right Bousfield delocalization

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###Context###
#### Model category theory
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[[!include model category theory - contents]]
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#### Locality and descent
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[[!include descent and locality - contents]]
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#Contents#
* table of contents
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## Idea

A [[model category]] structure $M_1$ is a
[[right Bousfield localization]] of a model structure $M_2$ on the same
underlying [[category]] if $M_1$ and $M_2$ have the same [[fibrations]]
and the [[weak equivalences]] of $M_1$ contain those
of $M_2$

The notion of _right Bousfield delocalization_ reverses this relation: $M_1$ is a right Bousfield delocalization of $M_2$ if $M_2$ is a right Bousfield localization of $M_1$.

Of course, the nontrivial task here
is to establish interesting existence criteria
for right Bousfield delocalizations.

## Existence theorem

+-- {: .num_theorem}
###### Theorem (Corrigan-Salter)

If $M_1$ and $M_2$ are two [[cofibrantly generated model category]]
structures on the same category with coinciding classes of fibrations,
then there is a third cofibrantly generated model structure $M_3$ with the same fibrations and whose weak equivalences are the [[intersection]] of weak equivalences in $M_1$ and $M_2$. This model structure is a right Bousfield delocalization
of both $M_1$ and $M_2$.

=--

([Corrigan-Salter 15](#CS15))

## Related entries

* [[right Bousfield localization]]

* [[colocalization]]

## References

* {#CS15} [[Bruce Corrigan-Salter]], _Right delocalization of model categories_, [arXiv:1504.04545](http://arxiv.org/abs/1504.04545).


right Bousfield localization of model categories

## Idea

Right Bousfield localizations (alias **cellularizations**) of [[model categories]] are a way to encode [[coreflective localizations]] of [[(∞,1)-categories]] in the setting of [[model categories]].

Given a [[model category]] $M$,
the formal definition and elementary properties can be deduced from the definition and properties of [[left Bousfield localizations]] of the [[opposite category]] of $M$.

However, the typical existence theorems for right Bousfield localizations are not formulated in this manner, since typical conditions involve some form of [[local presentability]] of the underlying category, or an analogous condition, which typically do not hold for the [[opposite category]].

## Barwick's existence theorem

If $C$ is a [[tractable model category]], $D$ is an accessible accessibly embedded subcategory of $C$ that is closed under [[homotopy colimits]] in $C$.
Then there is a right Bousfield localization of $C$, which is a [[right model category]] (but not necessarily a [[model category]]), and its colocal objects are precisely the object of $D$.

## Related concepts

* [[left Bousfield localization of model categories]]

## References

The original article:

* {#Bousfield75} [[A. K. Bousfield]], *The localization of spaces with respect to homology*, Topology **14** (1975), 133–150 [doi:10.1016/0040-9383(75)90023-3]

Detailed discussion (including existence results for [[left proper model category|left proper]] [[cellular model category|cellular model categories]]):

* {#Hirschhorn02} [[Philip Hirschhorn]], Chapter 5 of: _[[Model Categories and Their Localizations]]_, AMS Math. Survey and Monographs 99 (2002) [[ISBN:978-0-8218-4917-0](https://bookstore.ams.org/surv-99-s/), [pdf toc](http://www.gbv.de/dms/goettingen/360115845.pdf), [pdf](http://www.maths.ed.ac.uk/~aar/papers/hirschhornloc.pdf)]

Existence results for [[combinatorial model categories]]:

* {#Barwick10} [[Clark Barwick]], *On left and right model categories and left and right Bousfield localizations*, Homology, Homotopy and Applications **12** 2 (2010) 245–320 [[doi:10.4310/hha.2010.v12.n2.a9](https://doi.org/10.4310/hha.2010.v12.n2.a9), [euclid:1296223884](https://projecteuclid.org/journals/homology-homotopy-and-applications/volume-12/issue-2/On-left-and-right-model-categories-and-left-and-right/hha/1296223884.full), subsuming: [arXiv:0708.2067](https://arxiv.org/abs/0708.2067), [arXiv:0708.2832](https://arxiv.org/abs/0708.2832), [arXiv:0708.3435](http://arxiv.org/abs/0708.3435)]

Right Bousfield localization specifically for [[stable model categories]] (such as [[model structures on spectra]]):

* D. Barnes and C. Roitzheim. Stable left and right Bousfield localisations. Glasg. Math. J., 56(1):13–42, 2014.

[[!redirects right Bousfield localization]]
[[!redirects right Bousfield localizations]]
[[!redirects right Bousfield localisation]]
[[!redirects right Bousfield localisations]]
~


right cancellative category > history

< [[cancellative category]]

filler to avoid the spam filter more filler


right-connected double category

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###Context###
#### 2-category theory
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\tableofcontents

## Idea

A [[double category]] consists of [[objects]], two [[classes]] of [[1-morphisms]] ([[horizontal morphism|horizontal and vertical]]), and [[2-morphisms]] between these.

The idea behind a right-connected double category is that each vertical morphism $f \colon A \nrightarrow B$ has an *[[underlying]]* horizontal morphism $U f \colon A \rightarrow B$.
Therefore, the vertical morphisms should be understood as horizontal morphisms with certain [[properties]] or equipped with additional [[structure]].
Correspondingly, the main examples of right-connected double categories arise from [[orthogonal factorization systems]] or [[algebraic weak factorization systems]], where the vertical morphisms are in the *right class* of the factorization system.

The concept of a right-connected double category is unrelated to the notion of a "[[connection on a double category]]".

## Definition

A [[double category]] $\mathbb{D} = (D_{0}, D_{1})$ is *right-connected* if its [[identity morphisms|identity]]-assigning map $id \colon D_{0} \rightarrow D_{1}$ is [[right adjoint]] to its [[codomain]]-assigning map $cod \colon D_{1} \rightarrow D_{0}$.

Dually, a double category $\mathbb{D} = (D_{0}, D_{1})$ is *left-connected* if its [[identity morphisms|identity]]-assigning map $id \colon D_{0} \rightarrow D_{1}$ is [[left adjoint]] to its [[domain]]-assigning map $dom \colon D_{1} \rightarrow D_{0}$.

\begin{remark}
This definition makes sense for any pseudo [[category object]] in a [[2-category]].
\end{remark}

Let $\mathbf{RcDbl}$ denote the [[2-category]] of right-connected double categories, *unitary* [[double functors]] (those which preserve vertical identities strictly), and [[vertical transformation|horizontal transformations]].

### Unpacking the definition

We can unpack the definition of right-connectedness as follows.
Let $\rho \colon 1_{D_{1}} \Rightarrow id \circ cod$ denote the [[unit of an adjunction|unit]] of the adjunction $cod \dashv id$, and let the [[counit of an adjunction|counit]] be the identity natural transformation by the axioms of an [[internal category]].
Given a vertical morphism $f \colon A \nrightarrow B$ in $\mathbb{D}$, the component of the unit may be depicted as follows:
\begin{tikzcd}
A
\arrow[rd, phantom, "\rho_{f}"]
\arrow[d, "f"', "\bullet"{anchor=center}]
\arrow[r, "Uf"]
& B
\arrow[d, "id_{B}", "\bullet"{anchor=center}]
\\
B
\arrow[r, "1_{B}"']
& B
\end{tikzcd}
The [[triangle identities]] of an adjunction imply that:

1. the bottom boundary of $\rho_{f}$ is an identity morphism;

2. the component of $\rho$ at an identity vertical morphism $id_{A} \colon A \nrightarrow A$ is the identity cell on $A$.

Given a cell $\alpha$ in $\mathbb{D}$, [[natural transformation|naturality]] of $\rho$ states that the cells
  \begin{tikzcd}
    A
    \arrow[r, "h"]
    \arrow[d, "\bullet"{anchor=center}, "f"']
    \arrow[rd, phantom, "\alpha"]
    & C
    \arrow[d, "\bullet"{anchor=center}, "g"]
    \arrow[r, "Ug"]
    \arrow[rd, phantom, "\rho_{g}"]
    & D
    \arrow[d, "\bullet"{anchor=center}, "id_{D}"]
    \\
    B
    \arrow[r, "k"']
    & D
    \arrow[r, "1_{D}"']
    & D
    \end{tikzcd}
and
    \begin{tikzcd}
    A
    \arrow[r, "Uf"]
    \arrow[d, "\bullet"{anchor=center}, "f"']
    \arrow[rd, phantom, "\rho_{f}"]
    & B
    \arrow[d, "\bullet"{anchor=center}, "id_{B}"]
    \arrow[r, "k"]
    \arrow[rd, phantom, "id_{k}"]
    & D
    \arrow[d, "\bullet"{anchor=center}, "id_{D}"]
    \\
    B
    \arrow[r, "1_{B}"']
    & B
    \arrow[r, "k"']
    & D
    \end{tikzcd}
are equal.

## Examples

\begin{example}\label{horizontal-double-category}
For each category $C$, the horizontal double category $\mathbb{H}(C)$ ---whose objects and horizontal morphisms come from $C$, and whose vertical morphisms and cells are identities --- is right-connected.

Viewing double categories as [[internal categories]] in $\mathbf{Cat}$, we can see that $\mathbb{H}(C) = (C, C)$ is the discrete internal category on $C$,
with [[identity morphisms|identity]]-assigning map and  [[codomain]]-assigning map are both the [[identity functor]] on $C$ and therefore adjoint to each other.
\end{example}

\begin{example}\label{double-category-of-squares}
For each category $C$, the [[double category of squares]] $\mathbb{S}q(C)$ is both right-connected and left-connected.

More generally, if $C$ is a category equipped with a [[wide subcategory]] $B$, the double category $\mathbb{S}q(C, B)$ -- whose objects and horizontal morphisms come from $C$, whose vertical morphisms comes from $B$, and whose cells are commutative squares -- is both right-connected and left-connected.
\end{example}

If we modify the above example to consider only [[pullback squares]] or [[pushout|pushout squares]], then it is no longer right-connected or left-connected; we must also modify the classes of vertical morphisms.

\begin{example}
Consider a category $C$ equipped with a wide subcategory $M$ of [[monomorphisms]] stable under pullback along morphisms in $C$.
The double category $\mathbb{P}b(C, M)$ --- whose objects and horizontal morphisms come from $C$, whose vertical morphisms comes from $M$, and whose cells are pullback squares --- is left-connected.

Dually, consider a category $C$ equipped with a wide subcategory $E$ of [[epimorphisms]] stable under pushout along morphisms in $C$.
The double category $\mathbb{P}o(C, E)$ --- whose objects and horizontal morphisms come from $C$, whose vertical morphisms comes from $E$, and whose cells are pushout squares --- is right-connected.
\end{example}

\begin{example}\label{split-epimorphisms}
Given a category $C$, let $\mathbb{S}plEpi(C)$ denote the double category whose objects and horizontal morphisms come from $C$, whose vertical morphisms are [[split epimorphisms]] with a chosen [[section]], and whose cells are squares
\begin{tikzcd}
A
\arrow[d, shift left = 2, "f"]
\arrow[r, "h"]
& C
\arrow[d, shift left = 2, "g"]
\\
B
\arrow[u, shift left = 2, "s"]
\arrow[r, "k"']
& D
\arrow[u, shift left = 2, "t"]
\end{tikzcd}
such that $k \circ f = g \circ h$ and $h \circ s = t \circ k$.
This double category is right-connected, since for each vertical morphism there is a cell:
\begin{tikzcd}
A
\arrow[d, shift left = 2, "f"]
\arrow[r, "f"]
& B
\arrow[d, shift left = 2, "1_{B}"]
\\
B
\arrow[u, shift left = 2, "s"]
\arrow[r, "1_{B}"']
& B
\arrow[u, shift left = 2, "1_{B}"]
\end{tikzcd}
If $C$ has pullbacks, the [[codomain]]-assigning map of this double category is not only a left adjoint, but also a [[Grothendieck fibration]], namely, the [[fibration of points]].
\end{example}

\begin{example}
The double category $\mathbb{S}plFib$ --- whose objects are [[categories]], whose horizontal morphisms are [[functors]], whose vertical morphisms are [[cleavage|split]] [[Grothendieck fibrations]], and whose [[2-morphisms]] are commutative squares which preserve chosen [[cartesian lifts]] --- is right-connected.
\end{example}

\begin{example}
The double category $\mathbb{L}ens$ --- whose objects are categories, whose horizontal morphisms are functors, and whose vertical morphisms are [[delta lenses]] --- is right-connected (see [Clarke, Section 3.3](#Clarke2022)).
\end{example}

The previous three examples (although Example \ref{split-epimorphisms} only if $C$ has coproducts) are all instances of the following.

\begin{example}\label{awfs}
For each [[algebraic weak factorization system]] $(R, L)$ on a category $C$, the double category $R$-$\mathbb{A}lg$ of $R$-algebras is right-connected.
\end{example}

\begin{example}\label{ofs}
Consider an [[orthogonal factorization system]] $(R, L)$ on a category $C$, where $R$ denotes the right class and $L$ denotes the left class.
Then the double category $\mathbb{S}q(C, R)$ is right-connected.
This is a special case of both Example \ref{double-category-of-squares} and
Example \ref{awfs}.
\end{example}

## Properties

\begin{proposition}
If a double category is right-connected, then it has all [[cotabulators|1-cotabulators]].
\end{proposition}

\begin{proposition}\label{double-functor-U}
If $\mathbb{D} = (D_{0}, D_{1})$ is a right-connected double category,
then there is a [[double functor]] $U \colon \mathbb{D} \rightarrow \mathbb{S}q(D_{0})$ with assignment given below.
\begin{tikzcd}
A
\arrow[r, "h"]
\arrow[d, "\bullet"{anchor=center}, "f"']
\arrow[rd, phantom, "\alpha"]
& C
\arrow[d, "\bullet"{anchor=center}, "g"]
& \phantom{X}
\arrow[d, phantom, "\longmapsto"]
& A
\arrow[r, "h"]
\arrow[d, "Uf"']
& C
\arrow[d, "Ug"]
\\
B
\arrow[r, "k"']
& D
& \phantom{X}
& B
\arrow[r, "k"']
& D
\end{tikzcd}
\end{proposition}
A proof can be found in [Bourke & Garner, Section 3.5](#BourkeGarner2016a).

Let $U_{1} \colon D_{1} \rightarrow Sq(D_{0})$ denote the functor underlying the double functor $U \colon \mathbb{D} \rightarrow \mathbb{S}q(D_{0})$ above.

\begin{proposition}
A right-connected double category is [[thin category|thin]] if and only if the functor $U_{1} \colon D_{1} \rightarrow Sq(D_{0})$ is [[faithful]].
\end{proposition}
In the terminology of [Bourke & Garner, Section 2.8](#BourkeGarner2016a), a thin right-connected double category $\mathbb{D} = (D_0, D_1)$ is an example of a *concrete double category* over $D_{0}$.

Let $Obj \colon \mathbf{RcDbl} \rightarrow \mathbf{Cat}$ denote the [[2-functor]] which assigns a double category $\mathbb{D} = (D_{0}, D_{1})$ to
its category $D_{0}$ of objects and horizontal morphisms.
Let $\mathbb{H} \colon \mathbf{Cat} \rightarrow \mathbf{RcDbl}$ be the $2$-functor which assigns each category to its horizontal double category (see Example \ref{horizontal-double-category}), and let $\mathbb{S}q \colon  \mathbf{Cat} \rightarrow \mathbf{RcDbl}$ be the $2$-functor which assigns each category to its double category of squares (see Example \ref{double-category-of-squares}).

\begin{proposition}
There is an [[adjoint triple]] of $2$-functors:
\begin{tikzcd}
\mathbf{RcDbl}
\arrow[r, "\mathrm{Obj}" description]
& \mathbf{Cat}
\arrow[l, phantom, shift right = 5, "\bot"]
\arrow[l, phantom, shift left = 5, "\bot"]
\arrow[l, "\mathbb{H}"', shift right = 8, hook']
\arrow[l, "\mathbb{S}q", shift left = 8, hook']
\end{tikzcd}
\end{proposition}
\begin{proof}(*Idea*)
The component of the counit of the adjunction $\mathbb{H} \dashv \mathrm{Obj}$ at a right-connected double category $\mathbb{D} = (D_{0}, D_{1})$ is given by the double functor $\mathbb{H}(D_{0}) \rightarrow \mathbb{D}$ determined by the pair of functors $(1 \colon D_{0} \rightarrow D_{0}, id \colon D_{0} \rightarrow D_{1})$.
The component of the unit of the adjunction $\mathrm{Obj} \dashv \mathbb{S}q$ is given in Proposition \ref{double-functor-U}.
\end{proof}

Let $\mathbf{AWFS}_{lax}$ denote the $2$-category of [[algebraic weak factorization systems]].
There is a $2$-functor $\mathbf{AWFS}_{lax} \rightarrow \mathbf{RcDbl}$ which assigns each algebraic weak factorisation system to its double category of $R$-algebras (see Example \ref{awfs}).
The following theorem characterises the [[essential image]] of the this $2$-functor.

\begin{theorem}
The $2$-functor $\mathbf{AWFS}_{lax} \rightarrow \mathbf{RcDbl}$ has in its essential image exactly those right-connected double categories
$\mathbb{D} = (D_0, D_1)$ for which the functor $U_{1} \colon D_{1} \rightarrow Sq(D_{0})$ is [[monadic|strictly monadic]].
\end{theorem}
The proof combines the results in Theorem 6 and Proposition 11 of [Bourke & Garner](#BourkeGarner2016a).

\begin{corollary}
An [[orthogonal factorization system]] correspond exactly those right-connected double categories
$\mathbb{D} = (D_0, D_1)$ for which the functor $U_{1} \colon D_{1} \rightarrow Sq(D_{0})$ is [[fully faithful]] and [[monadic|strictly monadic]].
\end{corollary}
This is essentially [Bourke & Garner, Proposition 3](#BourkeGarner2016a).

## References

The notion was first defined in Section 3.5 of:

 * {#BourkeGarner2016a} [[John Bourke]], [[Richard Garner]], _Algebraic weak factorisation systems I: Accessible AWFS_, Journal of Pure and Applied Algebra, **220**, 2016 ([arXiv:1412.6559](https://arxiv.org/abs/1412.6559), [doi:10.1016/j.jpaa.2015.06.002](https://doi.org/10.1016/j.jpaa.2015.06.002))

Further discussion of right-connected double categories appears in Section 2.5 of:

 * {#BourkeGarner2016b} [[John Bourke]], [[Richard Garner]], _Algebraic weak factorisation systems II: Categories of weak maps_, Journal of Pure and Applied Algebra, **220**, 2016 ([arXiv:1412.6560](https://arxiv.org/abs/1412.6560), [doi:10.1016/j.jpaa.2015.06.003](https://doi.org/10.1016/j.jpaa.2015.06.003))

The example of the double category of [[delta lenses]] first appeared in:

* {#Clarke2022} [[Bryce Clarke]], _The double category of lenses_, PhD thesis, Macquarie University, 2022. ([doi:10.25949/22045073.v1](https://doi.org/10.25949/22045073.v1))

[[!redirects right-connected double categories]]
[[!redirects left-connected double category]]
[[!redirects left-connected double categories]]


right-handed neutrino

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### Context
#### Physics
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[[!include physicscontents]]
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#### Fields and quanta
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[[!include fields and quanta - table]]
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#Contents#
* table of contents
{:toc}

## Idea

(...)

## Related concept

* [[neutrino]]

* [[ANITA experiment]]

## References

### General

See also

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

Attempt to explain the anomalies possibly seen by the [[ANITA experiment]] by sterile neutrinos:

* {#ChauhanMohanty18} Bhavesh Chauhan, Subhendra Mohanty, _A common leptoquark solution of flavor and ANITA anomalies_ ([arXiv:1812.00919](https://arxiv.org/abs/1812.00919))

[[!redirects right-handed neutrinos]]

[[!redirects sterile neutrino]]
[[!redirects sterile neutrinos]]


# Right/left fibrations of simplicial spaces

* table of contents
{: toc}

## Idea

The analogue for [[complete Segal space|(complete) Segal spaces]] of the notion of [[right/left Kan fibration]] for [[quasi-categories]].

## References

* P. B de Brito, *Segal Objects and the Grothendieck Construction*, 2016. [arXiv](https://arxiv.org/abs/1605.00706)

* Nima Rasekh, *Yoneda Lemma for Simplicial Spaces*, 2017. [arxiv](https://arxiv.org/abs/1711.03160)

* Nima Rasekh, *Cartesian Fibrations and Representability*, 2017.  [arxiv](https://arxiv.org/abs/1711.03670)

[[!redirects left fibration of simplicial spaces]]
[[!redirects right/left fibration of simplicial spaces]]
[[!redirects left/right fibration of simplicial spaces]]
[[!redirects right fibration of simplicial spaces]]

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[[!redirects left fibration of complete Segal spaces]]
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right/left Kan fibration

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###Context###
#### $(\infty,1)$-Category theory
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[[!include quasi-category theory contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

For ordinary [[categories]] there is the notion of

1. [[Grothendieck fibration]] between two categories,

2. and the special case of a [[fibration fibered in groupoids]].

The analog of this for [[quasi-categories]] are

1. [[Cartesian fibration]]s

2. the special case of left/right (Kan-) [[fibrations of quasi-categories]]

respectively.

## Definition

A [[morphism]] of [[simplicial set]]s $f : X \to S$ is a **left fibration** or **left Kan fibration** if it has the [[right lifting property]] with respect to all [[horn]] inclusions $\Lambda[n]_k \to \Delta[n]$ except possibly the right outer ones: $0 \leq k \lt n$.

It is a **right fibration** or **right Kan fibration** if its extends against all horns except possibly the left outer ones: $0 \lt k \leq n$.

So $X \to S$ is a left fibration precisely if for all commuting squares

$$
  \array{
    \Lambda[n]_{k} &\to& X
    \\
    \downarrow &{}^{\exists}\nearrow& \downarrow
    \\
    \Delta[n] &\to& S
  }
$$

for $n \in \mathbb{N}$ and $0 \leq k \lt n$, a diagonal lift exists as indicated.

Morphisms with the [[left lifting property]] against all left/right fibrations are called **left/right anodyne** maps.

Write

$$
  RFib(S) \subset sSet/S
$$

for the full [[SSet]]-[[subcategory]] of the [[overcategory]] of [[sSet]] over $S$ on those morphisms that are right fibrations.

This is a [[Kan complex]]-enriched category and as such an incarnation of the **[[(∞,1)-category]] of right fibrations**.
It is modeled by the [[model structure for left fibrations|model structure for right fibrations]]. For details on this see the discussion at [[(∞,1)-Grothendieck construction]].

## Motivation: ordinary fibrations in groupoids are right Kan fibrations {#GrothGrpdFibsAreRightKanFibs}

Ordinary [[fibration fibered in groupoids|categories fibered in groupoids]] have a simple characterization in terms of their [[nerve]]s. Let $N : Cat \to sSet$ be the [[nerve]] functor and for $p : E \to B$ a morphism in [[Cat]] (a [[functor]]), let $N(p) : N(E) \to N(B)$ be the corresponding morphism in [[sSet]].

Then

\begin{proposition}
\label{FibrationInGroupoidsAsRightKanFibrations}
A [[functor]] $p \colon E \to B$ is a  [[fibration fibered in groupoids|fibration in groupoids]] precisely if its [[nerve]] $N(p) \colon N(E) \to N(B)$ is a right Kan fibration of [[simplicial sets]].
\end{proposition}

To see this, first notice the following facts:

+-- {: .num_lemma }
###### Lemma 1

For $C$ a [[category]], the [[nerve]] $N(C)$ is [[coskeletal|2-coskeletal]]. In particular all $n$-spheres for $n \geq 3$ have unique fillers

$$
  \array{
    \partial \Delta[n] &\stackrel{\forall}{\to}& N(C)
    \\
    \downarrow & \nearrow_{\mathrlap{\exists !}}
    \\
    \Delta[n]
  }
    \;\;\;\;\;
    (n \geq 3)
$$

and (implied by that) all $n$-horns for $n \gt 3$ have fillers

$$
  \array{
    \partial \Lambda[n] &\stackrel{\forall}{\to}& N(C)
    \\
    \downarrow & \nearrow_{\mathrlap{\exists }}
    \\
    \Delta[n]
  }
  \;\;\;\;\;
  (n \gt 3)
  \,.
$$

=--

This is discussed at [[nerve]].

+-- {: .num_lemma }
###### Lemma 2

If $p : E \to B$ is an ordinary [[functor]], then $N(f) : N(E) \to N(B)$ is an [[inner fibration]], meaning that its has the [[right lifting property]] with respect to all _inner_ horn inclusions $\Lambda[n]_i \hookrightarrow \Delta[n]$ for $0 \lt i \lt n$.

=--

This is discussed at [[inner fibration]].

+-- {: .proof}
###### Proof of the proposition

From the above lemmas it follows that $N(p) : N(E) \to N(B)$ is a right fibration already precisely if it has the right lifting property with respect only to the three horn inclusions

$$
  \{  \Lambda[n]_n \hookrightarrow \Delta[n] | n = 1,2,3\}
  \,.
$$

So we check explicitly what these three conditions amount to

* $n=1$ -- The existence of all fillers

  $$
    \array{
      \Lambda[1]_1 = \Delta^{\{1\}} &\stackrel{e}{\to}& N(E)
      \\
      \downarrow & {}^{{\hat f}}\nearrow  & \downarrow^{\mathrlap{N(p)}}
      \\
      \Delta^{\{0 \to 1\}} &\stackrel{f}{\to}& N(B)
    }
  $$

  means that for all objects $e \in E$ and morphism $f : b \to p(e) $ in $B$, there exists a morphism $\hat f : \hat b \to e$ in $E$ such that $p(\hat f) = f$.

* $n=2$ -- The existence of fillers

  $$
    \array{
      \Lambda[2]_2 &\stackrel{e}{\to}& N(E)
      \\
      \downarrow & {}^{{\hat f}}\nearrow  & \downarrow^{\mathrlap{N(p)}}
      \\
      \Delta[2] &\stackrel{f}{\to}& N(B)
    }
  $$

  means that for all diagrams

  $$
    \array{
       && e_1
       \\
       &&& \searrow^{\mathrlap{\epsilon_{12}}}
       \\
       e_0 &&\stackrel{\epsilon_{02}}{\to}&& e_2
    }
  $$

  in $E$ and commuting triangles

  $$
    \array{
       && p(e_1)
       \\
       & {}^{\mathllap{f}}\nearrow&& \searrow^{\mathrlap{p(\epsilon_{12}})}
       \\
       p(e_0) &&\stackrel{p(\epsilon_{02})}{\to}&& p(e_2)
    }
  $$

  in $B$, there is a commuting triangle

  $$
    \array{
       && e_1
       \\
       &{}^{\mathllap{\hat f}}\nearrow&& \searrow^{\mathrlap{\epsilon_{12}}}
       \\
       e_0 &&\stackrel{\epsilon_{02}}{\to}&& e_2
    }
  $$

  in $E$, such that $p(\hat f ) = f$.

* $n=3$ -- ...

  Consider first the case of degenrate 3-simplices on $N(B)$,
  on 2-simplices as above.

  Suppose in the above situation two lifts $(\hat f)_1$ and $(\hat f)_2$
  are found. Together these yield a $\Lambda[3]_3$-horn in $N(E)$.
  The filler condition says this can be filled, which implies
  that $(\hat f)_1 = (\hat f)_2$.

  So the $n=3$-condition implies that the lift
  whose existence is guaranteed by the
  $n=2$-condition is unique.

  By similar reasoning one sees that this is all the $n=3$-condition yields.

In total, these three lifting conditions are precisely those
for a Grothendieck fibration in groupoids.

=--

## Properties

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

Under the operation of forming the [[opposite quasi-category]], left fibrations turn into right fibrations, and vice versa: if $p : C \to D$ is a left fibration then $p^{op} : C^{op} \to D^{op}$ is a right fibration.

Therefore it is sufficient to list properties of only one type of these fibrations, that for the other follows.

=--

### Homotopy lifting property

In classical [[homotopy theory]], a continuous map $p : E \to B$ of [[topological spaces]] is said to have the [[homotopy lifting property]] if it has the [[right lifting property]] with respect to all morphisms $Y \stackrel{(Id, 0)}{\to} Y \times I$ for $I = [0,1]$ the standard [[interval]] and every commuting diagram

$$
  \array{
    Y &\to& E
    \\
    \downarrow && \downarrow
    \\
    Y \times I &\to& B
  }
$$

there exists a lift $\sigma : Y \times I \to E$ making the two triangles

$$
  \array{
    Y &\to& E
    \\
    \downarrow &{}^\sigma\nearrow& \downarrow
    \\
    Y \times I &\to& B
  }
$$

commute. For $Y = *$ the [[point]] this can be rephrased as saying that the
universal morphism $E^I \to B^I \times_B E$ induced by the commuting square

$$
  \array{
    E^I &\to& E
    \\
    \downarrow && \downarrow
    \\
    B^I &\to& B
  }
$$

is an [[epimorphism]]. If it is even an [[isomorphism]] then the lift $\sigma$ exists _uniquely_ . This is the situation that the following proposition generalizes:

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

A morphism $p : X \to S$ of simplicial sets is a left fibration precisely if the canonical morphism

$$
  X^{\Delta[1]} \to X^{\{0\}} \times_{S^{\{0\}}} S^{\Delta^1}
$$

is a trivial Kan fibration.

=--

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

This is a corollary of the characterization of left anodyne morphisms in [Properties of left anodyne maps](#PropRightAnodyne) by [[Andre Joyal]], recalled in [[Higher Topos Theory|HTT, corollary 2.1.2.10]].

=--

By the [[pasting law for pullbacks]] this implies

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

Let $p : X \to S$ be a left fibration of simplicial sets, and $f : Y \to X$ be a morphism of simplicial sets. Then $f$ is a left fibration iff $p f$ is a left fibration

=--

### As fibrations in $\infty$-groupoids
 {#AsFibrationsInInfinityGroupoids}

The notion of right fibration of quasi-categories generalizes the notion of [[category fibered in groupoids]]. This follows from the following properties.

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

Over a [[Kan complex]] $T$, left fibrations $S \to T$ are automatically [[Kan fibration]]s.

=--

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

This appears as [[Higher Topos Theory|HTT, prop. 2.1.3.3]].

=--

As an important special case:

\begin{corollary}

For $C \to *$ a right (left) fibration over the [[point]], $C$ is a [[Kan complex]], i.e. an [[∞-groupoid]].

\end{corollary}

\begin{proof}

This is originally due to [[Andre Joyal]]. Recalled at [[Higher Topos Theory|HTT, prop. 1.2.5.1]].

\end{proof}

\begin{proposition}
 \label{LeftFibrationsArePreservedByPullback}
Right (left) fibrations are preserved by [[pullback]] in [[sSet]].
\end{proposition}
\begin{proof}
  This follows on general grounds, since they are defined by a right [[lifting property]] (see, e.g. [here](Introduction+to+Homotopy +Theory#ClosurePropertiesOfInjectiveAndProjectiveMorphisms))
\end{proof}

\begin{proposition}
  \label{PullbackOfLeftFibrationsIsHomotopyPullback}
  The pullback (in [[SimplicialSets]]) of a left or right fibration is a [[homotopy pullback]] in the [[Joyal model structure for quasi-categories]]
\end{proposition}

([Cisinski 2019, Cor. 5.3.6](#Cisinski19))

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

It follows that the fiber $X_c$ of every right fibration $X \to C$ over every point $c \in C$, i.e. the [[pullback]]

$$
  \array{
    X_c &\to& X
    \\
    \downarrow && \downarrow
    \\
    \{c\} &\to& C
  }
$$

is a [[Kan complex]].

=--

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

For $C$ and $D$ quasi-categories that are ordinary [[categories]] (i.e. simplicial sets that are [[nerve]]s of ordinary categories), a morphism $C \to D$ is a right fibration precisely if the correspunding ordinary [[functor]] exhibits $C$ as a [[category fibered in groupoids]] over $D$.

=--

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

This is [[Higher Topos Theory|HTT, prop. 2.1.1.3]].

=--

A canonical class of examples of a [[fibered category]] is the [[codomain fibration]]. This is actually a [[bifibration]]. For an ordinary category, a bifiber of this is just a set. For an $(\infty,1)$-category it is an $\infty$-groupoid. Hence fixing only one fiber of the bifibration should yield a fibration in $\infty$-groupoids. This is asserted by the following statement.

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

Let $p : K \to C$ be an arbitrary morphism to a [[quasi-category]] $C$ and let $C_{p/}$ be the corresponding [[over quasi-category|under quasi-category]]. Then the canonical propjection $C_{p/} \to C$ is a left fibration.

=--

Due to [[Andre Joyal]]. Recalled as [[Higher Topos Theory|HTT, prop 2.1.2.2]].

### (Left/)Right anodyne morphisms {#PropRightAnodyne}

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

The collection of left anodyne morphisms (those with [[left lifting property]] against left fibrations) is equivalently $LAn = LLP(RLP(LAn_0))$ for the following choices of $LAn_0$:

$LAn_0 =$

1. the collection of all left [[horn]] inclusions

  $\{ \Lambda[n]_{i} \to \Delta[n] | 0 \leq i \lt n \}$;

1. the collection of all inclusions of the form

   $$
     (\Delta[m] \times \{0\})
     \coprod_{\partial \Delta[m] \times \{0\}}
     (\partial \Delta[m] \times \Delta[1])
     \hookrightarrow
     \Delta[m] \times \Delta[1]
   $$

1. the collection of all inclusions of the form

   $$
     (S' \times \{0\})
     \coprod_{S \times \{0\}}
     (S \times \Delta[1])
     \hookrightarrow
     S' \times \Delta[1]
   $$

   for all inclusions of simplicial sets $S \hookrightarrow S'$.

=--

This is due to [[Andre Joyal]], recalled as [[Higher Topos Theory|HTT, prop 2.1.2.6]].

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

...

=--

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

For $i : A \to A'$ left-anodyne and $j : B \to B'$ a cofibration in the [[model structure for quasi-categories]], the canonical morphism

$$
  (A \times B') \coprod_{A \times B} (A' \times B)
  \to
  A' \times B'
$$

is left-anodyne.

=--

This appears as [[Higher Topos Theory|HTT, cor. 2.1.2.7]].

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

For $p : X \to S$ a left fibration and $i : A \to B$ a cofibration of simplicial sets, the canonical morphism

$$
  q : X^B \to X^A \times_{S^A} S^B
$$

is a left fibration. If $i$ is furthermore left anodyne, then it is an acyclic [[Kan fibration]].

=--

This appears as [[Higher Topos Theory|HTT, cor. 2.1.2.9]].

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

For $f : A_0 \to A$ and $g : B_0 \to B$ two inclusions of [[simplicial set]]s with $f$ left anodyne, we have that the canonical morphism

$$
  (A_0 \star B ) \coprod_{A_0 \star B_0} (A \star B_0)
  \to
  A \star B
$$

into the [[join of simplicial sets]] is left anodyne.

=--

This is due to [[Andre Joyal]]. It appears as [[Higher Topos Theory|HTT, lemma 2.1.4.2]].

+-- {: .num_prop}
###### Proposition
**(restriction of over-quasi-categories along left anodyne inclusions)**

Let $p : B \to S$ be a morphism of [[simplicial set]]s and $i : A \to B$ a left anodyne morphism, then the restriction morphism of [[over quasi-categories|under quasi-categories]]

$$
  S_{/p} \to S_{/p|_A}
$$

is an acyclic [[Kan fibration]].

=--

This is a special case of what appears as [[Higher Topos Theory|HTT, prop. 2.1.2.5]], which is originally due to [[Andre Joyal]].

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

Let $p : X \to S$ be a morphism of simplicial sets with [[section]] $s : S \to X$. If there is a fiberwise simplicial [[homotopy]] $X \times \Delta[1] \to S$ from $s \circ p $ to $Id_X$ then $s$ is left anodyne.

=--

This appears as [[Higher Topos Theory|HTT, prop. 2.1.2.11]].

## Related concepts

* [[Kan fibration]], [[anodyne morphism]]

* **right/left Kan fibration**, [[right/left anodyne map]]

  * [[model structure for left fibrations]]

  * [[universal right fibration]]

* [[inner fibration]]

* [[Cartesian fibration]]

* [[right/left fibration of simplicial spaces]]

## References

* {#Lurie09} [[Jacob Lurie]], Section 2.1 of: _[[Higher Topos Theory]]_, Annals of Mathematics Studies **170**, Princeton University Press 2009 ([pup:8957](https://press.princeton.edu/titles/8957.html), [pdf](https://www.math.ias.edu/~lurie/papers/HTT.pdf))

* {#AyalaFrancis17} [[David Ayala]], [[John Francis]], _Fibrations of $\infty$-Categories_ ([arXiv:1702.02681](https://arxiv.org/abs/1702.02681))

* {#Cisinski19} [[Denis-Charles Cisinski]], Section 3.4 and 5.3 of: _[[Higher Categories and Homotopical Algebra]]_,  Cambridge University Press 2019 ([doi:10.1017/9781108588737](https://doi.org/10.1017/9781108588737), [pdf](http://www.mathematik.uni-regensburg.de/cisinski/CatLR.pdf))

[[!redirects left/right Kan fibrations]]

[[!redirects left fibration]]
[[!redirects left fibrations]]
[[!redirects left Kan fibration]]
[[!redirects left Kan fibrations]]

[[!redirects right fibration]]
[[!redirects right fibrations]]
[[!redirects right Kan fibration]]
[[!redirects right Kan fibrations]]

[[!redirects left anodyne morphism]]
[[!redirects right anodyne morphism]]

[[!redirects left anodyne morphisms]]
[[!redirects right anodyne morphisms]]

[[!redirects left anodyne map]]
[[!redirects right anodyne map]]

[[!redirects left anodyne maps]]
[[!redirects right anodyne maps]]

[[!redirects right/left anodyne map]]


rigid analytic geometry

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

#Contents#
* table of contents
{:toc}

## Idea

_Rigid analytic geometry_ (often just "rigid geometry" for short) is a form of [[analytic geometry]] over a [[nonarchimedean field]] $K$ which considers [[spaces]] glued from [[polydiscs]], hence from [[maximal spectra]] of [[Tate algebras]] (quotients of a $K$-algebra of [[convergence|converging]] [[power series]]). This is in contrast to some modern approaches to [[non-Archimedean analytic geometry]] such as [[Berkovich spaces]] which are glued from Berkovich's [[analytic spectra]] and more recent Huber's [[adic spaces]].

The issue is that while the [[p-adic numbers]] are [[complete topological space|complete]] in the [[p-adic norm]], that [[topology]] is exotic: $\mathbb{Q}_p$ is a [[Stone space]], hence in particular a [[totally disconnected topological space]].

For that reason the naive idea of formulating [[p-adic geometry|p-adic]] analytic geometry in analogy to [[complex analytic geometry]] as modeled on domains in $\mathbb{Q}_p^n$, regarded with their [[subspace topology]], fails, as also all these domains are totally disconnected.

Instead there is ([Tate 71](#Tate71)) a suitable [[Grothendieck topology]] on such [[affinoid domains]] -- the _[[G-topology]]_ -- with respect to which there is a good theory of [[non-archimedean analytic geometry]] ("[[rigid analytic geometry]]") and hence in particular of [[p-adic geometry]]. Moreover, one may sensibly assign to a $p$-adic domain a [[topological space]] which _is_ well behaved (in particular locally connected and even locally contractible), this is the _[[analytic spectrum]]_ construction. The resulting topological spaces equipped with covers by [[affinoid domain]] under the [[analytic spectrum]] are called [[Berkovich spaces]].

According to [Kedlaya, p. 18](#Kedlaya), the terminology "rigid" is because...

> ... one develops everything "rigidly" by imitating the theory of [[schemes]] in [[algebraic geometry]], but using rings of [[convergence|convergent]] [[power series]] instead of [[polynomials]].

See also _[[global analytic geometry]]_.

## Cohomology

The related type of cohomology is called [[rigid cohomology]].

## Relation to perfectoid spaces

Any smooth rigid-analytic variety $X$ admits a cover $U_{i}\to X$ where the $U_{i}$'s are affinoid [[perfectoid spaces]] (corollary 4.7 of [Scholze12](#Scholze12)). Together with the fact that $H^{i}(U_{et},\mathcal{O}_{X}^{+})$ is almost zero for $i\geq 1$ when $U$ is an affinoid perfectoid space, this allows one to compute the cohomology $H_{et}^{i}(X,\mathcal{O}_{X}^{+})$. The Artin-Schreier sequence can then be used to compute the cohomology $H^{i}(X_{et},\mathbb{Z}/p\mathbb{Z})$ (see the discussion in section 5.7 of [Weinstein15](#Weinstein15)).

## Applications

* The solution by Raynaud and Harbater of Abyhankar's conjecture concerning fundamental groups of curves in positive characteristic uses the rigid analytic
GAGA theorems (whose proofs are very similar to Serre's proofs in the
complex-analytic case).

* Work of Kisin on modularity of Galois representations makes creative use of rigid-analytic spaces associated to [[Galois deformation rings]].

## Related concepts

* [[complex analytic space]]

* [[analytification]]

* [[global analytic geometry]]

* [[overconvergent global analytic geometry]]

## References

An original article is

* {#Tate71} [[John Tate]], _Rigid analytic spaces_, Invent. Math. __12__:257–289, 1971.

and for the construction of the generic fiber of formal schemes over the ring of integers of $K$

* Michel Raynaud, _Géométrie analytique rigide d'après Tate, Kiehl,⋯_, Table Ronde d'Analyse non archimédienne (Paris, 1972), pp. 319–327. Bull. Soc. Math. France, Mem. No. 39–40, Soc. Math. France, Paris, 1974, [MR470254](http://www.ams.org/mathscinet-getitem?mr=470254)

Introductions are in

* Johannes Nicaise, _Formal and rigid geometry: an intuitive introduction, and some applications_ ([pdf](http://gc83.perso.sfr.fr/GTIM/PDF%20GROUPE%20DE%20TRAVAIL/Nicaise/formal%20rigid%20Nicaise.pdf))

* [[Brian Conrad]], _Several approaches to non-Archimedean geometry_, [pdf](http://math.stanford.edu/~conrad/papers/aws.pdf)
* Peter Schneider, _Basic notions of rigid analytic geometry_, in: Galois representations in arithmetic algebraic geometry (Durham, 1996), 369–378, London Math. Soc. Lecture Note Ser. __254__, Cambridge Univ. Press 1998, [doi](http://dx.doi.org/10.1017/CBO9780511662010.010)

A comprehensive textbook account is in

* {#BoschGuntzerRemmert84} S. Bosch, U. Güntzer, [[Reinhold Remmert]], _[[Non-Archimedean Analysis]] -- A systematic approach to rigid analytic geometry_, Grundlehren der Mathem. Wissen. __261__, Springer 1984 ([MR0746961](http://www.ams.org/mathscinet-getitem?mr=0746961), [pdf](http://math.arizona.edu/~cais/scans/BGR-Non_Archimedean_Analysis.pdf))

Comparison of various spectra and topologies is in

* M. van der Put, P. Schneider, _Points and topologies in rigid geometry_, Math. Ann. 302 (1995), no. 1, 81–103, [MR96k:32070](http://www.ams.org/mathscinet-getitem?mr=1329448), [doi](http://dx.doi.org/10.1007/BF01444488)

Other accounts include

* Ahmed Abbes, _Éléments de Géométrie Rigide_, vol. I. _Construction et étude géométrique des espaces rigides_, Progress in Mathematics __286__, Birkhäuser 2011, 496 p.[book page](http://perso.univ-rennes1.fr/ahmed.abbes/egr.html)

* Siegfried Bosch, _Lectures on formal and rigid geometry_, Preprints of SFB Geom. Struk. Math. Heft 378, [pdf](http://wwwmath.uni-muenster.de/sfb/about/publ/heft378.pdf) (revised 2008)
* J. Fresnel, M. van der Put, _Rigid geometry and applications_, Birkhäuser (2004) [MR2014891](http://www.ams.org/mathscinet-getitem?mr=2014891)
* F. Denef, L. van den Dries, _$p$-adic and real subanalytic sets_, Ann. of Math. __128__ (1988) no. 1, 79–138 [MR951508](http://www.ams.org/mathscinet-getitem?mr=951508), [doi](http://dx.doi.org/10.2307/1971463)
* [[Yan Soibelman]], _On non-commutative analytic spaces over non-archimedean fields_, preprint IHES, [pdf](http://www.math.ksu.edu/~soibel/lapkin-spaces-2.ps)
* Hans Grauert, Reinhold Remmert, _Coherent analytic sheaves_, Springer 1984
* [[R. Cluckers]], L. Lipshitz, _Fields with analytic structure_, J. Eur. Math. Soc. __13__, 1147–1223, [pdf](http://wis.kuleuven.be/algebra/Raf/prints/JEMS.pdf)
and several articles (in various formalisms) in collection
* R. Cluckers, J. Nicaise, J. Sebag (Editors), _Motivic integration and its interactions with [[model theory]] and non-archimedean Geometry_, 2 vols.
London Mathematical Society Lecture Note Series __383__, __384__
* Peter Schneider, _Points of rigid analytic varieties_, J. Reine Angew. Math. 434 (1993), 127–157, [MR94b:14017](http://www.ams.org/mathscinet-getitem?mr=1195693), [doi](http://dx.doi.org/10.1515/crll.1993.434.127)

See also

* [[Kiran Kedlaya]], _$p$-Adic differential equations_ ([pdf](http://www-math.mit.edu/~kedlaya/18.787/compiled.pdf))
  {#Kedlaya}

The relation to perfectoid spaces can be found in

* [[Peter Scholze]], _$p$-adic Hodge theory for rigid-analytic varieties_ ([arXiv:1205.3463] (https://arxiv.org/abs/1205.3463)) {#Scholze12}

There is also a survey of the above in chapter 5 of

* Jared Weinstein, _Reciprocity laws and Galois representations: recent breakthroughs_ ([pdf] (https://math.bu.edu/people/jsweinst/CEB/BAMS.pdf)) {#Weinstein15}

category: geometry

[[!redirects rigid analytic geometries]]

[[!redirects rigid analytic space]]
[[!redirects rigid analytic spaces]]

[[!redirects rigid geometry]]
[[!redirects rigid geometries]]

[[!redirects non-archemedean analysis]]
[[!redirects non-archimedean analysis]]


rigid body dynamics

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

#Contents#
* table of contents
{:toc}

## Idea

The [[mechanics]] of _rigid bodies_ in [[Cartesian space]].

Up to the dynamics of the [[center of mass]], this is the special case of [[Hamiltonian dynamics on Lie groups]] for the case of the [[special orthogonal group]] $SO(n)$.

Often this is considered (only) for $n = 3$, which is the case pertaining to rigid bodies in observable nature, hence using [[SO(3)]].

## Related concepts

* [[non-relativistic particle]]

* [[Hamiltonian dynamics on Lie groups]]

  * **rigid body dynamics**

    * [[angular velocity]]

    * [[angular momentum]]

    * [[moment of inertia]]

* [[solid state physics]]

## References

A general introduction is in section 1.1d of

* [[Theodore Frankel]], _[[The Geometry of Physics - An Introduction]]_

A discussion from the more general perspective of [[Hamiltonian dynamics on Lie groups]] is in section 4.4 of

* [[Ralph Abraham]], [[Jerrold Marsden]], _[[Foundations of Mechanics]]_

* [[Vladimir Arnol'd]], _[[Mathematical methods of classical mechanics]]_, Springer 1989 (section 28)

* [[Landau-Lifschitz]], vol.1, Mechanics

* R. Abraham, J. Marsden: The Foundations of Mechanics, Benjamin Press, 1967, Addison-Wesley, 1978; large pdf 86 Mb free at [CaltechAuthors](http://resolver.caltech.edu/CaltechBOOK:1987.001)

A discussion of rigid body dynamics as a special case of the general [[Euler-Arnold equation]] is at

* [[Terence Tao]], _The Euler-Arnold equation_ ([web](http://terrytao.wordpress.com/2010/06/07/the-euler-arnold-equation/))

References from the point of view of [[Geometric Algebra]] include

* [[David Hestenes]], _Rotational dynamics with geometric algebra_ ([web](http://adsabs.harvard.edu/full/1983CeMec..30..133H))

* Terje Vold, _An introduction to geometric algebra with an application to rigid body mechanics_ ([pdf](http://www.helsinki.fi/kemia/fysikaalinen/opetus/matnum/Lectures/2009/Vold_1.pdf))

Discussion of ([[geometric quantization|geometric]]) [[quantization]] of rigid bodies is in

* M. Modugno, C. Prieto, R. Vitolo, _Geometric aspects of the quantization of a rigid body_, in B. Kruglikov, V. Lychagin, E. Straume, _Differential Equations – Geometry, Symmetries and Integrability_, Proceedings of the 2008 Abel Symposium, Springer, 275-285. ([pdf](http://poincare.unisalento.it/vitolo/vitolo_files/publications/proceedings/14c_quanrig-abel.pdf))

[[!redirects rigid body]]
[[!redirects rigid bodies]]


rigid cohomology

Rigid cohomology is certain extension of [[crystalline cohomology]] to schemes which are not necessarily proper or smooth, introduced in

* Pierre Berthelot, (1986), _Géométrie rigide et cohomologie des variétés algébriques de caractéristique p_, Mémoires de la Société Mathématique de France. Nouvelle Série (23): 7–32

It is also extending from the affine case the [[Monsky-Washnitzer cohomology]].

The corresponding theory of coefficients is given by [[overconvergent isocrystals]] and more generally [[arithmetic D-modules]].

* Kiran S. Kedlaya, Rigid cohomology and its coefficients, slides, [pdf](http://folk.uio.no/rognes/yff/kedlaya.pdf), workshop: p-adic Geometry and Homotopy Theory, Loen, 2009

* Kiran Kedlaya, lecture 20, [pdf](http://kskedlaya.org/18.787/rigid-cohom.pdf) from a course on algebraic geometry

* wikipedia [rigid cohomology](http://en.wikipedia.org/wiki/Rigid_cohomology)

* Denis-Charles Cisinski, Frédéric Déglise, _Mixed Weil cohomologies_, [arxiv/0712.3291](http://arxiv.org/abs/0712.3291)

* Bernard le Stum, Rigid cohomology, Cambridge Tracts in Mathematics, 172 (2007)

* David Brown, _Rigid cohomology for algebraic stacks_, Virginia 2010, [slides](http://www.math.wisc.edu/~brownda/slides/rigidCohStacksAMSVirginia.pdf)

Warning: rigid cohomology in contemporary sense should *not* be confused with the topic of 1977 MacKenzie's paper _Rigid cohomology for topological groupoids_ ([pdf](http://kchmackenzie.staff.shef.ac.uk/publications/RCTG.pdf))

See also [[arithmetic D-modules]].


rigid monoidal category

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###Context###
#### Monoidal categories
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[[!include monoidal categories - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _rigid (monoidal) category_, also called an _autonomous (monoidal) category_ is a kind of [[category with duals]]. In particular, a rigid monoidal category is a category in which every object $A$ has a dual object $A^*$. Depending on whether these duals exist on the right or the left we call the category left or right rigid. Sometimes _rigid_ refers to categories with duals on both sides.

It is important to note that every right/left rigid category can be fully embedded into a category with duals on both sides, by repeatedly applying the double dual functor and taking a colimit (see e.g. [Selinger](#Selinger2011), Proposition 4.7).

## Definition

A [[monoidal category]] is **right rigid** (or right autonomous) if every [[object]] has a right [[dualizable object|dual]]. Similarly, a monoidal category is **left rigid** (or left autonomous) if every object has a left dual. A monoidal category is autonomous (sometimes called rigid) if duals exist on both sides.

Conventions differ regarding which type of duals are which. One convention is as follows: a _right dual_ of an object $A$ in a monoidal category $\mathscr{C}$ is an object $A^*$ equipped with evaluation/co-evaluation maps

$$\text{ev}_{A}:A\otimes A^*\to 1,\,\, \text{coev}_{A}:1\to A^*\otimes A$$

where $1$ is the tensor unit, subject the the [[triangle identities]] (snake diagrams)

$$(\text{ev}_A\otimes \text{id}_A)\circ (\text{id}_A\otimes \text{coev}_{A})=\text{id}_{A},$$

$$(\text{id}_{A^*}\otimes \text{ev}_{A})\circ ( \text{coev}_{A}\otimes \text{id}_{A^*})=\text{id}_{A^*}.$$

A left dual is an object $A^*$ paired with morphisms $\text{ev}_A:A^*\otimes A\to 1$ and $\text{coev}_A:1\to A\otimes A^*$ satisfying the dual conditions.

## Examples

The canonical example of a rigid category is the category $\text{Vec}_k$ of finite dimensional [[vector space |vector spaces]] over a field $k$. Namely, given a vector space $V$ the standard dual $V^*=\text{Hom}_k(V,k)$ serves as a categorical dual for $V$. Explicitely we have the following:

\begin{proposition} Let $k$ be a field. For every vector space $V\in \text{Vec}_k$, define the maps

$$\text{ev}_V: V\otimes V^*\to k $$
$$(v,\varphi)\mapsto \varphi(v)$$

and

$$\text{coev}_V: k\to V^*\otimes V$$
$$x\mapsto x\cdot \sum_{i\in I}\varphi_i\otimes v_i$$

where $\{v_i\}_{i\in I}$ is a basis for $V$, and $\varphi_i$ is a dual basis. That is, $\varphi_i$ is defined by

$$\varphi_i(v_j)=
\begin{cases}
1 & i=j\\
0 & \text{otherwise}.
\end{cases}$$

These formulas give well defined $\text{Vec}_k$-morphisms. Moreover, the satisfy the snake identities and hence give $\text{Vec}_k$ the structure of a (right) rigid category.
\end{proposition}

\begin{proof} To begin, we verify that the definition of $\text{coev}$ is independent of basis. Suppose $\{\tilde{v}_i\}_{i\in I}$ is a different choice of basis. There is a change of basis matrix $c=(c_{i,j})_{(i,j)\in I\times I}$ between them. It is straightforward to see that the dual bases are related by the change of basis matrix $c^{-1}$. Expanding, we thus find that

$$\sum_{i}\tilde{v}_i\otimes \tilde{\varphi}_i=\sum_{i}\left(\sum_{j}c_{i,j}\cdot v_i\right)\otimes \left(\sum_{k}(c^{-1})_{k,i}\cdot \varphi_i\right)$$

$$=\sum_{j,k} \left(\sum_{i}c_{i,j}(c^{-1})_{k,i}\right)\cdot v_i\otimes \varphi_i.$$

The definition of the inverse matrix says that $\sum_{i}c_{i,j}(c^{-1})_{k,i}=\begin{cases}1 & j=k \\ 0 &\text{otherwise}\end{cases}$. Hence, we conclude that

$$\sum_{i}\tilde{v}_i\otimes \tilde{\varphi}_i=\sum_{i}v_i\otimes \varphi_i,$$

so $\coev_{V}$ is well defined as desired. We now show that the rigidity diagrams commute. Going around the square, we find that this is equivalent to the condition that $1\otimes w=\sum_{i}\varphi_i(w)\otimes v_i$ for all $w\in V$. This follows from choosing a basis such that $v_i=w$ for some $i$. Then, $\varphi_i(w)\otimes v_i=w$ and all the other terms are zero, making the equality obvious.
\end{proof}

Another key example is that of endofunctor categories Given any category $\mathscr{C}$, the hom-space of functors $\text{Fun}(\mathscr{C},\mathscr{C})$ is a monoidal category, with the monoidal structure coming from composition. Duals correspond to [[adjoint functor]] as can be stated formally below:

\begin{proposition} Let $\mathscr{C}$ be a category, and let $F\in \text{Fun}(\mathscr{C},\mathscr{C})$ be an endofunctor. Let $F^*$ be the right adjoint of $F$. $F^*$ is a right dual for $F$, under the natural transformations

$$\text{ev}_{F}:F\otimes F^*\to \text{id}_{\mathscr{C}}$$

$$F(F^*(A))\xrightarrow{\text{id}_{F^*(A)}} A,\,\, A\in \mathscr{C}$$

where we here make the implicit identification

$$\text{Hom}_{\mathscr{C}}(F(F^*(A)),A)\cong\text{Hom}_{\mathscr{C}}(F^*(A),F^*(A))$$

using adjointness, and

$$\text{coev}_{F}:\text{id}_{\mathscr{C}}\to F\otimes F^*$$

$$A\xrightarrow{\text{id}_{F(A)}} F^*(F(A)),\,\, A\in \mathscr{C}$$

where we here make the implicit identification

$$\text{Hom}_{\mathscr{C}}(A,F^*(F(A)))\cong\text{Hom}_{\mathscr{C}}(F(A),F(A))$$

using adjointness. Conversely, if $F^*$ is a right dual for an endofunctor $F$ then $F^*$ is a right adjoint for $F$.
\end{proposition}
\begin{proof} Verifying the details is straightforward.
\end{proof}

## Graphical language

Rigid categories are best expressed using the language of [[string diagrams]]. That is, we can write the rigidity axioms as

\begin{imagefromfile}
"file_name": "string-naive.png",
"width": 300,
"unit": "px"
\end{imagefromfile}

Even better, we often fix graphical notation for $\text{ev}$ and $\text{coev}$ which makes these axioms even simpler to write out:

\begin{imagefromfile}
"file_name": "string-notation-rigidity.png",
"width": 300,
"unit": "px"
\end{imagefromfile}

We can now express rigidity as follows:

\begin{imagefromfile}
"file_name": "string-better.png",
"width": 300,
"unit": "px"
\end{imagefromfile}

Rigidity is thus graphically the statement that string diagrams can be "straightened out". This approach is taken, for example, in [Selinger](#Selinger2011).

## Remarks

Note that this definition only asserts the existence of the dual objects. It does _not_ assert that specific duals have been chosen. However, the choice of duals is unique up to unique isomorphism, justifying reference to '[[the]]' dual of an object; in fact, this extends to a [[contravariant functor|contravariant]] [[anafunctor]] ${}^*\colon M \to M$.  (Using the [[axiom of choice]] to pick duals for every object at once, we can make this into a [[strict functor]].) This functor is defined explicitly by the formula

$$f^*:A^*\xrightarrow{\text{coev}_B\otimes \text{id}_{A^*}} B^*\otimes B\otimes A^*\xrightarrow{\text{id}_{B^*}\otimes f\otimes \id_{A^*}} B^*\otimes A\otimes A^* \xrightarrow{\text{id}_{B^*}\otimes \text{ev}_A} B^*$$

This formula is clearer when written graphically:

\begin{imagefromfile}
"file_name": "duals-graphical.png",
"width": 150,
"unit": "px"
\end{imagefromfile}

Additionally, this definition does not assert that the right dual of an object is isomorphic to its left dual: this need not be the case in general, though it is true in a [[braided monoidal category]], and thus automatically also in a [[symmetric monoidal category]] (this last fact can be considered an algebraic form of the "Whitney trick" for knots; see this [MO discussion](http://mathoverflow.net/questions/162677/why-is-a-braided-left-autonomous-category-also-right-autonomous/162693#162693)). Left duals and right duals are also isomorphic in a [[semisimple category]]. Note that a rigid monoidal category which is also symmetric is sometimes called [[compact closed category|compact closed]], or simply "compact".

In practice, algebraic geometers are the most frequent users of the term 'rigid', and they focus on the symmetric monoidal case, so they ignore the difference between right and left duals.

## Properties

### Tannaka duality

The statement of [[Tannaka duality]] for associative algebras says that rigid monoidal categories equipped with a [[fiber functor]] are [[categories of modules]] over a [[Hopf algebra]].

[[!include structure on algebras and their module categories - table]]

### Free rigid monoidal categories

The inclusion of the 2-category of monoidal categories into the 2-category
of rigid monoidal categories admits a left [[2-adjoint functor]] $L$.

Furthermore, the unit of the adjunction is a strong monoidal [[fully faithful functor]], i.e., any monoidal category $C$ admits
a fully faithful strong monoidal functor $C\to L(C)$,
where $L(C)$ is a rigid monoidal category.

See Theorems 1 and 2 in [Delpeuch](#Delpeuch).

## References

* N. Saavedra Rivano, "Catégories Tannakiennes." _Bulletin de la Société Mathématique de France_ 100 (1972): 417-430. [EuDML](https://eudml.org/doc/87193)

* {#Delpeuch} [[Antonin Delpeuch]], _Autonomization of monoidal categories_, [arXiv](https://arxiv.org/abs/1411.3827), [doi](https://dx.doi.org/10.4204/EPTCS.323.3).

* {#Selinger2011} [[Peter Selinger]], _A survey of graphical languages for monoidal categories_, New structures for physics, [arXiv](https://arxiv.org/abs/0908.3347)

## Related concepts

* [[star-autonomous category]]

* [[(∞,n)-category with duals]]

* [[linguistics|natural language syntax]]

[[!redirects rigid monoidal category]]
[[!redirects rigid monoidal categories]]
[[!redirects rigid category]]
[[!redirects rigid categories]]
[[!redirects autonomous monoidal category]]
[[!redirects autonomous monoidal categories]]
[[!redirects autonomous category]]
[[!redirects autonomous categories]]
[[!redirects left autonomous monoidal category]]
[[!redirects left autonomous monoidal categories]]
[[!redirects left autonomous category]]
[[!redirects left autonomous categories]]
[[!redirects right autonomous monoidal category]]
[[!redirects right autonomous monoidal categories]]
[[!redirects right autonomous category]]
[[!redirects right autonomous categories]]


rigid object

# Contents
* table of contents
{:toc}

## Definition

An [[object]] $x$ in a category $C$ is said to be **rigid** if its [[automorphism group]] is trivial, in other words if any [[isomorphism]] $f : x \cong x$ must be equal to the [[identity morphism]] $f = 1_x : x \to x$.

More generally, an object of an [[n-category]] (or [[(n,r)-category]], etc.) is **rigid** if its higher category $Aut(x)$ of automorphisms is [[terminal object|terminal]].

## Idea

Certain types of questions are made much more difficult to answer by the presence of non-trivial automorphisms (or "symmetries").  In such settings, sometimes it is helpful to first consider the question for rigid objects, and then try to extend the answer to non-rigid objects.  It can even be useful to artificially "point" or "root" the objects of the original category $C$ by moving to a larger category $C_\bullet$ (with a [[forgetful functor]] $C_\bullet \to C$) in which _all_ objects are rigid.

## Examples

* As trivial examples, any [[initial object]] or [[terminal object]] is rigid, as is every object of a [poset](https://ncatlab.org/nlab/show/partial+order#AsACategoryWithExtraProperties). (While trivial, such examples are also significant; e.g., [[universal properties]] may be formulated in terms of initial or terminal objects in suitable categories.)

* In [[graph theory]], a rigid object in the category of undirected graphs is called an _asymmetric graph_.

* Let $C$ be the category of [[transitive action|transitive]] [[G-sets]] for some group $G$.  Although a transitive $G$-set may in general have non-trivial automorphisms, these symmetries can be killed off by moving to the larger category $C_\bullet$ of _pointed_ transitive $G$-sets, whose objects are pairs of a transitive $G$-set $X$ equipped with an element $r \in X$, and whose morphisms $f : (X,r) \to (Y,s)$ are $G$-equivariant functions $f : X \to Y$ preserving the point $f(r) = s$.  Indeed, suppose $f : (X,r) \to (X,r)$ is any endomorphism of $(X,r)$, and let $x \in X$ be any element.  By the assumption that $G$ acts transitively on $X$, there exists $g \in G$ such that $x = g * r$.  But then by equivariance and preservation of the point we have that
$$f(x) = f(g*r) = g*f(r) = g*r = x.$$
Note that this example is relevant to the combinatorics of [[embedded graphs]] (see at [[combinatorial map]]).

* In [[point-set topology]], there are constructions of [[continua]] whose only [[continuous map|continuous]] [[endomorphisms]] are [[constant maps]] and the identity. Examples include so-called "Cook continua". Note that the interest is not merely in the pleasure of concocting exotic and pathological spaces: there is also some import for category theory, for instance in better understanding the problem of characterizing [[reflective subcategories]] of [[Top]]. See [Kannan and Rajagopolan](#Kannan) (and references therein) for some discussion.

* In the [[(∞,1)-category]] [[Grpd]], an [[Eilenberg-MacLane space]] $K(G,1)$ is rigid if $G$ has trivial [[center]] and also trivial [[outer automorphism]] group, since $Aut(K(G,1))$ is a homotopy 1-type with $\pi_0(Aut(K(G,1)))=Out(G)$ and $\pi_1(Aut(K(G,1)))=Z(G)$.  In particular, this is the case for $G=S_n$ the [[symmetric group]] on $n$ letters, where $n\ge 3$ and $n\neq 6$.  This allows us to construct an [[monomorphism in an (∞,1)-category|embedding]] of $\mathbb{N}$ into the [[object classifier]].

* A [[real closed field]] is a rigid object in the category [[Field]] of [[fields]].

## Related concepts

* [[groupoid cardinality]]

## References

For a discussion of the example of pointed transitive $G$-sets (among other things), see

* [[Qiaochu Yuan]], "The categorical exponential formula", blog post at Annoying Precision, November 4, 2015. ([web](https://qchu.wordpress.com/2015/11/04/the-categorical-exponential-formula/))

Rigid topological spaces are discussed in

* V. Kannan and M. Rajagopolan, _Constructions and applications of rigid spaces, I_, Adv. Math., Vol. 29 Iss. 1 (July 1978), 89-130. ([web](http://www.sciencedirect.com/science/article/pii/0001870878900063))
 {#Kannan}

[[!redirects rigid objects]]
[[!redirects rigid]]


rigidification of a stack

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

#Contents#
* table of contents
{:toc}

## Idea
 {#Idea}

Given a [[stack]] $\mathcal{S}$ over a [[site]] $\mathcal{C}$. One often wants to rigidify (kill off a flat subgroup of the [[inertia orbifold|inertia]]) in order to realize the stack as a [[gerbe]] over an [[algebraic space]].

Alternative idea:

Given a [[moduli stack]] classifying some kind of structure, one sometimes wants to "remove the automorphisms" inside it such as to be left with just a [[moduli space]]. This is sometimes called "rigidification". The archetypical example is the passage from the the groupoid of [[line bundles]] over a space to its [[decategorification]] given by the (set underlying the) [[Picard group]]. Doing this over all spaces means passing from the stack of line bundles to the _[[Picard scheme]]_. The general process of "rigidification" is supposed to be a mechanism that generalizes this process ([ACV, 5.1.1](#ACV)).

## Definition
 {#Definition}

We first give the simple general definition of rigidification

* [General definition](#GeneralDefinition)

Then we discuss specifically the case for [[algebraic stacks]] where one may add a bunch of technical assumptions

* [Definition for algebraic stacks](#DefinitionForAlgebraicStacks)

### General
 {#GeneralDefinition}

For $\mathbf{H}$ an [[(∞,1)-topos]] and $X \in \mathbf{H}$ any [[object]], write $\mathbf{Aut}(X) \in Grp(\mathbf{H})$ for its internal [[automorphism ∞-group]]. Consider a [[braided ∞-group]] $H \in BrGrp(\mathbf{H})$ and an [[∞-group]] [[homomorphism]]

$$
  \iota
  \;\colon\;
  \mathbf{B}H \to \mathbf{Aut}(X)
$$

of [[∞-groups]].
This defines an [[∞-action]] of $\mathbf{B}H$ on $X$, hence a [[fiber sequence]] in $\mathbf{H}$ of the form

$$
  \array{
    X &\to& X//\mathbf{B}H
    \\
    && \downarrow
    \\
    && \mathbf{B}^2 H
  }
  \;\;\;\;
  inside
  \;\;\;\;
  \array{
    X &\to& X//\mathbf{Aut}(X)
    \\
    && \downarrow
    \\
    && \mathbf{B} \mathbf{Aut}(X)
  }
  \,.
$$

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

The [[∞-quotient]] $X//\mathbf{B}H$ is what is sometimes called the "rigidification" of $X$, especially if $H$ is maximal such that there is a homomorphism $\mathbf{B}H \to \mathbf{Aut}(X)$.

=--

### For algebraic stacks
 {#DefinitionForAlgebraicStacks}

Let $X$ be a [[scheme]]. Let $\mathcal{S}\to X$ be an [[algebraic stack]] fibered in groupoids over $X$. Let $H$ be a finitely presented, [[separated morphism|separated]], [[group scheme]] over $X$ such that for each $\xi\in\mathcal{S}(T)$ there is an embedding $H(T)\to Aut_T(\xi)$ compatible with pullback.

It follows that $H$ must be [[abelian group|abelian]] (because $H(T)$ lies in the center of $Aut_T(\xi)$). The condition on $H$ is trivially satisfied whenever $\mathcal{S}$ is banded by $H$.

Define the $H$-rigidification of $\mathcal{S}$ to be $\mathcal{S}^H$. ([ACV, def. 5.1.4](#ACV)).

Theorem ([A-C-V, theorem 5.1.5](#ACV)): The space $\mathcal{S}^H$ exists such that there is a [[formally smooth morphism|smooth]] surjective finitely presented morphism of stacks $\mathcal{S}\to \mathcal{S}^H$ satisfying the following:

1. For any $\xi\in \mathcal{S}(T)$ with image $\eta\in \mathcal{S}^H(T)$, we have $H(T)$ lies in the kernel of $Aut_T(\xi)\to Aut_T(\eta)$.
2. The map $\mathcal{S}\to \mathcal{S}^H$ is universal with respect to stack morphisms satisfying (1).
3. If $T$ is the spectrum of an algebraically closed field, then $Aut_T(\eta)=Aut_T(\xi)/H(T)$.
4. A moduli space for $\mathcal{S}$ is also a moduli space for $\mathcal{S}^H$.

and if $\mathcal{S}$ is a [[Deligne-Mumford stack]], then $\mathcal{S}^H$ is also a Deligne-Mumford stack and $\mathcal{S}\to \mathcal{S}^H$ is [[formally etale morphism|etale]].

## Examples
 {#Examples}

We discuss some examples. First, to get rid of all distraction introduced by the dependence on objects of a [[site]] of definition, we consider the special case where the underlying site is the point, hence where stacks are just plain [[groupoids]] -- _geometrically [[discrete groupoids]]_ for emphasis.

* [For a geometrically discrete groupoid](#ForAGeometricallyDiscreteGroupoid)

Then we discuss aspects of regidification for [[algebraic stacks]]

* [For an algebraic stack](#ForAnAlgebraicStack).

### For a geometrically discrete groupoid
 {#ForAGeometricallyDiscreteGroupoid}

If $\mathbf{H} = $ [[∞Grpd]] and $X \in \infty Grpd$ is a [[1-truncated]] object, hence just a [[groupoid]], then $\mathbf{Aut}(X)$ is its [[automorphism 2-group]]. Its [[objects]] are naturally identified with those [[functors]] $\alpha \colon X \to X$ that are [[equivalence of groupoids|equivalences]], and its [[morphisms]] with the [[natural isomorphisms]] $g \colon \alpha \to \beta$ between these. In particular if $\alpha = \beta = id$ is the identity automorphism, then such a $g$ is a [[function]] which to each [[object]] $\xi \in X$ assigns an [[automorphism]] $g_\xi \colon \xi \to \xi$ in $X$ such that for each [[morphism]] $\phi \colon \xi \to \eta$ in $X$ the naturality square

$$
  \array{
    \xi &\stackrel{\phi}{\to}& \eta
    \\
    \downarrow^{\mathrlap{g_\xi}} && \downarrow^{\mathrlap{g_\eta}}
    \\
    \xi &\stackrel{\phi}{\to}& \eta
  }
  \,.
$$

[[commuting diagram|commutes]].

Now for $H$ an [[abelian group]] there is the [[delooping]] [[groupoid]] $\mathbf{B}H$ which has a single object and $H$ as the group of morphisms from that object to itself. Both $\mathbf{Aut}(X)$ and $\mathbf{B}H$ are [[2-groups]] in this case. A homomorphism of 2-groups

$$
  \iota
   \;\colon\;
  \mathbf{B}H
  \to
  \mathbf{Aut}(X)
$$

has to send the essentially unique point of $\mathbf{B}H$ to the [[identity]] [[functor]] $id_X$ and is hence
equivalently a [[function]] that sends each element $g \in G$ to a [[natural isomorphism]] $g \colon id_X \to id_X$, hence a function $g_{(-)}$ that sends each object $\xi \in X$ to a morphism $g_\xi \colon \xi \to \xi$ in $X$, such that the above diagram commutes. Moreover, this being a [[2-group]] [[homomorphism]] means that for $g_1, g_2 \in H$ two elements, they are sent to the composite $(g_2)_\xi\circ (g_1)_\xi$ in $X$.

In other words, we have a [[functor]]

$$
  \rho \colon X \times \mathbf{B}H \to X
  \,,
$$

which takes a pair of objects $(\xi,\ast)$ to $\xi$, takes a pair of morphisms of the form $(id_\xi, \ast \stackrel{g}{\to} \ast)$ to $(\xi \stackrel{g_\xi}{\to} \xi)$ and takes a pair of morphisms of the form $(\xi \stackrel{\phi}{\to} \eta, id_\ast)$ to $(\xi \stackrel{\phi}{\to} \eta)$; and which satisfies the action property,

$$
  \array{
      X \times \mathbf{B}H \times \mathbf{B}H
      &\stackrel{\rho \times id_{\mathbf{B}H}}{\to}&
      X \times \mathbf{B}H
      \\
      {}^{\mathllap{id_X \times \cdot_{\mathbf{B}H}}}\downarrow &\swArrow& \downarrow^{\rho}
      \\
      X \times \mathbf{B}G &\stackrel{\rho}{\to}& X
  }
  \,.
$$

In fact, with the groupoids explicitly presented the way we have discussed them, the [[natural transformation]] filling this [[diagram]] is the [[identity]] and hence we have exhibited the [[∞-action]] of the [[2-group]] $\mathbf{B}H$ on the groupoid $X$ by an ordinary [[action]]. More precisely, under passing to [[nerves]] of [[groupoids]] we have exhibited it as the [[action]] of a [[simplicial group]] on a [[Kan complex]], which is just a [[simplicial object|simplicial diagram]] of ordinary actions of ordinary groups on plain sets. Since these are [[simplicial skeleton|2-coskeletal]] simplicial sets (being the nerves of just 1-groupoids), it is sufficient to consider them just in degrees 0,1,2. So then we have the following simplicial diagram of ordinary groups acting on ordinary sets

$$
  \left(
    \array{
      X_1 \times_{X_0} X_1
      \\
      \downarrow \downarrow \downarrow
      \\
      X_1
      \\
      \downarrow \downarrow

      \\
      X_0
    }
  \right)
  \times
  \left(
    \array{
      H \times H
      \\
      \downarrow \downarrow \downarrow
      \\
      H
      \\
      \downarrow \downarrow
      \\
      \ast
    }
  \right)
  \to
  \left(
    \array{
      X_1 \times_{X_0} X_1
      \\
      \downarrow \downarrow \downarrow
      \\
      X_1
      \\
      \downarrow \downarrow

      \\
      X_0
    }
  \right)
  \,.
$$

In degree 0 this is the identity map $(\xi,\ast) \mapsto \xi$, in degree 1 it is (with the symbols as above) the map $(\phi,g) \mapsto g_\eta \circ \phi = \phi \circ g_\xi$ and so on.

Finally, the [[∞-quotient]] $X//\mathbf{B}H$ of an [[∞-action]] of an [[∞-group]] presented as an ordinary action of a [[simplicial group]] on a [[Kan complex]] this way is presented by the [[Borel construction]], namely the ordinary [[quotient]] of [[simplicial sets]]

$$
  X \times_{\mathbf{B}H} \mathbf{E}\mathbf{B}H
  \coloneqq
  (X \times \mathbf{E}\mathbf{B}H)/\mathbf{B}H
$$

(where now all symbols stand for the corresponding simplicial sets as described above). Here

$$
  \mathbf{E} \mathbf{B}H
  \coloneqq
  (\mathbf{B}H)^{\Delta^1} \times_{\mathbf{B}G} *
$$

is a model for the total space of the [[universal principal infinity-bundle|universal principal 2-bundle]] over $\mathbf{B}H$.

So the [[Kan complex]] $X \times_{\mathbf{B}H} \mathbf{E}\mathbf{B}H$ presents the "rigidification" of $X$ with respect to the chosen $\iota \colon \mathbf{B}H \to \mathbf{Aut}(X)$.

### For an algebraic stack
 {#ForAnAlgebraicStack}

The standard example is the $\mathbb{G}_m$-rigidification of the [[Picard scheme|Picard stack]]. Suppose $X/k$ is an irreducible [[variety]] over a [[field]]. One can say that the failure of the Picard stack, $\mathcal{Pic}_X$ to be representable comes from the fact that objects in fiber categories have automorphisms by the multiplicative group, so we would like to kill this group.

As pointed out in _[[Picard scheme]]_, the relative Picard scheme is the [[sheafification]] of $\mathcal{Pic}_X$ and [[representable]]. Moreover $\mathcal{Pic}_X\to Pic_X$ is a $\mathbb{G}_m$-gerbe, so $\mathbb{G}_m$ satisfies the conditions to rigidify.

By the [[universal property]], the rigidification is exactly $Pic_X$, so in this case we see that the sheafification and the rigidification by the inertia are the same.

## References

* [[Dan Abramovich]], Alessio Corti, and [[Angelo Vistoli]], _Twisted Bundles and Admissible Covers_ ([arXiv:0106211](http://arxiv.org/abs/math/0106211))
 {#ACV}

* Matthieu Romagny, _Group Actions on Stacks and Applications_, Michigan Math. J. Volume 53, Issue 1 (2005), 209-236 ([Project Euclid](http://projecteuclid.org/euclid.mmj/1114021093)).

category: algebraic geometry
[[!redirects rigidification]]


rigidification of quasi-categories

+-- {: .rightHandSide}
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###Context###
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
#### Enriched category theory
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[[!include enriched category theory contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Both [[quasi-categories]] as well as [[sSet-enriched categories]] serve as models for [[(infinity,1)-categories|$(\infty,1)$-categories]] (exhibited by the [[Joyal model structure for quasi-categories]] and the [[Dwyer–Kan–Bergner model structure on sSet-enriched categories]], respectively) and in fact as [[Quillen equivalence|Quillen equivalent]] models (see the *[[relation between quasi-categories and simplicial categories]]*) whose [[right Quillen functor]] is the [[homotopy coherent nerve]]-operation. Its [[left adjoint]]
$$
  \mathfrak{C}
     \;\colon\;
  sSet \longrightarrow sSet\text{-}Cat
$$
[[Lurie (2009), pp. 20](#Lurie09)] thus encodes any [[quasi-category]] equivalently in an [[sSet]]-[[enriched category]]. Since in the latter model, but not in the former, [[horizontal composition]] is a strictly [[associativity|associative]] and [[unitality|unital]] operation, one may think of $\mathfrak{C}(-)$ as (partially) "rigidifying" the data in a quasi-category, in the sense of [[semi-strict infinity-category|semi-strictification of $\infty$-categories]], whence it is also known as the operation of *rigidification of quasi-categories* [[Dugger & Spivak (2011a)](#DuggerSpivak11a)].

## Definition

### The Joyal rigidification functor

The __Joyal rigidification functor__ $\mathscr{C}$ is defined as the [[left adjoint]] of the Cordier–Vogt [[homotopy coherent nerve functor]] $N$:

\[
  \label{RigidificationAsLeftAdjointToHomotopyCoherentNerve}
  sSet
  \underoverset
    {\underset{N}{\longleftarrow}}
    {\overset{\mathfrak{C}}{\longrightarrow}}
    {\;\;\; \bot \;\;\;}
  sSet\text{-}Cat
\]

This notion is briefly mentioned and attributed to [[André Joyal]] by [Bergner (2007), above Theorem 7.8](#Bergner07); the first detailed account is due to [Lurie (2009), §1.1.5 and §2.2](#Lurie09).

### The Dugger–Spivak rigidification functor

The __Dugger–Spivak rigidification functor__ [[Dugger & Spivak (2011a)](#DuggerSpivak11a)]
provides a more explicit model for the same [[(∞,1)-functor]], by virtue of writing down an explicit formula that does not use colimits.
The resulting functor is connected to the Joyal functor by a zigzag of natural weak equivalences.

Specifically, given a [[simplicial set]] $S$ (not necessarily fibrant in the [[Joyal model structure]]),
we construct the Dugger–Spivak rigidification $\mathfrak{C}^{\mathrm{nec}}S$ as the following [[simplicial category]]:

Objects are vertices of $S$.

The [[simplicial set]] of morphisms $x\to y$ is the nerve of the category of __necklaces__ in $S$ (introduced by [[Hans-Joachim Baues]]).
A necklace is a simplicial map
$$\Delta^{n_1}\vee \cdots \vee \Delta^{n_k}\to S,$$
where $\vee$ glues the final vertex of the preceding simplex to the initial vertex of the following simplex.
Morphisms are commutative triangles of simplicial maps that preserve the initial and final vertex of the entire necklace.

Composition is defined by concatenating necklaces.
The resulting functor from simplicial sets admits
a zigzag of weak equivalences to the Joyal rigidification functor.

## Properties
 {#Properties}

### Respect for products

The rigidification operation does not strictly preserve [[Cartesian products]] (cf. *[[products of simplicial sets]]*) but it does so up to equivalence:

\begin{proposition}\label{PreservationOfProducts}
For $S,\, S' \,\in\, sSet$, the [[natural transformation]]
$$
  \mathfrak{C}(S \times S')
  \overset{
    \big(
      \mathfrak{C}(pr_S)
      ,\,
      \mathfrak{C}(pr_{S'})
    \big)
  }{\longrightarrow}
  \mathfrak{C}(S) \times \mathfrak{C}(S')
$$
(induced from the [[projections]] $S \overset{pr_S}{\leftarrow} S \times S' \overset{pr_{S'}}{\rightarrow} S'$)
is a [[Dwyer-Kan equivalence]].
\end{proposition}
[[Lurie (2009), Cor. 2.2.5.6](#Lurie09); [Dugger & Spivak (2011a), Prop. 6.2](#DuggerSpivak11a)]

### Relation to Dwyer-Kan groupoids

Let
\[
  \label{LocalizationOfsSetCategoriesToSSetGroupoids}
  L
    \,\colon\,
  sSet\text{-}Cat
  \longrightarrow
  sSet\text{-}Grp
\]
denote the functor from [[sSet-enriched categories]] to [[sSet]]-[[enriched groupoids]] (Dwyer-Kan [[simplicial groupoids]]) which is degreewise given by the [[localization]] operation [[left adjoint]] to the [[full subcategory]] inclusion [[Grpd]] $\hookrightarrow$ [[Cat]].

Let
\[
  \label{DwyerKanFundamentalGroupoid}
  \mathcal{G}
  \,\colon\,
  sSet
    \longrightarrow
  sSet\text{-}Grpd
    \hookrightarrow
  sSet\text{-}Cat
\]
denote the [[Dwyer-Kan fundamental simplicial groupoid]]-construction with values in [[sSet]]-[[enriched groupoids]] (Dwyer-Kan [[simplicial groupoids]]), here to be regarded among [[sSet-enriched categories]].

\begin{proposition}\label{ComparisonWithDKGroupoids}
  For $S \,\in\, sSet$ there is a [[natural transformation]]
$$
  L \circ \mathfrak{C}(S)
  \longrightarrow
  \mathcal{G}(S)
$$
which is a [[Dwyer-Kan equivalence]]
(from the localization (eq:LocalizationOfsSetCategoriesToSSetGroupoids) of the rigidification (eq:RigidificationAsLeftAdjointToHomotopyCoherentNerve) to the Dwyer-Kan fundamental groupoid (eq:DwyerKanFundamentalGroupoid)).
\end{proposition}
[[Minichiello, Rivera & Zeinalian (2023), Thm. 1.1 (Cor. 4.2)](#MinichielloRiveraZeinalian23)]

## Related concepts

* [[homotopy coherent nerve]]

* [[relation between quasi-categories and simplicial categories]]

* [[Dwyer-Kan fundamental simplicial groupoid]]

## References

The construction is briefly mentioned and attributed to [[André Joyal]] in:

* {#Bergner07} [[Julie Bergner]], Theorem 7.8 in: *A survey of $(\infty,1)$-categories*, in: [[John Baez]], [[Peter May]] (eds.),  _[[Towards Higher Categories]]_ The IMA Volumes in Mathematics and its Applications **152**, Springer (2007) [[arXiv:math/0610239](http://arxiv.org/abs/math/0610239), [doi:10.1007/978-1-4419-1524-5_2](https://doi.org/10.1007/978-1-4419-1524-5_2)]

First detailed discussion (not using a name beyond the symbol "$\mathfrak{C}$") is due to:

* {#Lurie09} [[Jacob Lurie]], §1.1.5 and §2.2 of: _[[Higher Topos Theory]]_, Annals of Mathematics Studies **170**, Princeton University Press (2009) [[pup:8957](https://press.princeton.edu/titles/8957.html), [pdf](https://www.math.ias.edu/~lurie/papers/HTT.pdf)]

Further discussion and the terminology "rigidification" is due to

* {#DuggerSpivak11a} [[Dan Dugger]], [[David Spivak]], *Rigidification of quasi-categories*, Algebr. Geom. Topol. **11** (2011) 225-261 [[arXiv:0910.0814](http://arxiv.org/abs/0910.0814), [doi:10.2140/agt.2011.11.225](http://dx.doi.org/10.2140/agt.2011.11.225)]

aimed at discussing [[quasi-categorical hom-spaces]]:

* {#DuggerSpivak11b} [[Dan Dugger]], [[David Spivak]], *Mapping spaces in quasi-categories*, Algebr. Geom. Topol. **11** (2011) 263-325 [[arXiv:0911.0469](http://arxiv.org/abs/0911.0469), [doi:10.2140/agt.2011.11.263](http://dx.doi.org/10.2140/agt.2011.11.263)]

See also:

* [[Emily Riehl]], *On the structure of simplicial categories associated to quasi-categories*, Math. Proc. Camb. Phil. Soc. **150** (2011) 489-504 [[arXiv:0912.4809](https://arxiv.org/abs/0912.4809), [doi:10.1017/S0305004111000053](https://doi.org/10.1017/S0305004111000053)]

* {#MinichielloRiveraZeinalian23} [[Emilio Minichiello]], [[Manuel Rivera]], [[Mahmoud Zeinalian]], §3.2 in: *Categorical models for path spaces*, Advances in Mathematics **415** (2023) 108898 [[arXiv:2201.03046](https://arxiv.org/abs/2201.03046), [doi:10.1016/j.aim.2023.108898](https://doi.org/10.1016/j.aim.2023.108898)]


rigidity theorem for elliptic genera

#Contents#
* table of contents
{:toc}

## Idea

(...)

## Related concepts

* [[elliptic genus]]

* [[equivariant elliptic genus]]

## References

The statement, with a [[string theory]]-motivated plausibility argument, is due to

* {#Witten87} [[Edward Witten]], _Elliptic genera and quantum field theory_, Comm. Math. Phys. Volume 109, Number 4 (1987), 525-536.  ([euclid:cmp/1104117076](http://projecteuclid.org/euclid.cmp/1104117076))

The first [[proof]] was given in

* [[Clifford Taubes]], _$S^1$-actions and elliptic genera_, Comm. Math. Phys. Volume 122, Number 3 (1989), 455-526 ([euclid:cmp/1104178471](https://projecteuclid.org/euclid.cmp/1104178471))

* [[Raoul Bott]], [[Clifford Taubes]], _On the Rigidity Theorems of Witten_, Journal of the American Mathematical Society Vol. 2, No. 1 (Jan., 1989), pp. 137-186 ([doi:10.2307/1990915](https://doi.org/10.2307/1990915))

Review in

* [[Raoul Bott]], _On the Fixed Point Formula and the Rigidity Theorems of Witten_, Lectures at Cargése 1987. In: ’t Hooft G., Jaffe A., Mack G., Mitter P.K., Stora R. (eds) _Nonperturbative Quantum Field Theory_. NATO ASI Series (Series B: Physics), vol 185. Springer (1988) ([doi:10.1007/978-1-4613-0729-7_2](https://doi.org/10.1007/978-1-4613-0729-7_2))

Further proofs:

* [[Friedrich Hirzebruch]], _Elliptic Genera of Level $N$ for Complex Manifolds_, In: Bleuler K., Werner M. (eds) _Differential Geometrical Methods in Theoretical Physics_ NATO ASI Series (Series C: Mathematical and Physical Sciences), vol 250. Springer (1988) ([doi:10.1007/978-94-015-7809-7_3](https://doi.org/10.1007/978-94-015-7809-7_3))

* I. M. Krichever, _Generalized elliptic genera and Baker-Akhiezer functions_, Mathematical Notes of the Academy of Sciences of the USSR 47, 132–142 (1990) ([doi:10.1007/BF01156822](https://doi.org/10.1007/BF01156822))

* [[Kefeng Liu]], _On modular invariance and rigidity theorems_, J. Differential Geom.
Volume 41, Number 2 (1995), 343-396 ([euclid:jdg/1214456221](https://projecteuclid.org/euclid.jdg/1214456221))

* [[Kefeng Liu]], _On elliptic genera and theta-functions_, Topology Volume 35, Issue 3, July 1996, Pages 617-640 (doi:10.1016/0040-9383(95)00042-9)

* [[Anand Dessai]], Rainer Jung, _On the Rigidity Theorem for Elliptic Genera_, Transactions of the American Mathematical Society Vol. 350, No. 10 (Oct., 1998), pp. 4195-4220 (26 pages) ([jstor:117694](https://www.jstor.org/stable/117694))

* [[Ioanid Rosu]], _Equivariant Elliptic Cohomology and Rigidity_, American Journal of Mathematics 123 (2001), 647-677 ([arXiv:math/9912089](https://arxiv.org/abs/math/9912089))

[[!redirects rigidity theorem for equivariant elliptic genera]]


Rikard von Unge

* [webpage](http://www.muni.cz/people/33259)

## related $n$Lab entries

* [[sigma model]], [[supersymmetry]]

* [[Poisson-Lie T-duality]], [[T-duality]]

* [[supergravity]]

* [[Starobinsky model of cosmic inflation]]

category: people


Rinat Kashaev

* [webpage](https://www.unige.ch/math/folks/kashaev/)

## Selected writings

* {#Kashaev95} [[Rinat Kashaev]], _A Link Invariant from Quantum Dilogarithm_,  Modern Physics Letters AVol. 10, No. 19, pp. 1409-1418 (1995) ([arXiv:q-alg/9504020](https://arxiv.org/abs/q-alg/9504020))

* {#Kashaev95} [[Rinat Kashaev]], _The Hyperbolic Volume Of Knots From The Quantum Dilogarithm_ Lett. Math. Phys. 39 (1997) 269-275 ([arXiv:q-alg/9601025](https://arxiv.org/abs/q-alg/9601025))

* {#KashaevTirkkonen99} [[Rinat Kashaev]], O. Tirkkonen, _Proof of the volume conjecture for torus knots_, Journal of Mathematical Sciences (2003) 115: 2033 ([arXiv:math/9912210](http://arxiv.org/abs/math/9912210))

  (on the [[volume conjecture]])

## Related $n$Lab entries

* [[volume conjecture]]

category: people


Ring

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

#Contents#
* table of contents
{:toc}

## Definition

_Ring_ is the [[category]] of [[rings]] (with [[unital ring|unit]]) and ring [[homomorphisms]] (that preserve the unit).

A _ring_ is a [[monoid]] [[internalization|in]] [[Ab]], where $Ab$ is the category of [[abelian group|abelian groups]].  So, $Ring$ is an example of a category of [[internal monoids]].

## Properties

### Epi/Monomorphisms

For more see at [[Stacks Project]], _[10.106 Epimorphisms of rings](http://stacks.math.columbia.edu/tag/04VM)_.

Every [[surjective function|surjective]] [[homomorphism]] of rings is an [[epimorphism]] in $Ring$, but not every epimorphism is surjective.

A counterexample:

\begin{proposition}
  \label{InclusionOfIntegersIntoRationalsIsEpimorphismOfRings}

In [[unital ring|unital]] [[Rings]], the canonical inclusion $\mathbb{Z} \overset{i}{\hookrightarrow} \mathbb{Q}$ of the [[integers]] into the [[rational numbers]] is an [[epimorphism]].

\end{proposition}

\begin{proof}

Since every [[rational number]] is the product of an [[integer]] with the multiplicative inverse of an integer

$$
  \frac{a}{b} \,=\,  a \cdot b^{-1}
  \;\;
  \in
  \;
  \mathbb{Q}
$$

and since [[unital ring|unital]] [[ring homomorphism]]

$$
  \mathbb{Z}
  \overset{\;\; i \;\;}{\hookrightarrow}
  \mathbb{Q}
  \underoverset
  {\;\;g\;\;}
  {\;\;f\;\;}
  {\rightrightarrows}
  R
  \,.
$$

preserve multiplicative inverses, $f\left( a/b \right) = f(a) \cdot \big(f(b)\big)^{-1}$, it follows that any pair $(f,g)$ of [[parallel morphisms]] on $\mathbb{Q}$ are equal as soon as they take equal value on the integers.

\end{proof}

## Related categories

* [[Mod]]

* [[CRing]]

* [[Rng]]

* [[Rig]]

* [[Mon]], [[CMon]]

## References

* [[Stacks Project]], _[10.106 Epimorphisms of rings](http://stacks.math.columbia.edu/tag/04VM)_

[[!redirects category of rings]]

[[!redirects Rings]]

category: category