Pascal Lambrechts

* [webpage](https://uclouvain.be/fr/repertoires/pascal.lambrechts)

## Selected writings

On [[graph complex]]-models for the [[rational homotopy type]] of [[configuration spaces of points]]:

* [[Pascal Lambrechts]], [[Victor Tourtchine]], _Homotopy graph-complex for configuration and knot spaces_, Transactions of the AMS, Volume 361, Number 1, January 2009, Pages 207–222 ([arxiv:math/0611766](https://arxiv.org/abs/math/0611766))

* {#LambrechtsVolic14} [[Pascal Lambrechts]], [[Ismar Volić]], _Formality of the little N-disks operad_, Memoirs of the American Mathematical Society; no. 1079, 2014  ([arXiv:0808.0457](https://arxiv.org/abs/0808.0457), [doi:10.1090/memo/1079](http://dx.doi.org/10.1090/memo/1079))

  ([[formality of the little n-disk operad]])

* {#CamposIdrissiLambrechtsWillwacher18} [[Ricardo Campos]], [[Najib Idrissi]], [[Pascal Lambrechts]], [[Thomas Willwacher]], _Configuration Spaces of Manifolds with Boundary_ ([arXiv:1802.00716](https://arxiv.org/abs/1802.00716))

## Related $n$Lab entries

* [[configuration space of points]]

* [[Fulton-MacPherson operad]]

* [[little n-disk operad]],

* [[formality of the little n-disk operad]]

* [[semi-algebraic manifold]]

category: people


pascallothar


Pascual Jordan

Pascual Jordan was one of the founders of [[quantum mechanics]]. For instance [[Jordan algebras]] of [[quantum observables]] are named after him.

A brief biography of his scientific life is in the appendix of

* Ivan Todorov, _"Quantization is a mystery"_,  Bulg. J. Phys. 39 (2012) 107-149 ([arXiv:1206.3116](http://arxiv.org/abs/1206.3116))

See also his [Wikipedia entry](http://en.wikipedia.org/wiki/Pascual_Jordan)

## Selected writings

Early discussion of [[quantum mechanics]] (and apparently introducing that term):

* [[Max Born]], [[Pascual Jordan]], *Zur Quantenmechanik*, Zeitschrift für Physik **34** (1925) 858–888 [[doi:10.1007/BF01328531](https://doi.org/10.1007/BF01328531)]

and with emphasis on the [[Born rule]] and [[quantum probability]]:

* [[Pascual Jordan]], *Über eine neue Begründung der Quantenmechanik*, Zeitschrift für Physik **40** (1927) 809–838 [[doi:10.1007/BF01390903](https://doi.org/10.1007/BF01390903)]

Introducing [[Jordan algebras]]:

* {#Jordan32} [[Pascual Jordan]], *Über eine Klasse nichtassociativer hyperkomplexer Algebren*, Nachr. Ges. Wiss. Göttingen (1932) 569-575 [[eudml:59403](https://eudml.org/doc/59403), [[Jordan-Algebren.pdf:file]]]

* {#JordanvNeumannWigner34} [[Pascual Jordan]], [[John von Neumann]], [[Eugene Wigner]], *On an algebraic generalization of the quantum mechanical formalism*, Ann. Math. **35** (1934) 29-64 [[jstor:1968117](https://www.jstor.org/stable/1968117), [doi:10.1007/978-3-662-02781-3_21](https://doi.org/10.1007/978-3-662-02781-3_21)]

## Related entries

* [[Jordan algebra]]

category: people


Pasquale Marra

* [personal page](https://www.ms.u-tokyo.ac.jp/~pmarra/)

* [GoogleScholar page](https://scholar.google.it/citations?user=i0DxtkgAAAAJ&hl=en)

## Selected writings

On potential [[experiment|experimental]] realization of "[[Majorana zero modes]]" in [[solid state physics]]:

* [[Pasquale Marra]], *Majorana nanowires for topological quantum computation: A tutorial*, J. of Applied Physics **132** (2022) 231101   [[arXiv:2206.14828](https://arxiv.org/abs/2206.14828), [doi:10.1063/5.0102999](https://doi.org/10.1063/5.0102999)]

category: people


pasting diagram

+-- {: .rightHandSide}
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### Context
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

Given an ordinary [[category]] $C$, a pasting diagram in $C$ is a sequence of composable morphisms in $C$

$$
  a_1 \stackrel{f_1}{\to} a_2 \stackrel{f_2}{\to}
  \cdots
  \stackrel{f_n}{\to} a_{n+1}
  \,.
$$

We think of these arrows as not yet composed, but _pasted_ together at their objects, such as to form a composable sequence, and then say the _value_ of the sequence is the composite morphism in $C$. Or, we could say that a pasting diagram is a specified decomposition of whatever morphism $f$ it evaluates to, thus breaking down $f$ into morphisms $f_i$ which, in practice, are usually "more basic" than $f$ relative to some type of structure on $C$. For example, if $C$ is a [[monoidal category]], the $f_i$ might be instances of associativity isomorphisms.

Pasting decompositions become more elaborate in higher categories. An example of a pasting diagram in a (let's say strict) 2-category is a pasting of two squares

$$
  \array{
    a &\to& b &\to& c
    \\
    \downarrow &\swArrow& \downarrow &\swArrow& \downarrow
    \\
    d &\to& e &\to& f
  }
  \,.
$$

To evaluate this diagram as a single 2-morphism, we read this diagram explicitly as the vertical composite of the 2-morphism

$$
  \array{
    a &\to& b &\to& c
    \\
    && \downarrow &\swArrow& \downarrow
    \\
     && e &\to& f
  }
$$

with the 2-morphism

$$
  \array{
    a &\to& b &&
    \\
    \downarrow &\swArrow& \downarrow &&
    \\
    d &\to& e &\to& f
  }
  \,.
$$

(Notice that the two halves of the boundary of each of these two 2-morphisms are themselves 1-dimensional pasting diagrams.) Similarly, there can be situations where one pastes together other shapes (triangles, pentagons, etc.), and there may be multiple paths on the way to resolving the diagram into a single 2-morphism, but the idea is that all such paths evaluate to the same 2-morphism, at least in a strict 2-category. General theorems which refer to the uniqueness of pastings are called _pasting theorems_.

On the other hand, the following diagram is _not_ a pasting diagram:

$$
  \array{
    a &\to& b &\to& c
    \\
    \downarrow &\neArrow& \downarrow &\swArrow& \downarrow
    \\
    d &\to& e &\to& f
  }
  \,.
$$

Thus, formal definitions of pasting diagram include conditions which impose a consistent "directionality" of the cells so they can be pasted together.

In an [[n-category]] a pasting diagram is similarly a collection of [[k-morphism|n-morphisms]] with a prescribed way how they are to fit together at their boundaries. There are a number of ways of formalizing pasting diagrams, depending partly on the constituent shapes one allows, and also on the technical hypotheses that allow proofs of pasting theorems, which include directionality conditions but usually also "loop-freeness" conditions to ensure there exists a unique way to resolve or evaluate the diagram. (N.B.: usually a completely unambiguous evaluation is only possible in a strict $n$-category; in a weak $n$-category, one allows uniqueness up to a "contractible space of choices".)

Despite some technical differences among the formalizations, the core idea throughout is that the overall geometric shapes of pasting diagrams should be (contractible) $n$-dimensional polyhedra, broken down into smaller polyhedral cells which come equipped with directionality or orientations, that can be sensibly pasted together as in the above descriptions once the cells have been assigned values in an $n$-category.

## Notions of pasting diagrams

Various formalisms for pasting diagrams have been proposed. They include

* [[pasting scheme|pasting schemes]] (Johnson)

* [[parity complex|parity complexes]] (Street)

* [[directed complex|directed complexes]] (Steiner)

* [[generalized parity complex|generalized parity complexes]] (Forest)

Each of these formalisms involve graded sets $\{C_n\}_{n \geq 0}$ together with maps $\partial^+_n: C_{n+1} \to P(C_n)$, $\partial^-_n: C_{n+1} \to P(C_n)$. This goes under various names; here we call it a **parity structure**. It should be thought of as assigning to each "cell" of dimension $n+1$ a collection of positive boundary cells and negative boundary cells in dimension $n$. The formalisms above are distinguished by the choice of axioms on parity structures, but there is definite kinship among them.

Also related are various notions of categories of shapes, including

* [[test category]]

* [[direct category]]

* [[compositions in cubical sets]]

## Examples

* [[Toda bracket]]

## Related concepts

* [[pasting law]] for [[pullbacks]]/[[homotopy pullbacks]]

## References

The notion of pasting in a [[2-category]] was introduced in

* [[Jean Benabou]], _Introduction to bicategories_, Reports of the Midwest Category Seminar, Lecture Notes in Mathematics, **47**, 1967. ([doi:10.1007/BFb0074299](https://doi.org/10.1007/BFb0074299))

A reasonably self-contained, formal, and illustrated seven-page introduction to pasting diagrams in the context of weak 2-categories, and their relation to [[string diagrams]] is given in Section A.4 of:

* [[Chris Schommer-Pries]], _The Classification of Two-Dimensional Extended Topological Field Theories_, 2011. ([arXiv:1112.1000v2](https://arxiv.org/abs/1112.1000v2))

A survey discussion of pasting in [[2-categories]] is in

* [[John Power]], _2-Categories_ ([pdf](http://www.brics.dk/NS/98/7/BRICS-NS-98-7.pdf))

from definition 2.10 on. Details are in

* [[John Power]], _A 2-categorical pasting theorem_ , Journal of Algebra, **129**, 1990. [doi:10.1016/0021-8693(90)90229-H]

Dominic Verity gave a bicategorical pasting theorem in

* [[Dominic Verity]], _Enriched categories, internal categories, and change of base_, PhD thesis, Cambridge University, 1992. Reprinted in TAC ([link](http://www.tac.mta.ca/tac/reprints/articles/20/tr20abs.html)).

A definition and discussion of pasting diagrams in [[strict omega-categories]] is in

* [[Sjoerd Crans]], _Pasting presentation for Omega-categories_ ([web](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.48.4342))

* [[Sjoerd Crans]], _Pasting schemes for the monoidal biclosed structure on $\omega$-$\mathbf{Cat}$_ ([web](http://home.tiscali.nl/secrans/papers/thten.html), [ps](http://home.tiscali.nl/secrans/papers/thten.ps.gz), [[thten.pdf|pdf:file]])

The notion of pasting scheme used by Crans was introduced by Johnson,

* [[Michael Johnson]], _The combinatorics of n-categorical pasting_, Journal of Pure and Applied Algebra, **62**, 1989. [doi:10.1016/0022-4049(89)90136-9]

Other notions of pasting presentations have been given by Street and by Steiner,

* [[Ross Street]], _Parity complexes_, Cahiers de Topologie et Géométrie Différentielle Catégoriques, **32**, 1991. ([link](http://www.numdam.org/item/?id=CTGDC_1991__32_4_315_0))

* [[Ross Street]], _Parity complexes : corrigenda_, Cahiers de Topologie et Géométrie Différentielle Catégoriques, **35**, 1994. ([link](http://www.numdam.org/item/?id=CTGDC_1994__35_4_359_0))

* [[Richard Steiner]], _The algebra of directed complexes_, Applied Categorical Structures, **1**, 1993. ([doi:0.1007/BF00873990](https://doi.org/10.1007/BF00873990))

These notions were compared, and a new generalized one introduced, in

* [[Simon Forest]], _Unifying notions of pasting diagrams_, Higher Structures, **6**, 2022 ([arxiv:1903.00282](https://arxiv.org/abs/1903.00282), [doi:10.21136/HS.2022.01](https://doi.org/10.21136/HS.2022.01))

For an online link to the notion of directed complex, see

* [[Sjoerd Crans]] and [[Richard Steiner]], _Presentations of omega-categories by directed complexes_, Journal of the Australian Mathematical Society, **63**, 1997. ([doi:10.1017/S1446788700000318](https://doi.org/10.1017/S1446788700000318))

There is also

* [[Michael Makkai]], _Computads and 2 dimensional pasting diagrams_ ([pdf](http://www.math.mcgill.ca/makkai/2dcomputads/2DCOINS1.pdf))

For a cubical approach to multiple compositions and other references see the paper

* [[Philip J. Higgins]], _Thin elements and commutative shells in cubical omega-categories_, Theory and Applications of Categories, **14**, 2005. ([link](http://www.tac.mta.ca/tac/volumes/14/4/14-04abs.html))

A formal approach to [[pasting diagrams]] in [[Gray categories]]:

* {#DiVittorio2023} [[Nicola Di Vittorio]], _A Gray-categorical pasting theorem_, Theory and Applications of Categories **39** 5 (2023) 150-171 [[tac:39-05](http://www.tac.mta.ca/tac/volumes/39/5/39-05abs.html)]

[[!redirects pasting diagrams]]
[[!redirects pasting]]


pasting law for pullbacks

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

#Contents#
* table of contents
{:toc}

## Idea

In [[category theory]], the *pasting law* or *pullback lemma* is a statement about (de-)composition of [[pullback]]/[[pushout]] [[diagrams]].

## Statement

### General pasting law

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

Let $\mathcal{C}$ be a [[category]] or more generally an [[(∞,1)-category]]. Consider a [[commuting diagram]] in $\mathcal{C}$ of the following shape:

\begin{tikzcd}
  {}
  \ar[r]
  \ar[d]
  &
  {}
  \ar[r]
  \ar[d]
  &
  {}
  \ar[d]
  \\
  {}
  \ar[r,]
  &
  {}
  \ar[r]
  &
  {}
\end{tikzcd}

Then:

1. if the right square is a pullback, then the total rectangle is a pullback precisely if the left square is a pullback.

1. if the left square is a [[pushout]], then the total rectangle is a pushout precisely if the right square is a pushout.

=--

For **proof** see:

* for [[category theory]]: at _[pullback -- pasting law](pullback#Pasting)_; also e.g. [Bauer 2012](#Bauer12);

* for [[(∞,1)-category theory]]: at _[(∞,1)-limit -- pushout pasting law](limit+in+a+quasi-category#PushoutPasting)_

\linebreak

In general, the implications in the above result do require the hypothesis (e.g. in the pullback case that the right square is a pullback).  However, in some cases this can be omitted:

\begin{prop}
Suppose we have a diagram of the above shape

$$
  \array{
    x & \longrightarrow & y & \longrightarrow & z
    \\
    \downarrow && \downarrow && \downarrow
    \\
    u & \longrightarrow & v & \longrightarrow & w
  }
$$

in which the total rectangle (consisting of $x,z,u,w$) is a pullback, and moreover the induced map $y\to v\times z$ is a [[monomorphism]].  Then the left-hand square (consisting of $x,y,u,v$) is also a pullback.
\end{prop}

### Reverse pasting law

In a [[regular category]] this also works in the other direction, if the bottom left morphisms is a [[regular epimorphism]]:
\begin{proposition}
\label{WrongWayPastingLawInRegularCategory}
**(reverse pasting law)**
\linebreak
  In a [[regular category]], consider a [[commuting diagram]] of the form
\begin{tikzcd}
  {}
  \ar[r]
  \ar[d]
  \ar[dr,phantom,"\mbox{\tiny(pb)}"]
  &
  {}
  \ar[r]
  \ar[d]
  &
  {}
  \ar[d]
  \\
  {}
  \ar[r, ->>]
  &
  {}
  \ar[r]
  &
  {}
\end{tikzcd}
where

1. the left square is a [[pullback]];

1. the bottom left morphism is a [[regular epimorphism]].

Then the right square is a pullback iff the total rectangle is.
\end{proposition}
(e.g. [Gran 2021, Lem. 1.15](#Gran21), see also [Carboni, Janelidze, Kelly and Paré 1997, Lemma 4.6](#CarboniJanelidzeKellyParé97), [Garner and Lack 2012, Lemma 2.2](#GarnerLack12))

For a more general statement that holds in categories that are not necessarily [[regular]], see [Przybylek (2013)](#Przybylek13).

Prop. \ref{WrongWayPastingLawInRegularCategory} implies the $\infty$-category theoretic reverse pasting law, at least in good cases:

\begin{proposition}\label{ReversePastingLawForInfinityGroupoids}
**(reverse pasting law for $\infty$-pullbacks of $\infty$-groupoids)**
\linebreak
  The reverse pasting law (Prop. \ref{WrongWayPastingLawInRegularCategory}) holds also for [[homotopy pullbacks]] of [[infinity-groupoids|$\infty$-groupoids]] as soon as the bottom left morphism is an [[effective epimorphism in an (infinity,1)-category|effective epimorphism in this $\infty$-category]].
\end{proposition}
\begin{proof}
  By the discussion at *[[homotopy pullback]]* and using the [[classical model structure on simplicial sets]], we may model the situation by a diagram of [[simplicial sets]] where both bottom morphisms are [[Kan fibrations]], and then need to show that the ordinary reverse pasting law applies.

Now observe that:

1. an effective epimorphism of $\infty$-groupoids is a [[surjection]] on [[connected components]] (by [this Prop.](effective+epimorphism+in+an+infinity1-category#EffectiveEpisOfInfinityGroupoids));

1. a Kan fibration which resolves a surjection on connected components is degreewise surjective (by [this Prop.](Kan+fibration#EveryPiZeroEpiIsResolvedByASurjectiveKanFibration));

1. a degreewise surjection of simplicial sets is an [[epimorphism]] (by [this Prop.](SimpSet#EpimorphismsOfSimplicialSetsDetectedDegreewise)),

   hence is a [[regular epimorphism]] in a [[regular category]] (since [[sSet]] is a [[topos]]).

Hence the bottom left morphism in our diagram of simplicial sets is a regular epimorphism and the claim follows by Prop. \ref{WrongWayPastingLawInRegularCategory}.
\end{proof}

### Further related statements

Another related statement involves a pair of rectangles and equalizers.

+-- {: .num_prop}
###### Proposition
Suppose $\mathcal{C}$ is any [[category]] with [[equalizers]] and that we have a diagram of the following shape:

$$
  \array{
    x & \longrightarrow & y & \rightrightarrows & z
    \\
    \downarrow && \downarrow && \downarrow
    \\
    u & \longrightarrow & v & \rightrightarrows & w
  }
$$

such that the vertical arrows are all [[monomorphisms]], the squares on the right are serially [[commutative diagram|commutative]], and the lower row is an [[equalizer]].  Then the upper row is an equalizer if and only if the left square is a [[pullback]].
=--

## References
 {#References}

### In 1-category theory

Discussion in [[1-category|1-]][[category theory]]:

#### Ordinary pasting law

Statements of the pasting law in textbooks, typically leaving the proof to the reader:

* [[Saunders MacLane]], Ex. 8 on p. 72 in: *[[Categories for the Working Mathematician]]*, Graduate texts in mathematics, Springer 1971 ([doi:10.1007/978-1-4757-4721-8](https://link.springer.com/book/10.1007/978-1-4757-4721-8))

* [[Jiri Adamek]], [[Horst Herrlich]], [[George Strecker]], Prop. 11.10 in: *[[Abstract and Concrete Categories]]*, John Wiley and Sons, New York (1990) reprinted as: Reprints in Theory and Applications of Categories **17** (2006) 1-507 ([tac:tr17](http://www.tac.mta.ca/tac/reprints/articles/17/tr17abs.html), [book webpage](http://katmat.math.uni-bremen.de/acc/), [pdf](http://katmat.math.uni-bremen.de/acc/acc.pdf))

A proof is spelled out in:

* {#Bauer12} [[Andrej Bauer]], *The pullback lemma in gory detail*, 2012 ([pdf](http://math.andrej.com/wp-content/uploads/2012/05/pullback.pdf), [[Bauer_PullbackLemma.pdf:file]])

#### Reverse pasting law

The reverse pasting law is discussed in:

* {#CarboniJanelidzeKellyParé97} [[Aurelio Carboni]], [[George Janelidze]], [[Max Kelly]], [[Robert Paré]], Lemma 4.6 of: *On localization and stabilization for factorization systems*, Appl. Categ. Structures **5** (1997) 1-58 [[doi:10.1023/A:1008620404444](https://doi.org/10.1023/A:1008620404444)]

* {#GarnerLack12} [[Richard Garner]], [[Steve Lack]], Lemma 2.2 in: *On the axioms for adhesive and quasiadhesive categories*, Theory and Applications of Categories, **27**  3 (2012) 27-46 [[arXiv:1108.2934](http://arxiv.org/abs/1108.2934), [tac:27-03](http://www.tac.mta.ca/tac/volumes/27/3/27-03abs.html)]

* {#Gran21} [[Marino Gran]], *An introduction to regular categories*, in: New Perspectives in Algebra, Topology and Categories,  Coimbra Mathematical Texts, Springer (2021) [[arXiv:2004.08964](https://arxiv.org/abs/2004.08964), [ISBN:978-3-030-84319-9](https://www.springerprofessional.de/en/new-perspectives-in-algebra-topology-and-categories/19764096)]

* {#Przybylek13} [[Michal Przybylek]], _The other pullback lemma_ (2013) [[arXiv:1311.2974](https://arxiv.org/abs/1311.2974)]

cf. [MO:143070](https://mathoverflow.net/q/143070/381)

### In model category theory

Discussion in [[model category]]-theory

for [[right proper model categories]]:

* {#Hirschhorn02} [[Philip Hirschhorn]], Prop. 13.3.15 of: _Model Categories and Their Localizations_, AMS Math. Survey and Monographs Vol 99 (2002) ([ISBN:978-0-8218-4917-0](https://bookstore.ams.org/surv-99-s/), [pdf toc](http://www.gbv.de/dms/goettingen/360115845.pdf), [pdf](http://www.maths.ed.ac.uk/~aar/papers/hirschhornloc.pdf))

and for general model categories:

* [[Alisa Govzmann]], [[Damjan Pištalo]], [[Norbert Poncin]], Prop. 8 in: *Indeterminacies and models of homotopy limits* [[arXiv:2109.12395](https://arxiv.org/abs/2109.12395)]

### In $(\infty,1)$-category theory

Discussion in [[(infinity,1)-category theory|$(\infty,1)$-category theory]]

* [[Jacob Lurie]], Lemma 4.4.2.1 in *[[Higher Topos Theory]]*

[[!redirects pasting law for pullbacks]]
[[!redirects pasting law for pushouts]]

[[!redirects pasting law for homotopy pullbacks]]
[[!redirects pasting law for homotopy pushouts]]

[[!redirects pasting law]]
[[!redirects pullback lemma]]


pasting scheme

+-- {: .rightHandSide}
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### Context
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
=--
=--

# Contents
* table of contents
{:toc}

## Idea

Pasting schemes are one possibility to give a rigorous treatment of  the (older) notational device of [[pasting diagrams]].

## Notions of pasting scheme

More than one definition of *pasting scheme* has been used to give a justification to the practice of [[pasting diagrams]].

For example, there are pasting schemes in the sense of [Johnson 1987](#Johnson1987), and there are pasting schemes in the sense of [Power 1990](#Power1990).

### Power's definition

Power's notion is based on planar embeddings of [[quivers]] $Q$ that satisfy the following conditions:

* $Q$ is [[finite set|finite]] and [[connected object|connected]];

* For the free category $F(Q)$ generated by $Q$, every [[endomorphism]] is an [[identity map|identity]] ($Q$ has no directed cycles);

* There are vertices $s, t$ of $Q$ (the *source* and *sink*, respectively) such that for every vertex $v$, the hom-sets $F(Q)(s, v)$ and $F(Q)(v, t)$ are [[inhabited set|inhabited]] (there exist directed paths in $Q$ from $s$ to $v$ and from $v$ to $t$).

We will call such quivers *progressive*, in rough analogy with progressive [[string diagrams]].

There is a restricted [[geometric realization]] functor

$$Quiv = Set^{(0 \rightrightarrows 1)^{op}} \stackrel{skel}{\to} Set^{\Delta^{op}} \stackrel{R}{\to} Space$$

(into a [[convenient category of topological spaces]] $Space$) which we again denote as $R: Quiv \to Space$.

+-- {: .num_defn}
###### Definition
**(after Power)**
A *pasting scheme* consists of a progressive [[quiver]] $Q$ together with an subspace embedding $i: R(Q) \hookrightarrow \mathbb{R}^2$, such that $i$ is a $C^1$ embedding on each edge interior.
=--

To each pasting scheme $(Q, i)$ we may associate a 2-[[computad]] $C = Comp(Q, i)$. Part of the computad structure is easily described:

* The underlying quiver of $C$ is $Q$;

* The elements of $C_2$ (the 2-cells) are the bounded [[connected components]] of $\mathbb{R}^2 \setminus i(R(Q))$.

To complete the description of the computad, we need to define the source and target $\sigma(c), \tau(c) \in Mor(F(Q))$ of a 2-cell $c$. For each edge $e \in Q_1$ such that $i(e)$ lies in the boundary of $c$, and each interior point $x \in e$, we let $t(x) = i'(x)$ be the tangent vector, and $n(x)$ the inward-pointing normal (pointing into $c$); the $C^1$ embedding condition ensures that these make sense. Then say that $e$ is a *negative edge* of $c$ if

$$\det \binom{t(x)}{n(x)} \lt 0$$

(the cell $c$ is "to the right" as $i$ is traversed along $e$) and a *positive edge* if

$$\det \binom{t(x)}{n(x)} \gt 0$$

(the cell $c$ is to the left).

Power shows that under the progressivity conditions, for each $c \in C_2$ there are unique distinct vertices $u, v$ such that the negative edges of $c$ form a directed path $\alpha \in F(Q)(u, v)$ from $u$ to $v$, and the positive edges of $c$ form a directed path $\beta \in F(Q)(u, v)$. We then define $\sigma(c) = \alpha$ and $\tau(c) = \beta$.

*Power's pasting theorem* may be stated in the following form:

+-- {: .num_theorem}
###### Theorem
For every pasting scheme $(Q, i)$, the free 2-category on the 2-computad $Comp(Q, i)$ has exactly one 2-morphism $\gamma$ whose $0$-domain is the source of $Q$, whose $0$-codomain is the sink of $Q$, whose $1$-domain of $\gamma$ is the directed path consisting of **positive** edges of the unique unbounded component of $\mathbb{R}^2 \setminus i(R(Q))$, and whose $1$-codomain is the directed path consisting of **negative** edges of that component.
=--

## Related concepts

* [[pasting diagram]]

* [[pasting law]] for [[pullbacks]]/[[homotopy pullbacks]]

## References

The notion of pasting in a [[2-category]] was introduced in

* [[Jean Bénabou]], _Introduction to bicategories_, in _Lecture Notes in Mathematics_ Vol. 47, pp. l-77, Springer-Verlag, New York/Berlin, (1967)

* [[Michael Johnson]]: _Pasting Diagrams in $n$-Categories with Applications to Coherence Theorems and Categories of Paths_, Doctoral Thesis, University of Sydney, 1987
{#Johnson1987}

* [[Michael Johnson]]: _The Combinatorics of $n$-Categorical Pasting_, Journal of Pure and Applied Algebra 62 (1989)
{#Johnson1989}

* [[John Power]]: _A 2-Categorical Pasting Theorem_, Journal of Algebra 129 (1990)
{#Power1990}

Other notions of pasting presentations have been given by Street ([[parity complexes]]) and by Steiner,

* [[Ross Street]], _Parity complexes_, Cahiers Top. Géom Diff. Catégoriques 32 (1991), 315-343. ([link](http://archive.numdam.org/ARCHIVE/CTGDC/CTGDC_1991__32_4/CTGDC_1991__32_4_315_0/CTGDC_1991__32_4_315_0.pdf)) Corrigenda, Cahiers Top. Géom Diff. Catégoriques 35 (1994), 359-361. ([link](http://archive.numdam.org/ARCHIVE/CTGDC/CTGDC_1994__35_4/CTGDC_1994__35_4_359_0/CTGDC_1994__35_4_359_0.pdf))

* Richard Steiner, The algebra of directed complexes, Appl. Cat. Struct. 1 (1993), 247-284.

[[!redirects pasting schemes]]


path

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###Context###
#### Topology
+--{: .hide}
[[!include topology - contents]]
=--
=--
=--

# Paths
* table of contents
{: toc}

## Definitions

In [[topology]], a (parametrised, oriented) __path__ in a [[space]] $X$ is a map (a [[morphism]] in an appropriate [[category]] of [[spaces]], such as a [[continuous function]] between [[topological space]]) to $X$ from the [[topological interval]] $\mathbb{I} = [0,1]$.

A __path from $a$ to $b$__ is a path $f$ such that $f(0) = a$ and $f(1) = b$.
An __unparametrised path__ is an [[equivalence class]] of paths, such that $f$ and $g$ are equivalent if there is an [[monotone function|increasing]] [[automorphism]] $\phi$ of $\mathbb{I}$ such that $g = f \circ \phi$.  An __unoriented path__ is an equivalence class of paths such that $f$ is equivalent to $(x \mapsto f(1 - x))$.

If $P$ is a path, then its __reverse path__[^1], denoted $\overline{P}$, is defined to be the composite $P \circ ( t\mapsto 1-t )$.
The operation $P\mapsto\overline{P}$ is called _path reversal_.

[^1]: Cf. e.g. [[Introduction to Topology -- 2]], or also [Section 2.1](#tomDieck2008); beware that that reference, (0) like many others, uses the term "inverse path", even though the operation of concatenation of paths does not *in and of itself* yield a [[strict category|strict groupoid]], in which $\overline{P}$ would be an inverse, and (1) that it uses $a$ and $b$ for the endpoints of the _interval_, not the endpoints of the paths in the space $X$, and (2) that it uses $P^-$ instead of $\overline{P}$, which however is less suited for notational iterating (compare $\overline{\overline{P}}=P$ with $(P^-)^-=P$), and that (3)  the 2008 edition has a typo: " $w(1-t)$ " in loc. cit., when   *inverse path* gets defined, should be $u(1-t)$.

A __Moore path__ is defined like a path, except for having another domain: replace $[0,1]$ with the interval $[0,n]$ for some [[natural number]] (or, more commonly, any non-negative [[real number]]) $n$.  All of these variations can be combined, of course.  (For unoriented paths, one usually says 'between $a$ and $b$' instead of 'from $a$ to $b$'.  Also, a Moore path from $a$ to $b$ has $f(n) = b$ instead of $f(1) = b$.  Finally, there is not much difference between unparametrised paths and unparametrised Moore paths, since we may interpret $(t \mapsto n t)$ as a reparametrisation $\phi$.)

In [[graph theory]], a __path__ is a [[list]] of edges, each of which ends where the next begins.  Actually, this is a special case of the above, if we use Moore paths and interpret $[0,n]$ as the linear graph with $n + 1$ vertices and $n$ edges; in this way, the other variations become meaningful.  (However, as the only directed graph automorphism of $[0,n]$ is the [[identity morphism|identity]], parametrisation is trivial for directed graphs and equivalent to orientation for undirected graphs.  Note that a non-Moore path is simply an edge, one of the fundamental ingredients of a graph.)

## Concatenation

Given a Moore path $f$ from $a$ to $b$ and a Moore path $g$ from $b$ to $c$, the __concatenation__ of $f$ and $g$ is a Moore path $f ; g$ or $g \circ f$ from $a$ to $c$.  If the domain of $f$ is $[0,m]$ and the domain of $g$ is $[0,n]$, then the domain of $f ; g$ is $[0,m+n]$, and
$$ (f ; g)(x) \coloneqq \left \{ \array { f(x) & \quad x \leq m \\ g(x-m) & \quad x \geq m .} \right . $$
In this way, we get a ([[strict category|strict]]) [[category]] whose [[objects]] are [[global element|points]] in $X$ and whose [[morphisms]] are Moore paths in $X$, with concatenation as [[composition]].  This category is called the __[[Moore path category]]__.

Often we are more interested in a [[quotient category]] of the Moore path category.  If we use unparametrised paths (in which case we may use paths with domain $\mathbb{I}$ if we wish), then we get the __unparametrised [[path category]]__.  If $X$ is a [[smooth space]], then we may additionally identify paths related through a [[thin homotopy]] to get the __[[path groupoid]]__.  Finally, if $X$ is a [[continuous space]] and we identify paths related through any (endpoint-preserving) [[homotopy]], then we get the __[[fundamental groupoid]]__ of $X$.

In graph theory, the Moore path category is known as the __[[free category]]__ on the graph.

## Related entries

* [[path-connected topological space]]

* [[locally path-connected topological space]]

* [[loop]]

* [[path space]],

* [[fundamental group]], [[fundamental groupoid]]

* [[fundamental theorem of covering spaces]]

* [homotopy of smooth paths relative to their endpoints](smooth+homotopy#HomotopyOfSmoothPathsRelativeToTheirEndpoints)

## References

E.g.

* {#tomDieck2008} [[Tammo tom Dieck]], _Algebraic Topology_, European Mathematical Society, 2008

(...)

[[!redirects path]]
[[!redirects paths]]

[[!redirects Moore path]]
[[!redirects Moore paths]]

[[!redirects path concatenation]]
[[!redirects path concatenations]]

[[!redirects concatenation of paths]]
[[!redirects concatenrations of paths]]

[[!redirects reverse path]]
[[!redirects reverse paths]]

[[!redirects path reversal]]
[[!redirects path reversion]]
[[!redirects path reversals]]
[[!redirects path reversions]]

[[!redirects constant path]]
[[!redirects constant paths]]


path algebra > history

< [[path algebra]]
[[!redirects path algebra -- history]]


path category

> This article is about the notion of *path category* in ordinary category theory. For the notion of path category in [[model category theory]] and [[homotopy theory]], see *[[category with path objects]]*.

----

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### Contex
#### Category theory
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[[!include category theory - contents]]
=--
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# Contents
* table of contents
{:toc}

## Idea

One can view the concept of a [[morphism]] or arrow in a [[category]] as an extreme abstraction of the concept of a spatial trajectory. It therefore comes with no surprise that (notions of) paths stemming from various conceptualizations of 'space' like e.g. [[quivers]] or [[topological spaces]] arrange themselves in appropriate categories thereby giving rise to several _different notions_ of **path category**.

## Free category on a directed graph

There is a [[forgetful functor]] from [[small category|small]] [[strict category|strict]] [[categories]] to [[quivers]]. This forgetful functor has a [[left adjoint]], giving the **[[free category]]** or **path category** of a quiver, whose [[objects]] are the vertices of the quiver.  The [[morphisms]] from $a$ to $b$ in this free category are *not* merely the arrows from $a$ to $b$ in the quiver but instead are [[lists]] of the form $(a_n,f_n,a_{n-1},\ldots,a_{1},f_1,a_0)$ where $n \geq 0$ is a [[natural number]], $a_0,a_1,\ldots,a_n$ are vertices of the graph, $a = a_0$, $b = a_n$,  and for all $0 \lt i \leq n$, $f_i\colon a_{i-1} \to a_i$ is an edge from $a_{i-1}$ to $a_i$.  The [[composition]] is given by the [[concatenation]]
$$(a_n,f_n,a_{n-1},\ldots,a_{1},f_1,a_0)\circ (b_m,g_m,a_{m-1},\ldots,b_{1},g_1,b_0) := (a_n,f_n,a_{n-1},\ldots,a_{1},f_1,a_0= b_m,g_m,a_{m-1},\ldots,b_{1},g_1,b_0)$$
whenever $a_0 = b_m$, and the [[target]] and [[source]] maps are given by $s(a_n,f_n,a_{n-1},\ldots,a_{1},f_1,a_0)=a_0$ and $t(a_n,f_n,a_{n-1},\ldots,a_{1},f_1,a_0) = a_n$. One informally writes $f$ for the morphism $(b,f,a)\colon a \to b$ in the free category and the [[identity morphism|identities]] of the free category are $id_a = (a,a)$; thus
$$ f_n \circ f_{n-1} \circ \ldots \circ f_1 = (t(f_n),f_n,t(f_{n-1}),\ldots,t(f_1),f_1,s(f_1))\quad .$$

### Remark

With free objects, one is often primarily interested in taking quotients whence a free category over a graph is usually a somewhat auxiliary gadget its main interest lying in the role it plays in defining [[category of fractions|categories of fractions]] (cf. [[Gabriel-Zisman]] 1967 pp.1,6; probably the original source of the path category in this sense).

But similar techniques apply to various notions of graphs or more restricted classes of categories with forgetful functors to graphs permitting to syntactically generate various notions of free categories with extra structure and in some of these cases it occurs that the corresponding free category (of paths) is indeed interesting in itself e.g. the [[free topos]] arises this way and the syntactic derivations of context free grammars arrange themselves into such categories of paths as explored in Walters ([1989a](#Walters1),[1989b](#Walters2)) or Latch ([1991](#Latch)).

## Path category of a space

Given a [[topological space]] $X$ (or some other kind of [[space]] with a notion of maps from [[intervals]] into it), there are various ways to obtain a [[small category|small]] [[strict category|strict]] [[category]] whose objects are the points of $X$ and whose morphisms are [[paths]] in $X$. This is also often called a _path category_.

* In particular, for every topological space there is its [[fundamental groupoid]] whose morphisms are [[homotopy]] classes of paths in $X$.

* If $X$ is a [[directed space]] there is a notion of path category called the [[fundamental category]] of $X$.

* When $X$ is a [[smooth space]], there is a notion of path category where less than homotopy of paths is divided out: just _thin_ homotopy. This yields a notion of [[path groupoid]].

* If _parameterized_ paths are used, there is a way to get a category of paths without dividing out any equivalence relation: this is the [[Moore path category]].

## Arrow category

Given a [[category]] $X$, the [[functor category]] $[I,X]$ for $I$ the [[interval category]] might be called a "directed path category of $X$" (similar to [[path space]]). However, this functor category is referred to instead as the [[arrow category]] of $X$ and sometimes even denoted $Arr(X)$.

## Related concepts

* [[concatenation]]

* [[free groupoid]]

* [[free diagram]]

* [[free monoid]], [[free operad]]

* [[free topos]]

## References

The [[free category]] on a graph has a section in

* [[Francis Borceux]], _Handbook of categorical Algebra vol. 1_ , Cambridge UP 1994.

or in

* {#MacLane} [[Saunders MacLane]], _Categories for the Working Mathematician_ , Springer 1998.

The following is another source for this, even an open source:

* {#BarrWells}[[Michael Barr]] and [[Charles Wells]], _Category Theory for Computing Science_ [PDF](http://www.math.mcgill.ca/triples/Barr-Wells-ctcs.pdf)

The path categories of context free grammars are explored in

* {#Latch} Dana Latch, _The connection between the fundamental groupoid and a unification algorithm for syntactil algebras (extended abstract)_ , Cah. Top. Géom. Diff. Cat. **XXXII** (1991).  ([link](http://www.numdam.org/item?id=CTGDC_1991__32_3_203_0))

* {#Walters1} [[Bob Walters]], _The free category with products on a multigraph_ , JPAA **62** (1989). ([link](http://www.sciencedirect.com/science/article/pii/0022404989901527))

* {#Walters2} [[Bob Walters]], _A note on context-free languages_ , JPAA **62** (1989). ([link](http://www.sciencedirect.com/science/article/pii/0022404989901515))

Since the perspective on categories as 'graphs with operations' is closely related to their 'logico-deductive' side, it figures prominently in the work of [[Jim Lambek]] and [[Phil Scott]] (cf. the references at [[free topos]]).

[[!redirects path category]]
[[!redirects path categories]]


path groupoid

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### Context
#### Differential geometry
+--{: .hide}
[[!include synthetic differential geometry - contents]]
=--
#### $\infty$-Lie theory
+-- {: .hide}
[[!include infinity-Lie theory - contents]]
=--
=--
=--

# Contents
* automatic table of contents goes here
{: toc}

## Idea

For $X$ a smooth space, there are useful refinements of the [[fundamental groupoid]] $\Pi_1(X)$ which remember more than just the [[homotopy]] class of [[paths]], i.e. whose morphisms are (piecewise, say) smooth paths in $X$ modulo an equivalence relation still strong enough to induce a groupoid structure, but weaker than dividing out homotopies relative to endpoints.

## Definition

Let $X$ be a smooth [[manifold]].

+-- {: .un_defn}
###### Definition
For $\gamma_1, \gamma_2  : [0,1] \to X$ two smooth maps, a **[[thin homotopy]]** $\gamma_1 \Rightarrow \gamma_2$ is a smooth [[homotopy (as an operation)|homotopy]], i.e. a smooth map
$$
  \Sigma : [0,1]^2 \to X
$$
with

* $\Sigma(0,-) = \gamma_1$
* $\Sigma(1,-) = \gamma_2$
* $\Sigma(-,0) = \gamma_1(0) = \gamma_2(0)$
* $\Sigma(-,1) = \gamma_1(1) = \gamma_2(1)$

which is _thin_ in that it doesn't sweep out any surface: every $2$-form pulled back to it vanishes:

* $\forall B \in \Omega^2(X)\colon \Sigma^* B = 0$.
=--

+-- {: .un_defn}
###### Definition
A path $\gamma\colon [0,1] \to X$ has **[[sitting instants]]** if there is a neighbourhood of the boundary of $[0,1]$ such that $\gamma$ is locally constant restricted to that.
=--

+-- {: .un_defn}
###### Definition
The **path groupoid** $P_1(X)$ is the [[diffeological groupoid]] that has

* $Obj(P_1(X)) = X$
* $P_1(X)(x,y) = \{$thin-homotopy classes of paths $\gamma\colon x \to y $ with sitting instants$\}$.

Composition of paths comes from concatenation and reparameterization of representatives. The quotient by thin-homotopy ensures that this yields an associative composition with inverses for each path.
=--

This definition makes sense for $X$ any [[generalized smooth space]], in particular for $X$ a [[sheaf]] on [[Diff]].

Moreover, $P_1(X)$ is always itself naturally a groupoid [[internal category|internal to]] [[generalized smooth spaces]]: if $X$ is a [[Chen space]] or [[diffeological space]] then $P_1(X)$ is itself internal to that category.  However, even if $X$ is a manifold, $P_1(X)$ will not be a manifold, see [[smooth structure of the path groupoid]] for details.

There are various generalizations of the path groupoid to [[n-groupoids]] and [[∞-groupoids]]. See

* [[path n-groupoid]],

* [[path ∞-groupoid]].

## Remarks

If $G$ is a Lie group, then internal (i.e. smooth) functors from the path groupoid to the one-object Lie groupoid corresponding to $G$ are in bijection to $Lie(G)$-valued differential forms on $X$. With gauge transformations regarded as morphisms between Lie-algebra values differential forms, this extends naturally to an equivalence of categories
$$
  [P_1(X), \mathbf{B}G] \simeq \Omega^2(X, Lie(G))
$$
where on the left the [[functor category]] is the one of internal (smooth) functors.

More generally, smooth [[anafunctors]] from $P_1(X)$ to $\mathbf{B}G$ are canonically equivalent to smooth $G$-principal bundles on $X$ with [[connection]]:
$$
  Ana(P_1(X), \mathbf{B}G) \simeq G Bund_\nabla(X)
  \,.
$$

See also

* [[Atiyah Lie groupoid]].

## References

For "path groupoids" in [[simplicial sets]], aka *[[Dwyer-Kan loop groupoids]]* see there, such as:

* [[André Joyal]], [[Myles Tierney]], *On the theory of path groupoids*, Journal of Pure and Applied Algebra **149** 1 (2000) 69-100 [doi:10.1016/S0022-4049(98)00164-9]

[[!redirects path goupoid]]
[[!redirects path groupoids]]

[[!redirects smooth path groupoid]]
[[!redirects smooth path groupoids]]


path infinity-groupoid

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#### Cohesive homotopy theory
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#Contents#
* table of contents
{:toc}

## Idea

The path [[∞-groupoid]] $\Pi(X)$ of a [[generalized smooth space]] $X$ is a geometric version of the [[fundamental ∞-groupoid]] of $X$: its [[1-morphisms]] form the geometric space of [[paths]] in $X$, and so forth.

For more see at:

* *[[shape via cohesive path ∞-groupoid]]*

Its [[truncations]] to lower categorical degree yield

* [[path groupoids]]

* [[path n-groupoids]].

For details see at _[[shape via cohesive path ∞-groupoid]]_.

[[!redirects  path infinity-groupoids]]
[[!redirects  path ∞-groupoid]]
[[!redirects  path ∞-groupoids]]


path infinity-groupoid in a lined topos

> see [[path ∞-groupoid]]


path integral

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#### Quantum systems
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#### Quantum field theory
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#### Measure and probability theory
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[[!include measure theory - contents]]
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#### Integration theory
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[[!include integration theory - contents]]
=--
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> under construction

#Contents#
* table of contents
{:toc}

## Idea

The notion of **path integral** originates in and is mainly used in the context of [[quantum mechanics]] and [[quantum field theory]], where it is a certain operation supposed to model the notion of [[quantization]].

The idea is that the quantum propagator -- in [[FQFT]] the value of the functor $U : Cob \to Vect$ on a certain [[cobordism]] -- is given by an [[integral kernel]] $U : \psi \mapsto \int K(-,y) \psi(y) d\mu$ where $K(x,y)$ is something like the integral of the exponentiated [[action functional]] $S$ over all field configurations $\phi$ with prescribed boundary data $x$ and $y$. Formally one writes

$$
  K(x,y) = \int \exp(i S(\phi))\; D\phi
$$

and calls this the **path integral**. Here the expression $D \phi$ is supposed to allude to a [[measure space|measure integral]] on the space of all $\phi$. The main problem with the path integral idea is that it is typically unclear what this measure should be, or, worse, it is typically clear that no suitable such measure does exist.

The name _path integral_ originates from the special case where the system is the [[sigma model]] describing a particle on a target space manifold $X$. In this case a field configuration $\phi$ is a path $\phi : [0,1] \to X$ in $X$, hence the integral over all field configurations is an integral over all paths.

The idea of the path integral famously goes back to [[Richard Feynman]], who motivated the idea in [[quantum mechanics]]. In that context the notion can typically be made precise and shown to be equivalent to various other [[quantization]] prescriptions.

The central impact of the idea of the path integral however is in its application to [[quantum field theory]], where it is often taken in the physics literature as the _definition_ of what the quantum field theory encoded by an [[action functional]] should be, disregarding the fact that in these contexts it is typically quite unclear what the path integral actually means, precisely.

Notably the [[Feynman perturbation series]] summing over [[Feynman graphs]] is motivated as one way to make sense of the path integral in quantum field theory and in practice usually serves as a _definition_ of the perturbative path integral.

## Realizations

We start with stating the elementary description of the [[Feynman-Kac formula]] as traditional in physics textbooks in

* [Elementary description in quantum mechanics](#ElementaryDescription).

Then we indicate the more abstract formulation of this in terms of [[integration]] against the [[Wiener measure]] on the space of paths (for the Euclidean path integral) in

* _[As an integral against the Wiener measure](#AsAnIntegralAgainstTheWienerMeasure)_.

Then we indicate a formulation in [[perturbation theory]] and [[BV-formalism]] in

* _[Perturbatively in the BV-formalism](#PerturbativelyInBVFormalism)_

### Elementary description in quantum mechanics
 {#ElementaryDescription}

A simple form of the path integral is realized in [[quantum mechanics]], where it was originally dreamed up by [[Richard Feynman]] and then made precise using the [[Feynman-Kac formula]].  (Most calculations in practice are still done using [[perturbation theory]], see the section _[Perturbatively in BV-formalism](#PerturbativelyInBVFormalism)_ below).

The [[Schrödinger equation]] says that the rate at which the phase of an energy eigenvector rotates is proportional to its energy:
\[ i \hbar \frac{d}{dt} \psi = H \psi. \]
Therefore, the probability that the system evolves to the final state $\psi_F$ after evolving for time $t$ from the initial state $\psi_I$ is
\[ \langle \psi_F|e^{-iHt}|\psi_I\rangle.\]
Chop this up into time steps $\Delta t = t/N$ and use the fact that
\[\int_{-\infty}^{\infty}|q\rangle\langle q| = 1\]
to get
\[ \langle \psi_F| e^{-iH\Delta t} \left(\int_{-\infty}^{\infty} |q_{N-1} \rangle \langle q_{N-1}| dq_{N-1}\right) e^{-iH\Delta t} \left(\int_{-\infty}^{\infty} |q_{N-2} \rangle \langle q_{N-2}| dq_{N-2}\right) e^{-iH\Delta t} \cdots e^{-iH\Delta t} \left(\int_{-\infty}^{\infty} |q_1 \rangle \langle q_1| dq_1\right) e^{-iH\Delta t} |\psi_I\rangle \]
\[ = \int_{q_1} \cdots \int_{q_{N-2}} \int_{q_{N-1}} \langle \psi_F| e^{-iH\Delta t} |q_{N-1} \rangle \langle q_{N-1}| e^{-iH\Delta t} |q_{N-2} \rangle \langle q_{N-2}|  e^{-iH\Delta t} \cdots e^{-iH\Delta t} |q_1 \rangle \langle q_1| e^{-iH\Delta t} |\psi_I\rangle dq_{N-1} dq_{N-2} \cdots dq_1 \]
Assume we have the free Hamiltonian $H=p^2/2m.$  Looking at an individual term $\langle q_{n+1}| e^{-iH\Delta t} |q_{n} \rangle,$ we can insert a factor of 1 and solve to get
\[ \array{\langle q_{n+1}| e^{-iH\Delta t} \left(\int_{-\infty}^{\infty} \frac{dp}{2\pi}|p\rangle \langle p|\right)|q_{n} \rangle &=& \int_{-\infty}^{\infty} \frac{dp}{2\pi} e^{-ip^2\Delta t/2m} \langle q_{n+1}|p\rangle \langle p|q_{n} \rangle \\ &=& \int_{-\infty}^{\infty} \frac{dp}{2\pi} e^{-ip^2\Delta t/2m} e^{ip(q_{n+1}-q_n)} \\ &=& \left(\frac{-i 2\pi m}{\Delta t}\right)^{\frac{1}{2}} e^{i \Delta t (m/2)[(q_{n+1}-q_n)/\Delta t]^2}.} \]
Defining
\[\int Dq = \lim_{N \to \infty} \left(\frac{-i 2\pi m}{\Delta t}\right)^{\frac{N}{2}} \prod_{n=0}^{N-1} \int dq_n,\]
and letting $\Delta t \to 0, N \to \infty,$ we get
\[ \langle \psi_F|e^{-iHt}|\psi_I\rangle = \int Dq e^{i \int_0^t dt \frac{1}{2}m \dot{q}^2}. \]

For arbitrary Hamiltonians $H = \frac{p^2}{2m} + V(x),$ we get
\[ \array{\langle \psi_F|e^{-iHt}|\psi_I\rangle &=& \int Dq e^{i \int_0^t dt \frac{1}{2}m \dot{q}^2 - V(x)} \\ &=& \int Dq e^{i\int_0^t\mathcal{L}(\dot{q},q) dt} \\ &=& \int Dq e^{iS(q)}, } \]
where $S(q)$ is the [[action]] functional.

\begin{remark}\label{OperatorOrderAndTimeOrder}
**(operator product order and time-order)**
\linebreak
  Under the path integral, the order of the product of [[linear operators]] (such as $x \cdot \frac{d}{d x}$ as opposed to $\frac{d}{d x} x \cdot$) corresponds to *temporal ordering* of their [[observable]] values ([Feynman 1942 p. 35](#Feynman42), [1948 p. 381](#Feynman48),  reviewed in [Nagaosa 1999, pp. 33](#Nagaosa99); [Ong](#Ong)).

Conversely the product of [[observable]] values in the path integral corresponds to the [[time-ordered product]] of the corresponding [[linear operators]], eg. [Polchinski 1998 (A.1.17)](#Polchinski98); [Rischke 2021 (5.63)](#Rischke21).
\end{remark}

### As an integral against the Wiener measure
 {#AsAnIntegralAgainstTheWienerMeasure}

More abstractly, the Euclidean path integral for the [[quantum mechanics]] of a [[charged particle]] may be defined by [[integration]] the gauge-coupling action again the [[Wiener measure]] on the space of paths.

Consider a [[Riemannian manifold]] $(X,g)$ -- hence a [[background field]] of [[gravity]] -- and a [[connection on a bundle|connection]] $\nabla : X \to \mathbf{B}U(1)_{conn}$  -- hence an [[electromagnetic field|electromagnetic]] [[background gauge field]].

The gauge-coupling [[interaction]] term is given by the [[parallel transport]] of this connection

$$
  \exp(i S)
  \coloneqq
  \exp(2\pi i \int_{(-)} [(-),\nabla] )
  \colon
  [I, X]_{x_0,x_1} \to Hom(E_{x_0}, E_{x_1})
  \,,
$$

where $E \to X$ is the [[complex line bundle]] which is [[associated bundle|associated]] to $\nabla$.

The [[Wiener measure]] $d\mu_W$ on the space of stochastic paths in $X$,we may write suggestively write as

$$
  d\mu_W = [\exp(-S_{kin})D\gamma]
$$

for it combines what in the physics literature is the [[kinetic action]] and a canonical measure on paths.

(This is a general phenomenon in formalizations of the process of [[quantization]]: the [[kinetic action]] (the [[free field theory]]-part of the [[action functional]]) is absorbed as part of the integration [[measure]] against with the remaining [[interaction]] terms are integrated.  )

Then one has
(e.g. [Norris92, theorem (34)](#Norris), [Charles 99, theorem 6.1](#Charles99)):

the [[integral kernel]] for the time evolution propagator is

$$
  U(x_0,x_1) = \int_{\gamma} tra(\nabla)(\gamma) \, [\exp(-S_{kin}(\gamma)) D\gamma]
  \,,
$$

hence the [[integration]] of the [[parallel transport]]/[[holonomy]] against the [[Wiener measure]].

(To make sense of this one first needs to extend the [[parallel transport]] from smooth paths to stochastic paths, see the references [below](#ReferencesForChargedParticle).)

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

This "holonomy integrated against the Wiener measure" is the path integral in the form in which it notably appears in the _[[worldline formalism]]_ for computing [[scattering amplitudes]] in [[quantum field theory]]. See ([Strassler 92, (2.9), (2.10)](#Strassler92)). Notice in particular that by the discussion there this is the correct [[Wick rotation|Wick rotated]] form: the [[kinetic action]] is not a [[complex phase]] but a real exponential $\exp(- S_{kin})$ while the [[gauge field|gauge]] [[interaction]] term (the [[holonomy]]) is a complex phase (locally $\exp(i \int_\gamma A)$).

=--

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

From the point of view of [[higher prequantum field theory]] this means that the path integral sends a [[correspondence]] in the [[slice (infinity,1)-topos]] of [[smooth infinity-groupoids]] over the [[delooping]] groupoid $\mathbf{B}U(1)$

$$
  \array{
    && [I,X]
    \\
    & {}^{(-)|_0}\swarrow && \searrow^{(-)|_1}
    \\
    X && \swArrow_{\exp(i S)} && X
    \\
    & {}_{\mathllap{\chi(\nabla)}}\searrow && \swarrow_{\mathrlap{\chi(\nabla)}}
    \\
    && \mathbf{B}U(1)
  }
$$

(essentially a [[prequantized Lagrangian correspondence]]) to another [[correspondence]], now in the slice over the [[stack]] (now an actual [[2-sheaf]]) $\mathbb{C}\mathbf{Mod}$ of [[modules]] over the [[complex numbers]], hence of [[complex vector bundles]]:

$$
  \array{
    && X \times X
    \\
    & {}^{p_1}\swarrow && \searrow^{p_2}
    \\
    X && \swArrow_{\int_{\gamma}\exp(i S(\gamma)) [\exp(-S_{kin}(\gamma))D\gamma]} && X
    \\
    & {}_{\mathllap{\rho(\chi(\nabla))}}\searrow && \swarrow_{\mathrlap{\rho(\chi(\nabla))}}
    \\
    && \mathbb{C}\mathbf{Mod}
   \,.
  }
$$

=--

For more discussion along these lines see at _[[motivic quantization]]_.

### Perturbatively for free field theory in BV-formalism
 {#PerturbativelyInBVFormalism}

[[BV-BRST formalism]] is a means to formalize the path integral in [[perturbation theory]] as the passage to [[cochain cohomology]] in a _[[quantum BV-complex]]_. See at _[The BV-complex and homological integration](BV-BRST+formalism#HomologicalIntegration)_ for more details.

[[!include action (physics) - table]]

## The path integral in the bigger picture

Ours is the age whose central fundamental theoretical physics question is:

>_What is [[quantum field theory]]_?

A closely related question is:

> _What is the path integral_ ?

After its conception by [[Richard Feynman]] in the middle of the 20th century It was notably [[Edward Witten]]'s achievement in the late 20th century to make clear the vast potential for fundamental physics and pure math underlying the concept of the quantum field theoretic path integral.

And yet, among all the aspects of QFT, the notion of the path integral is the one that has resisted attempts at formalization the most.

While [[FQFT|functorial quantum field theory]] is the formalization of the properties that the _locality_ and the _sewing law_ of the path integral is demanded to have -- whatever the path integral is, it is a process that in the end yields a [[functor]] on a [[(infinity,n)-category of cobordisms]] -- by itself, this sheds no light on what that procedure called "path integration" or "path integral quantization" is.

The single major insight into the right [[higher category theory|higher categorical]] formalization of the path integral is probably the idea indicated in

* [[Dan Freed]]

  * _Quantum groups from path integrals_ ([arXiv:q-alg/9501025](https://arxiv.org/abs/q-alg/9501025))

  * _[[Higher Algebraic Structures and Quantization]]_ ([arXiv:hep-th/9212115](http://arxiv.org/abs/hep-th/9212115))

which says that

* it is wrong to think of the _action functional_ that the path integral integrates over as just a _function_: it is a higher categorical object;

* accordingly, the path integral is not something that just controls the numbers or linear maps assigned by a $d$-dimensional [[quantum field theory]] in dimension $d$: also the assignment to higher codimensions is to be regarded as part of the path integral;

  * notably: the fact that quantum mechanics assigns a (Hilbert) space of sections of a vector bundle to codimension 1 is to be regarded as due to a _summing operation_ in the sense of the path integral, too: the space of sections of a vector bundle is the continuum equivalent of the direct sum of its fibers

More recently, one sees attempts to formalize this observation of Freed's, notably in the context of the [[cobordism hypothesis]]:

* [[geometric infinity-function theory]] is used to compute at least something like a path integral in codimension 1 and 2 in the context of [[sigma-model]] QFT;

* see [[path integral as a pull-push transform]]

* and something similar or is indicated in section 3 and section 6 of

  * [[Dan Freed]], [[Mike Hopkins]], [[Jacob Lurie]], [[Constantin Teleman]], _[[Topological Quantum Field Theories from Compact Lie Groups]]_ ([arXiv:0905.0731](http://arxiv.org/abs/0905.0731))

based on material (on categories of "families") in _[[On the Classification of Topological Field Theories]]_ .

## Related concepts

* [[Feynman-Kac formula]]

* [[S-matrix]]

* [[Dyson formula]]

* [[fermionic path integral]]

* [[principle of extremal action]]

* [[integration over infinite-dimensional manifolds]]

* [[cohomological integration]], [[BV-BRST quantization]]

* [[Schwinger-Dyson equation]]

## References
 {#References}

### General

The original articles:

* {#Dirac33} [[Paul A. M. Dirac]], *The Lagrangian in Quantum Mechanics*, Physikalische Zeitschrift der Sowjetunion, Band **3**, Heft 1 (1933) 64–72 [[pdf](https://www.informationphilosopher.com/solutions/scientists/dirac/Lagrangian_1933.pdf)], reprinted in [Brown 2005](#Brown05) [[doi:10.1142/9789812567635_0003](https://doi.org/10.1142/9789812567635_0003)]

* {#Feynman42} [[Richard P. Feynman]], *The Principles of Least Action in Quantum Mechanics*, PhD thesis (1942), reprinted in [Brown 2005](#Brown05) [[doi:10.1142/9789812567635_0001](https://doi.org/10.1142/9789812567635_0001)]

* {#Feynman48} [[Richard P. Feynman]], *Space-Time Approach to Non-Relativistic Quantum Mechanics*, Rev. Mod. Phys. **20** (1948) 367 [[doi:10.1103/RevModPhys.20.367](https://doi.org/10.1103/RevModPhys.20.367), [pdf](http://www-f1.ijs.si/~ramsak/km1/feynman.pdf)]

reprinted in:

* {#Brown05} [[Laurie M. Brown]], *Feynman's Thesis — A New Approach to Quantum Theory*, World Scientific (2005) [[doi:10.1142/5852](https://doi.org/10.1142/5852)]

and then the original monograph:

* [[Richard P. Feynman]], [[Albert R. Hibbs]], _Quantum Mechanics and Path Integrals_, New York: McGraw-Hill (1965)

* [[Daniel F. Styer]], *The Errors of Feynman and Hibbs* [[pdf](https://www2.oberlin.edu/physics/dstyer/FeynmanHibbs/ErrorsOfFeynmanHibbsX.pdf)]

* [[Daniel F. Styer]], [[Richard P. Feynman]], [[Albert R. Hibbs]], *Quantum Mechanics and Path Integrals: Emended Edition*, Dover (2010) [[ISBN:0486477223](https://store.doverpublications.com/0486477223.html), [pdf](http://www-f1.ijs.si/~ramsak/km1/FeynmanHibbs.pdf)]

Textbook accounts:

* [[John W. Negele]], [[Henri Orland]], *Functional Integral Formulation*, §2.2 in: *Quantum Many-Particle Systems*, Westview Press (1988, 1998) [[doi:10.1201/9780429497926](https://doi.org/10.1201/9780429497926)]

* {#Kleinert90} [[Hagen Kleinert]], *Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets*, World Scientific (1990, 1994, 2003, 2006, 2009) [[doi:10.1142/7305](https://doi.org/10.1142/7305), [pdf](https://hagenkleinert.de/documents/pi/HagenKleinert_PathIntegrals.pdf)]

* {#Polchinski98} [[Joseph Polchinski]], *A short course on path integrals*,  Appendix A in: _[[String theory]]_ part I: *An Introduction to the Bosonic String*, Cambridge University Press (1998) [[doi:10.1017/CBO9780511816079](https://doi.org/10.1017/CBO9780511816079), [pdf](http://www.nucleares.unam.mx/~alberto/apuntes/polchinski1.pdf)]

* {#Nagaosa99} [[Naoto Nagaosa]], *Quantization with Path Integral Methods*, Section 2 of: *Quantum Field Theory in Condensed Matter Physics*, Texts and Monographs in Physics, Springer (1999) [[doi:10.1007/978-3-662-03774-4_2](https://doi.org/10.1007/978-3-662-03774-4_2)]

* G. Johnson, M. Lapidus, _The Feynman integral and Feynman's operational calculus_, Oxford University Press (2000)

* [[Barry Simon]], _Functional integration and quantum physics_ AMS Chelsea Publ. (2005)

* [[Pierre Cartier]], [[Cécile DeWitt-Morette]], _Functional integration: action and symmetries_, Cambridge Monographs on Mathematical Physics (2006) [[ISBN:9780521143578](https://www.cambridge.org/ae/academic/subjects/physics/theoretical-physics-and-mathematical-physics/functional-integration-action-and-symmetries?format=PB#contentsTabAnchor)]

* Daisuke Fujiwara, _Rigorous Time Slicing Approach to Feynman Path Integrals_ (2017) Springer [[doi:/10.1007/978-4-431-56553-6](https://dx.doi.org/10.1007/978-4-431-56553-6)]

See also:

* {#Ong} Yen Chin Ong, *Where is the Commutation Relation Hiding in the Path Integral Formulation?* [[pdf](https://physicstravelguide.com/_media/quantum_theory/path-integral.pdf), [[Ong-CommutationRelationInPathIntegral.pdf:file]]]

Further lecture notes:

* R. Rosenfelder, _Path Integrals in Quantum Physics_ [[arXiv:1209.1315](http://arxiv.org/abs/1209.1315)]

* {#Rischke21} [[Dirk Rischke]], *Path Integrals in Quantum Mechanics*, lecture notes (2021) [[pdf](https://itp.uni-frankfurt.de/~drischke/Script_Path_Integrals_GU2021.pdf)]

Discussion in [[constructive quantum field theory]]:

* [[James Glimm]], [[Arthur Jaffe]], *[[Glimm-Jaffe|Quantum physics: a functional integral point of view]]*, Springer (1981, 1987) [[doi:10.1007/978-1-4612-4728-9](https://doi.org/10.1007/978-1-4612-4728-9)]

* Simon, _Functional Integration in Quantum Physics_ (AMS, 2005)

* [[Sergio Albeverio]], Raphael Høegh-Krohn, Sonia Mazzucchi. _Mathematical theory of Feynman path integrals - An Introduction_, 2 nd corrected and enlarged edition,
Lecture Notes in Mathematics, Vol. 523. Springer, Berlin, 2008 ([ZMATH](href="http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:pre05233292&format=complete))

* Sonia Mazzucchi, _Mathematical Feynman Path Integrals and Their Applications_,
World Scientific, Singapore, 2009.

The [[worldline]] path integral as a way to compute [[scattering amplitudes]] in [[QFT]] was understood in

* [[Matthew Strassler]], _Field Theory Without Feynman Diagrams: One-Loop Effective Actions_, Nucl. Phys. B385:145-184,1992 ([arXiv:hep-ph/9205205](http://arxiv.org/abs/hep-ph/9205205))
 {#Strassler92}

### Stochastic integration theory

The following articles use the integration over [[Wiener measures]] on [[stochastic processes]] for formalizing the path ingegral.

* [[James Norris]], _A complete differential formalism for stochastic calculus in manifolds_, Séminaire de probabilités de Strasbourg,  26 (1992), p. 189-209 ([NUMDAM]( http://www.numdam.org/item?id=SPS_1992__26__189_0))
 {#Norris92}

* Vassili Kolokoltsov, _Path integration: connecting pure jump and Wiener processes_ ([pdf](http://www2.warwick.ac.uk/fac/sci/statistics/staff/academic-research/kolokoltsov/minnesota.pdf))

* Bruce Driver, Anton Thalmaier, _Heat equation derivative formulas for vector bundles_, Journal of Functional Analysis 183, 42-108 (2001) ([pdf](http://www.math.ucsd.edu/~bdriver/DRIVER/Papers/Drivers_Papers/A26-Heat-Derivative-Formula.pdf))

### For charged particle/path integral of holonomy functional
 {#ReferencesForChargedParticle}

The following articles discuss (aspects of) the path integral for the [[charged particle]] coupled to a [[background gauge field]], in which case the path integral is essentially the integration of the [[holonomy]]/[[parallel transport]] functional against the [[Wiener measure]].

* Marc Arnaudon and Anton Thalmaier, _Yang--Mills fields and random holonomy along Brownian bridges_, Ann. Probab. Volume 31, Number 2 (2003), 769-790. ([Euclid](http://projecteuclid.org/euclid.aop/1048516535))

* [[Mikhail Kapranov]], _Noncommutative geometry and path integrals_, in _Algebra, Arithmetic and Geometry_, Birkhäuser Progress in Mathematics 27 (2009) ([arXiv:math/0612411](http://arxiv.org/abs/math/0612411))

* [[Christian Bär]], [[Frank Pfäffle]], _Path integrals on manifolds by finite dimensional approximation_,  J. reine angew. Math., (2008), 625: 29-57. ([arXiv:math.AP/0703272](http://arxiv.org/abs/math.AP/0703272))

* Dana Fine, Stephen Sawin, _A Rigorous Path Integral for Supersymmetric Quantum Mechanics and the Heat Kernel_ ([arXiv:0705.0638](http://arxiv.org/abs/0705.0638))

* Florian Hanisch, Matthias Ludewig, _A Rigorous Construction of the Supersymmetric Path Integral Associated to a Compact Spin Manifold_, ([arXiv:1709.10027](https://arxiv.org/abs/1709.10027))

A discussion for [[phase spaces]] equipped with a [[Kähler polarization]] and a [[prequantum line bundle]] is in

* Laurent Charles, _Feynman path integral and Toeplitz Quantization_, Helv. Phys. Acta **72** (1999) 341., ([pdf](http://ipht.cea.fr/DocsphtV2/articles/t98/093/public/publi.pdf))
 {#Charles99}

following [Norris 92, theorem (34)](#Norris92).

### More

Other references on mathematical aspects of path integrals include

* [[Pierre Cartier]], [[Cecile DeWitt-Morette]], _Functional integration: action and symmetries_ ([ZMATH](http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:1122.81004&f
ormat=complete))

* {#HenneauxTeitelboim} [[Marc Henneaux]], [[Claudio Teitelboim]], _[[Quantization of Gauge Systems]]_, Princeton University Press 1992

* [[Nikolai Reshetikhin]], _Lectures on quantization of gauge systems_, [arxiv/1008.1411](http://arxiv.org/abs/1008.1411)

* [[Edward Witten]], _A new look at the path integral of quantum mechanics_, ([arxiv/1009.6032](http://arxiv.org/abs/1009.6032))

Detailed rigorous discussion for [[quadratic Hamiltonians]] and for [[phase space]] paths in in

* {#RobbinSalamon93} [[Joel Robbin]],  [[Dietmar Salamon]], _Feynman path integrals on phase space and the metaplectic representation_ in [[Dietmar Salamon]] (ed.), _Symplectic Geometry_, LMS Lecture Note series 192 (1993) ([[RobbinSalamonMetaplectic.pdf:file]])

Discussion of quantization of [[Chern-Simons theory]] via a [[Wiener measure]] is in

* Adrian P. C. Lim, _Chern-Simons Path Integral on $\mathbb{R}^3$ using Abstract Wiener Measure_ ([pdf](http://www.math.cornell.edu/~pclim/Docs/papers/CSabe01.pdf))

Lecture notes on [[quantum field theory]], emphasizing mathematics of the Euclidean path integrals and the relation to statistical physics are at

* [[AJ Tolland]], _[Wilsonian QFT for Mathematicians](http://www.math.sunysb.edu/~ajt/Teaching/560spring2011/)_

MathOverflow questions: [mathematics-of-path-integral-state-of-the-art](http://mathoverflow.net/questions/19495/mathematics-of-path-integral-state-of-the-art),[path-integrals-outside-qft](http://mathoverflow.net/questions/20393/path-integrals-outside-qft), [doing-geometry-using-feynman-path-integral](http://mathoverflow.net/questions/19490/doing-geometry-using-feynman-path-integral), [path-integrals-localisation](http://mathoverflow.net/questions/17577/path-integrals-localisation), [finite-dimensional-feynman-integrals](http://mathoverflow.net/questions/31966/finite-dimensional-feynman-integrals), [the-mathematical-theory-of-feynman-integrals](http://mathoverflow.net/questions/24823/the-mathematical-theory-of-feynman-integrals)

* [[Theo Johnson-Freyd]], _The formal path integral and quantum mechanics_, J. Math. Phys. __51__, 122103 (2010) [arxiv/1004.4305](http://arxiv.org/abs/1004.4305),  [doi](http://dx.doi.org/10.1063/1.3503472); _On the coordinate (in)dependence of the formal path integral_, [arxiv/1003.5730](http://arxiv.org/abs/1003.5730)

[[!redirects path integrals]]

[[!redirects Feynman path integral]]
[[!redirects Feynman path integrals]]

[[!redirects path-integral]]
[[!redirects path-integrals]]

[[!redirects Feynman path-integral]]
[[!redirects Feynman path-integrals]]


path integral as a pull-push transform

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Functorial quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The [[path integral]] in [[quantization]] can be understood -- to the extent and in the cases that it can be understood at all -- as an [[integral transform]] induced from a [[span]] that is given by [[hom-space]]s.  According to the general reasoning of [[integral transforms on sheaves]] this means that it is given by a _pull-push_ operation through spans.

## Examples

### The quantum particle

The archetypical example of the [[path integral]] is that for the [[sigma model]] that describes the [[quantum mechanics]] of the [[particle]] propagating on the [[target space]] $X = \mathbb{R}$ --  the [[line]].

In a finite approximation, one considers for $N \in \mathbb{N}$ paths consisting on $N$ discrete steps: let $I_N := \{0, 1, \cdots, N\}$ be the set of $N+1$ elements. Regarding this as an abstract discrete [[cobordism]] of $N$ steps, we consider the [[cospan]]

$$
  \array{
    && I_N
    \\
    & {}^{\mathllap{in}}\nearrow && \nwarrow^{\mathrlap{out}}
    \\
    * &&&& *
  }
  \,,
$$

where $in : * \to I_N$ is the inclusion of the first elements 0, and $out : * \to I_N$ the inclusion of the last element, $N$.

The mapping space $X^{I_N} \simeq X \times X \times \cdots \times X$ is the _space of paths_ -- the space of paths in $X = \mathbb{R}$ consisting of $n$ linear steps. Homming the above cospan into $X$ produces the span

$$
  \array{
    && X^{I_N}
    \\
    & {}^{\mathllap{X^{in}}}\swarrow && \searrow^{\mathrlap{X^{out}}}
    \\
    X \simeq X^* &&&& X^* \simeq X
  }
  \,.
$$

In the canonical coordinates on $X^{I_N}$ a path $\gamma \in X^{I_N}$ is parameterized as

$$
  \gamma = (x_0, x_1, \cdots, x_N)
  \,.
$$

The [[action functional]] that encodes the dynamics of the free particle on $X$ is the [[differential form|differential form]] on $X^{I_N}$ given by

$$
  \exp(i S) := \exp(i \sum_{i = 1}^N \frac{m}{2}(\frac{x_i -  x_{i-1}}{1/N})^2) d x_0 \wedge d x_1 \wedge \cdots \wedge d x_{N-1}
  \in
  \Omega^{N}(X^{I_n})
  \,.
$$

Denote by

$$
  (X^{in})^* : \Omega^\bullet(X) \to \Omega^\bullet(X^{I_N})
$$

the pullback of differential forms along $X^{in}$, and by

$$
  (X^{out})_* := \int_{X^{I_N}/X} : \Omega^\bullet(X^{I_N}) \to \Omega^{\bullet - N}(X)
$$

the "pushforward": the [[fiber integration]] of differential forms.

Then then standard path integral for the particle is given by

1. pulling back functions along $X^{in}$;

1. taking their wedge product with $\exp(i S)$;

1. pushing the result forward along $X^{out}$.

This gives the map

$$
   (X^{out})_* \circ exp(i S) \wedge (-) \circ (X^{in})^*
   :
   \Omega^0(X) \stackrel{(X^{in})^*}{\to} C^\infty(X^{I_n}) \stackrel{\exp(i S)\wedge}{\to}
   \Omega^{N}(X^{I_N}) \stackrel{(X^{out})_*}{\to}
   \Omega^0(X)
$$

that acts by

$$
  (\psi \in C^\infty(X))
  \mapsto
  \left(
   x \mapsto
  \int_{X^{I_N}/X}
     \psi(x_0) \exp(i \frac{m}{2} \sum_{i = 1}^N (\frac{x_i - x_{i-1}}{1/N})^2 )
     d x_0 \wedge \cdots \wedge d x_{N-1}
  \right)
  \,.
$$

This is the standard expression for the path integral of the particle on the line, at the approximation of $N$ discrete steps.

### Gromov-Witten theory

* [[Gromov-Witten theory]]

### String topology

On the [[singular homology]] of [[smooth manifold]]s and other [[topological space]]s, pullback operations can be defined by [[Thom isomorphism]]s and [[fiber integration]] ("[[Umkehr map]]s"). Together with the canonically defined push-forward of singular cycles, this yields a definition of pull-push transformations on singular homology.

It was realized in ([CohenGodin](#CohenGodin)) and ([Godin](#Godin)) that such pull-push operations define a 2-dimensional [[HQFT]] whose [[space of states]] is the singular complex, and that the [[correlator]]s of the thus defined [[FQFT]] are the Chas-Sullivan [[string topology]] operations. See there for more details.

* [[string topology]]

### Geometric $\infty$-function theory

Every perfect [[derived stack]] in [[dg-geometry]] forms the [[target space]] for a pull-push transform on the [[stable (infinity,1)-category]] of [[quasicoherent sheaves]] and yields a 2-dimensional [[TQFT]]. For details on this see _[[geometric infinity-function theory]]_ .

## Related concepts

* [[motivic quantization]]

* [[integral transforms on sheaves]]

## References

The pull-push nature of the path integral was originally amplified somewhat implicitly in

* [[Dan Freed]]

  * _Quantum groups from path integrals_ ([arXiv:q-alg/9501025](http://xxx.lanl.gov/abs/q-alg/9501025))

  * _Higher algebraic structures and quantization_ ([arXiv:hep-th/9212115](http://arxiv.org/abs/hep-th/9212115))

and fully explicitly in

* [[Dan Freed]], _Twisted K-theory and the Verlinde ring_ ([pdf slides](http://www.ma.utexas.edu/users/dafr/Andrejewski%20Lectures.html))

The description of [[string topology]] operations as an [[HQFT]] defined by pull-push transforms was originally realized in

* [[Ralph Cohen]], [[Veronique Godin]],
  _[[A Polarized View of String Topology]]_

* Hirotaka Tamanoi, _Loop coproducts in string topology and triviality of higher genus TQFT operations_ (2007) ([arXiv:0706.1276](http://arxiv.org/abs/0706.1276))
 {#Tamanoi07}

A detailed discussion and generalization to the open-closed [[HQFT]] in the presence of a single space-filling [[brane]] is in

* [[Veronique Godin]], _Higher string topology operations_ (2007)([arXiv:0711.4859](http://arxiv.org/abs/0711.4859))
 {#Godin}

For more see the references at _[[motivic quantization]]_.

[[!redirects path integral as a pull-push transformation]]


path integral quantization

+-- {: .rightHandSide}
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###Context###
#### Quantum systems
+--{: .hide}
[[!include quantum systems -- contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

[[quantization]] by means of the [[path integral]].

Rigorous constructions go by the term _[[constructive quantum field theory]]_

## Related concepts

* [[S-matrix]]

## References

See at *[path integral -- references](path+integral#References)*.

[[!redirects path integral quantizations]]


path n-groupoid

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $\infty$-Lie theory
+--{: .hide}
[[!include infinity-Lie theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _path $n$-groupoid_  $P_n(X)$ of a smooth space (or [[generalized smooth space]]) $X$ is a [[diffeological n-groupoid]] which is

* a generalization of the [[path groupoid]] to higher categorical dimension

* a truncation of sorts of an [[path ∞-groupoid]] of $X$.

Its [[k-morphisms|j-morphisms]] are given by (possibly equivalence classes of) $j$-dimensional smooth paths in $X$, i.e. usually smooth maps $\gamma : D^j \to X$. Composition is by gluing of such maps.

## Path 1-groupoid

See [[path groupoid]].

## Path 2-groupoid

Definitions of path 2-groupoids as strict [[2-groupoids]] internal to [[diffeological spaces]] appear (at least) in

* [[John Baez]], [[Urs Schreiber]], _Higher gauge theory_ ([arXiv](http://arxiv.org/abs/math/0511710))

* [[Urs Schreiber]], [[Konrad Waldorf]], _Smooth functors vs. differential forms_ ([arXiv](http://arxiv.org/abs/0802.0663))

* [[João Faria Martins]], [[Roger Picken]], _On 2-dimensional holonomy_ ([arXiv](http://arxiv.org/abs/0710.4310))

For the underlying notion of [[fundamental 2-groupoid]] see there.

## Path 3-groupoid

A realization of the path [[3-groupoid]] as a [[Gray-groupoid]] internal to [[diffeological spaces]] appears in

* [[João Faria Martins]], [[Roger Picken]], _The fundamental Gray 3-groupoid of a smooth manifold and local 3-dimensional holonomy based on a 2-crossed module_ ([arXiv](http://arxiv.org/abs/0907.2566))

[[!redirects path n-groupoids]]

[[!redirects path 2-groupoid]]
[[!redirects path 2-groupoids]]

[[!redirects path 3-groupoid]]
[[!redirects path 3-groupoids]]


path space

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Topology
+--{: .hide}
[[!include topology - contents]]
=--
#### Mapping space
+--{: .hide}
[[!include mapping space - contents]]
=--
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[topology]] the _path space_ of a [[topological space]] $X$ is a _topological space of all continuous [[paths]]_ in $X$.

In [[topological homotopy theory]] the path space construction serves to exhibit [[homotopies]] in the guise of _[[right homotopies]]_. This situation generalizes to many other [[model categories]] and one speaks more generally of _[[path space objects]]_ in this case.

For exposition in the context of [[point-set topology]] see at _[[Introduction to Topology -- 1]] around [this example](Introduction+to+Topology+--+1#PathSpace)_.

For exposition in the context of [[topological homotopy theory]] see at _[[Introduction to Homotopy Theory]]_ around [this definition](Introduction+to+Homotopy+Theory#TopologicalPathSpace).

## Definition

For $X$ a [[topological space]], then its _path space_ is the [[mapping space]] $X^{[0,1]}$, out of the [[topological interval]] into $X$, i.e. the set of [[continuous functions]] $\gamma \;\colon\; [0,1] \to X$ equipped with the [[compact-open topology]].

The two endpoint inclusions $\ast \colon [0,1]$ and the unique projection $[0,1] \to \ast$ induce continuous functions

$$
  X
    \overset{}{\longrightarrow}
  X^{[0,1]}
   \overset{X^{(const_0,const_1)}}{\longrightarrow}
  X \times X
$$

(inclusion of constant paths and endpoint evaluation of paths).

## Properties

### Relation to loop space

The [[fiber product]] of the projection with the [[diagonal]] on $X$ is the [[free loop space]] $\mathcal{L}X$ of $X$:

$$
  \array{
    \mathcal{L}X &\longrightarrow& X^{[0,1]}
    \\
    \downarrow &(pb)& \downarrow^{\mathrlap{X^{(const_0,const_1)}}}
    \\
    X &\underset{\Delta_X}{\longrightarrow}& X \times X
  }
$$

If $X$ is equipped with a choice of basepoint $x \colon \ast \to X$ (making it a [[pointed topological space]]), then the further [[fiber product]] with this basepoint inclusion is the based [[loop space]] $\Omega_x X$:

$$
  \array{
    \Omega_x X &\longrightarrow& \mathcal{L}X &\longrightarrow& X^{[0,1]}
    \\
    \downarrow &(pb)& \downarrow &(pb)& \downarrow^{\mathrlap{X^{(const_0,const_1)}}}
    \\
    \ast &\underset{x}{\longrightarrow}& X &\underset{\Delta_X}{\longrightarrow}& X \times X
  }
$$

## Related concepts

* [[loop space]]

* [[path groupoid]];

* [[fundamental groupoid]];

* [[topological complexity]]

* [[path space object]]

* [[identity type]]

[[!redirects path space]]
[[!redirects path spaces]]


path space object

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

# Contents
* table of contents
{: toc}

## Idea

A _path space object_ in [[homotopy theory]] is an [[object]] in a [[homotopical category]] that behaves for many purposes as the [[topology|topological]] [[path space]] does in [[topological homotopy theory]].

## Definition
 {#Definition}

\begin{definition}\label{PathSpaceObject}
**(path space object)**
\linebreak
For $\mathcal{C}$ a [[category with weak equivalences]] and with [[binary products]], a **path space object of/for an [[object]] $X$** of $\mathcal{C}$ is a factorization of the [[diagonal]] morphism $X \stackrel{(Id, Id)}{\to} X \times X$ into the [[product]] as

$$
  X
    \xrightarrow{\;\;\; s \;\;\;}
  Paths_X
    \xrightarrow{\;\;\; (d_0, d_1) \;\;\;}
  X \times X
$$

such that $s$ is a [[weak equivalence]] ([Quillen 1967, §I.1](#Quillen67)).

Moreover ([Dwyer & Spalinski 1995, §4.12](#DwyerSpalinski95)):

If $\mathcal{C}$ in addition has the structure of a [[fibration category]] then one speaks, furthermore, of a **good path space object** if $(d_0,d_1)$ is a [[fibration]].

If $\mathcal{C}$ furthermore has the structure of a [[model category]] then one speaks of a **very good path space object** if $(d_0,d_1)$ is a [[fibration]] and $s$ is a [[cofibration]] (hence an [[acyclic cofibration]]).
\end{definition}

\begin{remark}
In Def. \ref{PathSpaceObject} one interprets

1. a ([[generalised element|generalised]]) [[global element|element]] of $X^I$ as a [[path]] in $X$;

1. $d_0, d_1$ as the maps that send a path to its start- or endpoint, respectivelyl

1. $s$ as the map that sends a point to the path [[constant function|constant]] on that point.

\end{remark}

\begin{remark}\label{InModelCategories}
**(in model categories)**
  In any [[model category]], the [[factorization system|factorization axioms]] applied to the [[diagonal maps]] immediately imply that every object has a path space object, and in fact a "very good" one. (See [below](#InModelCategories).)

The very good path space objects in [[locally cartesian closed model categories]] serve as [[categorical semantics]] for the [[identity types]] in [[dependent type theory]] ([[homotopy type theory]]).
\end{remark}

\begin{remark}
In the presence of a (good, very good) [[interval object]]

$$
  \ast \sqcup \ast \to I \to \ast
  \,,
$$

the [[exponential objects]] of the form $X^I$ are (good, very good) path space objects (at least for the evident corresponding definition of "interval object").

In particular, in a [[convenient category of topological spaces]], with $I = [0,1]$ the standard [[closed interval]], the [[compact-open topology|mapping space]] $X^{[0,1]}$ is the standard [[path space]] and is a path object in the general sense of Def. \ref{PathSpaceObject}.
\end{remark}

## Examples

### In model categories
 {#InModelCategories}

If $C$ is a [[model category]] then the factorization axiom ensures that for every object $X \in C$ there is a factorization of the diagonal

$$
  X \stackrel{\simeq}{\to} X^I \stackrel{(d_0, d_1)}{\to} X \times X
$$

with the additional property that $X^I \to X \times X$ is a fibration.

If $X$ itself is fibrant, then the projections $X \times X \to X$ are fibrations and moreover by 2-out-of-3 applied to the diagram

$$
  \array{
    && X^I
    \\
    & {}^{\mathllap{s}}\nearrow && \searrow^{\mathrlap{d_i}}
    \\
    X &&\stackrel{Id}{\to}&& X
  }
$$

are themselves weak equivalences $X^I \stackrel{\simeq}{\to} X$. This is a key property that implies the [[category of fibrant objects|factorization lemma]].

If moreover the [[small object argument]] applies in the model category $C$, then such factorizations, and hence path objects, may be chosen functorially: such that for each morphism $X \to Y$ the factorizations fit into a [[commuting diagram]]

$$
  \array{
     X &\stackrel{\simeq}{\to} &X^I &\to & X \times X
    \\
    \downarrow && \downarrow && \downarrow
    \\
    Y & \stackrel{\simeq}{\to} & Y^I &\to & Y \times Y
  }
$$

### In simplicial model categories

If $C$ is a [[simplicial model category]], then the [[power]]ing over [[sSet]] can be used to explicitly construct functorial path objects for fibrant objects $X$: define $X \to X^I \to X \times X$ to be the [[power|powering]] of $X$ by the morphisms

$$
  \Delta[0] \coprod \Delta[0]
  \stackrel{d_0, d_1}{\hookrightarrow} \Delta[1]
  \stackrel{\simeq}{\to}
  \Delta[0]
$$

in $sSet_{Quillen}$. Notice that the first morphism is a cofibration and the second a weak equivalence in the standard [[model structure on simplicial sets]] and that all objects are cofibrant.

Since by the axioms of an [[enriched model category]] the [[power|powering functor]]

$$
  (-)^{(-)} : sSet^{op} \times C \to C
$$

sends cofibrations and acyclic cofibrations in the first argument to fibrations and acyclic fibrations inif the second argument is fibrant, and since this implies by the [[category of fibrant objects|factorization lemma]] that it then also preserves weak equivalences between cofibrant objects, it follows that $X^{\Delta[1]}$ is indeed a path object with the extra property that also the two morphisms $X^{\Delta[1]} \to X$ are acyclic fibrations.

## Related notions

### Right homotopies

Path objects are used to define a notion of [[right homotopy]] between morphisms in a category. Thus they capture aspects of [[higher category theory]] in a $1$-categorical context.

### Loop space objects

From a path space object may be derived [[loop space object]]s.

## References

The general definition in [[model categories]] is due to:

* {#Quillen67} [[Daniel Quillen]], § I.1, Def. 4, p. 9 (15 of 165) in: _Axiomatic homotopy theory_ in: _[[Homotopical Algebra]]_, Lecture Notes in Mathematics 43, Springer 1967 ([doi:10.1007/BFb0097438](https://doi.org/10.1007/BFb0097438))

The terminology of "good" and "very good" path space objects appears in:

* {#DwyerSpalinski95} [[William Dwyer]], [[Jan Spalinski]], §4.12 in: *[[Homotopy theories and model categories]]* ([[DwyerSpalinski_HomotopyTheories.pdf:file]])

  in: [[Ioan Mackenzie James|I. M. James]], *[[Handbook of Algebraic Topology]]*, North Holland 1995 ([ISBN:9780080532981](https://www.elsevier.com/books/handbook-of-algebraic-topology/james/978-0-444-81779-2), [doi:10.1016/B978-0-444-81779-2.X5000-7](https://doi.org/10.1016/B978-0-444-81779-2.X5000-7))

Lecture notes:

* *[[Introduction to Homotopy Theory]]*, around [this Def.](Introduction+to+Homotopy+Theory#PathAndCylinderObjectsInAModelCategory)

[[!redirects path object]]
[[!redirects path objects]]
[[!redirects path space object]]
[[!redirects path space objects]]

[[!redirects good path space object]]
[[!redirects good path space objects]]

[[!redirects very good path space object]]
[[!redirects very good path space objects]]

[[!redirects stable path object]]
[[!redirects stable path objects]]

[[!redirects cocylinder object]]
[[!redirects cocylinder objects]]

[[!redirects path fibration]]
[[!redirects path fibrations]]


path type

\tableofcontents

## Disambiguation

There are a few different types which can be called *path types* or *path space types* in [[dependent type theory]]:

* The [[identity type]] $x =_A y$ for type $A$ and elements $x:A$ and $y:A$, which is the type of paths between $x:A$ and $y:A$;

* The [[dependent sum type]] $\sum_{x:A} \sum_{y:A} x =_A y$, which is the type of all paths in $A$; reflexivity is given by the [[diagonal function]];

* The [[function type]] $\mathbb{I} \to A$ from the [[interval type]] $\mathbb{I}$ to type $A$; reflexivity is given by the [[constant functions]] in $\mathbb{I} \to A$ - this parallels the topological definition of [[path spaces]] as continuous functions out of the [[unit interval]];

* In [[cubical type theory]], the [[cubical path type]] $\mathrm{path}_A(x, y)$ for type $A$ and elements $x:A$ and $y:A$ - this is notated similarly to the identity type but also behaves as function types from the predefined interval primitive in addition to the behavior of identity types.

The [[categorical semantics]] of path types in any of the above forms is the [[path space object]].

## Related concepts

* [[heterogeneous path type]]

* [[identity type]]

* [[cubical path type]]

* [[loop type]]

category: disambiguation

[[!redirects path type]]
[[!redirects path types]]

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

[[!redirects path space type]]
[[!redirects path space types]]

[[!redirects weak path space type]]
[[!redirects weak path space types]]
[[!redirects strict path space type]]
[[!redirects strict path space types]]


Patricia Johann

* [website](https://cs.appstate.edu/~johannp/)

category: people


Patricia Ritter

* [G+ page](https://plus.google.com/117011416665127979744/posts)

On the equivalence of [[M2-brane 3-algebras]] to [[Lie algebra representations]] over [[metric Lie algebras]]:

* {#MFFMER08} [[Paul de Medeiros]], [[José Figueroa-O'Farrill]], [[Elena Méndez-Escobar]], [[Patricia Ritter]], _On the Lie-algebraic origin of metric 3-algebras_, Commun. Math. Phys. 290:871-902, 2009 ([arXiv:0809.1086](http://arxiv.org/abs/0809.1086))

## related $n$Lab pages

* [[M2-brane 3-algebra]]

* [[Nambu mechanics]]

* [[membrane matrix model]]

category: people


Patrick Blackburn

[[!redirects Blackburn]]
Patrick Blackburn is a Directeur de Recherche (Director of Research) for INRIA at LORIA (Nancy).  He works in ([[modal logic|modal]]) [[formal logic]] and its applications in cognitive science, [[computer science]], and ppphilosophy]].

His website is [here](http://www.loria.fr/~blackbur/).

## Selected writings

On [[modal logic]]:

* {#BlackburnDeRijkeVenema01} [[Patrick Blackburn]], [[Maarten de Rijke]], [[Yde Venema]], *Modal Logic*, Cambridge Tracts in Theoretical Computer Science **53** (2001) [[doi:10.1017/CBO9781107050884](https://doi.org/10.1017/CBO9781107050884)]

* {#BlackburnvanBenthemWolter07} [[Patrick Blackburn]], [[Johan van Benthem]], [[Frank Wolter]] (eds.), *The Handbook of Modal Logic*, Elsevier Amsterdam (2007) [[book webpage](https://cgi.csc.liv.ac.uk/~frank/MLHandbook/), [publisher page](https://www.sciencedirect.com/bookseries/studies-in-logic-and-practical-reasoning/vol/3/suppl/C)]

category: people


Patrick Dehornoy

Patrick Dehornoy (born 1952) is a French mathematician who is widely known for his work on the ordering of the [[braid group]].[^fine]

[^fine]: Dehornoy is sometimes considered to be the first discoverer of such an ordering in the early 90s, but in view of Langmaack's work in the late 60s this needs some qualification.

## Links

* [Wikipedia entry](http://en.wikipedia.org/wiki/Patrick_Dehornoy)

* [homepage](http://www.math.unicaen.fr/~dehornoy)

* [list of publications](http://www.math.unicaen.fr/~dehornoy/papers.html)

## References

* P. Dehornoy, _Braid groups and left distributive operations_ , Transactions AMS **345** no.1 (1994) pp.115–150.

* H. Langmaack, _Verbandstheoretische Einbettung von Klassen unwesentlich verschiedener
Ableitungen in die Zopfgruppe_ , Computing **7** no.3-4 (1971) pp.293-310.

category:people


Patrick du Val

* [Wikipedia entry](https://en.wikipedia.org/wiki/Patrick_du_Val)

## selected writings

on [[ADE singularities]]

* {#duVal1934I} [[Patrick du Val]], _On isolated singularities of surfaces which do not affect the conditions of adjunction. I_, Proceedings of the Cambridge Philosophical Society, 30 (4): 453–459  (1934a) ([doi:10.1017/S030500410001269X](https://doi.org/10.1017/S030500410001269X))

* {#duVal1934II} [[Patrick du Val]], _On isolated singularities of surfaces which do not affect the conditions of adjunction. II_, Proceedings of the Cambridge Philosophical Society, 30 (4): 460–465 (1934) ([doi:10.1017/S0305004100012706](https://doi.org/10.1017/S0305004100012706))

* {#duVal1934III} [[Patrick du Val]], _On isolated singularities of surfaces which do not affect the conditions of adjunction. III_, Proceedings of the Cambridge Philosophical Society, 30 (4): 483–491 (1934) ([doi:10.1017/S030500410001272X](https://doi.org/10.1017/S030500410001272X))

on [[finite subgroups of O(4)]]:

* {#DuVal65} [[Patrick du Val]], _Homographies, Quaternions and Rotations_, Oxford Mathematical Monographs, Clarendon Press (1964)

  also(?): Journal of the London Mathematical Society, Volume s1-40, Issue 1 (1965) ([doi:10.1112/jlms/s1-40.1.569b](https://doi.org/10.1112/jlms/s1-40.1.569b))

## related $n$Lab entries

* [[du Val singularity]]

* [[finite subgroups of O(4)]]

category: people


Patrick Hayden

* [webpage](http://www.cs.mcgill.ca/~patrick/)
* [wikipedia entry](http://en.wikipedia.org/wiki/Patrick_Hayden_%28scientist%29)

## Selected writings

* [[Daniel Harlow]], [[Patrick Hayden]], _Quantum Computation vs. Firewalls_ 2013 ([arXiv:1301.4504](http://arxiv.org/abs/1301.4504))

  (on the [[black hole firewall problem]] -- points out the significance of the so-called "overlap criterion," and suggests a way around the firewall paradox by arguing that the criterion should be relaxed)

## Related $n$Lab entries

* [[black hole firewall problem]]

category: people


Patrick Iglesias-Zemmour

* [personal webpage](http://math.huji.ac.il/~piz/)

* [diffeology.net](http://diffeology.net/)

## Selected writings

On [[thermodynamics]] and [[general relativity]]:

* {#IglesiasSouriau84} [[Patrick Iglesias-Zemmour]], [[Jean-Marie Souriau]] _Heat, cold and Geometry_, in: M. Cahen et al (eds.) Differential geometry and mathematical physics, 37-68, D. Reidel 1983 ([web](http://www.jmsouriau.com/Heat_Cold_And_Geometry_1983.htm), [pdf](http://www.jmsouriau.com/Publications/JMSouriau-PIglesias-HeatColdAndGeometry1983.pdf), [doi:978-94-009-7022-9_5](https://doi.org/10.1007/978-94-009-7022-9_5))

* [[Patrick Iglesias-Zemmour]], _Essai de «thermodynamique rationnelle» des milieux continus_, Annales de l'I.H.P. Physique théorique, Volume 34 (1981) no. 1,  p. 1-24 ([numdam:AIHPA_1981__34_1_1_0](http://www.numdam.org/item/AIHPA_1981__34_1_1_0))

On [[diffeological spaces]]:

* {#IglesiasZemmour85} [[Patrick Iglesias-Zemmour]], _Fibrations difféologiques et Homotopie_,  Dissertation (1985) ([web](http://math.huji.ac.il/~piz/Site/The%20Articles/D9DD15EE-6993-4CA3-8B9B-4FC1DEF4A418.html), [pdf](http://math.huji.ac.il/~piz/documents/TheseEtatPI.pdf), [[IglesiasZemmourFibrationsDiffeologiques1985.pdf:file]])

* {#PIZ} [[Patrick Iglesias-Zemmour]], _Diffeology_, Mathematical Surveys and Monographs, AMS (2013) ([web](http://math.huji.ac.il/~piz/Site/The%20Book.html), [publisher](http://www.ams.org/bookstore-getitem/item=SURV-185)).
Reprint (revised version) by Beijing WPC (2022), World Map Technology, ([pdf] (http://www.icloud.com/iclouddrive/08bIgfV5NT8F8oBo-Ipqoz1kg#Diffeology), [publisher] (https://mp.weixin.qq.com/s?__biz=MjM5NzE3Nzc4MQ==&mid=2664511235&idx=2&sn=a2b55b7fc0e88a3ae4142a8f53077055)).

On the [[de Rham theorem]] over [[diffeological spaces]]:

* [[Patrick Iglesias-Zemmour]], *Une cohomologie de Čech pour les espaces differentiables et sa relation a la cohomologie de De Rham* (1988) [[pdf](https://math.huji.ac.il/~piz/documents/UCDCPLEDESRALCDDR.pdf), [[IglesiasZemmour-CechDeRham.pdf:file]]]

* [[Patrick Iglesias-Zemmour]], *Čech–De Rham bicomplex in diffeology*, Israel Journal of Mathematics (2023) 1–38 [[doi:10.1007/s11856-023-2486-8](https://doi.org/10.1007/s11856-023-2486-8)]

* [[Emilio Minichiello]], *The Diffeological Čech-de Rham Obstruction* [[arXiv:2401.09400](https://arxiv.org/abs/2401.09400)]

On [[orbifolds]] regarded as geometric local quotient [[diffeological spaces]]:

* {#IKZ10} [[Patrick Iglesias-Zemmour]], [[Yael Karshon]], Moshe Zadka, _Orbifolds as diffeologies_, Transactions of the American Mathematical Society 362 (2010), 2811-2831 ([arXiv:math/0501093](https://arxiv.org/abs/math/0501093))

On the [[moment map]] in [[diffeology]]:

* {#IZ10} [[Patrick Iglesias-Zemmour]], *The moment map in diffeology*, Memoirs of the American Mathematical Society (2010) $[$[web](http://math.huji.ac.il/~piz/Site/The%20Articles/084469EE-D890-4CAA-8A13-1C655A76CF63.html), [ISBN: 978-1-4704-0586-1](https://bookstore.ams.org/memo-207-972)$]$

On [[manifolds with boundaries and corners]] as [[diffeological spaces]], and on [[differential forms]] on these:

* {#GurerIZ19} [[Serap Gürer]],  [[Patrick Iglesias-Zemmour]], _Differential forms on manifolds with boundary and corners_, Indagationes Mathematicae, Volume 30, Issue 5, September 2019, Pages 920-929 ([doi:10.1016/j.indag.2019.07.004](https://doi.org/10.1016/j.indag.2019.07.004))

On [[orbifolds]] as [[stratified space|stratified]] [[diffeological spaces]]:

* [[Serap Gürer]], [[Patrick Iglesias-Zemmour]], *Orbifolds as stratified diffeologies*, Differential Geometry and its Applications **86** (2023) 101969 [[doi:10.1016/j.difgeo.2022.101969](https://doi.org/10.1016/j.difgeo.2022.101969)]

On [[Noncommutative Geometry]] and [[diffeology]]:

* [[Patrick Iglesias-Zemmour]], Jean-Pierre Laffineur, *Noncommutative geometry and diffeology: The case of orbifolds*, Journal of Noncommutative Geometry, volume 12, issue 4 (2018). [doi 10.4171/jncg/319] (https://doi.org/10.4171/jncg/319).

* [[Patrick Iglesias-Zemmour]], Elisa Prato, *Quasifolds, diffeology and noncommutative geometry*, Journal of Noncommutative Geometry, volume 15, issue 2 (2021). [doi 10.4171/jncg/419] (https://doi.org/10.4171/jncg/419).

## Related entries

* [[diffeological space]]

* [[moment map]]

* [[manifolds with boundaries and corners]]

* [[orbifold]]

* [[Noncommutative Geometry]]

[[!redirects Patrick Iglesias-Zemmour]]
[[!redirects piz]]

category: people


Patrick Koppenburg

* [webpage](https://www.nikhef.nl/~pkoppenb/)

## Selected writings

Early proposal to investigate [[flavour anomalies]] in [[B-meson]] decays, specifically at the [[LHCb]]-experiment:

* {#Koppenburg07} [[Patrick Koppenburg]], _Selection of $B_u \to \ell \ell K$ at LHCb and Sensitivity to $R_K$_, 2007 ([cern:1027442](cds.cern.ch/record/1027442), [pdf](cds.cern.ch/record/1027442/files/lhcb-2007-034.pdf))

List of [[hadrons]] dicovered at the [[LHC experiment]]:

* {#Koppenburg21} [[Patrick Koppenburg]] for the [[LHCb collaboration]]: *List of hadrons observed at the LHC* (2021) [[pdf](https://cds.cern.ch/record/2749030/files/FIGURE-2021-001.pdf), [webpage](https://www.nikhef.nl/~pkoppenb/particles.html), [cds:2749030](https://cds.cern.ch/record/2749030)]

On [[B meson]]-[[decays]]:

* [[Patrick Koppenburg]], [[Zdenek Dolezal]], [[Maria Smizanska]], _Rare decays of b hadrons_, Scholarpedia, 11(6):32643 ([doi:10.4249/scholarpedia.32643](http://dx.doi.org/10.4249/scholarpedia.32643))

* {#Koppenburg21} [[Patrick Koppenburg]], *Penguin B decays*, 2021 ([cds:2767155](https://cds.cern.ch/record/2767155), [pdf](https://cds.cern.ch/record/2767155/files/Koppenburg-20210519-Usach.pdf))

* {#Koppenburg20} [[Patrick Koppenburg]], _Beauty 2019 -- Conference summary_ ([arXiv:2001.11796](https://arxiv.org/abs/2001.11796))

Review of the [[flavour anomalies]]:

* {#GuadagnoliKoppenburg22} Diego Guadagnoli, [[Patrick Koppenburg]], *Lepton-flavor violation and lepton-flavor-universality violation in b and c decays* [[arXiv:2207.01851](https://arxiv.org/abs/2207.01851)]

Updated plots:

* {#KoppenburgAnomalies} [[Patrick Koppenburg]], *Flavour Anomalies* [[anomalies.htlm](https://www.nikhef.nl/~pkoppenb/anomalies.html)]

## Related $n$Lab entries

* [[LHCb]]

* [[flavour anomaly]]

category: people


Patrick Massot

* [website](https://www.imo.universite-paris-saclay.fr/~patrick.massot/en/)

## Selected writings

* [[Kevin Buzzard]], [[Johan Commelin]], [[Patrick Massot]], *Formalising perfectoid spaces*, CPP 2020: Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, Pages 299-312 ([doi:10.1145/3372885.3373830](https://doi.org/10.1145/3372885.3373830), [arXiv:1910.12320](https://arxiv.org/abs/1910.12320))

category:people


Patrick Meessen

* [personal page](http://www.unioviedo.es/hepth/people/Patrick/index.html)

* [GoogleScholar page](https://scholar.google.com/citations?user=MkD-idd6KjUC&hl=en)

## Selected writings

On the [[Wess-Zumino-Witten model]]:

* [[Patrick Meessen]], *Strings Moving on Group Manifolds: The WZW Model* [[pdf](http://www.unioviedo.es/hepth/people/Patrick/fysica/zooi/WZW_ClassMunoz.pdf), [[Meessen-WZWModel.pdf:file]]]

On [[D8-branes]] as [[black branes]] in [[massive type IIA string theory]]:

* {#JanssenMeessenOrtin99} [[Bert Janssen]], [[Patrick Meessen]], [[Tomas Ortin]], _The D8-brane tied up: String and brane solutions in massive type IIA supergravity_, Phys. Lett. B453 (1999) 229-236 ([spire:494174](http://inspirehep.net/record/494174), doi:10.1016/S0370-2693(99)00315-9)

category: people


Patrick Morandi

* [webpage](http://sierra.nmsu.edu/morandi/)

## Selected writings

On [[error correcting codes]] and [[algebraic curves]]:

* [[Patrick Morandi]], _Error Correcting Codes and Algebraic Curves_ , lecture notes New Mexico State University 2001. ([pdf](http://www.math.nmsu.edu/~pmorandi/math601f01/LectureNotes.pdf))

On [[wallpaper groups]]:

* [[Patrick Morandi]], _The Classification of Wallpaper Patterns: From Group Cohomology to Escherís Tessellations_ ([pdf](http://sierra.nmsu.edu/morandi/notes/Wallpaper.pdf))

category: people

[[!redirects Patrick J. Morandi]]