently, the set $\hom(U R, U R)$ carries a commutative $R$-algebra structure under the pointwise operations, and there is a unique $R$-algebra map $R[x] \to \hom(U R, U R)$ that sends '$x$' to the [[identity map]]. The [[value]] of a polynomial $p \in R[x]$ under this map is then the corresponding polynomial function $U R \to U R$.

This conflation of polynomials with polynomial functions is often forgivable, particularly in those cases where the map $R[x] \to \hom(U R, U R)$ is [[injective function|injective]] (so that 'polynomials' are identified with certain types of functions). Of course the map *won't* be injective if $R$ is [[finite set|finite]], to give one example. But in analysis, where we consider functions on $\mathbb{R}$ or $\mathbb{C}$, the conflation is familiar and rarely cause for concern. The conflation may also be responsible for certain notational artifacts, such as the common (and useful!) practice of writing $p(x)$ for polynomials $p$.
=--

With these preliminary remarks out of the way, we recall some of the more syntactic considerations with an example.

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

The set of **polynomials** in one
[[variable]] $z$ with [[coefficients]] in $R$, also called the set of **univariate polynomials**, is the set $R[\mathbb{N}]$ of all [[formal linear combinations]] on elements $n \in \mathbb{N}$, thought of as powers $z^n$ of the variable $z$. As a string of symbols, a polynomial is frequently represented by a form like

$$
  a_n z^n + \cdots + a_1 z + a_0
  \,,
$$

where $n$ is an arbitrary [[natural number]] and $a_0, \dots, a_n \in R$, subject to the usual fine print (where we work [[quotient set|modulo]] the [[congruence]] generated by equations of the form

$$
  0 z^{n+1} + a_n z^n + \cdots + a_1 z + a_0 = a_n z^n + \cdots + a_1 z + a_0 $$

so that we ignore coefficients of zero). (The **degree** of a polynomial is the maximum $n$ for which $a_n$ is nonzero, in which case the **leading term** of the polynomial is $a_n z^n$. A polynomial is **constant** if its degree is $0$. The degree of the zero polynomial may be left chastely undefined, although for some purposes it may be convenient to define it as $-1$ or as $-\infty$. Even $0$ is possible if one is prepared to observe some fine print. *Chacun à son goût*.)

This set is equipped with an $R$-module structure (where formal linear combinations are added and scalar-multiplied as usual) and also with the structure of a ring, in fact a [[commutative algebra]] over $R$, denoted $R[z]$
and called the **polynomial ring** or **ring of polynomials**, with ring multiplication

$$
  R[z] \cdot R[z] \to R[z]
$$

the unique one that bilinearly extends the multiplication of monomials given by

$$
  z^k \cdot z^l = z^{k+l}
  \,.
$$

=--

Thus, one way to construct a polynomial ring is first to construct the free commutative monoid generated by a set $X$ (the monoid of monomials), and then to construct the free $R$-module generated by the underlying set of that monoid, extending the monoid multiplication to a ring multiplication by bilinearity.

In addition to the ring structure, there is a further operation $R[z] \times R[z] \to R[z]$ which may be described as "substitution" or "composition"; see Remark \ref{substitution} below for a general description (which applies in fact to any [[Lawvere theory]]).

+-- {: .num_defn}
###### Definition
In the case of univariate polynomials, since the function type $R[z] \to R[z]$ is a [[function algebra|function $R$-algebra]] with the [[commutative algebra|commutative $R$-algebra]] [[homomorphism]] to constant functions $C:R \to (R[z] \to R[z])$, there exists a commutative $R[z]$-algebra homomorphism $i:R[z] \to (R[z] \to R[z])$ inductively defined by $i(s(r)) \coloneqq C(r)$ for $r \in R$ and $i(z) \coloneqq id_{R[z]}$. The __substitution__ or __composition__ of univariate polynomials $(-) \circ (-):R[z] \times R[z] \to R[z]$ is the [[uncurrying]] of $i$, $p \circ q \coloneqq i(p)(q)$ for $p \in R[z]$ and $q \in R[z]$.
=--

Moreover, there is a noncommutative analogue of polynomial ring on a set $X$, efficiently described as the free $R$-[[module]] generated by the (underlying set of the) free [[monoid]] on $X$. This carries also a ring structure, with ring multiplication induced from the monoid multiplication. A far-reaching generalization of this construction is given at [[distributive law]].

Finally: polynomial algebras may be regarded as [[graded algebras]] (graded over $\mathbb{N}$). Specifically: let us regard $R[X]$ as the free $R$-module generated by (the underlying set of) the free [[commutative monoid]] $F(X)$. The monoid homomorphism $F(!): F(X) \to F(1) \cong \mathbb{N}$ induced by the unique function $!: X \to 1$ gives an $\mathbb{N}$-fibering of $F(X)$ over $\mathbb{N}$, with typical fiber $F(X)_n$ whose elements are called *monomials of degree $n$*. Then the _homogeneous component_ of degree $n$ in $R[X]$ is the $R$-submodule generated by the subset $F(X)_n \subset F(X)$. The elements of this component are called *homogeneous polynomials of degree $n$*.

## Properties

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

The polynomial ring $R[z]$ is the [[free construction|free]] $R$-[[commutative algebra|algebra]] on one generator (the variable $z$).

=--

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

By the definition of [[free objects]] one needs to check that algebra [[homomorphisms]]

$$
  f : R[z] \to K
$$

to another algebra K are in [[natural bijection]] with [[functions]] of [[sets]]

$$
  \bar f : * \to K
$$

from the [[singleton]] to the set underlying $K$. Take $\bar f \coloneqq f(z)$. Using $R$-linearity, this is directly seen to yield the desired bijection.

=--

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

Similarly, the set of polynomials in any given [[set]] of variables with coefficients in $R$ is the [[free construction|free]] commutative $R$-algebra on that set of generators; see [[symmetric power]] and [[symmetric algebra]].

=--

+-- {: .num_remark #substitution}
###### Remark
As usual in the study of universal algebra via [[Lawvere theories]], there is an [[operad]] whose $n^{th}$ component $C_n$ is the free algebra $R[x_1, \ldots, x_n]$, and whose operadic multiplication is given by maps
$$C_k \times C_{n_1} \times \ldots \times C_{n_k} \to C_n$$
($n = n_1 + \ldots + n_k$) that take a tuple of elements $(p; q_1, \ldots, q_k)$ to $p(q_1(x), \ldots, q_k(x))$. Formally, it takes this tuple to the value of $p$ under the unique algebra map $R[x_1, \ldots, x_k] \to R[x_1, \ldots, x_n]$ that extends the mapping $x_j \mapsto i_j(q_j)$. Here the $i_j: R[x_1, \ldots, x_{n_j}] \to R[x_1, \ldots, x_n]$ are appropriate coproduct inclusions (in the category of commutative rings), where $i_j(x_l) = x_{n_1 + \ldots + n_{j-1} + l}$. A particularly important case of substitution is the case $k=1$ and $n_1 = 1$, where the map $R[x] \times R[x] \to R[x]$ is ordinary substitution $(p, q) \mapsto p(q(x))$. This is a special case of the more general notion of [[Tall-Wraith monoid]].
=--

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

In case $R$ is an [[integral domain]], the [[field of fractions]] of $R[z]$ is the [[field]] $R(z)$ of [[rational fractions]].

=--

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

The underlying $R$-[[module]] of the polynomial ring $R[z]$ on one generator is the [[natural numbers object]] in the category [[RMod]] of $R$-modules.

=--

### As differential algebras

Polynomial rings on one generator also have the structure of a [[differential algebra]].

+-- {: .num_defn}
###### Definition
For every univariate polynomial ring $R[z]$, one could [[inductive definition|inductively define]] a function called a __[[derivative]]__ or __[[derivation]]__
$$\frac{\partial}{\partial z}: R[z] \to R[z]$$
such that

* for all polynomials $p \in R[z]$ and $q \in R[z]$ and constant polynomials $a \in R[z]$ and $b \in R[z]$,
$$\frac{\partial}{\partial z}(a \cdot p + b \cdot q) = a \cdot \frac{\partial}{\partial z}(p) + b \cdot \frac{\partial}{\partial t}(q)$$
* for all polynomials $p \in R[z]$ and $q \in R[z]$,
$$\frac{\partial}{\partial z}(p \cdot q) = p \cdot \frac{\partial}{\partial z}(q) + \frac{\partial}{\partial z}(p) \cdot q$$
* and
$$\frac{\partial}{\partial z}(z) = 1$$
Thus the univariate polynomial ring $R[z]$ is a [[differential algebra]].
=--

+-- {: .num_prop}
###### Proposition
In the multivariate polynomial ring $R[z_1, z_2 \ldots z_n]$, there is a [[derivation]]
$$\frac{\partial}{\partial z_i}: R[z_1, z_2 \ldots z_n] \to R[z_1, z_2 \ldots z_n]$$
for each $1 \leq i \leq n$ called a **[[partial derivative]]**.
=--

+-- {: .num_prop}
###### Proposition
Since $R$ is power-associative, for every positive integer $n \in \mathbb{Z}_+$,
$$\frac{\partial}{\partial z}z^{n+1} = j(n) \circ z^{n}$$
where $j:\mathbb{Z}_+ \to \mathbb{Z} \to R[z]$ is the canonical function from the positive [[integers]] to $R[x]$.
=--

+-- {: .num_prop}
###### Proposition
If $R$ is an [[integral domain]], then for all constant polynomials $a \in R[z]$,
$$\frac{\partial}{\partial x}a = 0$$
=--

+-- {: .num_defn}
###### Definition
Given a polynomial $p \in R[z]$, we define the __set of antiderivatives__ of $p$ to be the [[fiber]] of the derivative at $p$:
$$\mathrm{antiderivatives}(p) \coloneqq \left\{q \in R[z] \bigg| \frac{\partial}{\partial z}\left(q\right) =_{R[z]} p\right\}$$
=--

### As domains

In the case where $R = k$ is a [[field]], the polynomial ring $k[x]$ has a number of useful properties. One is that it is a [[Euclidean domain]], where the degree serves as the Euclidean function:

+-- {: .num_lemma}
###### Lemma
Let $R$ be a commutative ring. Given $f, g \in R[x]$ where the leading coefficient of $g$ is a unit (e.g., if $g$ is a [[monic polynomial]]), there are unique $q, r \in k[x]$ such that $f = q \cdot g + r$ and $\deg(r) \lt \deg(g)$.
=--

+-- {: .proof}
###### Proof
If $\deg(f) \lt \deg(g)$, then $q = 0$ and $r = f$ will serve. Otherwise we may argue by [[induction]] on $\deg(f)$, where if $a_m x^m$ is the leading term of $f$ and $b_n x^n$ the leading term of $g$, then $h(x) = f(x) - a_m b_n^{-1} x^{m-n}g(x)$ has lower degree than $f(x)$. This proves existence. Uniqueness is clear, since if $q \cdot g + r = q' \cdot g + r'$ and $q \neq q'$, we have $\deg((q - q')g) = \deg(r' - r) \lt \deg(g)$ which is impossible; then $r = r'$ quickly follows from $q = q'$.
=--

+-- {: .num_cor}
###### Corollary
If $k$ is field, then $k[x]$ is a [[Euclidean domain]]. As a result, $k[x]$ is a [[principal ideal domain]], and therefore a [[unique factorization domain]].
=--

See [[Euclidean domain]] for a proof.

+-- {: .num_cor}
###### Corollary
For any commutative ring $R$, if $a \in R$ is a [[root]] of $p(x) \in R[x]$, i.e., if the value of the polynomial function $p(a)$ is $0$, then $p(x)$ is of the form $(x - a)q(x)$.
=--

+-- {: .proof}
###### Proof
Since $x - a$ is monic, we may write $p(x) = (x - a)q(x) + r$ where $\deg(r) \lt 1$, whence $r$ is a constant. By evaluating the polynomial function at $x = a$, the term $r$ is $0$.
=--

This observation may be exploited in various neat ways. One is that if $p(x)$ is a polynomial, then $p(y) = p(x) + (y - x)q(x, y)$ for some unique $q(x, y) \in k[x, y]$. A consequence is that the Lawvere theory of commutative $k$-algebras is a [[Fermat theory]]. The [[derivative]] of $p$ may be defined to be $q(x, x) \in k[x]$.

## Examples

* [[Hermite polynomial]]

* [[Jones polynomial]], [[Alexander polynomial]] ([[knot theory]])

* [[Wick polynomial]]

## Related concepts

* [[monomial]]

* [[power series]], [[formal power series]]

* [[constant polynomial]]

* [[monic polynomial]]

* [[Lorentzian polynomial]]

* [[tensor algebra]]

* [[affine line]], [[affine space]]

* [[Tate algebra]], [[polydisk]]

* [[variety]], [[Diophantine equation]]

* [[spectral polynomial algebra]]

* [[polynomial Poisson algebra]]

* [[polynomial function]]

* [[polynomial functor]]

* [[composition ring]]

[[!redirects polynomial]]
[[!redirects polynomials]]

[[!redirects univariate polynomial]]
[[!redirects univariate polynomials]]

[[!redirects polynomial ring]]
[[!redirects polynomial rings]]
[[!redirects polynomial algebra]]
[[!redirects polynomial algebras]]

[[!redirects free commutative algebra]]
[[!redirects free commutative algebras]]


polynomial (∞,1)-functor

> This page is about a [[categorification]] of the notion of [[polynomial functor]].  For the notion of "polynomial $\infty$-functor" used in [[Goodwillie calculus]], see [[n-excisive (∞,1)-functor]].

# Polynomial $(\infty,1)$-functors

* table of contents
{: toc}

## Idea

A **polynomial $(\infty,1)$-functor** is a [[categorification]] of the notion of [[polynomial functor]].

## Definition

Recall that a map of spaces $f\colon I\to J$
induces an adjoint triple
$$f_!\dashv f^*\dashv f_*,$$
where $f^*\colon S/J\to S/I$ is the base change functor.

A __polynomial $(\infty,1)$-functor__ is a functor $S/I\to S/J$
equivalent to a functor of the form $t_! p_* s^*$,
where
$$I \leftarrow E \to B\to J$$
are maps of spaces.

Polynomial functors are closed under compositions ([GHK, Theorem 2.1.8](#GHK)).

A functor $F\colon S/I\to S/J$
is polynomial
if and only if it is [[accessible functor|accessible]] and preserves weakly contractible limits,
the latter referring to limits indexed by categories whose [[nerve]]
is a weakly contractible [[simplicial set]],
see ([GHK, Theorem 2.2.3(ii)](#GHK)).

Recall that an $(\infty,1)$-functor $F\colon C\to D$
is a local right adjoint functor
if for any object $X\in C$ the induced functor
$$C/X\to D/F(X)$$
is a [[right adjoint functor]].

A functor $F\colon S/I\to S/J$
is polynomial if and only if it is a [[local right adjoint]] functor,
see ([GHK, Theorem 2.2.3(iii)](#GHK)).

## Related pages

* [[(∞,1)-operad]]

## References

{#GHK} [[David Gepner]], [[Rune Haugseng]], [[Joachim Kock]], _∞-Operads as Analytic Monads_, ([arXiv:1712.06469](https://arxiv.org/abs/1712.06469))

[[!redirects polynomial ∞-functors]]
[[!redirects polynomial (∞,1)-functor]]
[[!redirects polynomial (∞,1)-functors]]
[[!redirects polynomial infinity-functor]]
[[!redirects polynomial infinity-functors]]
[[!redirects polynomial (infinity,1)-functor]]
[[!redirects polynomial (infinity,1)-functors]]


polynomial (∞,1)-functor > history

< [[n-excisive (∞,1)-functor]]


polynomial function

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

# Contents
* table of contents
{: toc}

## Definition ##

### In commutative rings ###

#### Without scalar coefficients ####

Let $R$ be a [[commutative ring]]. A **polynomial function** is a a [[function]] $f:R \to R$ such that

* $f$ is in the image of the [[function]] $j:R^* \to (R \to R)$ from the [[free monoid]] $R^*$ on $R$, i.e. the set of lists of elements in $R$, to the [[function algebra]] $R \to R$, such that

  * $j(\epsilon) = 0$, where $0$ is the zero function.
  * for all $a \in R^*$ and $b \in R^*$, $j(a b) = j(a) + j(b) \cdot (-)^{\mathrm{len}(a)}$, where $(-)^n$ is the $n$-th power function for $n \in \mathbb{N}$
  * for all $r \in R$, $j(r) = c_r$, where $c_r$ is the constant function whose value is always $r$.

* $f$ is in the image of the canonical [[ring homomorphism]] $i:R[x] \to (R \to R)$ from the polynomial ring in one indeterminant $R[x]$ to the [[function algebra]] $R \to R$, which takes constant polynomials in $R[x]$ to constant functions in $R \to R$ and the indeterminant $x$ in $R[x]$ to the identity function $\mathrm{id}_R$ in $R \to R$

#### With scalar coefficients ####

For a [[commutative ring]] $R$, a **polynomial function** is a [[function]] $f:R \to R$ with a [[natural number]] $n \in \mathbb{N}$ and a function $a:[0, n] \to R$ from the set of natural numbers less than or equal to $n$ to $R$, such that for all $x \in R$,

$$f(x) = \sum_{i:[0, n]} a(i) \cdot x^i$$

where $x^i$ is the $i$-th [[power function]] for multiplication.

### In non-commutative algebras ###

For a [[commutative ring]] $R$ and a $R$-[[associative unital algebra|non-commutative algebra]] $A$, a $R$-**polynomial function** is a [[function]] $f:A \to A$ with a [[natural number]] $n \in \mathbb{N}$ and a function $a:[0, n] \to R$ from the set of natural numbers less than or equal to $n$ to $R$, such that for all $x \in A$,

$$f(x) = \sum_{i:[0, n]} a(i) x^i$$

where $x^i$ is the $i$-th [[power function]] for the (non-commutative) multiplication.

## See also

* [[real polynomial function]]

* [[quadratic function]]

* [[rational zero theorem]]

* [[fundamental theorem of algebra]]

* [[polynomial]]

* [[degree of a polynomial]]

## References

* Wikipedia, _[Polynomial function](https://en.wikipedia.org/wiki/Polynomial_function)_

[[!redirects polynomial functions]]

[[!redirects polynomial map]]
[[!redirects polynomial maps]]
[[!redirects polynomial mapping]]
[[!redirects polynomial mappings]]


polynomial functor

> This entry is about a notion in [[category theory]]. For a different notion of the same name in ([[stable homotopy theory|stable]]) [[homotopy theory]] see at _[[Goodwillie calculus]]_.

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

# Polynomial functors
* table of contents
{: toc}

## Idea

The concept of _polynomial functor_ is a [[categorification]] of that of _[[polynomial]]_.

Polynomial endo-functors are used to encode a class of [[inductive types]] called _[[W-types]]_, and also as the underlying data of [[polynomial monads]].

## Definition

Let $C$ be a [[locally cartesian closed category]].  A **polynomial** is a diagram
$$ W \overset{f}{\leftarrow} X \overset{g}{\to} Y \overset{h}{\to} Z $$
in $C$.  The corresponding **polynomial functor** is the composite

$$
  C/W
   \overset{f^*}{\to}
  C/X
   \overset{\Pi_g}{\to}
  C/Y
   \overset{\Sigma_h}{\to}
  C/Z
  \,,
$$

where $\Pi_g$ and $\Sigma_h$ are the [[dependent product]] and [[dependent sum]] operations, right and left adjoint respectively to [[pullback#PullbackFunctor|pullback functors]] $g^*$ and $h^*$.

When $W=Z$, this is a **polynomial endofunctor**.

Sometimes this general notion is called a **dependent polynomial functor** (also **typed** or **colored** ([FMLS, p. 68](#FMLS))), with "polynomial (endo)functor" reserved for the "one-variable" case, when $W=Z=1$ is the [[terminal object]].

The data $f$, $g$, and $h$ that specify a polynomial functor is sometimes referred to as a **container** (or an **indexed container**, with *container* reserved for the case $W=Z=1$).  Other times *container* is used as a synonym for "polynomial functor".  Sometimes the data $f,g,h$ are instead referred to as a **polynomial** to distinguish them from the "polynomial functor" they determine.

If $g$ is an identity, the functor is sometimes called a **linear functor** or a **linear polynomial functor**.  Note that this notion makes sense even if $C$ is not locally cartesian closed; all it needs are [[pullbacks]].  More generally, we can make sense of polynomial functors in any category with pullbacks if we restrict $g$ to be an [[exponentiable morphism]].

## Examples

### On sets
 {#ExamplesOnSets}

For $C$ = [[Set]] the polynomial functor induced from a
[[function]] $g$

$$
  \ast \stackrel{}{\leftarrow} X \stackrel{g}{\longrightarrow} Y
  \stackrel{}{\longrightarrow} \ast
$$

is given by

$$
  S \mapsto P_g(S) = \coprod_{y \in Y} S^{X_y}
  \,,
$$

where $X_y$ is the [[fiber]] of $g$ over $y \in Y$, and where the [[exponential object]] $S^{X_y}$ is the [[function set]] of [[functions]] from the fiber to $S$. The [[cardinality]] of the set on the right is

$$
  {\vert P_g(S)\vert}
   =
  \sum_{y \in Y} {\vert S\vert}^{\vert X_y\vert}
$$

and it is in this sense that the concept of polynomial functor is a kind of [[categorification]] of that of [[polynomial]].

On the other hand the dependent polynomial functor associated to

$$
  A \stackrel{p_1}{\leftarrow} X \stackrel{id}{\longrightarrow} X
  \stackrel{p_2}{\longrightarrow} B
$$

acts by

$$
  (S_a)_{a \in A}
  \mapsto
  \left(\coprod_{a \in A} S_a \times X_{a b} \right)_{b \in B}
  \,.
$$

Under [[cardinality]] this becomes [[matrix multiplication]] acting on [[vectors]] (with entries in the [[natural numbers]]). So in this case the dependent polynomial functor is a linear functor of several variables, an [[integral transform]].

### In a general lextensive category
 {#ExamplesOnLextensiveCats}

Taking the fibres in the previous example to be finite, uniformly bounded (in the sense that all the fibres have size at most $N$), we can partition the set $Y$ as $\coprod_{n \leq N} \{y\in Y\mid |g^{-1}(y)| = n \} = \coprod_{n \leq N} Y_n$. Then, after choosing isomorphisms $g^{-1}(y) \simeq \{1,\ldots,|g^{-1}(y)|\}$ for each $y\in Y$, the polynomial endofunctor $P_g$ is naturally isomorphic to the functor
$$
  S \mapsto  \coprod_{n=0}^N Y_n\times S^n\,,
$$
as the hom-set $S^{\{1,\ldots,n\}} \simeq S^n$.
Thus we have a "literal polynomial" in the object argument of the functor. Such literal polynomials make sense as endofunctors of any [[lextensive category]], for example any [[pretopos]]. However, this does not automatically mean that these endofunctors are actually _polynomial_ endofunctors, strictly speaking.

+--{: .num_prop}
###### Proposition
Literal polynomial endofunctors on an [[lextensive category]] are polynomial endofunctors.
=--

+-- {: .proof}
###### Proof
Note that in an arbitrary lextensive category there is little guarantee that any dependent products exist. However, the right adjoint to pullback functor associated to an iterated [[codiagonal]] $\coprod_{k=1}^n A\to A$ does exist, for any lextensive category. And this is all that is needed.

Consider, then, the map $g$ defined as
$$
  \coprod_{n=1}^N \left(\coprod_{k=1}^n Y_n \right) \stackrel{\coprod_{n=1}^N \nabla_{Y_n}}{\longrightarrow} \coprod_{n=0}^N Y_n
$$
The pullback along $\nabla_{Y_n}$ of an arbitrary map $f\colon T\to Y_n$ is $\coprod_{k=1}^n T \stackrel{\coprod_{k=1}^n f}{\longrightarrow} \coprod_{k=1}^n Y_n$. The dependent product along $\nabla_{Y_n}$ of an arbitrary map can be written as a composite of dependent product functors, which are themselves coproducts of dependent product along the ordinary codiagonal $Y_n \sqcup Y_n \to Y_n$, and along identity maps. The dependent product of $S\sqcup T \stackrel{f\sqcup g}{\longrightarrow} Y_n \sqcup Y_n$ along the codiagonal can be easily checked to be $S\times_{Y_n} T \to Y_n$. So we know that the functor $\Pi_{\nabla_{Y_n}}$ exists, and it remains to calculate what the polynomial functor associated to $g$ is.

This can be done summand by summand, i.e. for each $n=1,\ldots, N$, since the final dependent sum merely takes the coproduct of each dependent product. So the only thing to double check is that, for each $n$,
$$
  \Pi_{\nabla_{Y_n}} (\coprod_{k=1}^n Y_n \times S \to \coprod_{k=1}^n Y_n) = (Y_n\times S^n \to Y_n)\,,
$$
which follows from the description of the pullback along $\nabla_{Y_n}$ and the defining properties of products, coproducts, and their interactions in a lextensive category.
There is a boundary case, namely that corresponding to $n=0$, which amounts to dependent product along $0 \to Y_0$. But in an extensive category, initial objects are strict, and so we need to take the dependent product of $0\times S = 0 \to 0$, which is then just the identity map on $Y_0$.

The final dependent product then gives the output of the polynomial functor, which is then the literal polynomial we started with, namely $\coprod_{n=0}^N Y_n\times S^n$.

=--

In the above proof, the only place that the existence of arbitrary pullbacks in a lextensive category was used is in constructing the general dependent product along the codiagonal. Obviously one still needs finite products, just to get off the ground. For the purposes of expressing the literal polynomial as a polynomial functor, this dependent product only need be applied to the projection map $(Y\sqcup Y) \times S \to (Y\sqcup Y)$. Thus while $\Pi_{\nabla_{Y_n}}$ may not exist in a general _extensive_ category with binary products, the composite $\Pi_{\nabla_{Y_n}}\circ (Y\sqcup Y)^*$ does. The extent to which such an endofunctor deserves to be called a polynomial endofunctor is up for debate.

In a lextensive category with _countable_ coproducts (and the corresponding compatibility with pullbacks), for instance an [[infinitary pretopos]] one can extend the above example to endofunctors of the form
$$
  S \mapsto  \coprod_{n=0}^\infty Y_n\times S^n\,,
$$
with practically identical proof and no additional assumptions.

## The 2-category of polynomial functors

Any polynomial functor, as defined above, is automatically equipped with a [[tensorial strength]], when the slice categories of $C$ are regarded as tensored over $C$ in the canonical way.  The following theorem is proven in [Gambino--Kock](#GK):

+--{: .num_theorem}
###### Theorem
There is a [[bicategory]] whose objects are the objects of $C$, whose morphisms from $W$ to $Z$ are diagrams of the form
$$ W \overset{f}{\leftarrow} X \overset{g}{\to} Y \overset{h}{\to} Z, $$
and whose 2-morphisms are diagrams of the form
$$
\array{ & & X & \to & Y \\
  & \swarrow & \uparrow & & \mathllap{id}\uparrow & \searrow\\
  W && X' \times_{Y'} Y & \to & Y && Z\\
  &\nwarrow & \downarrow & & \downarrow & \nearrow \\
  && X' & \to & Y'. }
$$
This bicategory is equivalent to the 2-category whose objects are slice categories of $C$, whose morphisms are polynomial functors regarded as strong functors, and whose 2-morphisms are strength-respecting natural transformations.
=--

In particular, "being polynomial" is a mere [[stuff, structure, property|property]] of a strong functor between slice categories.  That is, the data $f,g,h$ are uniquely determined, up to isomorphism, by the strong functor they generate.  (Note, though, that the property of "being polynomial" depends on a prior identification of the domain and codomain as being slice categories of some specified ambient category $C$.  In particular, a functor $C/Z \to C/Z$ might be polynomial when its domain and codomain are regarded as slice categories of $C$, but not when they are regarded as slices of $C/Z$ over $1$ --- this happens when $W=Z$ but $h g \neq f$.)

Note that the above bicategory contains, as a locally full sub-bicategory, the usual bicategory of [[spans]].  Thus, as a special case, the bicategory of spans is equivalent to the 2-category of "linear" polynomial functors.  Both of these are instances of [[Lack's coherence theorem]].

There is a particular subclass of the 2-morphism in this bicategory that is also interesting: a 2-morphism corresponds to a [[cartesian natural transformation]] if and only if the map $X' \times_{Y'} Y \to X$ is an isomorphism.  Since anything isomorphic to a pullback is a pullback, in this case the diagram can be drawn more simply by omitting the upper square and merely asking that the lower square *be* a pullback.

Furthermore, this bicategory is actually the horizontal bicategory of a [[double category]], indeed a [[framed bicategory]], in which the vertical arrows are the arrows of $C$, and the cells are diagrams as above but allowing also morphisms $W\to W'$ and $Z\to Z'$ on the left and right.

## Related topics

* Polynomial endofunctors are important in the definition of [[W-types]] in categories.

* Polynomial functors are a special case of [[parametric right adjoint|parametric right adjoints]].

* Polynomial functors can be defined using [[exponentiable morphisms]] in a category that may not be locally cartesian closed.  See also [[distributivity pullback]].

* Kripke frames $(R,S)$ (with a transition relation $R$ of arity $2$) as studied in [[modal logic]] are [[coalgebra]]s for the power-set functor $P$. Kripke frames for a more general *modal similarity type* $t$ are a coalgebras of a functor of the form $X\mapsto \product_{d\in t} P(S^{arity(d)})$. Kripke models are coalgebras of functor $K:X\mapsto P(Prop)\times P(X)$ where $Prop$ is the set of propositional variables of the logic in consideration. In particular all the functors appearing here are polynomial functors. So, at least in some aspects, the study of modal logics reduces to the study of (certain) polynomial functors.

## Examples

* [[symmetric polynomial]], [[elementary symmetric polynomial]]

## Related entries

* [[polynomial monad]]

* [[integral transform]]

* [[Tambara functor]]

* [[polynomial (infinity,1)-functor]]

## References

The standard reference, which considers dependent (multivariate) polynomial functors, is:

* {#GK} [[Nicola Gambino]] and [[Joachim Kock]], _Polynomial functors and polynomial monads_, (2009) ([arXiv:0906.4931](http://arxiv.org/abs/0906.4931)).

The relation of plain polynomial functors to [[trees]] is discussed in

* {#Kock} [[Joachim Kock]], _Polynomial functors and trees_, (2009)  ([arXiv:0807.2874](http://arxiv.org/abs/0807.2874))

The category, $Poly$, of one-variable polynomial functors on $Set$ and its application to [[dynamical systems]] is considered in

* [[David Spivak]], _Poly: An abundant categorical setting for mode-dependent dynamics_, ([arXiv:2005.01894](https://arxiv.org/abs/2005.01894))

* {#FMLS} [[Eric Finster]], [[Samuel Mimram]], Maxime Lucas, Thomas Seiller, _A Cartesian Bicategory of Polynomial Functors in Homotopy Type Theory_ ([arXiv:2112.14050](https://arxiv.org/abs/2112.14050)).

Talks on polynomial functors are available at

* _Workshop on Polynomial Functors_, Topos Institute,
15–19 March 2021. ([website](https://topos.site/p-func-2021-workshop/))

Monograph on polynomial functors in view of [[categorical systems theory]]:

* {#SpivakPoly} [[David Spivak]], [[Nelson Niu]], _Polynomial Functors: A General Theory of Interaction_ [[webpage](https://topos.site/poly-course/), [pdf](https://topos.site/poly-book.pdf)]

Generalization to [[homotopy theory]] and [[higher category theory]] is discussed in

* [[Joachim Kock]], _Data types with symmetries and polynomial functors over groupoids_,  28th Conference on the Mathematical Foundations of Programming Semantics (Bath, June 2012); in Electronic Notes in Theoretical Computer Science.  ([arXiv:1210.0828](http://arxiv.org/abs/1210.0828))

* [[David Gepner]], [[Rune Haugseng]], [[Joachim Kock]], _∞-Operads as Analytic Monads_, ([arXiv:1712.06469](https://arxiv.org/abs/1712.06469))

* {#Weber14} [[Mark Weber]], _Operads as polynomial 2-monads_ ([arXiv:1412.7599](http://arxiv.org/abs/1412.7599))

* [[Benno van den Berg]], [[Ieke Moerdijk]], _W-types in Homotopy Type Theory_ ([arXiv:1307.2765](http://arxiv.org/abs/1307.2765))
 {#vdBergMoerdijk13}

See also

* [[Joachim Kock]], [[André Joyal]], [[Michael Batanin]], [[Jean-François Mascari]], _Polynomial functors and opetopes_, Advances in Mathematics, vol 224, 2010, pp 2690--2737. ([arXiv:0706.1033](http://arxiv.org/abs/0706.1033))
{#KJBM}

* [[Yde Venema]], _Algebras and Coalgebras_, §6 (p.332-426) in Blackburn, van Benthem, Wolter, _Handbook of modal logic_, Elsevier, 2007.

* [[Mark Weber]], *Polynomials in categories with pullbacks*, Theory and Applications of Categories, Vol. 30, 2015, No. 16, pp 533-598. ([journal](http://tac.mta.ca/tac/volumes/30/16/30-16abs.html))

* Charles Walker, *Universal properties of polynomials*, ([arXiv:1806.10477](https://arxiv.org/abs/1806.10477))

* [[Ross Street]], *Polynomials as spans*, Cahiers de topologie et géométrie différentielle catégoriques, Vol. LXI-2 (2020), pp 113-153 ([pdf](http://cahierstgdc.com/wp-content/uploads/2020/04/STREET-LXI-2.pdf))

[[!redirects polynomial functor]]
[[!redirects polynomial functors]]
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[[!redirects polynomial endofunctors]]

[[!redirects dependent polynomial functor]]
[[!redirects dependent polynomial functors]]
[[!redirects dependent polynomial endofunctor]]
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[[!redirects linear polynomial functor]]
[[!redirects linear polynomial functors]]
[[!redirects linear polynomial endofunctor]]
[[!redirects linear polynomial endofunctors]]

[[!redirects linear dependent polynomial functor]]
[[!redirects linear dependent polynomial functors]]
[[!redirects linear dependent polynomial endofunctor]]
[[!redirects linear dependent polynomial endofunctors]]

[[!redirects dependent linear polynomial functor]]
[[!redirects dependent linear polynomial functors]]
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[[!redirects dependent linear polynomial endofunctors]]

[[!redirects container]]
[[!redirects containers]]


polynomial monad

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

A **polynomial monad** is a [[monad]] on a [[category]] $C$ whose underlying [[endofunctor]] is a [[polynomial functor]] and whose unit and multiplication are [[cartesian natural transformations]].  This is of course equivalent to being a monad in the [[bicategory]] of polynomial functors and cartesian transformations.

## Examples

A basic example is the free-monoid monad ([Gambino-Kock 09, Example 1.9](#GambinoKock09)). This is exhibited by the polynomial $1\leftarrow \mathbb{N}^\prime\rightarrow  \mathbb{N}\rightarrow 1$ where the middle arrow is such that for all $n\in \mathbb{N}$ its fiber over $n$ has cardinality $n$.

One can construct the free monad on a polynomial endofunctor, and it is a polynomial monad.

## Properties

### Parametric right adjointness

Every polynomial monad is a [[p.r.a. monad]].

### Cartesianness

Since all polynomial functors preserve [[pullbacks]], a polynomial monad is necessarily a [[cartesian monad]].  Note that cartesian transformations between polynomial functors also have an explicit description: they are given by diagrams
$$ \array{
  & & X & \to & Y\\
  & \swarrow & & & & \searrow \\
  W && \downarrow && \downarrow && Z \\
  & \nwarrow & & & & \nearrow \\
  & & X' & \to & Y'
  }$$
in which the middle square is a pullback.  Thus, all the data of a polynomial monad can be described very concretely.  This gives a convenient way to produce cartesian monads, and indeed very many cartesian monads arising in practice are in fact polynomial.  Polynomial monads have close relations with a number of other notions.

### Relation to clubs

Polynomial monads (on $C$ rather than a slice of it) are, in some sense, the most natural base monads for the theory of [[clubs]].  Although the club construction can be performed over any cartesian monad, if this monad is polynomial, then all clubs over it are also polynomial, since polynomial functors are a [[sieve]] with respect to cartesian transformations.  Moreover, the club monoidal structure can be seen as just an instance of the explicit construction of composition for polynomials (see 1.11 of ([Gambino-Kock](#GambinoKock09))).

### Relation to generalized multicategories

The bicategory of polynomial functors and cartesian transformations is in fact the horizontal bicategory of a [[double category]] $\underline{Poly}$, whose vertical arrows are the morphisms of $C$ and whose cells are diagrams like
$$ \array{
  W & \leftarrow & X & \to & Y & \to & Z \\
  \downarrow && \downarrow && \downarrow &&\downarrow\\
  W' & \leftarrow & X' & \to & Y' & \to & Z'
}$$
in which the *middle* square (only) is a pullback.  Suppose $W'=Z'=1$, so the bottom polynomial is determined by a single morphism $g':X'\to Y'$, and that $W=Z$.  Then to give the rest of the diagram is to give an object $Y$, a morphism $Y\to Z$, and a morphism $X' \times_{Y'} Y \to Z$.  By adjointness, the latter is equivalent to giving a morphism $Y\to \Pi_{g'}(Z\times X')$; but this codomain is just $T Z$, where $T$ is the polynomial functor determined by $g'$.

Thus, a polynomial equipped with a cartesian transformation to $1 \leftarrow X' \to Y' \to 1$ is exactly a "$T$-span" such as arises in the theory of [[generalized multicategories]].  A similar argument shows that if $T$ is a polynomial monad, then the slice double category of $\underline{Poly}$ over $g'$ is equivalent to the double category of $T$-spans (a.k.a. the "horizontal Kleisli" double category).  Therefore, polynomial monads cartesianly-over $T$, being the horizontal monads in this double category, are precisely $T$-multicategories.  (Such polynomial monads on $C$ itself, rather than any slice, are the $T$-clubs mentioned above; in general a $T$-multicategory $T X_0 \leftarrow X_1 \to X_0$ is identified with the monad it induces on $C/X_0$.)

### Relation to object classifiers

Polynomial monads have a natural interpretation in terms of [[object classifiers]].  Specifically, given any polynomial $1 \leftarrow B \xrightarrow{g} A \to 1$, generating a polynomial functor $P_{1,g,1}$, we can consider the class $S_g$ of all pullbacks of $g$.  A cartesian unit $Id \to P_{1,g,1}$ says that $id_1 : 1\to 1$ is a pullback of $g$, and therefore so are all identities.  Similarly, the composite $P_{1,g,1} \circ P_{1,g,1}$ involves the classifying object for a pair of composable $S_g$-morphisms, and so a cartesian multiplication $P_{1,g,1} \circ P_{1,g,1} \to P_{1,g,1}$ tells us that $S_g$ is closed under composition.

Thus, a polynomial monad (on $C \cong C/1$) can be regarded as a morphism $g$ together with a coherent way to make $S_g$ into a category.  More precisely, consider the slice category $Cart(C)/g$, where $Cart(C)$ is the category whose objects are morphisms of $C$ and whose morphisms are pullback squares.  This comes with source and target functors $Cart(C)/g \rightrightarrows C$.  To make $P_{1,g,1}$ into a polynomial monad is then equivalent to giving unit and composition functors enhancing $Cart(C)/g \rightrightarrows C$ to a [[double category]] such that the forgetful map $Cart(C)/g \to Cart(C)$ is a double functor.

+--{: .standout}
This claim could stand some independent verification.
=--

For example, consider the free monoid monad above determined by $g:\mathbb{N}\prime \to \mathbb{N}$.  To exhibit a function as a pullback of this $g$ is to say that its fibers are finite and have been equipped with bijections to canonical finite sets, which we may as well think of as giving them linear orders.  To give the monad structure on this functor is equivalent to noting that every identity map has ordered finite fibers, and the composite of functions with ordered finite fibers again has ordered finite fibers, in a way that is associative, unital, and stable under pullback.

Other interesting examples include:

* The monad for symmetric strict monoidal categories on $Cat$: its $S_g$ consists of discrete (op)fibrations with ordered finite fibers in which all reindexing functions are bijections.
* The monad for cartesian strict monoidal categories on $Cat$: its $S_g$ consists of discrete fibrations with ordered finite fibers.
* The monad for cocartesian strict monoidal categories on $Cat$: its $S_g$ consists of discrete opfibrations with ordered finite fibers.

The class of discrete [[Conduche functors]] with ordered finite fibers is *almost* an example; it wants to be classified by the bicategory $FinSpan$ of spans between finite sets, but that is not an object of $Cat$.  Thus, a polynomial monad in $Cat$ determined by a discrete Conduche functor with ordered finite fibers is a reasonable substitute for the nonexistent notion of "$FinSpan$-club".

Identifying the classes of morphisms corresponding to standard polynomial monads like these also tells us how to identify when a polynomial monad is induced by a [[club]] (or more generally a [[generalized multicategory]]) over them: when the $g$ for that monad belongs to the appropriate class, compatibly.  For instance, if $S$ is the monad for cartesian strict monoidal categories, then a polynomial monad on $Cat$ is an $S$-club just when the morphism $g$ in its defining polynomial is a discrete fibration with ordered finite fibers, and its composition and unit respect that structure.

## References

A fairly comprehensive discussion of the notion is due to

* [[Nicola Gambino]], [[Joachim Kock]], _Polynomial functors and polynomial monads_, (2009);  ([arXiv:0906.4931](http://arxiv.org/abs/0906.4931))
 {#GambinoKock09}

The [[homotopy theory]] of [[algebra over a monad|algebras]] over polynomial monads is in

* [[Michael Batanin]], [[Clemens Berger]], _Homotopy theory for algebras over polynomial monads_ ([arXiv:1305.0086](http://arxiv.org/abs/1305.0086))
 {#BataninBerger13}

An application to the theory of [[opetopes]] in discussed in

* [[Joachim Kock]], [[André Joyal]], [[Michael Batanin]], [[Jean-François Mascari]] (2010); Polynomial functors and opetopes; Advances in Mathematics, vol 224, pages 2690--2737, [arXiv:0706.1033](http://arxiv.org/abs/0706.1033)
{#KJBM}, §4.3-4.6, where the main result is that the polynomial monad of stable opetopes is a least fixpoint of the Baez-Dolan construction for polynomial monads.

An extension of type theory with a universe of associative and unital polynomial monads to define [[opetopic type theory| opetopic types]] in order to encode fully coherent algebraic structures is in

* {#AlliouxFinsterSozeau20} [[Antoine Allioux]], [[Eric Finster]], [[Matthieu Sozeau]], _Types are internal infinity-groupoids_ ([PDF](https://hal.inria.fr/hal-03133144/document))

Polynomial monads as a "generalization of small categories" for homotopy theory are studied in

* [[Michael Batanin]], Florian De Leger; _Grothendieck's homotopy theory, polynomial monads and delooping of spaces of long knots_, 2017: [arxiv](https://arxiv.org/abs/1712.00904)

The relation between polynomial monads, object classifiers, and models of (homotopy) type theory is studied in

* [[Steve Awodey]],  _Natural models of homotopy type theory_, 2017: [arxiv](https://arxiv.org/abs/1406.3219)

* [[Steve Awodey]], Clive Newstead; _Polynomial pseudomonads and dependent type theory_, 2018: [arxiv](https://arxiv.org/abs/1802.00997)

* Clive Newstead, _Algebraic models of dependent type theory_, 2021: [arxiv](https://arxiv.org/abs/2103.06155)

* [[Steve Awodey]], _Algebraic type theory_, [slides from HoTT 2023](https://hott.github.io/HoTT-2023/slides/awodey.pdf)

[[!redirects polynomial monads]]


polynomial observable

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#### Algebraic Quantum Field Theory
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#Contents#
* table of contents
{:toc}

## Idea

Give a [[field theory]] (for instance a [[Lagrangian field theory]]) whose [[field bundle]] is a [[vector bundle]], then the [[space of sections]] of the field bundle, hence the [[space of field histories]], is canonically a [[vector space]] and hence it makes sense to consider [[observables]] which are [[linear functions]], or [[quadratic functions]] of the [[field histories]], etc. A [[sum]] of such is a _polynomial observable_.

Since linear smooth observables are [[compactly supported distributions]] (see at _[[distributions are the smooth linear functionals]]_) polynomial observables are sums of [[diagonals]] of [[distributions of several variables]].

If all these distributional coefficients are [[non-singular distributions]] one speaks of _regular observables_.

Of all the distributional coefficients at order $k$ satisfy the condition that their [[wave front sets]] exclude the subsets where all $k$ [[wave vectors]] are all in the [[future cone]] or all in the [[past cone]], then one speaks of _[[microcausal observables]]_.

[[!include perturbative observables -- table]]

## Definition

+-- {: .num_defn #PolynomialObservable}
###### Definition
**([[polynomial observable]])**

Let $E \overset{fb}{\to}$ be [[field bundle]] which is a [[vector bundle]]. An [[off-shell]] _[[polynomial observable]]_ is a [[smooth function]]

$$
  A
  \;\colon\;
  \Gamma_\Sigma(E)
    \longrightarrow
  \mathbb{C}
$$

on the [[on-shell]] [[space of sections]] of the [[field bundle]] $E \overset{fb}{\to} \Sigma$ (space of field histories) which may be expressed as

$$
  A(\Phi)
  \;=\;
  \alpha^{(0)}
  +
  \int_\Sigma
    \alpha^{(1)}_a(x) \Phi^a(x)
   \,
  dvol_\Sigma(x)
  +
  \int_\Sigma \int_\Sigma
   \alpha^{(2)}_{a_1 a_2}(x_1, x_2) \Phi^{a_1}(x_1) \Phi^{a_2}(x_2)
  \,dvol_\Sigma(x_1)
  \, dvol_\Sigma(x_2)
  +
  \cdots
  \,,
$$

where

$$
  \alpha^{(k)} \in \Gamma'_{\Sigma^k}\left((E^\ast)^{\boxtimes^k_{sym}} \right)
$$

is a [[compactly supported distribution]] [[distribution of two variables|of k variables]] on the $k$-fold graded-symmetric [[external tensor product of vector bundles]] of the [[field bundle]] with itself.

Write

$$
  PolyObs(E) \hookrightarrow Obs(E)
$$

for the [[subspace]] of off-shell polynomial observables onside all off-shell [[observables]].

Let moreover $(E,\mathbf{L})$ be a [[free field theory|free]] [[Lagrangian field theory]] whose [[equations of motion]] are [[Green hyperbolic differential equations]]. Then an _[[on-shell]] polynomial observable_ is the [[restriction]] of an off-shell polynomial observable along the inclusion of the [[on-shell]] [[space of field histories]] $\Gamma_{\Sigma}(E)_{\delta_{EL}\mathbf{L} = 0} \hookrightarrow \Gamma_\Sigma(E)$. Write

$$
  PolyObs(E,\mathbf{L}) \hookrightarrow Obs(E,\mathbf{L})
$$

for the subspace of all on-shell polynomial observables inside all on-shell [[observables]].

By [this prop.](Green+hyperbolic+partial+differential+equation#DistributionsOnSolutionSpaceAreTheGeneralizedPDESolutions) restriction yields an [[isomorphism]] between polynomial on-shell observables and polynomial off-shell observables modulo the image of the [[differential operator]] $P$:

$$
  PolyObs(E,\mathbf{L})
    \underoverset{\simeq}{\text{restriction}}{\longleftarrow}
  PolyObs(E)/im(P)
  \,.
$$

=--

Various special cases:

+-- {: .num_defn #LinearObservable}
###### Definition
**([[linear observable]])**

A _[[linear observable]]_ is a [[polynomial observable]] (def. \ref{PolynomialObservable}) all whose [[coefficients]] except possible for $\alpha^{(1)}$ are [[zero]].

=--

+-- {: .num_defn #RegularObservables}
###### Definition
**([[regular observable]])**

A _[[regular observable]]_ is a [[polynomial observable]] (def. \ref{PolynomialObservable}) all whose [[coefficients]] $\alpha^{(k)}$ are [[non-singular distributions]].

=--

+-- {: .num_defn}
###### Definition
**([[microcausal observable]])**

For $\Sigma$ a [[spacetime]], hence a [[Lorentzian manifold]] with [[time orientation]], then a _[[microcausal observable]]_ is a [[polynomial observable]] (def. \ref{PolynomialObservable}) such that each [[coefficient]] $\alpha^{(k)}$ has [[wave front set]] excluding those points  where all $k$ [[wave vectors]] are in the [[future cone]] or all in the [[past cone]].

=--

## Examples
 {#Examples}

+-- {: .num_example }
###### Example
**([[field observables]])**

If the [[field bundle]] is a [[vector bundle]], then the [[field observables]]

$$
  \mathbf{\Phi}^a(x) \;\colon\; \Phi \mapsto \Phi^a(x)
$$

are [[linear observables]] (def. \ref{LinearObservable}).

=--

+-- {: .num_example #PolynomialLocalObservablesArePolynomialObservables}
###### Example
**([[polynomial]] [[local observables]] are [[polynomial observables]])**

A [[local observable]] which comes form a [[variational differential form|horizontal differential form]] which is a [[polynomial]] in the fields and their jets times the [[volume form]] on [[spacetime]] is a polynomial observable.

These happen to be also [[microcausal observables]] ([this example](microcausal+functional#CompactlySupportedPolynomialLocalDensities)).

=--

## Related concepts

[[polynomial]] [[local observables]] $\hookrightarrow$ [[microcausal observables]] $\hookrightarrow$ [[polynomial observables]] $\hookrightarrow$ [[observables]]

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

## References

See the references at _[[microcausal observable]]_.

[[!redirects polynomial observables]]

[[!redirects linear observable]]
[[!redirects linear observables]]

[[!redirects regular observable]]
[[!redirects regular observables]]

[[!redirects regular linear observable]]
[[!redirects regular linear observables]]

[[!redirects regular polynomial observable]]
[[!redirects regular polynomial observables]]


polynomial Poisson algebra

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#### Symplectic geometry
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#### Formal geometry
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#Contents#
* table of contents
{:toc}

## Idea

It is a classical fact that
the [[universal enveloping algebra]] of a [[Lie algebra]] provides
a [[formal deformation quantization]] of the corresponding [[Lie-Poisson structure]]
(example \ref{DeformationQuantizationOfLiePoissonStructuresByUniversalEnvelopingAlgebras} below).
Remarkably, this statement generalizes to more general polynomial Poisson algebras (def. \ref{PolynomialPoissonAlgebra} below)
for a suitable generalized concept of universal enveloping algebra (def. \ref{DeformationQuantizationOfLiePoissonStructuresByUniversalEnvelopingAlgebras} below):
it is _always_ true up to third order in $\hbar$, and sometimes to higher order ([Penkava-Vanhaecke 00, theorem 3.2](#PenkavaVanhaecke00), prop. \ref{UniversalEnvelopingAlgebraProvidesDeformationQuantizationAtLeastToOrder3} below).
In particular it also holds true for restrictions of [[Poisson bracket Lie algebras]] to their [[Heisenberg Lie algebras]]
(example \ref{DeformationQuantizationOfLiePoissonStructuresByUniversalEnvelopingAlgebras} below).

## Definition

+-- {: .num_defn #PolynomialPoissonAlgebra}
###### Definition
**([[polynomial]] [[Poisson algebra]])**

A [[Poisson algebra]]  $((A,\cdot), \{-,-\})$
is called a _polynomial Poisson algebra_ if the underlying [[commutative algebra]] $(A,\cdot)$
is a [[polynomial algebra]], hence a [[symmetric algebra]]

$$
  Sym(V) \coloneqq T(V)/(x \otimes y - y \otimes x \vert x,y \in V)
$$

on some [[vector space]] $V$. Here

$$
  T(V) \coloneqq \underset{n \in \mathbb{N}}{\oplus} V^{\otimes^n}
$$

denotes the [[tensor algebra]] of $V$. We write

$$
  \mu \;\colon\; T(V) \longrightarrow Sym(V)
$$

for the canonical [[projection]] map (which is an algebra [[homomorphism]]) and

$$
  \sigma \;\colon\; Sym(V) \longrightarrow T(V)
$$

for its [[linear map|linear]] inverse (symmetrization, which is not in general an algebra [[homomorphism]]).

Notice that by its bi-[[derivation]] property the Poisson bracket
on a polynomial Poisson algebra is fixed by its restriction to linear elements

$$
  \{-,-\} \;\colon\; V \otimes V \longrightarrow Sym(V)
  \,.
$$

=--

+-- {: .num_defn #UniversalEnvelopingAlgebraOfPolynomialPoissonAlgebra}
###### Definition
**(universal enveloping algebra of polynomial Poisson algebra)**

Given a polynomial Poisson algebra $(Sym(V), \{-,-\})$ (def. \ref{PolynomialPoissonAlgebra}),
say that its _universal enveloping algebra_ $\mathcal{U}(V,\{-,-\})$ is the [[associative algebra]] which is the [[quotient]] of
the [[tensor algebra]] of $V$ with a [[power series|formal variable]] $\hbar$ adjoined
by the two-sided ideal which is generated by the the $\hbar$-Poisson bracket relation on linear elements:

$$
  \mathcal{U}(V,\{-,-\})
  \;\coloneqq\;
  T(V)/( x \otimes y - y \otimes x - \hbar \{x,y\} \vert x,y \in V )
  \,.
$$

This comes with the quotient projection linear map which we denote by

$$
  \rho \;\colon\; T(V)[ [ \hbar ] ] \longrightarrow \mathcal{U}(V,\hbar\{-,-\})
  \,.
$$

=--

([Penkava-Vanhaecke 00, def. 3.1](#PenkavaVanhaecke00))

The combined linear projection maps from def. \ref{PolynomialPoissonAlgebra} and def. \ref{UniversalEnvelopingAlgebraOfPolynomialPoissonAlgebra}
we denote by

$$
  \tau \coloneqq \rho \circ (\sigma/[ [ \hbar ] ]) \;\colon\; Sym(V)[ [ \hbar ] ] \longrightarrow \mathcal{U}(V,\{-,-\})
  \,.
$$

## Examples

+-- {: .num_example #PolynomialLiePoissonStructure}
###### Example
**([[Lie-Poisson structure]] on affine algebraic variety)**

Let $(C^\infty(\mathbb{R}^n), \pi)$ be a [[Poisson manifold]] whose
underlying manifold is a [[Cartesian space]] $\mathbb{R}^n$. Then the restriction
of its Poisson algebra $( C^\infty(\mathbb{R}^n, \cdot), \pi^{i j} \partial_i(-) \cdot \partial_j(-)  )$
to the [[polynomial functions]] $\mathbb{R}[x^1, \cdots, x^n ] \hookrightarrow C^\infty(\mathbb{R}^n)$
is a polynomial Poisson algebra according to def. \ref{PolynomialPoissonAlgebra}.

In particular if $(\mathfrak{g}, [-,-])$ is a [[Lie algebra]] and
$(\mathfrak{g}^\ast, \{-,-\})$ the corresponding [[Lie-Poisson structure|Lie-Poisson manifold]],
then the corresponding polynomial Poisson algebra is $(Sym(\mathfrak{g}), \{-,-\})$
where the restriction of the Poisson bracket to linear polynomial elements coincides with the [[Lie bracket]]:

$$
  \{x,y\} = [x,y]
  \,.
$$

=--

+-- {: .num_example #UniversalEnvelopingAlgebraOfLieAlgebra}
###### Example
**([[universal enveloping algebra]] of [[Lie algebra]])**

In the case of a polynomial [[Lie-Poisson structure]] $(Sym(\mathfrak{g}), [-,-])$ (example \ref{PolynomialLiePoissonStructure})
the universal enveloping algebra $\mathcal{U}(\mathfrak{g},[-,-])$ from def. \ref{UniversalEnvelopingAlgebraOfPolynomialPoissonAlgebra}
(for $\hbar = 1$) coincides with the standard [[universal enveloping algebra]] of the [[Lie algebra]] $(\mathfrak{g}, [-,-])$.

=--

## Properties

### Formal deformation quantization via universal enveloping algebra

+-- {: .num_prop #UniversalEnvelopingAlgebraProvidesDeformationQuantizationAtLeastToOrder3}
###### Proposition
**(universal enveloping algebra provides [[deformation quantization]] at least up to order 3)**

Let $( Sym(V), \{-,-\} )$ be a polynomial Poisson algebra (def. \ref{PolynomialPoissonAlgebra})
such that the canonical linear map to its universal enveloping algebra (def. \ref{UniversalEnvelopingAlgebraOfPolynomialPoissonAlgebra})
is [[injective function|injective]] up to order $n \in \mathbb{N}\cup \{\infty\}$

$$
  \tau/(\hbar^{n+1}) \;\colon\; Sym(V)[ [ \hbar ] ]/(\hbar^{n+1}) \hookrightarrow \mathcal{U}(V,\hbar\{-,-\})/(\hbar^{n+1})
  \,.
$$

Then the restriction of the product on $\mathcal{U}(V,\hbar\{-,-\})/(\hbar^{n+1})$ to $Sym(V)/(\hbar^{n+1})$
is a [[deformation quantization]] of $(Sym(V), \{-,-\})$ to order $n$ (hence a genuine deformation quantization in the case that $n = \infty$).

Moreover, this is _always_ the case for $n = 3$, hence for every polynomial Poisson algebra its universal enveloping algebra
always provides a deformation quantization of order $3$ in $\hbar$.

=--

([Penkava-Vanhaecke 00, theorem 3.2 with section 2](#PenkavaVanhaecke00))

+-- {: .num_example #DeformationQuantizationOfLiePoissonStructuresByUniversalEnvelopingAlgebras}
###### Example
**([[formal deformation quantization]] of [[Lie-Poisson structures]] by universal enveloping algebras)**

In the following cases the map $\tau$ in prop. \ref{UniversalEnvelopingAlgebraProvidesDeformationQuantizationAtLeastToOrder3}
is injective to arbitrary order, hence in these cases the universal enveloping algebra
provides a genuine [[deformation quantization]]:

1. the case that the Poisson bracket is [[linear Poisson structure|linear]] in that restricts as

   $$
     \{-,-\} \;\colon\; V \otimes V \longrightarrow V \hookrightarrow Sym(V)
     \,.
   $$

   This is the case of the [[Lie-Poisson structure]] from example \ref{PolynomialLiePoissonStructure}
   and the universal enveloping algebra that provides it deformation quantization is the
   standard one (example \ref{UniversalEnvelopingAlgebraOfLieAlgebra}).

1. more generally, the case that the Poisson bracket restricted to linear elements has linear and
   constant contribution in that it restricts as

   $$
     \{-,-\} \;\colon\; V \otimes V \longrightarrow \mathbb{R} \oplus V \hookrightarrow Sym(V)
     \,.
   $$

   This includes notably the Poisson structures induced by [[symplectic vector spaces]],
   in which case the restriction

   $$
     \{-,-\} \;\colon\; (\mathbb{R} \oplus V) \otimes (\mathbb{R} \oplus V) \longrightarrow (\mathbb{R} \oplus V)
   $$

   is the [[Lie bracket]] of the associated [[Heisenberg Lie algebra]].

=--

This is ([Penkava-Vanhaecke 00, p. 26](#PenkavaVanhaecke00))
The first statement in itself is a classical fact (reviewed e.g. in [Gutt 11](#Gutt11)).

## References

* {#PenkavaVanhaecke00} [[Michael Penkava]], [[Pol Vanhaecke]], _Deformation quantization of Polynomial Poisson algebras_, Journal of Algebra 227, 365-393 2000 ([arXiv:math/9804022](https://arxiv.org/abs/math/9804022))

[[!redirects polynomial Poisson algebras]]


polytope

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## Idea

The notion of _polytope_ is the generalization of the notion of [[polygon]] to arbitrary [[dimension]]s.

* a convex 0-polytope is the [[point]]

* a convex 1-polytope is the [[interval]]

* a convex 2-polytope is a [[polygon]].

* a convex 3-polytope is a [[polyhedron]].

In [[algebraic topology]], the term “polyhedron” refers to a [[polyhedral space]].  See there for more information.

## Related concepts

* [[polygon]]

* [[scissors congruence]]

[[!redirects polytopes]]


Pontrjagin character

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#Contents#
* table of contents
{:toc}

## Idea

The _Pontrjagin character_ is the analog for [[KO]]-[[topological K-theory|-theory]] of what the [[Chern character]] is for [[KU]]-[[topological K-theory|-theory]]. Like the latter may be expressed as a [[series]] of [[Chern classes]], so the Pontrjagin character is expressed as a series of [[Pontrjagin classes]].

Equivalently, the Pontrjagin character is the [[Chern-Dold character]] on [[KO]].

## References

* {#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))

* {#ConnerFloyd66} [[Pierre Conner]], [[Edwin Floyd]], p. 53 in: _[[The Relation of Cobordism to K-Theories]]_, Lecture Notes in Mathematics __28__ Springer 1966 ([doi:10.1007/BFb0071091](https://link.springer.com/book/10.1007/BFb0071091), [MR216511](http://www.ams.org/mathscinet-getitem?mr=216511))

* [[Werner Greub]], [[Stephen Halperin]], [[Ray Vanstone]], _Pontrjagin, Pfaffian, and Chern Classes_, Section 9.6 in: _Lie groups, principal bundles and characteristic classes_, Volume 2 of: _[[Connections, Curvature, and Cohomology]]_, Pure and Applied Mathematics Volume 47, Part B, 1972, Pages 420-476 (doi:doi.org/10.1016/S0079-8169(08)62879-2)

* Mitsunori Imaoka, Kouji Kuwana, _Stably extendible  vector bundles over  the quaternionic projective spaces_, Hiroshima Math. J. 29  (1999),  273-279 ([euclid:hmj/1206125008](https://projecteuclid.org/euclid.hmj/1206125008))

* [[Kiyoshi Igusa]], _Pontrjagin classes and higher torsion of sphere bundles_, pp. 21-29 in: R. Penner et al. (eds.) _Groups of Diffeomorphisms_ , Advanced Studies in Pure Mathematics 52 2008   ([euclid:aspm/1543447476](https://projecteuclid.org/euclid.aspm/1543447476))

* Manuel Krannich, Jens Reinhold, Section 2.4 of: _Characteristic numbers of manifold bundles over surfaces with highly connected fibers_,  Journal of the London Mathematical Society, June 2020.  ([arXiv:1807.11539](https://arxiv.org/abs/1807.11539), [doi:10.1112/jlms.12344](https://doi.org/https://doi.org/10.1112/jlms.12344))

* [[Daniel Grady]], [[Hisham Sati]], Section 2.1 of: _Differential KO-theory: constructions, computations, and applications_ ([arXiv:1809.07059](https://arxiv.org/abs/1809.07059))

See also:

* [[eom]], _[Pontryagin character](https://encyclopediaofmath.org/wiki/Pontryagin_character)_.

[[!redirects Pontrjagin characters]]

[[!redirects Pontryagin character]]
[[!redirects Pontryagin characters]]