pseudo-torsor

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#### Representation theory
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#### Bundles
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## Idea

In the definition of [[torsors]] and [[principal bundles]] one deals with [[group action]] [[action objects|objects]] $G \times P \overset{\rho}{\longrightarrow} P$ (generally [[over category|over]] some base object $X$) whose [[shear map]] is an [[isomorphism]]:

\[
  \label{ShearMapForBundles}
  \array{
    G \times P
    &
    \underoverset
      {\simeq}
      { (\rho, pr_2) }
      {\longrightarrow}
    &
    P \times_X P
    \\
    (g,p) &\mapsto& \big( \rho(g)(p), p \big)
    \,.
  }
\]

Now *if* $P$ is [[inhabited set|inhabited]] ([[fiber]]-wise over $X$), this implies a ([[fiber]]-wise) [[free action|free]] and [[transitive action|transitive]] (hence [[regular action|regular]]) action, which is typically what is understood to characterize [[torsors]]/[[principal bundles]].

However, the condition alone that the [[shear map]] (eq:ShearMapForBundles) be an isomorphism makes sense (and is then automatically satisfied) also for locally [[empty set|empty]] $P$, meaning for [[fibers]] of $P$ being [[strict initial objects]] ([[internalization|internal]] to the ambient [[category]]). If that case is meant to be included, one speaks, following [[Alexander Grothendieck|Grothendieck]], alternatively of:

1. a *formally principal* action ([Grothendieck 60, p. 312 (15 of 30)](#Grothendieck60))

1. a *pseudo-torsor* ([Grothendieck 67, EGA IV.4, 16.5.15](#EGAIV4))

1. a *formally principal homogeneous* action (ibid. & [Grothendieck 71, p. 9 (293)](#Grothendieck71)).

Examples of references using this terminology: [StacksProject](#StacksProject), [Moret-Bailly 13, slide 5](#MoretBailly13), [BGA 13, p. 73-74](#BGA13).

## Examples

* *[[equivariant principal bundles]]*

  Even if a $G$-[[equivariant principal bundle]] is an actual [[torsor]] ([[internalization|in]] [[topological G-spaces]] [[slice category|sliced]] over a given base, [SS 2021, Def. 2.1.3](equivariant+bundle#SatiSchreiber21)) its [[fixed loci]] will generally only be (similarly internal) pseudo-torsors. (For more see [SS 2021, Rem. 2.18](equivariant+bundle#SatiSchreiber21)).

## Properties

\begin{proposition}\label{PseudoPrincipalProjectionIsEffectiveIfItIsQuotient}
  [[internalization|Internal to]] some
  ambient [[category]] $\mathcal{C}$
  with [[finite limits]], let

  * $G \,\in\, Grp(\mathcal{C})$ be a [[group object]],
  * $P \,\in\, G Act(\mathcal{C})$ an [[action object]],
  * $(P \to X) \,\in\, G PsTor(\mathcal{C}_{/X})$ a [[formally principal bundle]].

Then the following are equivalent:

1. $P \to X$ is the $G$-[[quotient]] [[coprojection]];

1. $P \to X$ is an [[effective epimorphism]].

\end{proposition}
\begin{proof}
  The first condition is equivalent to
  $$
    P \times_X P \rightrightarrows P \to X
  $$
  being a [[coequalizer]], the second to
  $$
    P \times G \rightrightarrows P \to X
  $$
  being a coequalizer. But the pseudo-principality condition
  says that we have an [[isomorphism]] (the [[shear map]])
  $$
    P \times_X P \simeq P \times G
  $$
  which identifies these two [[diagrams]].
\end{proof}

## Related concepts

* [empty heap](heap#empty)

* [[invertible semigroup]]

* [[associative quasigroup]]

## References

The term seems to be due to:

* {#Grothendieck60} [[Alexander Grothendieck]], p. 312 (15 of 30) in: *Technique de descente et théorèmes d’existence en géométrie algébrique. I. Généralités. Descente par morphismes fidèlement plats*, Séminaire N. Bourbaki exp. no190 (1960) 299-327 [[numdam:SB_1958-1960__5__299_0](http://www.numdam.org/item/?id=SB_1958-1960__5__299_0)]

* {#Grothendieck67} [[Alexander Grothendieck]], 16.5.15 in: *[[Éléments de géométrie algébrique]]* : IV. *Étude locale des schémas et des morphismes de schémas (Quatrième partie)*, Publications mathématiques de l’I.H.É.S., tome  32 (1967), p. 5-361 ([numdam:PMIHES_1967__32__5_0](http://www.numdam.org/item/PMIHES_1967__32__5_0))

* {#Grothendieck71} [[Alexander Grothendieck]], p. 9 of _Exemples et Complements_ ([doi:10.1007/BFb0058666](https://link.springer.com/chapter/10.1007/BFb0058666), [pdf](https://link.springer.com/content/pdf/10.1007%2FBFb0058666.pdf)), which is p. 293 in:

  [[Alexander Grothendieck]], [[Michèle Raynaud]], *Revêtements étales et groupe fondamental* ([[SGA 1]]),  Lecture Notes in Mathematics **224**, Springer 1971 ([arXiv:math/0206203](https://arxiv.org/abs/math/0206203), [doi:10.1007/BFb0058656](https://link.springer.com/book/10.1007/BFb0058656))

See also:

* {#StacksProject} [[The Stacks Project]], *Pseudo G-Torsor* ([Definition 03AH](https://stacks.math.columbia.edu/tag/03AH), [Definition 0498](https://stacks.math.columbia.edu/tag/0498))

* {#MoretBailly13} [[Laurent Moret-Bailly]], slide 5 of: *Principal Bundles over Valued Fields*, Oberwolfach 2013 ([pdf](https://perso.univ-rennes1.fr/laurent.moret-bailly/exposes/GMBMFOWeb.pdf), [[MoretBaillyPrincipalBundlesValuedFields.pdf:file]])

* {#BGA13} Alessandra Bertapelle, Cristian D. Gonzalez-Aviles, *The Greenberg functor revisited*, European Journal of Mathematics volume 4, pages 1340–1389 (2018) ([arXiv:1311.0051](https://arxiv.org/abs/1311.0051), [doi:10.1007/s40879-017-0210-0](https://doi.org/10.1007/s40879-017-0210-0))

[[!redirects pseudo-torsors]]

[[!redirects pseudo torsor]]
[[!redirects pseudo torsors]]

[[!redirects formally principal homogeneous space]]
[[!redirects formally principal homogeneous spaces]]
[[!redirects formally-principal homogeneous space]]
[[!redirects formally-principal homogeneous spaces]]

[[!redirects formally principal bundle]]
[[!redirects formally principal bundles]]
[[!redirects formally-principal bundle]]
[[!redirects formally-principal bundles]]

[[!redirects formally principal action]]
[[!redirects formally principal actions]]
[[!redirects formally-principal action]]
[[!redirects formally-principal actions]]


pseudoaction

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## Idea
**Pseudoactions** are [[categorification|categorified]] [[actions]]. In other words, they are coherent actions of [[pseudomonoids]].

## Examples
1. A pseudomonoid in $(\mathbf{Cat}, 1, \times)$ is exactly a [[monoidal category]]. The pseudoaction of a monoidal category on a category is an [[actegory]].
1. A pseudomonoid in $(\mathbf{Fib}, 1, \times)$ is a [[monoidal fibration]]. Pseudoactions of a monoidal fibration on a fibration are considered in ([Vasilakopoulou '18](#Vas18)).
1. A pseudomonad is a pseudomonoid in the Gray monoid of endomorphisms of an object in a [[Gray-category]] (see ([Marmolejo '99](#MarmolejoDL), Definition 3.1)). A [[pseudoalgebra]] of a [[pseudomonad]] is a pseudoaction of the pseudomonad on the functor picking out the carrier of the pseudoalgebra.

## See also
* [[categorification]]
* [[pseudomonoid]]
* [[module]]

## References
* {#Vas18} [[Christina Vasilakopoulou]], _On enriched fibrations_, Cahiers de Geometrie et Topologie Differentielle, vol. LIX-4, 2018, ([cahiers](http://cahierstgdc.com/wp-content/uploads/2018/10/Vasilakopoulou-LIX-4.pdf)), ([arXiv:1801.01386](https://arxiv.org/abs/1801.01386))

* {#MarmolejoDL} [[Francisco Marmolejo]], *Distributive laws for pseudomonads*, Theory and Applications of Categories, **5** 5 (1999) 81-147 [[tac](http://tac.mta.ca/tac/volumes/1999/n5/5-05abs.html)]


pseudoadjunction

A **pseudoadjunction** is a kind of weak [[2-adjunction]].  Sometimes the word is used interchangeably with [[biadjunction]] to mean a "fully weak" notion of 2-dimensional adjunction.  Other times, a "pseudoadjunction" refers to the weakest kind of 2-adjunction definable in a [[Gray-category]], which when specialized to the prototypical Gray-category $Gray$ consists of strict 2-categories and strict 2-functors, but pseudonatural transformations.

## Related concepts

* [[adjoint functor]]
* [[2-adjunction]]
* [[biadjunction]]

## References

* [[Jacques Penon]], *Sous-catégorie classifiée d’une 2-catégorie*, 1974.

* [[Steve Lack]], *A coherent approach to pseudomonads*, 2000.

* I. J. Le Creurer, F. Marmolejo, E. M. Vitale, _Beck's theorem for pseudo-monads_, J. Pure Appl. Algebra __173__ (2002), no. 3, 293--313, [MR2003f:18004](http://www.ams.org/mathscinet-getitem?mr=2003f:18004), doi

[[!redirects pseudoadjunctions]]
[[!redirects pseudo-adjunction]]
[[!redirects pseudo-adjunctions]]
[[!redirects pseudoadjoint]]
[[!redirects pseudoadjoints]]
[[!redirects pseudoadjoint functor]]
[[!redirects pseudoadjoints]]
[[!redirects pseudoadjoint functors]]
[[!redirects pseudo-adjoint]]
[[!redirects pseudo-adjoints]]
[[!redirects pseudo-adjoint functor]]
[[!redirects pseudo-adjoints]]
[[!redirects pseudo-adjoint functors]]


pseudoalgebra for a 2-monad

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# Contents
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## Idea

If $T$ is a [[2-monad]] on a [[2-category]] $K$, then a **pseudoalgebra** for $T$ is a 2-dimensional version of an [[algebra over a monad]] which satisfies the laws only up to coherent isomorphism.

## Definition

A pseudo $T$-algebra is the same as a [[lax algebra for a 2-monad|lax algebra]] whose constraint 2-cells are invertible.

## Examples

...

## Normalization

A pseudoalgebra is said to be **normal** or **normalized** if its unit constraint isomorphism is an identity.

While making a pseudoalgebra strict is quite difficult, usually making it normal is quite easy, and many pseudoalgebras arising naturally are normal.  For instance, for the strict 2-monad $T$ whose strict algebras are strict monoidal categories and whose pseudoalgebras are unbiased non-strict monoidal categories, the unit constraint says that the "1-ary tensor product" $\otimes(x)$ is isomorphic to $x$ itself.  Clearly in most cases it is most sensible to *define* the 1-ary tensor product to *be* $x$, so that the pseudoalgebra is normal.

This situation is fairly general: if $T$ is a strict 2-monad for which the components of the unit $\eta_X \colon X\to T X$ are [[isocofibrations]], then any pseudoalgebra structure can be modified to a normalized one on the same underlying object.

## Coherence theorems

One way to state a coherence theorem is to say that every pseudoalgebra for a given 2-monad is equivalent to a strict one, perhaps in a structured way.  See [coherence theorems](/nlab/show/coherence+theorem#2monads).

[[!redirects pseudoalgebra for a 2-monad]]
[[!redirects pseudoalgebras for a 2-monad]]
[[!redirects pseudoalgebras for 2-monads]]
[[!redirects algebra for a 2-monad]]
[[!redirects algebras for a 2-monad]]
[[!redirects algebras for 2-monads]]
[[!redirects pseudoalgebra]]
[[!redirects pseudoalgebras]]


pseudocircle

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## Idea

The _pseudocircle_ is a [[finite topological space]] which is [[weak homotopy equivalence|weakly homotopy equivalent]] to the standard [[circle]].

In other word, for the purposes of [[homotopy theory]] (up to weak homotopy equivalence), it is equivalent to the circle, yet it is a purely finite combinatorial creature.

## Definition

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###### Definition

The **pseudocircle** $\mathbb{S}$ is the [[topological space]]

* whose [[underlying set]] is a $4$-set, say $\{l,r,t,b\}$ (for the 'left side', 'right side', 'top point', and 'bottom point' of the circle)

* and whose [[topological structure]], given as the collection of [[open subsets]], is
  $$
    \{\{l,r,t,b\}, \{l,r,t\}, \{l,r,b\}, \{l,r\}, \{l\}, \{r\}, \empty\}
    \,.
  $$

=--

That is the topology generated by the [[topological base|base]] $\{\{l,r,t\}, \{l,r,b\}, \{l\}, \{r\}\}$ (which is in fact the unique minimal base and furthermore the unique minimal subbase).

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###### Remark

As a [[frame]], this topology is a [[subframe]] of the [[frame of opens]] of the usual circle, where the names of $l$, $r$, $t$, and $b$ are taken literally.

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These also name a [[partition]] of the standard [[circle]], and this gives a [[quotient map]] from the circle to the pseudocircle; this map is the promised weak homotopy equivalence, see [below](#RelationToCircle).

## Properties

### Relation to the standard circle
 {#RelationToCircle}

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###### Proposition

The function of sets

$$
  f : S^1 \to \mathbb{S}
$$

from the standard [[circle]] to the pseudocircle, which sends

* a point in the open left half of $S^1$ (thought of under the standard [[embedding]] into $\mathbb{R}^2$) to the point $l \in \mathbb{S}$;

* a point in the open right half of $S^1$ to $r \in \mathbb{S}$;

* the top point of $S^1$ to $t \in \mathbb{S}$;

* the bottom point of $S^1$ to $b \in \mathbb{S}$

is a [[continuous function]]. Moreover, it is a [[weak homotopy equivalence]].

=--

See the proof of the general statement at _[finite topological space - properties](finite+topological+space#Properties)_.

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

The only continuous functions in the other direction, $\mathbb{S} \to S^1$ are the constant maps.

=--

## References

* Michael McCord, _Singular homology groups and homotopy groups of finite topological spaces_, Duke Math. J. Volume 33, Number 3 (1966), 465-474. ([EUCLID](http://projecteuclid.org/euclid.dmj/1077376525))
 {#McCord}

* Wikipedia, _[Pseudocircle](http://en.wikipedia.org/wiki/Pseudocircle)_


pseudocode

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## Idea

In [[computer science]], by *pseudo-code* one means semi-formal writing which is sufficiently close to code in an actual [[programming language]] to be readily turned into such if desired, but which may be more pleasant or more convenient to read and write for human communication.

## References

See also:

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

[[!redirects pseudo-code]]
[[!redirects pseudo code]]


pseudocompact module

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## Definition

A [[topological module]] over a [[topological ring]]
is __pseudocompact__ if it is [[isomorphism|isomorphic]]
to the [[limit]] in the [[category]] of [[topological modules]]
of [[discrete topological space|discrete]] [[topological modules]] of finite length
(meaning there is an upper bound on the length of
increasing chains of submodules).

Equivalently, a pseudocompact module $M$
is a complete Hausdorff [[topological module]] that
has a [[basis of neighborhoods]] of 0
consisting of open submodules $P$ such that
$M/P$ has finite length.

## Properties

Any pseudocompact module is a [[linearly compact module]].

## Related concepts

* [[linearly compact module]]

* [[pseudocompact ring]]

## References

Pseudocompact modules were introduced in

* [[Pierre Gabriel]], _Des catégories abéliennes_,
Bulletin de la S. M. F., tome 90 (1962), 323-448,
[http://www.numdam.org/item?id=BSMF_1962__90__323_0](NUMDAM)

[[!redirects pseudocompact modules]]
[[!redirects pseudo-compact module]]
[[!redirects pseudo-compact modules]]


pseudocompact ring

A __pseudocompact ring__ is a complete Hausdorff
[[topological ring]], $R$, which admits a base at $0$ of two-sided open ideals $I$ for each of which $R/I$ is an Artinian ring. Equivalently, it is a regular [[pro-object]] in Artinian rings.

More generally let $R$ be a commutative pseudocompact ring. A complete Hausdorff topological ring $\Lambda$ will be called a __pseudocompact algebra__ over $R$ if

(i)  $\Lambda$ is an $R$ algebra in the usual sense, and

(ii) $\Lambda$ admits a system of open neighbourhoods of 0 consisting of two-sided ideals $I$ such that $\Lambda/I$ has finite length as an $R$ module.

##Motivations and applications

1. Grothendieck developed the theory of [[formal group]]s over pseudocompact rings.

1. The completed [[group algebra]]s of [[profinite group]]s are naturally pseudocompact algebras, usually considered over finite fields or the [[profinite completion]] of the ring of integers.

##Related ideas

* [[linear topological ring]]

##References

The homological algebra of such rings and the corresponding modules are discussed in

* A. Brumer, _Pseudocompact algebras, profinite groups and class formations_,  J. Algebra __4__ (1966) 442-470, [MR202790](http://www.ams.org/mathscinet-getitem?mr=202790), doi [pdf](http://deepblue.lib.umich.edu/bitstream/handle/2027.42/33410/0000811.pdf?sequence=1)

category: algebra
[[!redirects pseudocompact rings]]
[[!redirects pseudocompact algebra]]
[[!redirects pseudocompact algebras]]


pseudodifferential operator

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## Idea

Pseudodifferential operators generalize [[differential operators]] and are of similar importance to the theory of [[partial differential equations]] as [[Schwartz distributions]], see also [[microlocal analysis]].

## Definition

Let $X \subset \mathbb{R}^n$ be open.
A pseudodifferential operator is a [[Fourier integral operator]] of the form

$$
A: C^{\infty}_0 (X) \to \mathcal{D}'(X)
$$

$$
 A u(x) = \frac{1}{(2 \pi)^n} \int \int e^{i (x-y) \theta} a(x, y, \theta) u(y) dy d\theta
$$

where the function $a$, called the **symbol** of the pseudodifferential operator $A$, belongs to the space $S^m_{\rho, \delta}(X \times X \times \mathbb{R}^n)$ defined below.

If $\mathcal{F}$ denotes the [[Fourier transform]] a short hand notation for this definition is $A u = \mathcal{F}^{-1} (a \, \mathcal{F}u)$, put in words: Fourier transform $u$, multiply with $a$ and transform back.

## Definition of symbols

If one allows arbitrary functions as symbols there will be no control of the behaviour of the associated pseudodifferential operators, of course. In order to get a theory where these operators are, for example, continuous with respect to the standard topologies on the [[topological vector space]]s that they are defined on, some assumptions have to be made. Different levels of generality of the theory correspond to different assumptions about the symbols. One standard symbol space is defined as follows:

Let $X \subset \mathbb{R}^n$ be open, $0 \le \rho \le 1, 0 \le \delta \le 1, m \in \mathbb{R}, n \in \mathbb{N}, n \neq 0$.

* definition: $S^m_{\rho, \delta}$ is the space of all $a \in C^{\infty}(X \times \mathbb{R}^n)$ such that for all compact $K \Subset X$ and all $\alpha, \beta \in \mathbb{N}^n$ there is a constant $C_{K, \alpha, \beta}$ such that
$$
        | \partial^{\alpha}_x \, \partial^{\beta}_{\theta} \, a(x, \theta)| \leq C_{K, \alpha, \beta} (1 + | \theta |)^{m - \rho | \beta | + \delta | \alpha |}
$$
The space $S^m_{\rho, \delta}$ is called the space of **symbols of order $m$ and of type $(\rho, \delta)$**.

It is easy to see that every space $S^m_{\rho, \delta}$ is a Fréchet space: every $X \subset \mathbb{R}^n$ open has a compact exhaustion, that is an increasing sequence $(K_i)$ with each $K_i \Subset X$ compact such that $\bigcup_{i = 1}^{\infty} K_i = X$, and one can define a countable family of seminorms via

$$
P_{K_i, \alpha, \beta} = \operatorname{sup}_{(x, \theta) \in K_i \times \mathbb{R}^n} \frac{ | \partial^{\alpha}_x \, \partial^{\beta}_{\theta} \, a(x, \theta)|}{(1 + | \theta |)^{m - \rho | \beta | + \delta | \alpha |}}
$$

The space of symbols of order $- \infty$ is defined to be
$$
    S^{- \infty} = \bigcap_{m \in \mathbb{R}} S^m_{\rho, \delta}
$$

Conversly the symbols of order $\infty$ are defined by

$$
    S^{\infty} = \bigcup_{m \in \mathbb{R}} S^m_{\rho, \delta}
$$

Symbols of order $- \infty$ are often called **smoothing** and their operators **smoothing operators**. The reason for this is that their pseudodifferential operators map distribution spaces into spaces of smooth functions, for example:

## Examples

### Every differential operator is a pseudodifferential operator

We restrict ourselfes to one dimension for simplicity, let $D := -i \frac{d}{dx}$ and write a differential operator $P$ as
$$
  P(x, D) := \sum_{k = 0}^n f_n(x) D^k
$$
with given functions $f_n$. Then $P$ is a pseudodifferential operator with symbol $a(x, y, \theta) = P(x, \theta)$. The symbol of a differential operator therefore is a polynom in $\theta$, which motivates a part of the definition of symbol classes below: We expect that the growth of the symbol in $\theta$ is polynomial at most, and the degree of the bounding polynomial decreases by $1$ if we apply differentiation in $\theta$ to the symbol.

## Properties

### Smoothing theorem

The pseudodifferential operator of a smoothing symbol maps $\mathcal{E}'$, the [[dual vector space|dual space]] of $C^{\infty}$, into $\mathcal{S}$, the Schwartz space of rapidly decreasing smooth functions.

### Propagation of singularities theorem

* [[properly supported pseudo-differential operator]]

* [[propagation of singularities theorem]]

## References

* Wikipedia on [pseudodifferential operator] (http://en.wikipedia.org/wiki/Pseudo-differential_operator)

An elementary and short introduction can be found here:

* Xavier Saint Raymond: _Elementary Introduction to the Theory of Pseudodifferential Operators_ ([ZMATH entry] (http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:0847.47035&format=complete))
* Sylvain Carpentier, Alberto De Sole, [[Victor G. Kac]], _Some algebraic properties of differential operators_, [arxiv/1201.1992](http://arxiv.org/abs/1201.1992)
* [[Lars Hörmander]], _The analysis of linear partial differential operators_, in 4 vols.: I. Distribution theory and Fourier analysis, II. Differential operators with constant coefficients, III. Pseudo-differential operators, IV. Fourier integral operators.

[[!redirects pseudodifferential operators]]

[[!redirects pseudo-differential operator]]
[[!redirects pseudo-differential operators]]


pseudoform

# Contents
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## Idea

A _pseudoform_ is a [[differential form]] that has been [[differential form|twisted]] by the [[pseudoscalar]] bundle.

Concretely, this means that when written in two (local) systems of
coordinates, the form transforms by an additional factor of $-1$
when the coordinate systems have opposite [[orientations]].
For a top-dimension pseudoform, this has the effect that at any given point,
the form has a consistent sign regardless of the [[orientation]] of the coordinates.

An $n$-pseudoform is the fundamental object of
[[integration of differential forms|integration on an $n$-manifold]].
In particular the [[volume]] (or area, or length) is a pseudoform.

## Definition

\begin{definition}
A __pseudoform__ on a [[manifold]] $X$ is a [[section]] of the
[[tensor product]] $\Lambda \otimes \Psi$, where

* $\Lambda$ is the [[exterior algebra]] of the [[cotangent bundle]] on $X$, and

* $\Psi$ is the [[pseudoscalar]] bundle on $X$.

\end{definition}

By comparison, an untwisted [[differential form]] is a [[section]] of
the [[exterior algebra]] $\Lambda$ itself.

## Properties

See also discussion under ["Twisted and vector-valued forms"](https://ncatlab.org/nlab/show/differential+form#twisted) at
[[differential form]].

### Correspondence to untwisted forms given orientation

For a pseudoform $\omega$ on an [[oriented manifold]] with orientation $o$, multiplying $\omega$ by the [[pseudoscalar field]] $1_o$ whose value at orientation $o$ is everywhere $1$ gives an untwisted form $\omega_o = 1_o \wedge \omega$. Similarly an untwisted form $\alpha$ corresponds to a pseudoform $1_o \wedge \alpha$.

As a result, on an [[orientable]] manifold, any pseudoform $\omega$ corresponds to two untwisted forms, one for each orientation, which differ only in sign. Conversely, on an orientable manifold any untwisted form corresponds to two pseudoforms differing only in sign.
Even when the manifold is [[nonorientable]], this correspondence holds locally because each point has an orientable neighborhood.

When working on a manifold with a chosen orientation, many authors conflate a pseudoform $\omega$ with the corresponding untwisted form $\omega_o$ and vice versa.

### Integration

When [[integration of differential forms|integrating differential forms]]
on an $n$-manifold, the most fundamental is the integration of $n$-pseudoforms.

This corresponds to how in multivariable calculus, the formula for
change of variables has a factor of the [[absolute value]] of the
[[determinant]] (just like for an $n$-pseudoform), and not of the
determinant itself (as one would have for an untwisted $n$-form).

With an orientation, one can of course also integrate (untwisted) $n$-forms;
this can be seen as using the orientation to convert the
$n$-form to an $n$-pseudoform and integrating that.

More generally one can integrate a $p$-pseudoform on an $n$-manifold $X$
if one has a $p$-[[submanifold]] $U$ to integrate over
and a [[pseudoorientation]] of $U$ in $X$.  See [[integration of
differential forms]].

## Examples

### Top-dimensional examples

The [[volume form|volume]] (or area, or length) on an $n$-manifold is an $n$-pseudoform.  On an [[oriented manifold]] one often speaks of a [[volume form]] as an untwisted $n$-form.

The [[absolute value of a form|absolute value]] of an untwisted $n$-form (on an $n$-manifold) is an $n$-pseudoform,
as is the [[absolute value of a pseudoform|absolute value]] of an $n$-pseudoform.

Alternatively, both of these examples can be thought of as [[absolute differential forms]]. On an $n$-manifold, an absolute $n$-form is equivalent to an $n$-pseudoform.

### Electromagnetism

In [[electromagnetism]], the current form $j$, measuring flux of [[electric charge|charge]],
is a 2-pseudoform.  This can be seen by considering how one integrates
it: you need a 2-surface together with a notion of which way _through_
the surface is forward and which is backward, or in other words
a [[pseudooriented]] surface.

By contrast the magnetic field strength $B$ is an untwisted 2-form.
To integrate $B$ over a surface $U$, one needs not a [[pseudoorientation]]
but an [[orientation]] of $U$ itself, identifying a direction
of circulation within the surface rather than a transverse direction
through the surface.

Traditionally in electromagnetism these look very similar because one
freely converts an orientation to a pseudoorientation and vice versa.
The tell when one does so is that it relies on an arbitrary choice of
orientation of the ambient 3-dimensional space, namely the [[right-hand rule]].
Comparing the right-hand rule to a left-hand rule, one gets the same physical meaning by integrating $j$ over the same surface with the same [[pseudoorientation]] (hence opposite [[orientations]]), confirming that the pseudoorientation is primary and $j$ is properly a pseudoform.
With $B$, one needs the same orientation on the surface (hence opposite pseudoorientations), confirming that the orientation is primary and $B$ is an untwisted form.

## References

Many useful explanations by [[Toby Bartels]] and [[John Baez]] in
[this long Usenet
thread](https://groups.google.com/g/sci.physics.research/c/aiMUJrOjE8A/m/0361hct91VsJ). In
particular:

* [Toby on absolute values, and getting a volume from a metric, 2001-12-23](https://groups.google.com/g/sci.physics.research/c/aiMUJrOjE8A/m/DiAumLzLdCcJ)

* [Toby on examples of pseudo- and untwisted forms in electromagnetism, 2002-02-08](https://groups.google.com/g/sci.physics.research/c/aiMUJrOjE8A/m/0361hct91VsJ)

[[!redirects differential pseudoform]]
[[!redirects differential pseudoforms]]
[[!redirects differential pseudo-form]]
[[!redirects differential pseudo-forms]]
[[!redirects differential pseudo form]]
[[!redirects differential pseudo forms]]
[[!redirects pseudoforms]]
[[!redirects pseudo-form]]
[[!redirects pseudo-forms]]
[[!redirects pseudo form]]
[[!redirects pseudo forms]]
[[!redirects p-pseudoform]]
[[!redirects p-pseudoforms]]
[[!redirects n-pseudoform]]
[[!redirects n-pseudoforms]]
[[!redirects pseudo-p-form]]
[[!redirects pseudo-p-forms]]
[[!redirects pseudo-n-form]]
[[!redirects pseudo-n-forms]]
[[!redirects pseudo p-form]]
[[!redirects pseudo p-forms]]
[[!redirects pseudo n-form]]
[[!redirects pseudo n-forms]]


pseudofunctor

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

# Pseudofunctors
* table of contents
{: toc}

## Idea

A **pseudofunctor** is a specific algebraic notion of _weak [[2-functor]]_ between [[bicategory|bicategories]] (including [[strict 2-category|strict 2-categories]]), i.e. a [[2-functor]] which preserves [[composition]] and [[identity morphism|identities]] of [[1-morphisms]] only up to [[coherent]] specified 2-[[isomorphism]]. (In contrast to _[[strict 2-functors]]_.)

In general, there is not much reason to say "pseudofunctor" instead of "functor," since by far the most important type of functor between arbitrary bicategories is weak.  However, if the domain and codomain are known to be [[strict 2-category|strict 2-categories]] (including ordinary $1$-[[1-category|categories]]), it can be helpful to say "pseudofunctor" or "weak functor" to emphasize that it is not a [[strict 2-functor]].  Note that if the codomain is a $1$-category, then there is no difference.

Pseudo or weak functors are also to be distinguished from [[lax functor]]s and [[lax functor|oplax functor]]s, which preserve identities and composition only up to a  transformation in one direction or the other, which may be non-invertible.

An older terminology, which should probably be avoided at all costs, uses "homomorphism of bicategories" for a weak functor and "morphism of bicategories" for a lax one.

## Definition

Given [[bicategories]] $C$ and $D$, a __pseudofunctor__ (or __weak $2$-functor__, or just __functor__) $P:C \to D$ consists of:

*  A function
$$
P:Ob_C\to Ob_D.
$$
*  For each [[hom-category]] $C(x,y)$ in $C$, a [[functor]]
$$
P_{x,y}\colon C(x,y) \rightarrow D(P_x,P_y).
$$
*  For each object $x$ of $C$, an invertible [[2-morphism]] ($2$-cell)
$$
P_{\id_x}\colon \id_{P_x} \Rightarrow P_{x,x}(\id_x).
$$
*  For each triple $x,y,z$ of $C$-objects, an [[natural isomorphism|isomorphism]] (natural in $f\colon x \to y$ and $g\colon y \to z$)
$$
P_{f,g}\colon P_{x,y}(f) ; P_{y,z}(g) \Rightarrow P_{x,z}(f;g)
$$
*  For each hom-category $C(x,y)$,
   $$ \array {
                                  &                                         & \id_{P_x} ; P_{x,y}(f) \\
                                  & {}^{P_{\id_x};\id_{P_{x,y}(f)}}\swArrow &                        & \seArrow^{\lambda_{P_{x,y}(f)}} \\
      P_{x,x}(\id_x) ; P_{x,y}(f) &                                         &                        &                                 & P_{x,y}(f) \\
                                  & {}_{P_{x,x,y}(\id_x,f)}\seArrow         &                        & \neArrow_{P_{x,y}(\lambda_f)} \\
                                  &                                         & P_{x,y}(\id_x;f) \\
   } $$
   and
   $$ \array {
                                  &                                         & P_{x,y}(f) ; \id_{P_y} \\
                                  & {}^{\id_{P_{x,y}(f)};P_{\id_y}}\swArrow &                        & \seArrow^{\rho_{P_{x,y}(f)}} \\
      P_{x,y}(f) ; P_{y,y}(\id_y) &                                         &                        &                              & P_{x,y}(f) \\
                                  & {}_{P_{x,y,y}(f,\id_y)}\seArrow         &                        & \neArrow_{P_{x,y}(\rho_f)} \\
                                  &                                         & P_{x,y}(f;\id_y) \\
   } $$
   commute.
*  For each quadruple $w,x,y,z$ of $C$-objects,
   $$ \array {
      \big(P_{w,x}(f) ; P_{x,y}(g)\big) ; P_{y,z}(h)       & \overset{\alpha_{P_{w,x}(f),P_{x,y}(g),P_{y,z}(h)}}\Rightarrow & P_{w,x}(f) ; \big(P_{x,y}(g) ; P_{y,z}(h)\big) \\
      \mathllap{P_{w,x,y}(f,g);\id_{P_{y,z}(h)}}\Downarrow &                                                                & \Downarrow\mathrlap{\id_{P_{w,x}(f)};P_{x,y,z}(g,h)} \\
      P_{w,y}(f;g) ; P_{y,z}(h)                            &                                                                & P_{w,x}(f) ; P_{x,z}(g;h) \\
      \mathllap{P_{w,y,z}(f;g,h)}\Downarrow                &                                                                & \Downarrow\mathrlap{P_{w,x,z}(f,g;h)} \\
      P_{w,z}\big((f;g);h\big)                             & \underset{P_{w,z}(\alpha_{f,g,h})}\Rightarrow                  & P_{w,z}\big(f;(g;h)\big) \\
   } $$
   commutes.

A pseudofunctor is **strict** if the structural isomorphisms for identities of 1-cells and composites are identity 2-cells.

### Composition of Pseudofunctors

The composite of two pseudofunctors $P:\mathfrak{C}\to\mathfrak{D}$, $Q:\mathfrak{B}\to\mathfrak{C}$ is defined as follows:

1. Action on objects is given by function composition, so
$$
P\circ Q(X)=P(Q(X))
$$
1. Hom functors are given by functor composition, so
$$
(P\circ Q)_{XY}=P_{XY}\circ Q_{XY}
$$
1. For each object $X\in\mathfrak{B}$,
$$
(P\circ Q)_{id_X}=P(Q_{id_X})\circ P_{id_{Q(X)}}
$$
1. For each pair of composable arrows $f:Y\to Z$, $g:X\to Y\in\mathfrak{B}$,
$$
(P\circ Q)_{f,g}=P(Q_{f,g})\circ P_{Q(f),Q(g)}
$$

Coherence diagrams commute as a consequence of the coherence diagrams for $P$ and $Q$ commuting.

## Pseudofunctors versus lax functors

If we remove the requirement that $P_{\id_x}$ and $P_{x,y,z}(f,g)$ be invertible, then we have the definition of __[[lax functor]]__.  Thus, a pseudofunctor may be defined as a lax functor whose comparison constraints are invertible.

If in the definition of lax functor we reverse the direction of the constraints, then we have an __[[oplax functor]]__.  Thus, if we consider the *inverses* of the constraints of a pseudofunctor, we obtain an oplax functor.  Because there is little difference between specifying an invertible morphism and specifying its invertible inverse, one could equally well define a pseudofunctor to be an oplax functor whose constraints are invertible (i.e. reverse the direction of the isomorphisms $P_{\id_x}$ and $P_{x,y,z}(f,g)$ above), and in the literature one sometimes finds this definition instead.

Of course, in particular applications, one direction or the other may be slightly more "natural".  For instance, when a [[Grothendieck fibration]] $E\to B$ gives rise to a pseudofunctor $B^{op} \to Cat$, the natural comparison maps (induced by the universal property of [[cartesian arrows]]) go in the "lax direction".  Dually, when a Grothendieck opfibration $E\to B$ gives rise to a pseudofunctor $B\to Cat$, the natural comparison maps go in the "oplax direction".

## Strictification

For $C$ a strict 2-category, there is a universal procedure for replacing pseudofunctors $F \colon C \to Cat$ with strict 2-functors $\hat{F} \colon C \to Cat$, where "universal" is in the following sense:

+-- {: .num_prop}
###### Proposition
For any pseudofunctor $F \colon C \to Cat$, there is a strict 2-functor $\hat{F}\colon C\to Cat$ and a pseudonatural equivalence $\eta_F\colon F\to \hat{F}$.  Moreover, for any strict 2-functor $G \colon C \to Cat$, composing with $\eta_F$ yields an isomorphism between the category of pseudonatural transformations $F \to G$ and modifications between them, and the category of strict natural transformations $\hat{F} \to G$ and modifications between them.
=--

+-- {: .proof}
###### Proof (Sketch)
For each object $c$ of $C$, define $\hat{F}(c)$ to be the category whose objects are pairs $(f, x)$ with $f \colon d \to c$ in $C$ and $x \in Ob(F(d))$, and whose morphisms $(f, x) \to (f', x')$ are morphisms $F(f)(x) \to F(f')(x')$ in $F(c)$.  For $g \colon c \to c'$ in $C$, the functor $\hat{F}(g) \colon \hat{F}(c) \to \hat{F}(c')$ takes $(f, x)$ to $(g f, x)$; notice $\hat{F}(-)$ preserves compositions $g' \circ g$ strictly since composition in $C$ is strictly associative. (Strictly speaking we have only defined the functor $\hat{F}(g)$ at the object level. However, the extension to morphisms $\theta \colon F(f)(x) \to F(f)(x')$ is the obvious one, where

$$\hat{F}(g)(\theta) \coloneqq (F(g f)(x) \cong F(g)(F(f)(x)) \stackrel{F(g)(\theta)}{\to} F(g)(F(f)(x')) \cong F(g f)(x'))$$

preserves compositions $\theta' \circ \theta$ by naturality of the structural constraints $F(g)F(f) \cong F(g f)$.) Similarly, for 2-cells $\alpha: g \to g'$, the natural transformation $\hat{F}(\alpha) \colon \hat{F}(g) \to \hat{F}(g')$ is defined by taking its component at an object $(f, x)$ to be given by pasting in the first component: $F(\alpha \cdot f)(x) \colon F(g f)(x) \to F(g' f)(x)$.

One easily checks that $\hat{F}$ is pseudonaturally equivalent to $F$...
=--

The construction of the strictification is a special case of a general strictification construction due to [Power](#Pow). Later Steve Lack [showed](#Lack) that the strictifications obtained from Power's coherence theorem always have a universal property analogous to that of the result above.

We also have the following result:

+-- {: .num_prop}
###### Proposition
Given bicategories $C$ and $D$, for any pseudofunctor $F \colon C \to D$, there is a pseudofunctor $\hat{F}\colon C\to D$ that strictly preserves identity 1-morphisms, and a pseudonatural equivalence $\eta_F\colon F\to \hat{F}$.
=--

+-- {: .proof}
###### Proof
This is Proposition 5.2 of [Lack and Paoli](#LP).  Pseudofunctors that strictly preserve identity 1-morphisms are called
**normal**.
=--

## History

Historically the term 'pseudofunctor' was conceived by [[Grothendieck]] who weakened, around 1957, the concept of a [[contravariant functor]] from a [[1-category]] to [[Cat]], by effectively replacing the $1$-category Cat by the [[2-category]] $Cat$ and allowing (contravariant) functoriality up to coherent $2$-cells. This was recorded in his [[Bourbaki]] seminar on [[descent]] via pseudofunctors. Later in [[SGA1]] Grothendieck (with the assistance of [[Pierre Gabriel]]) replaced pseudofunctors in the treatment of descent by more invariant [[fibered categories]]. [[Benabou]], in his 1967 [treatise](#Ben) introducing [[bicategories]], generalized the pseudofunctors of Grothendieck to pseudofunctors between arbitrary bicategories but under the name 'homomorphism of bicategories'.

## Related concepts

* [[function]]

* [[functor]]

* [[2-functor]] / **pseudofunctor**

  * [[monoidal functor]]

* [[n-functor]]

* [[(∞,1)-functor]]

* [[(∞,n)-functor]]

## References

The notion of pseudo-functors from a [[1-category]] to the [[2-category]] [[Cat]] originates with:

* [[Alexander Grothendieck]], §A.1 in: *Technique de descente et théorèmes d’existence en géométrie algébrique. I. Généralités. Descente par morphismes fidèlement plats*, Séminaire [[N. Bourbaki]] exp. no190 (1960) 299-327 [[numdam:SB_1958-1960__5__299_0](http://www.numdam.org/item/?id=SB_1958-1960__5__299_0)]

(where it is still called "[[fibered categories]]", a term that eventually came to refer to the [equivalent incarnation](Grothendieck+construction#EquivalenceBetweenFiberedCategoriesAndPsuedofunctors) of pseudofunctors as their [[Grothendieck constructions]]).

Review:

* [[Angelo Vistoli]], Def. 3.10 in: *Grothendieck topologies, fibered categories and descent theory*, in: *[[Fundamental algebraic geometry -- Grothendieck's FGA explained]]*, Mathematical Surveys and Monographs **123**, Amer. Math. Soc. (2005) 1-104 [[ISBN:978-0-8218-4245-4](https://bookstore.ams.org/surv-123-s), [math.AG/0412512](http://arxiv.org/abs/math/0412512)]

The general notion of pseudo-functors as [[2-functors]] between [[2-categories]]

* {#Ben} [[Jean Bénabou]], *Introduction to Bicategories*, Lecture Notes in Mathematics **47** Springer (1967), pp.1-77 [[doi:10.1007/BFb0074299](http://dx.doi.org/10.1007/BFb0074299)]

* {#JohnsonYau20} [[Niles Johnson]], [[Donald Yau]], *2-Dimensional Categories*, Oxford University Press 2021 ([arXiv:2002.06055](http://arxiv.org/abs/2002.06055), [doi:10.1093/oso/9780198871378.001.0001](https://oxford.universitypressscholarship.com/view/10.1093/oso/9780198871378.001.0001/oso-9780198871378))

See also:

* {#Lack} [[Stephen Lack]], _Codescent objects and coherence_, JPAA Vol. 175 (2002), 223-241. ([web](http://www.sciencedirect.com/science/journal/00224049/175/1))

* [[Stephen Lack]] and [[Simona Paoli]], 2-nerves for bicategories, $K$-Theory 38 (2008), 153-175.   ([arxiv](https://arxiv.org/abs/math/0607271)).
{#LP}

* A. J. Power, _A general coherence result_, JPAA Vol. 57 Iss. 2 (1989), 165-173. ([web](http://www.sciencedirect.com/science/article/pii/0022404989901138))
{#Pow}

[[!redirects pseudofunctors]]
[[!redirects pseudo functor]]
[[!redirects pseudo functors]]
[[!redirects pseudo-functor]]
[[!redirects pseudo-functors]]
[[!redirects strict pseudofunctor]]
[[!redirects strict pseudofunctors]]

[[!redirects weak 2-functor]]
[[!redirects weak 2-functors]]

[[!redirects homomorphism of bicategories]]
[[!redirects homomorphisms of bicategories]]


pseudograph

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

#Contents#
* table of contents
{:toc}

## Idea

In [[graph theory]] a _pseudograph_ is a particular type of [[graph]]. A pseudograph is a [[set]] of [[vertices]] and for each unordered pair of vertices a set of [[edges]] between these.

A [[directed graph|directed]] pseudograph is also called a _[[quiver]]_.

The terminology is used in distinction to:

1. _[[multigraphs]]_, which are like pseudographs but not admitted to have "[[loops]]" in the sense of [[edges]] between a vertex and itself;

1. _[[simple graphs]]_, which are multigraphs with at most one edge between any unordered pair of distinct vertices.

Beware that the terminology is not completely consistent across different authors. Some authors may allows [[loops]] when they speak of multigraphs.

From yet another perspective, pseudographs are also like [[small categories|small]] [[dagger categories]] with [[identity morphisms]] and [[composition]] forgotten. Conversely, a [[small category|small]] [[dagger category]] may be regarded as a pseudograph equipped with [[extra structure]]. Formally this is witnessed by a [[forgetful functor]]

$$U\colon DagCat \to Pseudograph$$

where Pseudograph is the category of pseudographs and [[DagCat]] is the category of ([[small category|small]] [[strict category|strict]]) dagger categories.  Moreover, this forgetful functor has a [[left adjoint]]

$$F\colon Pseudograph \to DagCat $$

sending each pseudograph to the [[free object|free dagger category]] on that pseudograph.

## Definition

### Slick definition

A **pseudograph** is a [[quiver]] $G$ with a contravariant quiver homomorphism $(-)^\dagger:G \to G^\op$ from $G$ to the opposite quiver $G^\op$ which

1. is the identity on [[objects]],

1. is an [[involution]] $\dagger \circ \dagger = \mathrm{id}_G$.

### Nuts and bolts definition

A **pseudograph** is a [[quiver]] $(E, V, s, t)$ with a function $(-)^\dagger:E \to E$ such that for any edge $f:E$, $s(f) = t(f^\dagger)$, $t(f) = s(f^\dagger)$, and $(f^\dagger)^\dagger = f$.

## See also

* [[dagger category]]

## References

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

[[!redirects pseudographs]]


pseudogroup

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

#Contents#
* table of contents
{:toc}

## Idea
The definition of a [[smooth manifold]] is, as usual, given by

: A smooth $n$-manifold is a topological space $M$ equipped with an open cover $U_\alpha$ with homeomorphisms $\phi_\alpha: U_\alpha \to \mathbb{R}^n$ such that the transition maps $\phi_\beta \circ \phi_\alpha^{-1}: \phi_\alpha(U_\alpha \cap U_\beta) \to \phi_\beta(U_\alpha \cap U_\beta)$ are smooth.

If we replace $\mathbb{R}^n$ with some other topological space $X$ (eg. $\mathbb{C}^n$), and "smooth" with some other suitable restriction (eg. analytic), then we get a different kind of manifold. A pseudogroup is a structure we put on $X$ to specify what transition maps are allowed. The usage of pseudogroups to define the corresponding notion of manifolds is described at [[manifolds]].

## Definition

+-- {: .num_defn #Pseudogroup}
###### Definition
Let $X$ be a [[topological space]] (or [[locale]]) and let $H$ be the groupoid whose objects are the open subsets of $X$ and morphisms are the homeomorphisms between them. A **pseudogroup** (also called transformation pseudogroup) on $X$ is a [[wide subcategory|wide]] sub-groupoid $G$ of $H$ (ie. $G$ contains all open subsets of $X$) satisfying the [[sheaf]] property:

* If $g: U \to V$ is a homeomorphism and $U_\alpha$ is a cover of $U$, then $g$ is in $G$ if and only if each restriction $g|_{U_\alpha}: U_\alpha \to g(U_\alpha)$ is in $G$.

Note that in particular the restriction of a map in $G$ remains in $G$.
=--

+-- {: .num_example}
###### Example
We have the following correspondence between common pseudogroups and kinds of manifolds (the precise correspondence is defined in [[manifold]]):

| Topological space | Morphisms | Manifold |
|--|--|--|
| [[real n-dimensional space]] $\mathbb{R}^n$ | [[smooth maps]] | (n-dimensional) [[smooth manifolds]] |
| [[complex n-dimensional space]] $\mathbb{C}^n$ | [[analytic functions]] | (n-dimensional) [[complex manifolds]] |
| n-dimensional upper half-space $H^n = \{(x_1, \ldots, x_n) \in \mathbb{R}^n: x_1 \geq 0\}$ | [[smooth maps]] | [[manifolds with boundary]]|
| the $n$-[[cube]] $I^n = [0, 1]^n$ | [[smooth maps]] | [[manifolds with boundary|manifold with corners]] |

=--

+-- {: .num_defn #AbstractPseudogroup}
###### Definition
More generally, an **abstract pseudogroup** is a sub-[[inverse semigroup]] of the inverse semigroup of partial bijections of a set (or, even more generally, local automorphisms of an object in more general context).
=--

## Properties

### Relation to effective étale stacks

In the language of [[topological groupoids]]/[[Lie groupoids]] and their associated [[geometric stacks]], pseudogroups correspond to [[effective Lie groupoid|effective]] [[étale stacks]].  ([Lawson-Lenz 11](#LawsonLenz11), [Carchedi 12, section 2](#Carchedi12)).

## Related concepts

* [[manifold]]
* [[inverse semigroup]]

## References

The concept was introduced in

* {#Cartan05} [[Élie Cartan]], _Sur la structure des groupes infinis de transformation (suite)_ Ann. Sci. École Norm. Sup. (3), 22:219–308, 1905 ([Numdam](http://www.numdam.org/item?id=ASENS_1904_3_21__153_0))

* {#Cartan09} [[Élie Cartan]], _Les groupes de transformations continus, infinis, simples_. Annales Scientifiques de l'École Normale Supérieure 26: 93–161, 1909 ([Numdam](http://www.numdam.org/item?id=ASENS_1909_3_26__93_0))

* {#Golab39} St. Golab, _Über den Begriff der "Pseudogruppe von Transformationen"_ Mathematische Annalen 116: 768–780 (1939)

The history of the concept is reviewed in

* B. L. Reinhart

who attributes the term "pseudogroup" to

* Veblen, Whitehead 1932

The relation of pseudogroups to [[Lie groupoids]] originates with Haefliger, see at _[[Haefliger groupoid]]_.

Further discussion of the relation of psuedogroups to [[étale groupoids]]/[[étale stacks]] includes

* {#LawsonLenz11} [[Mark Lawson]], [[Daniel Lenz]], _Pseudogroups and their étale groupoids_ ([arXiv:1107.5511](http://arxiv.org/abs/1107.5511))

* {#Carchedi12} [[David Carchedi]], section 2 of _Étale Stacks as Prolongations_ ([arXiv:1212.2282](http://arxiv.org/abs/1212.2282))

A textbook account is in

* Alan Paterson, _Groupoids, Inverse Semigroups, and their Operator Algebras_

See also

* Wikipedia, _[Pseudogroup](http://en.wikipedia.org/wiki/Pseudogroup)_

Abstract pseudogroups are studied in  the [[inverse semigroup]] literature, e.g.

* [[Mark V. Lawson]], _Inverse semigroups: the theory of partial symmetries_, World Scientific, 1998.

[[!redirects pseudogroups]]
[[!redirects pseudogroup of transformations]]


pseudoholomorphic curve

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

#Contents#
* table of contents
{:toc}

## Idea

Given a smooth [[Riemann surface]] $\Sigma$ with complex structure $j$, and an [[almost complex manifold]] $M$ with almost complex structure $J$,
a __pseudoholomorphic__ curve (or J-holomorphic curve) is a smooth map
$f:\Sigma\to M$ whose differential commutes with the almost complex structure in the sense that the equation
$$
J\circ d f = d f\circ j
$$
holds. One can also consider a [[symplectic manifold]] $(N,\omega)$ instead of $M$, in which case one chooses an almost complex structure compatible with the symplectic form $\omega$.

A more general notion is of a pseudoholomorphic map.

Counting pseudoholomorphic curves with constraints to obtain topological invariants has been pioneered by [[Mikhail Gromov]] and later [[Andreas Floer]].

## References

Related $n$Lab entries include [[symplectic category]], [[Lagrangian correspondence]], [[Fukaya category]], [[Gromov-Witten invariant]], [[quantum cohomology]].

* S. K. Donaldson, _What is...a pseudoholomorphic curve?_, Notices AMS __52__ (9): pp.1026–1027, [pdf](http://www.ams.org/notices/200509/what-is.pdf)
* wikipedia [pseudoholomorphic curve](http://en.wikipedia.org/wiki/Pseudoholomorphic_curve)
* [[Mikhail Gromov]], _Pseudo-holomorphic curves in symplectic manifolds_, Inventiones Math. __82__ (1985) 307-347 [pdf](http://www.ihes.fr/~gromov/PDF/9%5B45%5D.pdf)
* [[Andreas Floer]], _Holomorphic curves and a Morse theory for fixed points of exact symplectomorphisms_, in: Aspects dynamiques et topologiques des groupes infinis de transformation de la
mécanique (Lyon, 1986), Travaux en Cours, vol. 25, pp. 49–60. Hermann, Paris, 1987
* [[Kenji Fukaya]], Kaoru Ono, _Floer homology and Gromov-Witten invariant over integer of general symplectic manifolds—summary_, in: Taniguchi Conference on Mathematics Nara '98,
Adv. Stud. Pure Math. __31__, 75–91. Math. Soc. Japan, Tokyo, 2001.
* Brett Parker, _Integral counts of pseudo-holomorphic curves_, [arxiv/1309.0585](http://arxiv.org/abs/1309.0585)

[[!redirects pseudo-holomorphic curve]]
[[!redirects pseudo-holomorphic curves]]
[[!redirects pseudoholomorphic curves]]
[[!redirects J-holomorphic curve]]
[[!redirects J-holomorphic curves]]


pseudoholomorphic vector bundle

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Bundles
+-- {: .hide}
[[!include bundles - contents]]
=--
#### Complex geometry
+--{: .hide}
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## Idea

The concept of _pseudoholomorphic vector bundle_ ([Bryant 05, def. 1](#Bryant05)) is like that of _[[holomorphic vector bundle]]_ but with [[complex structures]] relaxed to [[almost complex structures]].

## Related concepts

* [[almost Kähler geometric quantization]]

## References

* {#Bryant05} [[Robert Bryant]], _Remarks on the geometry of almost complex 6-manifolds_,  	The Asian Journal of Mathematics, vol. 10 no. 3 (September, 2006), pp. 561--606 ([arXiv:math/0508428](https://arxiv.org/abs/math/0508428), [Euclid](https://projecteuclid.org/euclid.ajm/1175789046))

* [[Boris Kruglikov]], _Tangent and normal bundles in almost complex geometry_ ([arXiv:math/0304165](https://arxiv.org/abs/math/0304165))

*  [[Boris Kruglikov]], _Symmetries of almost complex structures and pseudoholomorphic foliations_, International Journal of Mathematics 2014; Volume 25 (8) ([arXiv:1103.4404](https://arxiv.org/abs/1103.4404))

* Alexander D. Popov, _Hermitian-Yang-Mills equations and pseudo-holomorphic bundles on nearly Kaehler and nearly Calabi-Yau twistor 6-manifolds_,  	Nucl.Phys.B828:594-624, 2010 ([arXiv:0907.0106](https://arxiv.org/abs/0907.0106))

[[!redirects peusdoholomorphic vector bundles]]

[[!redirects pseudo-holomorphic vector bundle]]
[[!redirects pseudo-holomorphic vector bundles]]


pseudolattice ordered abelian group

[[!redirects pseudolattice ordered abelian groups]]
[[!redirects l-group]]
[[!redirects l-groups]]

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## Idea

A pseudolattice ordered abelian group or l-group is an [[ordered abelian group]] whose order forms a [[pseudolattice]].

## Definition

### With the join operation

A __pseudolattice ordered abelian group__ or __l-group__ is an [[abelian group]] $G$ with a binary [[join]] operation $(-)\vee(-):G \times G \to G$ such that $(G, \vee)$ is a [[commutative magma|commutative]] [[idempotent]] [[semigroup]], and

* for all $a \in G$, $b \in G$, $c \in G$, $a \vee b = b$ implies that $(a + c) \vee (b + c) = b + c$ and $(c + a) \vee (c + b) = c + b$

The meet is defined as

$$
  a \wedge b \coloneqq -(-a \vee -b),
$$

the ramp function is defined as

$$
  ramp(a) \coloneqq a \vee 0,
$$

and the absolute value is defined as

$$
  \vert a \vert \coloneqq a \vee -a
$$

The order relation is defined as in all pseudolattices: $a \leq b$ if $a = a \wedge b$.

### With the ramp function

The following [[algebraic theory|algebraic]] definition is from [[Peter Freyd]]:

A __pseudolattice ordered abelian group__ or __l-group__ is an [[abelian group]] $G$ with a function $ramp:G \to G$ such that for all $a$ and $b$ in $G$,

$$
  a = ramp(a) - ramp(-a)
$$

and

$$
  ramp(a - ramp(b)) = ramp(ramp(a) - ramp(b))
$$

The [[join]] $(-)\vee(-):G \times G \to G$ is defined as

$$
  a \vee b \coloneqq a + ramp(b - a)
$$

the [[meet]] $(-)\wedge(-):G \times G \to G$ is defined as

$$
  a \wedge b \coloneqq a - ramp(a - b)
$$

and the absolute value is defined as

$$
  \vert a \vert \coloneqq ramp(a) + ramp(-a)
$$

The order relation is defined as $a \leq b$ if $ramp(a - b) = 0$.

## Examples

All [[total order|totally ordered]] abelian groups, such as the [[integers]], the [[rational numbers]], and the [[real numbers]], are pseudolattice ordered abelian groups.

An example of a pseudolattice ordered abelian group that is not totally ordered is the abelian group of [[Gaussian integers]] with $ramp(1) \coloneqq 1$ and $ramp(i) \coloneqq i$.

## Related concepts

* [[abelian group]]

* [[ordered abelian group]]

* [[pseudolattice]]

* [[totally ordered abelian group]]

* [[pseudolattice ordered ring]]

## References

* [[Peter Freyd]], *Algebraic real analysis*, Theory and Applications of Categories, Vol. 20, 2008, No. 10, pp 215-306 ([tac:20-10](http://www.tac.mta.ca/tac/volumes/20/10/20-10abs.html))

* [[Henri Lombardi]], [[Claude Quitté]] (2010): *Commutative algebra: Constructive methods (Finite projective modules)* Translated by Tania K. Roblo, Springer (2015) ([doi:10.1007/978-94-017-9944-7](https://link.springer.com/book/10.1007/978-94-017-9944-7), [pdf](http://hlombardi.free.fr/CACM.pdf))


pseudolattice ordered ring

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## Idea

A pseudolattice ordered ring is an [[partially ordered ring]] whose partial order forms a [[pseudolattice]].

## Definition

A **psuedolattice ordered ring** is a [[preordered ring]] where the partial order $\lt$ is a [[pseudolattice]]: it has binary [[joins]] and [[meets]].

If the relation $\leq$ is only a [[preorder]], then the [[preordered ring]] $R$ is said to be a **pseudolattice preordered ring**.

### Essentially algebraic definition

The following [[essentially algebraic]] definition is adapted from the algebraic definition of [[pseudolattice ordered abelian group]] by [[Peter Freyd]]:

A __pseudolattice ordered ring__ is an [[ring]] $R$ with a function $ramp:R \to R$ such that for all $a$ and $b$ in $G$,

$$
  a = ramp(a) - ramp(-a)
$$

$$
  ramp(a - ramp(b)) = ramp(ramp(a) - ramp(b))
$$

and the following [[Horn clause]]:

$$
  ramp(a) = a \wedge ramp(b) = b \vdash ramp(a b) = a b
$$

An element $a$ in $R$ is __non-negative__ if $ramp(a) = a$. The Horn clause can then be stated as multiplication of non-negative elements is non-negative.

The [[join]] $(-)\vee(-):R \times R \to R$ is defined as

$$
  a \vee b \coloneqq a + ramp(b - a)
$$

the [[meet]] $(-)\wedge(-):R \times R \to R$ is defined as

$$
  a \wedge b \coloneqq a - ramp(a - b)
$$

and the absolute value is defined as

$$
  \vert a \vert \coloneqq ramp(a) + ramp(-a)
$$

The order relation is defined as in all pseudolattices: $a \leq b$ if $a = a \wedge b$.

## Examples

All [[total order|totally ordered]] rings, such as the [[integers]], the [[rational numbers]], and the [[real numbers]], are pseudolattice ordered rings.

## Related concepts

* [[ring]]

* [[ordered ring]]

* [[pseudolattice]]

* [[totally ordered ring]]

* [[pseudolattice ordered abelian group]]

## References

* [[Peter Freyd]], *Algebraic real analysis*, Theory and Applications of Categories, Vol. 20, 2008, No. 10, pp 215-306 ([tac:20-10](http://www.tac.mta.ca/tac/volumes/20/10/20-10abs.html))

[[!redirects pseudolattice ordered ring]]
[[!redirects pseudolattice ordered rings]]

[[!redirects pseudolattice preordered ring]]
[[!redirects pseudolattice preordered rings]]


pseudomodel stack

A notion of _pseudomodel stack_ was introduced by [[Nikolai Durov]] ([arXiv](http://arxiv.org/abs/0704.2030)) as a generalization of [[model stacks]], one which seems to be a more appropriate [[internalization]] of [[model categories]] in the setting of [[topoi]].

A [[stack]] $C$ over a [[site]] $S$ is a __pseudomodel stack__ if it is a [[flat stack]] and in every fiber five local classes of morphisms are distinguished: fibrations, acyclic fibrations, cofibrations, acyclic cofibrations and weak equivalences: and

(PM1) all fiber categories are [[finitely complete category|finitely complete]] and [[finitely cocomplete category|finitely cocomplete]],

(PM2) each distinguished class is local and stable under composition and global [[retracts]],

(PM3) (2-out-of-3) If $f$, $g$, $g\circ f$ are defined morphisms in a fixed fiber, then if any two of the three are weak equivalences then so is the third,

(PM5) (factorization) Any morphism can be globally factored as a cofibration followed by a fibration, where one can make either one chosen of the two acyclic,

(PM8) any acyclic fibration is both a fibration and weak equivalence, and any acyclic cofibration is both a cofibration and a weak equivalence (but not necessarily other ways around); any weak equivalence can be factored as an acyclic cofibration folowed by an acyclic fibration.

There is no lifting axiom! But the axioms are still self-dual.

Each fiber of a pseudomodel stack is a [[pseudomodel category]] (in another way, it is a pseudomodel stack over the [[point]]).

[[!redirects pseudomodel stacks]]


pseudomonic functor

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## Definition

A [[functor]] $F: C \to D$ is __pseudomonic__ if

1. it is [[faithful functor|faithful]]; that is, for any [[pair]] of [[objects]] $x,y\in C$ the component [[function]] $F \colon C(x,y) \to D(F x,F y)$ between [[hom-sets]] is injective, and

2. it is _full on isomorphisms_, meaning that for any [[pair]] of [[objects]] $x,y\in C$ the [[function]] $F \colon Iso_C(x,y) \to Iso_D(F x, F y)$ to the [[set]] of [[isomorphisms]] between them is [[surjective]] (hence [[bijective]]), where $Iso_C(x,y)$.

More generally, a [[1-morphism]] $f \colon C\to D$ in any [[2-category]] $K$ is called a **[[pseudomonic morphism]]** if the square analogous to that below is a [[2-pullback]], or equivalently if $K(X,C)\to K(X,D)$ is a pseudomonic functor for any $X$.

## Properties

Every [[fully faithful functor]] is pseudomonic, and every pseudomonic functor is [[conservative functor|conservative]], as well as [[essentially injective]].

A functor $F \colon C \to D$ is pseudomonic if and only if the [[commuting square]]

$$
  \array{
    C &\overset{Id}{\longrightarrow}& C
    \\
    \Big\downarrow\mathrlap{^{Id}}
    &&
    \Big\downarrow\mathrlap{{}^F}
    \\
    C &\underset{F}{\longrightarrow}& D
  }
$$

is a [[2-pullback]] in [[Cat]] (cf. the pullback characterization of 1-[[monomorphisms]], [here](monomorphism#BasicCharacterizationOfMonomorphisms)).

## Examples

An interesting example of the notion appears in the context of Joyal's [[combinatorial species]] of structures.

A *[[combinatorial species]]* is a [[functor]] from the [[category]] $Bij$ of [[finite sets]] and [[bijections]] between them to [[Set]], and the functors that are obtained by taking [[left Kan extensions]] of species along the [[subcategory]]-inclusion $I \colon Bij \to Set$ are called _analytic functors_. Now taking left Kan extensions along $I$ is pseudomonic, and this implies that the [[coefficients]] of an analytic functor are unique up to isomorphism.

Arguably, pseudomonic functors are precisely the functors for which it makes sense to say that $A$ is uniquely determined by $F A$ up to unique isomorphism.  However, we do not really need faithfulness for this; bijectivity on isos suffices.

## References

Formalization in [[cubical Agda]]:

* [[1lab]], *[Pseudomonic Functors](https://1lab.dev/Cat.Functor.Properties.html#pseudomonic-functors)*

[[!redirects pseudomonic functors]]
[[!redirects pseudomonic]]


pseudomonic morphism

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## Definition

A [[morphism]] $f\colon A\to B$ in a [[2-category]] $K$ is said to be **(representably) pseudomonic** if for all objects $X$, the induced [[functor]]
$$K(X,A) \to K(X,B)$$
is [[pseudomonic functor|pseudomonic]].  In [[Cat]], this is equivalent to $f$ being pseudomonic in the usual sense.

## Remarks

* Pseudomonic morphisms may also be called **(2,1)-monic** and said to make their [[source]] into a **(2,1)-subobject** of their [[target]].  See [[subcategory]] for discussion.

* Of course, any [[fully faithful morphism]] is also pseudomonic, and in particular any [[inverter]] or [[equifier]] is pseudomonic.

## Related pages

* [[subcategory]]
* [[fully faithful morphism]]
* [[conservative morphism]]
* [[faithful morphism]]

[[!redirects pseudomonic morphisms]]
[[!redirects (2,1)-monic morphism]]
[[!redirects (2,1)-monic morphisms]]
[[!redirects (2,1)-subobject]]
[[!redirects (2,1)-subobjects]]


pseudomonoid

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## Idea

The notion of **pseudomonoid** (sometimes also called a **monoidale**) in a [[monoidal 2-category]] is a [[categorification]] of the notion of a *[[monoid object]]* in a [[monoidal category]].

See [Street & Day (1997)](#StreetDay97)

The archetypical example are [[monoidal categories]], which are the pseudomonoids in the [[cartesian monoidal category|cartesian]] [[monoidal 2-category]] [[Cat]]. Similarly, [[monoidal enriched categories]] are pseudomonoids in [[VCat]].

Just as a monoid in a monoidal category $C$ can be equivalently defined as a [[monad]] in the corresponding one-object [[2-category]] $\mathbf{B}C$ (the [[delooping]] of $C$), so a pseudomonoid in a monoidal 2-category $C$ can equivalently be defined as a [[pseudomonad]] in the corresponding one-object [[3-category]] $\mathbf{B}C$.

## Variations

* A **map pseudomonoid** is a pseudomonoid whose multiplication and unit are [[bicategory of maps|maps]], i.e. [[left adjoints]].  This is a more appropriate notion for monoidal bicategories whose morphisms are [[profunctors]], since maps therein can be identified (modulo [[Cauchy completion]]) with [[functors]]. (However, it is typically better to work instead in a [[double category]], where one may distinguish between **tight pseudomonoids** and **loose pseudomonoids**.)

Other more special kinds of pseudomonoid are generalizations of special kinds of monoidal categories, including:

* braided pseudomonoids
* symmetric pseudomonoids
* balanced pseudomonoids
* closed pseudomonoids
* $\ast$-autonomous, a.k.a. [[Frobenius pseudomonoids]]
* compact closed (or autonomous) pseudomonoids

Eventually these should probably have their own pages.

## Properties

The 2-category of symmetric pseudomonoids in a symmetric monoidal 2-category has (weak) 2-coproducts given by the tensor product of underlying objects (analogously to how the category of [[commutative monoids]] in a monoidal category has coproducts given by the tensor product of the underlying objects).  This is proven in [Schaeppi, Appendix A](#Schaeppi).

## See also
* [[pseudoaction]]
* [[pseudomonad]]

## References

* {#StreetDay97} [[Ross Street]], [[Brian Day]], *Monoidal bicategories and Hopf algebroids*, Advances in Mathematics, **129** 1 (1997) 99-157 [[doi:10.1006/aima.1997.1649](https://doi.org/10.1006/aima.1997.1649)]

* {#Schaeppi} Daniel Schäppi, *Ind-abelian categories and quasi-coherent sheaves*, [arXiv](https://arxiv.org/abs/1211.3678), 2014.

* Dominic Verdon, *Coherence for braided and symmetric pseudomonoids*, [arXiv](https://arxiv.org/abs/1705.09354).

[[!redirects pseudomonoid]]
[[!redirects pseudomonoids]]
[[!redirects pseudomonoid object]]
[[!redirects pseudomonoid objects]]
[[!redirects monoidale]]
[[!redirects monoidales]]
[[!redirects map pseudomonoid]]
[[!redirects map pseudomonoids]]

[[!redirects braided pseudomonoid]]
[[!redirects braided pseudomonoids]]
[[!redirects symmetric pseudomonoid]]
[[!redirects symmetric pseudomonoids]]
[[!redirects balanced pseudomonoid]]
[[!redirects balanced pseudomonoids]]
[[!redirects closed pseudomonoid]]
[[!redirects closed pseudomonoids]]
[[!redirects autonomous pseudomonoid]]
[[!redirects autonomous pseudomonoids]]
[[!redirects compact closed pseudomonoid]]
[[!redirects compact closed pseudomonoids]]

[[!redirects pseudo-monoid]]
[[!redirects pseudo-monoids]]


pseudonatural transformation

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## Idea

A __pseudonatural transformation__ is a [[lax natural transformation]] whose $2$-cell components are all [[equivalence|invertible]].

## Definition

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

Given two [[2-functors]] $U, V: S \stackrel{\to}{\to} C$ between [[2-categories]], a **pseudonatural transformation** $\phi: U \to V$ is a rule that assigns to each [[object]] $s$ of $S$ a [[morphism]] $\phi(s): U(s) \to V(s)$ of $C$, and to each [[morphism]] $f: r \to s$ of $S$ an [[equivalence|invertible]] [[2-morphism]] $\phi(f)$ of $C$:
$$
\array{
U(r) & \stackrel{U(f)}{\to} & U(s) \\
\phi(r) \downarrow & \phi(f) \swArrow & \downarrow \phi(s) \\
V(r) & \underset{V(f)}{\to} & V(s)
}
$$
such that the following [[coherence law]]s are satisfied in  $C$ (throughout we leave the [[associator]]s and [[unitor]]s in $C$ implicit):

1. respect for composition: for all composable morphisms $r \stackrel{f}{\to} s \stackrel{g}{\to} t$ in $S$ we have an equality

   $$
   \array{
      && U(s)
      \\
      & {}^{\mathllap{U(f)}}\nearrow &\downarrow^{\phi(s)}& \searrow^{\mathrlap{U(g)}}
      \\
     U(r) &\swArrow_{\phi(f)}&V(s) &\swArrow_{\phi(g)}& U(t)
     \\
     {}^{\mathllap{\phi(r)}}\downarrow &{}^{V(f)}\nearrow&
      \Downarrow^{V(f,g)}
     &\searrow^{V(g)}& \downarrow^{\mathrlap{\phi(t)}}
     \\
     V(r) &&\underset{V( g\circ f)}{\to}&& V(t)
   }
    \;\;\;
    =
    \;\;\;
   \array{
     && U(s)
     \\
     & {}^{\mathllap{U(f)}}\nearrow &\Downarrow^{U(f,g)}& \searrow^{\mathrlap{U(g)}}
     \\
     U(r) &&\stackrel{U(g \circ f)}{\to}&& U(t)
     \\
     {}^{\mathllap{\phi(r)}}\downarrow &&
       \swArrow_{\phi(g \circ f )}
     && \downarrow^{\mathrlap{\phi(t)}}
     \\
     V(r) &&\underset{V(g \circ f)}{\to}&& V(t)
   }
     \,,
   $$

   of [[pasting]] [[2-morphisms]] as indicated, where $U(f,g)$ and $V(f,g)$ denote the compositors of the [[2-functor]]s $U$ and $V$,

1. respect for units, (...)

1. naturality

   for every [[2-morphism]]
   $$
     \array{
        && \stackrel{f}{\to}
        \\
        & \nearrow && \searrow
        \\
        r &&\Downarrow^{F}&& s
        \\
        & \searrow && \nearrow
        \\
        && \underset{g}{\to}
     }
   $$
   in $S$ an equality
   $$
     \array{
        && \stackrel{U(f)}{\to}
        \\
        & \nearrow &\Downarrow^{U(F)}& \searrow
        \\
        U(r) &&\stackrel{U(g)}{\to}&& U(s)
        \\
        {}^{\mathllap{\phi(r)}}\downarrow &&\swArrow_{\phi(g)}&& \downarrow^{\mathrlap{\phi(s)}}
        \\
        V(r) &&\underset{V(g)}{\to}&& V(s)
     }
     \;\;\;
       =
     \;\;\;
     \array{
       U(r) &&\stackrel{U(f)}{\to}&& U(s)
       \\
       {}^{\mathllap{\phi(r)}}\downarrow &&\swArrow_{\phi(f)}&& \downarrow^{\mathrlap{\phi(s)}}
       \\
       V(r) &&\stackrel{V(f)}{\to}&& V(s)
       \\
       & \searrow &\Downarrow^{V(F)}& \nearrow
       \\
       && \underset{V(g)}{\to}
     }
   $$
   in $C$.

=--

A pseudonatural transformation is called a **pseudonatural equivalence** if each component $\phi(s)$ is an [[equivalence]] in the 2-category $C$.  This is equivalent to $\phi$ itself being an equivalence in the 2-category $[S,C]$ of 2-functors, pseudonatural transformations, and [[modifications]].

## Related concepts

* [[homotopy]]

* [[transfor]]

  * [[natural transformation]]

    * [[dinatural transformation]], [[extranatural transformation]]

  * **pseudonatural transformation**

  * [[lax natural transformation]]

## References

A generalization to [[extranatural transformations]] can be found in

* Alexander S. Corner, _A universal characterisation of codescent objects_, [TAC](http://tac.mta.ca/tac/volumes/34/24/34-24abs.html) 2019.

Discussion of the globular approach can be found in

* [[Camell Kachour]]: Kamel Kachour, Définition algébrique des cellules non-strictes, Cahiers de Topologie et de Géométrie Différentielle Catégorique (2008), volume 1, pages 1–68.

* [[Camell Kachour]]: Steps toward the Weak ω-category of the Weak ω-categories in the globular setting, Published in : Categories and General Algebraic Structures with Applications (2015).

[[!redirects pseudonatural transformation]]
[[!redirects pseudonatural transformations]]
[[!redirects pseudo-natural transformation]]
[[!redirects pseudo-natural transformations]]
[[!redirects pseudo natural transformation]]
[[!redirects pseudo natural transformations]]
[[!redirects pseudonatural equivalence]]
[[!redirects pseudonatural equivalences]]
[[!redirects pseudo-natural equivalence]]
[[!redirects pseudo-natural equivalences]]
[[!redirects pseudo natural equivalence]]
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pseudoorientation

# Contents
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{: toc}

## Idea

A _pseudoorientation_ of a [[submanifold]] $U \subset X$ is an orientation of how the ambient [[manifold]] $X$ lies around the submanifold.

This contrasts with an [[orientation]] of the submanifold itself: neither a pseudoorientation nor an orientation for a submanifold necessarily provides the other, unless one has an orientation for the ambient manifold.

An orientation of $U$ may be visualized with a set of arrows lying within $U$. A pseudoorientation is properly visualized instead with a set of arrows passing through $U$, or circulating around it.

A pseudoorientation is sometimes also called a __transverse orientation__, and the term is also spelled __pseudo-orientation__ or __pseudorientation__.

## Definition

\begin{definition}
For a [[submanifold]] $U$ of a [[manifold]] $X$, a __pseudoorientation__ of $U$ is a map that, for each point $a$ on $U$, takes a local [[orientation]] of $X$ at $a$ to a local orientation of $U$ at $a$, continuously in $a$ and taking opposite orientations to opposite orientations.
\end{definition}

An equivalent, more explicitly geometrical definition justifies the synonym _transverse orientation_:

\begin{definition}
For a [[submanifold]] $U$ of a [[manifold]] $X$, a __pseudoorientation__ of $U$ is an [[orientation]] of the [[vector bundle]] $T_U(X) / T_U(U)$, a bundle on $U$ whose [[fiber]] at each $x \in U$ is the [[quotient vector space|quotient]] $T_x(X) / T_x(U)$ of the [[tangent space]] to $X$ at $x$ by the tangent space to $U$ at $x$.
\end{definition}

## Properties

### Relation to orientations

When one has an [[orientation]] of the containing manifold $X$, that gives a correspondence between orientations and pseudoorientations for any submanifold $U$. As a result, many treatments conflate orientations with pseudoorientations.

### Integration

A pseudoorientation is precisely the added structure one needs in order to [[integration of differential forms|integrate]] over $U$ a $k$-[[pseudoform]] on $X$. This is much like how if one has instead an (untwisted) $k$-[[differential form|form]] on $X$ and wants to integrate it over $U$, one needs an orientation of $U$. For details, see [[integration of differential forms]].

## Examples

### Canonical pseudoorientation for a top-dimension submanifold

When $k = n$, a local [[orientation]] of $X$ is the same thing as a local orientation of $U$. This correspondence makes a canonical pseudoorientation for $U$.

This canonical pseudoorientation is what makes the integration of $n$-[[pseudoforms]] the most fundamental kind of integration on a [[manifold]], as discussed at [[integration of differential forms]].

## Related concepts

* An [[absolute differential form]] can be integrated on an entirely unoriented [[submanifold]], with neither an [[orientation]] nor a pseudoorientation.

## References

Many useful explanations by [[Toby Bartels]] and [[John Baez]] in [this long Usenet thread](https://groups.google.com/g/sci.physics.research/c/aiMUJrOjE8A/m/0361hct91VsJ).

[[!redirects pseudoorientations]]
[[!redirects pseudooriented]]

[[!redirects pseudo-orientation]]
[[!redirects pseudo-orientations]]
[[!redirects pseudo-oriented]]

[[!redirects pseudorientation]]
[[!redirects pseudorientations]]
[[!redirects pseudoriented]]


pseudoradial space

## Definition ##

**Definition.** A topological space $X$ is called **pseudoradial**, if for any closed subset $A$ of $X$, $A$ is closed, if and only if, for any converging transfinite sequence of elements in $A$, it's limit is in $A$.


pseudoscalar

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## Idea

Given a [[symmetry group]] $G$ equipped with a [[homomorphism]] $G \overset{\phi}{\longrightarrow} O(s,t)$ to an [[orthogonal group]] (for instance a [[Pin group]]), a  _pseudoscalar_ is an element of the 1-dimensional [[linear representation]] (over a given [[ground field]] $k$)

$$
  \mathbf{1}_{{}_{sgn}}
  \;\in\;
  Rep_k(G)
$$

that is given by forming the [[determinant]] ([[sign representation]]):

\[
  \label{DeterminantRepresentation}
  \mathbf{1}_{{}_{sgn}}
  \;\;\coloneqq\;\;
  \left(
  \array{
    G \times k
    &\longrightarrow&
    k
    \\
    (g, c ) &\mapsto& det\big( \phi(g) \big) \cdot c
  }
  \right)
  \,.
\]

More generally, given a [[function]] with values in $\mathbf{1}_{{}_{sgn}}$, or yet more generally a [[section]] of a [[fiber bundle]] with [[typical fiber]] $\mathbf{1}_{{}_{sgn}}$, this whole function/section is often called a _pseudoscalar_; more precisely: a _pseudoscalar [[field (physics)|field]]_.
This as opposed to [[scalar fields]], which take values in the 1-dimensional [[trivial representation]] $\mathbf{1}$.

If the [[fiber bundle]] in question is a "[[canonical bundle]]"/[[determinant line bundle]] of a [[Riemannian manifold]] of [[dimension]] $n$, hence the top [[exterior power]] of a [[tangent bundle]] (i.e. top degree ([[Kähler differential form|Kähler]]) [[differential form]]-bundle) then such "pseudoscalar fields" of [[physics]] are what are called _[[densities]]_ in [[mathematics]].

## Related concepts

* The [[tensor product of representations]] of any type $X$ of representations with a [[determinant]]/[[sign representation]] (eq:DeterminantRepresentation) is often called a "pseudo-X representation". For instance of $X$ is "[[vector representation]]" then we get [[pseudovector representation]], etc.

* [[pseudoscalar meson]]

* [[scalar]], [[scalar field]]

* [[density]], [[determinant line bundle]]

## References

See also

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

[[!redirects pseudoscalars]]

[[!redirects pseudo-scalar]]
[[!redirects pseudo-scalars]]

[[!redirects pseudoscalar representation]]
[[!redirects pseudoscalar representations]]
[[!redirects pseudo-scalar representation]]
[[!redirects pseudo-scalar representations]]

[[!redirects pseudoscalar field]]
[[!redirects pseudoscalar fields]]
[[!redirects pseudo-scalar field]]
[[!redirects pseudo-scalar fields]]


pseudosimplicial object

Given a 2-category (strict 2-category or bicategory) $C$, the pseudosimplicial object in $C$ is a [[pseudofunctor]], that is a 2-functor $\Delta^{op}\to C$ in the pseudosense (associativity coherences preserved up to _invertible_ 2-cells),
and where the category of finite ordinals and monotone maps $\Delta$ is viewed as
a 1-category or equivalently a 2-category with only trivial 2-cells. For example, we
can consider pseudosimplicial categories.

See also [[pseudocoherence]].

[[!redirects pseudosimplicial category]]
[[!redirects pseudosimplicial categories]]
[[!redirects pseudosimplicial objects]]
[[!redirects pseudosimplicial bicategory]]
[[!redirects pseudosimplicial bicategories]]


pseudotopological space

# Pseudotopological spaces
* tic
{: toc}

## Idea

A _pseudotopological space_ or _Choquet space_ is a generalisation of a [[topological space]] based on the concept of convergent [[ultrafilter]] as fundamental.  This view relies on the [[ultrafilter theorem]] to guarantee enough ultrafilters; however, we can also describe a pseudotopological structure in terms of convergence of arbitrary [[filters]] satisfying certain properties.  In this respect, a pseudotopological space is a special kind of [[convergence space]].

The [[category]] $Ps Top$ of pseudotopological spaces is a [[quasitopos]] and may be thought of as a [[nice category of spaces]] that includes [[Top]] as a [[full subcategory]]. It is the minimal quasitopos containing the [[cartesian closed category]] of [[sequential spaces]] (cf [[Johnstone's topological topos]] Thm 10.2).
It is also a reflective subcategory of [[filter spaces]] (Thm 10.3).

## Definitions

A **pseudotopological space** is a [[set]] $S$ together with a [[relation]] $\to$ from $\mathcal{F}S$ to $S$, where $\mathcal{F}S$ is the set of [[filters]] on $S$; if $F \to x$, we say that $F$ __converges__ to $x$.  This must satisfy some axioms:

1. Centred: The principal ultrafilter $F_x = \{ A \;|\; x \in A \}$ at $x$ converges to $x$;
2. Isotone: If $F \subseteq G$ and $F \to x$, then $G \to x$;
3. Star property: If $F$ is a filter such that for every proper filter $G \supseteq F$ there exists a proper filter $H \supseteq G$ with $H \to x$, then $F \to x$.

A pseudotopological space is a special case of a [[convergence space]]; the star property is a stronger version of the filter property of a convergence space:

*  If $F \to x$ and $G \to x$, then $F \cap G \to x$.

Assuming the [[ultrafilter principle]] (a weak version of the [[axiom of choice]]), the star property can be expressed in terms of [[ultrafilters]]:

*  If $F$ is a filter such that every ultrafilter $U \supseteq F$ converges to $x$, then $F \to x$.

The property of isotony gives the converse, so $F \to x$ if and only if every ultrafilter refining $F$ converges to $x$.  Thus a pseudotopology consists precisely of a convergence relation between ultrafilters and points satisfying the single axiom that $F_x$ converges to $x$ for every $x$.

A [[subsequential space]] is a pseudotopological space that may be defined using only [[sequences]] instead of arbitrary nets/filters.

As with other convergence spaces, a filter $F$ __clusters__ at a point $x$ if there exists a proper filter $G$ such that $F \subseteq G$ and $G \to x$; given the ultrafilter principle, we may assume that $G$ is an ultrafilter.  Note that an ultrafilter clusters at $x$ iff it converges to $x$.

The definition can also be phrased in terms of [[nets]]; a net $\nu$ converges to $x$ if and only if its [[eventuality filter]] converges to $x$.

The morphisms of convergence spaces are the [[continuous functions]]; a function $f$ between pseudotopological spaces is __continuous__ if $F \to x$ implies that $f(F) \to f(x)$, where $f(F)$ is the filter generated by the [[filterbase]] $\{f(A) \;|\; A \in F\}$.  In this way, pseudotopological spaces form a [[concrete category]] $PsTop$, which is in fact a [[quasitopos]].

## Properties

The [[topological spaces]] can be characterized as the pseudotopological ones in which the convergence satisfies a certain [[associativity]] condition; see [[relational beta-module]].  In this way one can think of a topological space as a [[multicategory]] parametrized by ultrafilters; see [[generalized multicategory]].

In particular, note that a [[compact space|compact]] [[Hausdorff space|Hausdorff]] pseudotopological space is defined by a single function $\mathcal{U}S \to S$, where $\mathcal{U}S$ is the set of ultrafilters on $S$, such that the composite $S \to \mathcal{U}S \to S$ is the identity.  That is, it is an [[algebra for an endofunctor|algebra]] for the [[pointed endofunctor]] $\mathcal{U}$.  The compact Hausdorff _topological_ spaces (the [[compacta]]) are precisely the [[algebra for a monad|algebras]] for $\mathcal{U}$ considered as a [[monad]].

Every [[pretopological space]] is also a pseudotopological space; these may be characterised as the infinitely directed pseudotopological spaces.

## References

*  [[Mike Shulman]] (2008); Ultrafilters, Pseudotopological spaces, and [[Stone-Čech compactification]]; [pdf](http://home.sandiego.edu/~shulman/papers/pstonecech.pdf)

[[!redirects pseudotopological space]]
[[!redirects pseudotopological spaces]]
[[!redirects Choquet space]]
[[!redirects Choquet spaces]]


pseudovector representation

+-- {: .rightHandSide}
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#### Representation theory
+-- {: .hide}
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#Contents#
* table of contents
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## Idea

Given a [[symmetry group]] $G$ equipped with a [[homomorphism]] $G \to O(V)$ to an [[orthogonal group]] of some [[inner product space]] $V$ -- for instance a [[Pin group]] -- then its _pseudovector representation_ is the [[linear representation]] (over the given [[ground field]] $k$)

$$
  V_{{}_{sgn}}
  \;\in\;
  Rep_k(G)
$$

which is the [[tensor product of representations]] of the [[vector representation]] $V$ with the [[pseudoscalar representation]] ([[determinant]]-representation) $\mathbf{1}_{{}_{sgn}}$:

$$
  V_{{}_{sgn}}
  \;\coloneqq\;
  V \otimes \mathbf{1}_{{}_{sgn}}
$$

## Related concepts

* [[vector representation]]

* [[pseudoscalar]], [[density]]

* [[pseudovector meson]]

[[!redirects pseudovector representations]]
[[!redirects pseudo-vector representation]]
[[!redirects pseudo-vector representations]]

[[!redirects pseudovector]]
[[!redirects pseudovectors]]
[[!redirects pseudo-vector]]
[[!redirects pseudo-vectors]]


PU(ℋ)

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#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
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#### Operator algebra
+-- {: .hide}
[[!include AQFT and operator algebra contents]]
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#Contents#
* table of contents
{:toc}

## Idea

The [[projective unitary group]] on an infinite-dimensional [[separable Hilbert space|separable]] [[complex vector space|complex]] [[Hilbert space]] $\mathcal{H}$ is traditionally denoted $PU(\mathcal{H})$, being the [[quotient space|quotient]] of the [[unitary group]] [[U(ℋ)]] by its [[circle group|circle]] [[subgroup]] $S^1 \simeq S(\mathbb{C}) \simeq$ [[U(1)]].

## Properties

### General

\begin{proposition}\label{UHCircleFiberBundle}
  The [[U(1)]]-[[quotient space]] [[coprojection]] of [[U(ℋ)]]  over [[PU(ℋ)]] -- both in their strong [[operator topology]] -- is a [[circle group|circle]]-[[principal bundle]]:
$$
  \array{
    S^1 &\hookrightarrow& \mathrm{U}(\mathcal{H})
    \\
    && \big\downarrow
    \\
    && PU(\mathcal{H})
    \mathrlap{ \; \simeq \; \mathrm{U}(\mathcal{H})/S^1 }
  }
$$
\end{proposition}
([Simms 1970, Thm. 1](#Simms70))
\begin{remark}
  Prop. \ref{UHCircleFiberBundle} means in particular that $\mathrm{U}(\mathcal{H}) \xrightarrow{\;} PU(\mathcal{H})$ is [[locally trivial bundle|locally trivial]], hence that the [[coset space]] [[coprojection]] $\mathrm{U}(\mathcal{H}) \xrightarrow{\;} \mathrm{U}(\mathcal{H})/S^1$ admits [[local sections]]. See also at *[[coset space coprojection admitting local sections]]*.
\end{remark}

\begin{prop}\label{WellPointedInOperatorTopology}
  In its [[operator topology]] ([[U(ℋ)#WithOperatorTopology|here]]), $PU(\mathcal{H})$ is a [[well-pointed topological group]].
\end{prop}
([Hebestreit & Sagave 2020, p. 23](#HebestreitSagave19), using [Dardalat & Pennig 2016, Prop. 2.26](twisted+K-theory#DardalatPennig16))

### Homomorphisms into $PU(\mathcal{H})$

Since $\Gamma \,\coloneqq\,$ [[PU(ℋ)]] is not a (finite-dimensional) [[Lie group]], it falls outside the applicability of the general theorem that [[nearby homomorphisms from compact Lie groups are conjugate]].
Nevertheless, the conclusion still holds, at least for [[domain]] $G$ a [[discrete group|discrete]], hence [[finite group]]:

  The [[PU(ℋ)]][[G-space|-space]] of homomorphisms $Hom\big(G,\, PU(\mathcal{H})\big)$ is a [[disjoint union]] ([as here](nearby+homomorphisms+from+compact+Lie+groups+are+conjugate#eq:DisjointUnionOfOrbits)) of [[orbits]] of the [[conjugation action]].

  This is established in [Uribe & Lück 2013, Sec. 15](#UribeLueck13), [p. 38](https://arxiv.org/pdf/1304.4862.pdf#page=38).

## Related concepts

* [[U(ℋ)]]

* [[universal equivariant PU(H)-bundle|universal equivariant $PU(\mathcal{H})$-bundle]]

## Literature

General discussion:

* {#Simms70} [[David John Simms]], *Topological aspects of the projective unitary group*, Math. Proc. Camb. Phil. Soc. **68** 1 (1970) 57-60  ([doi:10.1017/S0305004100001043](https://doi.org/10.1017/S0305004100001043))

* {#UribeLueck13} [[Bernardo Uribe]], [[Wolfgang Lück]], Section 15 of: _Equivariant principal bundles and their classifying spaces_, Algebr. Geom. Topol. 14 (2014) 1925-1995 ([arXiv:1304.4862](https://arxiv.org/abs/1304.4862), [doi:10.2140/agt.2014.14.1925](http://dx.doi.org/10.2140/agt.2014.14.1925))

* [[Jesus Espinoza]], [[Bernardo Uribe]], *Topological properties of spaces of projective unitary representations*, Rev. Acad. Colombiana Cienc. Exact. Fís. Natur. 40 (2016), no. 155, 337-352 ([arXiv:1511.06785](https://arxiv.org/abs/1511.06785), [scielo:S0370-39082016000200013](http://www.scielo.org.co/scielo.php?script=sci_arttext&pid=S0370-39082016000200013), [doi:10.18257/raccefyn.317](https://doi.org/10.18257/raccefyn.317))

Concerning well-pointedness:

* {#HebestreitSagave19} [[Fabian Hebestreit]], [[Steffen Sagave]], p. 23 of: *Homotopical and operator algebraic twisted K-theory*, Mathematische Annalen **378** (2020) 1021-1059 ([arXiv:1904.01872](https://arxiv.org/abs/1904.01872), [doi:10.1007/s00208-020-02066-6](https://doi.org/10.1007/s00208-020-02066-6))

The $\mathbb{Z}/2$-graded projective unitary group, (reflecting the twists of [[twisted K-theory]] not just in degree 3 but also in degree 1):

* {#Parker88} [[Ellen Maycock Parker]], Prop. 2.2 of: *The Brauer Group of Graded Continuous Trace $C^\ast$-Algebras*, Transactions of the American Mathematical Society **308** 1 (1988) ([jstor:2000953](https://www.jstor.org/stable/2000953))

reviewed in

* [[Alan Carey]], [[Bai-Ling Wang]], p. 5 of: *Thom isomorphism and Push-forward map in twisted K-theory*, Journal of K-Theory **1** 2 (2008) 357-393  ([arXiv:math/0507414](https://arxiv.org/abs/math/0507414), [doi:10.1017/is007011015jkt011](https://doi.org/10.1017/is007011015jkt011))

[[!redirects PU(H)]]

[[!redirects projective unitary group on a Hilbert space]]

[[!redirects graded PU(H)]]


Puiseux series

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#Contents#
* table of contents
{:toc}

## Definition

Let $K$ be a [[field]]. A **Puiseux series** with [[coefficients]] in $K$ is a formal [[Laurent series]]

$$f(t) = \sum_{k \geq m} a_k t^{k/r} \qquad (1)$$

where $r$ is a positive [[integer]], $m$ is an integer, and each $a_k$ belongs to $K$. Somewhat more abstractly but also more meaningfully, if $\mathbb{N}_{\cdot}$ is the [[poset]] of positive integers ordered by divisibility (so that $1$ is the least element), then the field of Puiseux series is the [[filtered colimit]] of the diagram of fields

$$F: \mathbb{N}_{\cdot} \to Field$$

where each $F(n)$ is the field of [[Laurent series]] $K((t))$, but where $F(m) \to F(n)$ (in case $m|n$) is the field [[homomorphism]] taking $f(t)$ to $f(t^{n/m})$.

The field of Puiseux series carries a [[valuation ring|valuation]] whose values are in the ordered group of [[rational number|rationals]] $(\mathbb{Q}, +)$: for $f(t)$ as in (1), $v(f)$ is the least exponent $k/r$ for which $a_k \neq 0$.

## Puiseux-Newton expansions

Puiseux series were in essence considered by [[Isaac Newton]], who developed a method of expanding algebraic functions as Puiseux series, based on an analogue of [[Newton's method]] of approximating roots. Here is a sample theorem:

+-- {: .num_theorem #Puiseux}
###### Theorem
If $K$ is [[algebraically closed field|algebraically closed]] and has [[characteristic]] 0, then the field of Puiseux series over $K$ is the algebraic closure of the field of Laurent series over $K$.
=--

+-- {: .proof}
###### Proof (sketch)
It is enough to show that every degree $n$ extension $E$ of the field of Laurent series $K((x))$ is of the form $K((x^{1/n}))$. For this, it suffices that the integral closure $B$ of $K[ [x] ]$ in $E$ be of the form $K[ [x^{1/n}] ]$.

Generally speaking, let $A$ be a complete DVR ([[valuation ring|discrete valuation ring]]) with maximal ideal $\mathfrak{m}_A$ and residue class field $k_A$, and let $F$ be its field of fractions. Let $E$ be a degree $n$ extension of $F$, and let $B$ be the integral closure of $A$ in $E$. Then $B$ is also a complete DVR. We may write the ideal $\mathfrak{m}_A B$ of $B$ as $\mathfrak{m}_B^e$ where $e$ is the _ramification index_, and we have

$$n = deg_F E = rank_A B = rank_{A/\mathfrak{m}_A} B/\mathfrak{m}_A B = dim_{k_A} B/\mathfrak{m}_B^e = e \cdot dim_{k_A} k_B$$

where the last equation holds because $\mathfrak{m}_B^i/\mathfrak{m}_B^{i+1} \cong k_B$ as $k_B$-modules and therefore also as $k_A$-modules. In the case $A = K[ [x] ]$ where $k_A = K$, we have that $dim_{k_A} k_B = 1$ since $K$ is algebraically closed, and therefore $e = n$. In other words, $(x)B = \mathfrak{m}_B^n$, so we can write $x = u \pi^n$ where $\pi$ generates the maximal ideal of $B$ and $u$ is a unit of $B$.

The residue class $\bar{u} \in K$ has an $n^{th}$ root (again by algebraic closure); in fact a simple root since $char(K) = 0$. By [[Hensel's lemma]], this lifts to an $n^{th}$ root of $u$ in $B$. The element $u^{1/n} \pi$ is thus an $n^{th}$ root of $x$, and is a generator of the maximal ideal of $B$. Writing this element as $y = x^{1/n}$, the ring $K[ [y] ] = A[x^{1/n}] \hookrightarrow B$ is an $A$-submodule of full rank $n$ and integrally closed (being abstractly isomorphic to $A$, which is [[integral closure|integrally closed]] because it's a [[principal ideal domain]] and therefore a [[unique factorization domain]]), so that $K[ [y] ] = B$, as was to be shown.
=--

+-- {: .un_example}
###### Example
(Intend to solve for $y$ in $y^3 - x y + 1 = 0$ as a Puiseux series in $x$.)
=--

## Related pages

Other rings of generalized power series include:

* [[Novikov field]]
* [[Hahn series]]
* [[Ribenboim power series]]

Hahn series are a special kind of Ribenboim power series, but Puiseux and Novikov series are not.  However, they are all instances of the linearization of a [[finiteness space]].

## References

* wikipedia [Puiseux series](http://en.wikipedia.org/wiki/Puiseux_series)

The sketched proof of theorem \ref{Puiseux} was extracted from notes on a seminar by Boyarchenko on local class field theory:

* Mitya Boyarchenko, Kottwitz Seminar Lectures, (U. Michigan, Winter 2011). ([pdf](http://www.math.lsa.umich.edu/seminars/kottwitz/HieuNgoLocalCFT.pdf))

and the reader may refer to the classic text by Serre for a fuller treatment:

* Jean-Pierre Serre, _Local Fields_ (trans. Marvin J. Greenberg), Graduate Texts in Math. 67, Springer 1980.

For a noncommutative generalization see

* D. Grigoriev, _Analogue of Newton-Puiseux series for non-holonomic D-modules and factoring_, Moscow Math. J. __9__, 2009, 775--800, [pdf](http://logic.pdmi.ras.ru/~grigorev/pub/puiseux_journal.pdf); MPIM2008-6, [pdf](http://www.mpim-bonn.mpg.de/preblob/3537)

Other references

* Jan Kiwi, _Puiseux series dynamics of Quadratic rational maps_,  [arxiv/1106.0059](http://arxiv.org/abs/1106.0059)
* Luis Felipe Tabera, _On real tropical bases and real tropical discriminants_, [arxiv/1311.2211](http://arxiv.org/abs/1311.2211)

> We explore the concept of real tropical basis of an ideal in the field of real Puiseux series. We show explicit tropical bases of zero-dimensional real radical ideals, linear ideals and hypersurfaces coming from combinatorial patchworking. But we also show that there exist real radical ideals that do not admit a tropical basis. As an application, we show how to compute the set of singular points of a real tropical hypersurface. i.e. we compute the real tropical discriminant.

* Evelia García Barroso, Pedro Daniel González Pérez, Patrick Popescu-Pampu, _Variations on inversion theorems for Newton-Puiseux series_, [arxiv/1606.08029](http://arxiv.org/abs/1606.08029)

[[!redirects Newton-Puiseux series]]