SVG Editor HowTo

# An Initial SVG editor HowTo

1. Before creating an SVG using the editor, you need to choose where it will be placed in a page.  So first you need to be editing a page, then click in the source text where you want your SVG to appear.  The SVG editor will now launch.

3. To insert some mathematics, you need to use the "foreign object" tool.  The icon for this is a $\Phi$ (currently bottom on the menu at the left-hand side).  Click on this and then click where you want the mathematics to be.  The menu across the top will now change to the "foreign object" menu.  The icon with a pencil and some MathML code in the background brings up an editor for the content of this "foreign object".  At the moment, this has to be put in as true MathML.  There are several ways to generate this, the simplest being to install `itex2MML` on your computer and use that to parse itex.  A slightly more complicated route would be to use the [[Sandbox]] and write in the itex, send it off to the nLab for processing, then use the "View Source" option on your browser. But, most likely, the symbols you need already appear on the page you are editing. So just "View Source", and grab the snippet of MathML that you need.  You may have to adjust the size of the "foreign object" to be able to see all of your amazing mathematics.  Either drag the appropriate side of the blue box or manually adjust the numbers in the "foreign object" menu across the top. (In some browsers, the blue box is, confusingly, not anywhere near where the text actually is; that's a bug in your browser, and you may be able to fix it by updating to the latest version.) It's important to make sure that the boxes are big enough, text outside a box doesn't get shown.  Leave a little room around as things may be seen differently on different browsers.

4. The tool which you are using defaults to "select" so if you want to lay out the diagram roughly first and then go back and edit each piece, you have to keep selecting the corresponding tool each time.  Sometimes.

5. You put arrows on afterwards.  So draw the lines first, then select them and put arrows on as desired.

6. There are a couple of ways of making curved arrows.  One is to draw a straight line and convert it to a path.  So draw a straight line, select it, and click the button at the top that is a red quarter-circle with some blue bits.  That converts it to a path.  Ensure that it is selected and click it again (it can take a couple of clicks to get this right: the line should go cyan).  Now change the menu "Straight" to "Curve".  You can adjust the curviness by dragging the control points around.

7. Just because you've put a path in doesn't mean it will get shown.  Paths need to be "stroked" to be visible (or they can be filled).  If your paths aren't visible, make sure that the "stroke" settings (bottom of the screen) are something sensible.  Click "wireframe mode" (the "circle-square" icon at the top) if you think you've drawn something that you can't see.

category: meta


SVG hookdownarrow

<svg viewBox="-3.99994 -42.00003 7.99988 44.0 " width="8pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" height="44pt"><g transform="translate(0 2) scale(1 -1) translate(0 42)"><g stroke="#000"><g fill="#000"><g stroke-width=".4pt"><path d="m0-2.3v-37" fill="none"/><g transform="matrix(0 1 -1 0 0 -2.3)"><g stroke-dasharray="none" stroke-dashoffset="0pt"><g stroke-linecap="round"><path d="m0 0h.42c.98 0 1.7-.93 1.7-1.7 0-.9-.7-1.7-1.7-1.7" fill="none"/></g></g></g><g transform="matrix(0 -1 1 0 0 -39)"><g stroke-width=".4pt"><g stroke-dasharray="none" stroke-dashoffset="0pt"><g stroke-linecap="round"><g stroke-linejoin="round"><path d="m-2.4 3.2c.2-1.2 2.4-3 3-3.2-.6-.2-2.8-2-3-3.2" fill="none"/></g></g></g></g></g></g></g></g></g></svg>


SVG hookrightarrow

<svg viewBox="-1.99997 -3.99994 44.0 7.99988 " width="44pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" height="8pt"><g transform="translate(0 4) scale(1 -1) translate(0 4)"><g stroke="#000"><g fill="#000"><g stroke-width=".4pt"><path d="m2.3 0h37" fill="none"/><g transform="matrix(-1 0 0 -1 2.3 0)"><g stroke-dasharray="none" stroke-dashoffset="0pt"><g stroke-linecap="round"><path d="m0 0h.42c.98 0 1.7-.93 1.7-1.7 0-.9-.7-1.7-1.7-1.7" fill="none"/></g></g></g><g transform="matrix(1 0 0 1 39 0)"><g stroke-width=".4pt"><g stroke-dasharray="none" stroke-dashoffset="0pt"><g stroke-linecap="round"><g stroke-linejoin="round"><path d="m-2.4 3.2c.2-1.2 2.4-3 3-3.2-.6-.2-2.8-2-3-3.2" fill="none"/></g></g></g></g></g></g></g></g></g></svg>


SVG images

# SVG images
* tic
{: toc}

## Introduction

Using the inbuilt SVG editor is a good way of creating SVG images, particularly when the desired image should mix both SVG and MathML.  However, there are many other ways of creating SVG images, either using other graphics editors or by generating them from some program.  This page is intended as a place to collect examples of SVGs together with instructions on how to generate them.

To avoid this page getting too big, the SVGs and instructions themselves should be on separate pages.  This page is to be simply a list of links.

## SVGs in the nLab

### Knot Theory

* [[trefoil knot - SVG]]
* [[trefoil knot (2 bridge) - SVG]]
* [[Hopf link - SVG]]
* [[Whitehead link - SVG]]
* [[Borromean link - SVG]] (also redirects from [[Borromean rings - SVG]])
* [[Reidemeister move 1 - SVG]]
* [[Reidemeister move 2 - SVG]]
* [[Reidemeister move 3 - SVG]]
* [[figure 8 knot - SVG]] (also redirects from [[figure eight knot - SVG]])
* [[SVG skein double crossing]]
* [[SVG skein positive crossing]]
* [[SVG skein negative crossing]]
* [[SVG skein no crossing]]

### Functional Analysis

* The [[diagram of LCTVS properties]] was created using [Graphviz](http://www.graphviz.org) from [[second lctvs diagram dot source]]
* The diagram [[TVS relationships]] was created using [Graphviz](http://www.graphviz.org) from [[lctvs dot source]].

category: meta


SVG nearrow

<svg viewBox="-1.99997 -5.99991 44.0 51.99988 " width="44pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" height="52pt"><g transform="translate(0 46) scale(1 -1) translate(0 6)"><g stroke="#000"><g fill="#000"><g stroke-width=".4pt"><path d="m0 0l39 39" fill="none"/><g transform="matrix(.71 .71 -.71 .71 39 39)"><g stroke-width=".4pt"><g stroke-dasharray="none" stroke-dashoffset="0pt"><g stroke-linecap="round"><g stroke-linejoin="round"><path d="m-2.4 3.2c.2-1.2 2.4-3 3-3.2-.6-.2-2.8-2-3-3.2" fill="none"/></g></g></g></g></g></g></g></g></g></svg>


SVG neswarrow

<svg viewBox="-0.2 -0.2 40.40005 40.40005 " width="40.4pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" height="40.4pt"><g transform="translate(0 40.2) scale(1 -1) translate(0 .2)"><g stroke="#000"><g><g stroke-width=".4pt"><path d="m0.65051 0.65051l38.699 38.699" fill="none"/><g transform="matrix(-.7071 -.7071 .7071 -.7071 .65051 .65051)"><g stroke-width=".4pt"><g stroke-dashoffset="0pt"><g stroke-linecap="round"><g stroke-linejoin="round"><path d="m-2.4 3.2c0.2-1.2 2.4-3 3-3.2-0.6-0.2-2.8-2-3-3.2" fill="none"/></g></g></g></g></g><g transform="matrix(.7071 .7071 -.7071 .7071 39.35 39.35)"><g stroke-width=".4pt"><g stroke-dashoffset="0pt"><g stroke-linecap="round"><g stroke-linejoin="round"><path d="m-2.4 3.2c0.2-1.2 2.4-3 3-3.2-0.6-0.2-2.8-2-3-3.2" fill="none"/></g></g></g></g></g></g></g></g></g></svg>


SVG nwsearrow

<svg viewBox="-40.20006 -0.2 40.40005 40.40005 " width="40.4pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" height="40.4pt"><g transform="translate(0 40.2) scale(1 -1) translate(0 .2)"><g stroke="#000"><g><g stroke-width=".4pt"><path d="m-0.65051 0.65051l-38.699 38.699" fill="none"/><g transform="matrix(.7071 -.7071 .7071 .7071 -.65051 .65051)"><g stroke-width=".4pt"><g stroke-dashoffset="0pt"><g stroke-linecap="round"><g stroke-linejoin="round"><path d="m-2.4 3.2c0.2-1.2 2.4-3 3-3.2-0.6-0.2-2.8-2-3-3.2" fill="none"/></g></g></g></g></g><g transform="matrix(-.7071 .7071 -.7071 -.7071 -39.35 39.35)"><g stroke-width=".4pt"><g stroke-dashoffset="0pt"><g stroke-linecap="round"><g stroke-linejoin="round"><path d="m-2.4 3.2c0.2-1.2 2.4-3 3-3.2-0.6-0.2-2.8-2-3-3.2" fill="none"/></g></g></g></g></g></g></g></g></g></svg>


SVG pullback

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SVG rightarrow

<svg viewBox="-1.99997 -3.99994 44.0 7.99988 " width="44pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" height="8pt"><g transform="translate(0 4) scale(1 -1) translate(0 4)"><g stroke="#000"><g fill="#000"><g stroke-width=".4pt"><path d="m0 0h39" fill="none"/><g transform="matrix(1 0 0 1 39 0)"><g stroke-width=".4pt"><g stroke-dasharray="none" stroke-dashoffset="0pt"><g stroke-linecap="round"><g stroke-linejoin="round"><path d="m-2.4 3.2c.2-1.2 2.4-3 3-3.2-.6-.2-2.8-2-3-3.2" fill="none"/></g></g></g></g></g></g></g></g></g></svg>


SVG searrow

<svg viewBox="-5.99991 -42.00003 51.99988 44.0 " width="52pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" height="44pt"><g transform="translate(0 2) scale(1 -1) translate(0 42)"><g stroke="#000"><g fill="#000"><g stroke-width=".4pt"><path d="m0 0l39-39" fill="none"/><g transform="matrix(.71 -.71 .71 .71 39 -39)"><g stroke-width=".4pt"><g stroke-dasharray="none" stroke-dashoffset="0pt"><g stroke-linecap="round"><g stroke-linejoin="round"><path d="m-2.4 3.2c.2-1.2 2.4-3 3-3.2-.6-.2-2.8-2-3-3.2" fill="none"/></g></g></g></g></g></g></g></g></g></svg>


SVG short neswarrow

<svg viewBox="-0.2 -0.2 10.40001 10.40001 " width="10.4pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" height="10.4pt"><g transform="translate(0 10.2) scale(1 -1) translate(0 .2)"><g stroke="#000"><g><g stroke-width=".4pt"><path d="m0.32526 0.32526l9.3495 9.3495" fill="none"/><g transform="matrix(-.7071 -.7071 .7071 -.7071 .32526 .32526)"><g stroke-width=".31999pt"><g stroke-dashoffset="0pt"><g stroke-linecap="round"><g stroke-linejoin="round"><path d="m-1.2 1.6c0.1-0.60002 1.2-1.5 1.5-1.6-0.3-0.09999-1.4-0.99998-1.5-1.6" fill="none"/></g></g></g></g></g><g transform="matrix(.7071 .7071 -.7071 .7071 9.6748 9.6748)"><g stroke-width=".31999pt"><g stroke-dashoffset="0pt"><g stroke-linecap="round"><g stroke-linejoin="round"><path d="m-1.2 1.6c0.1-0.60002 1.2-1.5 1.5-1.6-0.3-0.09999-1.4-0.99998-1.5-1.6" fill="none"/></g></g></g></g></g></g></g></g></g></svg>


SVG short nwsearrow

<svg viewBox="-10.20001 -0.2 10.40001 10.40001 " width="10.4pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" height="10.4pt"><g transform="translate(0 10.2) scale(1 -1) translate(0 .2)"><g stroke="#000"><g><g stroke-width=".4pt"><path d="m-0.32526 0.32526l-9.3495 9.3495" fill="none"/><g transform="matrix(.7071 -.7071 .7071 .7071 -.32526 .32526)"><g stroke-width=".31999pt"><g stroke-dashoffset="0pt"><g stroke-linecap="round"><g stroke-linejoin="round"><path d="m-1.2 1.6c0.1-0.60002 1.2-1.5 1.5-1.6-0.3-0.09999-1.4-0.99998-1.5-1.6" fill="none"/></g></g></g></g></g><g transform="matrix(-.7071 .7071 -.7071 -.7071 -9.6748 9.6748)"><g stroke-width=".31999pt"><g stroke-dashoffset="0pt"><g stroke-linecap="round"><g stroke-linejoin="round"><path d="m-1.2 1.6c0.1-0.60002 1.2-1.5 1.5-1.6-0.3-0.09999-1.4-0.99998-1.5-1.6" fill="none"/></g></g></g></g></g></g></g></g></g></svg>


SVG skein double crossing

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 <g stroke-width="1.0pt">
 <g stroke="rgb(100.0%,0.0%,0.0%)">
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SVG skein negative crossing

<svg viewBox="-2.5 -2.5 61.90549 61.90549 " width="62pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" height="62pt"><g transform="translate(0 59) scale(1 -1) translate(0 2.5)"><g stroke="#000"><g fill="#000"><g stroke-width=".4pt"><g stroke-width="2pt"><g stroke="#fff"><g fill="#fff"><g stroke-width="5pt"><path d="m0 0l57 57" fill="none"/><g stroke-width="1pt"><g stroke="#f00"><path d="m0 0l57 57" fill="none"/></g></g></g><g stroke="#f00"><g fill="#f00"><g transform="matrix(.71 .71 -.71 .71 55 55)"><g transform="matrix(1 0 0 1 0 0)"><g stroke-dasharray="none" stroke-dashoffset="0pt"><g stroke-linejoin="miter"><path d="m-7.3 4.5l7.8-4.5-7.8-4.5z"/></g></g></g></g></g></g><g stroke-width="2pt"><g stroke="#fff"><g fill="#fff"></g></g></g></g></g></g><g stroke-width="2pt"><g stroke="#fff"><g fill="#fff"><g stroke-width="5pt"><path d="m57 0l-57 57" fill="none"/><g stroke-width="1pt"><g stroke="#f00"><path d="m57 0l-57 57" fill="none"/></g></g></g><g stroke="#f00"><g fill="#f00"><g transform="matrix(-.71 .71 -.71 -.71 1.7 55)"><g transform="matrix(1 0 0 1 0 0)"><g stroke-dasharray="none" stroke-dashoffset="0pt"><g stroke-linejoin="miter"><path d="m-7.3 4.5l7.8-4.5-7.8-4.5z"/></g></g></g></g></g></g><g stroke-width="2pt"><g stroke="#fff"><g fill="#fff"></g></g></g></g></g></g></g></g></g></g></svg>


SVG skein no crossing

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SVG skein positive crossing

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SVGsandbox

(free)


swampland

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

#Contents#
* table of contents
{:toc}

## Idea

In [[string theory]] the term _swampland_ had been introduced ([Vafa 05](#Vafa05)) to publicly highlight the basic fact that many [[effective quantum field theory]] [[vacuum state|vacua]] will not admit a [[UV-completion]] to a [[string theory vacuum]], hence that admitting a completion to a [[string theory vacuum]] is a strong constraint, hence that [[string theory]] predicts many more conditions to be satisfied by [[gauge groups]], [[field (physics)|field content]] and [[coupling constants]] than predicted by plain [[quantum field theory]].

The terminology was motivated since the collection of [[string theory vacua]] had previously come to be called the _[[landscape of string theory vacua]]_. The idea is to imagine the remaining [[effective field theory|EFTs]] not "in" this landscape to form a space "away from the landscape", whence the colorful imagery of a swampland.

More in detail, there is supposedly a map

\[
  \label{PointParticleLimitMap}
  \text{String Vacua}
  \overset{\;\;ppl\;\;}{\longrightarrow}
  \text{effective QFT Vacua}
\]

that takes [[string theory vacua]] to their low-energy approximation by [[vacuum state|vacua]] of [[effective quantum field theories]]. One way to possibly formalize this is to take the point-particle limit (ppl) of [[2d SCFTs]] to obtain [[spectral triples]] (as discussed at _[[2-spectral triple]]_) taking [[string]] [[worldsheets]] to [[Feynman diagrams]]:

> graphics grabbed from [Schubert 96](worldline+formalism#Schubert96)

Then with language of [[homological algebra]] or more generally of [[category theory]] we may begin to formalize the situation as follows:

1. the [[domain]] of (eq:PointParticleLimitMap) is the [[landscape of string theory vacua]];

1. the [[image]] of (eq:PointParticleLimitMap) is the landscape of corresponding [[effective quantum field theories|eQFTs]] that admit stringy [[UV-completion]];

1. the [[cokernel]] of (eq:PointParticleLimitMap) is the _swampland_;

1. the [[kernel]] of (eq:PointParticleLimitMap), or more generally its [[fiber]] over any [[effective quantum field theory|EFT]], is the space of different choices of stringy [[UV-completion]] of the same [[effective quantum field theory]].

Making this fully precise requires saying more about what the [[domain]] and [[codomain]] in (eq:PointParticleLimitMap) actually are, and in which ambient [[category]] (they will be some kind of [[moduli stacks]] in an ambient [[(∞,1)-category]] which may not quite be [[stable (∞,1)-category|stable]], whence "[[cokernel]]" may need to be interpreted in a non-abelian sense; but such details don't change the general idea here).

For example, part of what it means to specify a [[string theory vacuum]] is to declare the [[D-brane charge]] contained in this vacuum (subject to [[RR-field tadpole cancellation]] against [[O-plane]]-charges). In actual [[string theory]] this [[RR-field]] [[charge]] is supposed to be (see [there](D-brane#ReferencesKTheoryDescription)) a [[cocycle]] in some flavour of [[topological K-theory]] ([[twisted K-theory|twisted]] [[equivariant K-theory|equivariant]] [[differential K-theory|differential]] [[KR-theory]]), while its image in the underlying [[effective field theory]] is in the corresponding flavour of [[ordinary cohomology]]/[[de Rham cohomology]]. The map that relates the two incarnations of RR-field charge is the _[[Chern character]]_, which is what formalizes the map (eq:PointParticleLimitMap) in in the D-brane charge "sector" of the theory

$$
  \array{
    \text{String Vacua}
    &\overset{\;\;ppl\;\;}{\longrightarrow}&
    \text{effective QFT Vacua}
    \\
    \left\{
    \array{
      \text{D-brane RR-charge}
      \\
      \text{in K-theory}
    }
    \right\}
    &\overset{\text{Chern character}}{\longrightarrow}&
    \left\{
    \array{
      \text{flux forms in}
      \\
      \text{ordinary cohomology}
    }
    \right\}
  }
$$

The [[Chern character]] in general does have non-trivial [[cokernel]] ("swampland RR-fields") and [[kernel]] (choices of [[UV-completion]] of the effective RR-fields). In fact it fits not just in a [[short exact sequence]], but in the [[differential cohomology hexagon]] (see there for more) of [[K-theory]].

> graphics grabbed from [SS 18](#SS18)

In contrast to this example, the literature on the "swampland" phenomenon is currently dominated by informal hand-wavy arguments. Starting with [Ooguri-Vafa 06](#OoguriVafa06) is an attempt to guess semi-precise rules-of-thumb for recognizing [[effective field theory|EFTs]] in the swampland, now known as the _swampland conjectures_. Motivated by the re-opening of the question whether [[de Sitter spacetime]] actually appears in [[string theory vacua]] or not ([Danielsson-van Riet 18](#DanielssonVanRiet18)), these swampland conjecture currently revolve around bounds on the [[cosmological constant]] in relation to [[scalar fields]] in the theory.

An argument that much of the existing swampland literature does not even live up to the level of rigour of [[folklore]] string theory, and in fact that it is fundamentally flawed, is made in [Banks 19](#Banks19).

## Swampland conjectures
 {#SwamplandConjectures}

The following list of statements have come (around 2020-2021) to be widely discussed in a sector of the [[string phenomenology]] community which associates with the "swampland" imagery. They are now commonly referred to and known as  "conjectures", specifically "the swampland conjectures", but the terms used tend to remain vague/undefined and may change their intended meaning with context (not the least the notion of "state in quantum gravity", which however in most of these statements really means: classical solitonic [[black brane]]-solution in some [[supergravity]]-theory -- e.g. in Conj. \ref{ElectricWeakGravityConjecture}).

But some of these statements have been interpreted in special cases as more precise statements about [[moduli spaces]] of [[Calabi-Yau manifolds]], and in this form they may be closer to [[conjectures]] in the sense of established sense of [[mathematics]].

### No global symmetries conjecture

\begin{conjecture}
A theory coupled to [[gravity]] must have no global symmetries, which means that they are either broken or gauged.
\end{conjecture}

This includes global $p$-form symmetries as well as global discrete symmetries.

### Completeness hypothesis

\begin{conjecture}
A [[gauge theory]] coupled to [[gravity]] must contain physical states with all possible gauge charges consistent with [[Dirac charge quantization]].
\end{conjecture}

\begin{remark}
This immediately implies that any continuous [[gauge group]] (i.e. a [[Lie group]]) must be [[compact]].
However, this does not restrict discrete gauge groups, which can be non-compact (e.g. [[duality]] groups).
\end{remark}

### BPS completeness hypothesis

\begin{conjecture}
If a given charge can be populated by [[BPS states]], then the physical spectrum must contain a BPS state with this charge.
\end{conjecture}

### Weak gravity conjecture

#### Electric weak gravity conjecture

\begin{conjecture}\label{ElectricWeakGravityConjecture}
Given a $U(1)$-[[gauge theory]] weakly coupled to [[gravity]] with [[gauge coupling]] $g$, there must exist an electrically charged state with mass $m$ and charge $q$ satisfying

$$ m \leq \sqrt{2} g q M_p, $$

where $M_p$ is the [[Planck mass]].
\end{conjecture}

The electric weak gravity conjecture can be also formulated by stating that there must exist an electrically charged state satisfying

$$ \frac{Q}{m} \leq \frac{Q_{\mathrm{ext}}}{M_{\mathrm{ext}}} $$

where $Q=q g$ is the non-quantized charge, $Q_{\mathrm{ext}}$ and $M_{\mathrm{ext}}$ are charge and mass of an [[extremal black hole]].

#### Magnetic weak gravity conjecture

\begin{conjecture}
Given a $U(1)$-[[gauge theory]] weakly coupled to [[gravity]] with [[gauge coupling]] $g$, The cutoff scale $\Lambda$ of the [[effective field theory]] must be bounded from above by the [[gauge coupling]] as

$$ \Lambda \leq g M_p^{(d-2)/2}, $$

where $M_p$ is the [[Planck mass]] and $d$ is the [[dimension]].
\end{conjecture}

#### Convex hull weak gravity conjecture

\begin{conjecture}
Given a $U(1)^n$-[[gauge theory]] weakly coupled to [[gravity]], let $\vec{z}_i:=\vec{q}_i/m \in\mathbb{R}^n$ be the charge to mass ratio of a charged state $i$ and let $I$ be the set of charged states in the theory. Then, the [[convex hull|convex set]] of the set $\{\pm\vec{z}_i\}_{i\in\I}$ must contain the unit ball as measured by the kinetic matrix of the $U(1)^n$-[[gauge theory]].
\end{conjecture}

This condition is a stronger statement than requiring the weak gravity conjecture to be satised for each $U(1)$-[[gauge theory]].

#### Sublattice (or tower) weak gravity conjecture

(...)

### Swampland distance conjecture

\begin{terminology}
Consider the [[moduli space]] $\mathcal{M}$ of a theory coupled to [[gravity]], which is parametrized by the expectation values of some field $\phi^i$ that has no [[potential]]. Starting from any point $P \in \mathcal{M}$ there always exists another point $Q \in \mathcal{M}$ such that their [[geodesic distance]] $d(P,Q)$ is infinite. This is called *infinite field distance limit*.
\end{terminology}

\begin{example}
The decompactification limit in compactified [[string theory]] is the standard example of infinite field distance limit.
\end{example}

\begin{conjecture}
There exists an infinite tower of states with a mass scale $M$ that becomes exponentially light at any infinite field distance limit as

$$ M(Q) \sim M(P)e^{-c d(P,Q)}, $$

where $c$ is a positive constant.
\end{conjecture}

### Emergent string conjecture

\begin{conjecture}
Any infinite field distance limit is either

* a decompactification limit, or

* a limit in which a weakly coupled string becomes tensionless.

\end{conjecture}

This is a refinement of the *swampland distance conjecture* which explicits the nature of the tower of states.

### AdS distance conjecture

\begin{conjecture}
Any [[AdS]] [[vacuum]] has an infinite tower of states that becomes light in the flat spacetime limit $\Lambda\rightarrow 0$, satisfying

$$ m\sim|\Lambda|^c, $$

where $c$ is a constant.
\end{conjecture}

### AdS instability conjecture

\begin{conjecture}
Any [[non-supersymmetric vacuum]] is at best metastable (and eventually has to decay).
\end{conjecture}

### dS conjecture

\begin{conjecture}
The scalar [[potential]] $V$ of an [[effective field theory]] weakly coupled to gravity must satisfy the following bound on its derivatives:

$$ |\nabla V| \geq \frac{c}{M_p}V, $$

where $c\sim\mathcal{O}(1)$ is positive.
\end{conjecture}

The dS conjecture was further refined as it follows.

\begin{conjecture}
The previous bound needs to be imposed only if the condition
$$ \mathrm{min}(\nabla_i\nabla_j V) \leq -\frac{c'}{M_p^2}V, $$
for some $c'\sim\mathcal{O}(1)$, on the second derivative of the [[potential]] is violated.
\end{conjecture}

Thanks to this refinement, only proper dS minima are excluded and not general critical points.

### Trans-planckian censorship conjecture

(...)

### Asymptotic dS conjecture

\begin{conjecture}
The scalar [[potential]] $V$ of an [[effective field theory]] weakly coupled to [[gravity]] presents a _runaway behavior_ when approaching an infinite field distance point, i.e.

$$ |\nabla V| \geq c V, $$

where $c\sim\mathcal{O}(1)$.
\end{conjecture}

This is an asymptotic version of the *dS conjecture*.

### Swampland cobordism conjecture
 {#SwamplandCobordismConjecture}

From [McNama & Vafa 17](#McNamaVafa17):

\begin{conjecture}\label{OriginalSwamplandCobordismConjecture}
For any [[quantum gravity]] theory [[KK-compactification|compactified]] on a $d$-dimensional internal manifold, some kind of "quantum-gravity version" of a [[cobordism group]] must vanish in degree $d$. In heuristic symbols:

$$ \Omega^{\mathrm{QG}}_d = 0. $$
\end{conjecture}

\begin{remark}
A rigorous discussion of a possible role of [[cobordism cohomology]] in [[M-theory]], assuming [[Hypothesis H]], is in [Sati & Schreiber 21a](#SatiSchreiber21a); for relation to discussion of Conj. \ref{OriginalSwamplandCobordismConjecture} see [p. 83](https://arxiv.org/pdf/2103.01877.pdf#page=83) there.
\end{remark}

## References

### General

The terminology "swampland", in the context of [[string phenomenology]], originates with:

* {#Vafa05} [[Cumrun Vafa]], _The String Landscape and the Swampland_ ([arXiv:hepth/0509212](http://arxiv.org/abs/hepth/0509212))

Review and lecture notes:

* [[Eran Palti]], _The Swampland: Introduction and Review_, lecture notes ([arXiv:1903.06239](https://arxiv.org/abs/1903.06239))

* Marieke van Beest, José Calderón-Infante, Delaram Mirfendereski, Irene Valenzuela, _Lectures on the Swampland Program in String Compactifications_ ([arXiv:2102.01111](https://arxiv.org/abs/2102.01111))

Further discussion:

* {#OoguriVafa06} [[Hirosi Ooguri]], [[Cumrun Vafa]], _On the Geometry of the String Landscape and the Swampland_, Nucl.Phys.B766:21-33, 2007 ([arXiv:hep-th/0605264](https://arxiv.org/abs/hep-th/0605264))

* {#BrennanCartaVafa17} T. Daniel Brennan, Federico Carta, [[Cumrun Vafa]], _The String Landscape, the Swampland, and the Missing Corner_ ([arXiv:1711.00864](https://arxiv.org/abs/1711.00864))

* [[Ben Heidenreich]], [[Matthew Reece]], [[Tom Rudelius]], _Emergence and the Swampland Conjectures_ ([arXiv:1802.08698](https://arxiv.org/abs/1802.08698))

* Hee-Cheol Kim, Houri-Christina Tarazi, [[Cumrun Vafa]], _Four Dimensional $\mathcal{N}=4$ SYM and the Swampland_ ([arXiv:1912.06144](https://arxiv.org/abs/1912.06144))

See also

* Wikipedia, _Swampland_(physics)_

Beware that the landscape literature is presently dominated by non-rigorous hand-wavy [[string phenomenology]]:

* {#Banks19} [[Tom Banks]], _On the Limits of Effective Quantum Field Theory: Eternal Inflation, Landscapes, and Other Mythical Beasts_ ([arxiv:1910.12817](https://arxiv.org/abs/1910.12817))

  from pages 14-22:

  > these considerations lead to conclusions at odds with the seemingly similar arguments of [the [[swampland conjectures]]]. $[\cdots]$  Perturbative moduli space completely distorts the true nature of the class of consistent models.

  > It’s important to realize that the entire procedure just outlined for finding (meta) stable AdS minima of a non-perturbative effective potential is purely hypothetical and has no basis in well founded string theory calculations. $[\cdots].$

  > The hypothesis of the String Landscape is entirely based on low energy effective field theory ideas about finding "vacua" by minimizing an effective potential.  Everything that’s been said above indicates that this idea has no validity in genuine models of quantum gravity. $[\cdots]$

  > The most serious issue,  in  my  opinion,  is  the  contention  that  one  can  make  the  AdS  radius  much  larger  thanthe size of the compact manifold.  All well established examples of large radius AdS/CFT havea compact manifold of dimension 2 or greater whose radius is comparable to that of the AdSspace.  In Appendix A we’ll present an argument based on the properties of AdS black holes,that this is in fact necessary.

  > The next step in the construction of "realistic" models involves "adding an anti-brane to break supersymmetry and make the c.c.  positive".  This is supposed to be a small modification of the model, calculable in low energy effective field theory, and that seems manifestly incorrect. $[\cdots]$.

  > even if one believes that the construction of meta-stable dS models is reliable, there is no clear argument about  what  the  proper  observables  of  the  model  are  nor  that different  dS  constructions are part of the same model.  Neither is there an interpretation of these correlators as transition amplitudes in a quantum mechanical model. $[\cdots]$

  > The conclusion  that  effective  field  theorists  should  draw  from  this is that unlike super-symmetric string models in flat or AdS space-time, many of which have at least perturbative definitions as mathematical models obeying the axioms of quantum mechanics, all literature on the String Landscape is speculation based on the unfounded notion that all string models with a given amount of SUSY are part of one single model and that it makes sense to define an effective action that encompasses all string models. Every single non-perturbative construction of string models contradicts this claim $[\cdots]$

Identification of the swampland charge structure with the [[cokernel]] of the [[Chern character]]:

* {#SS18} [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:Lift of fractional D-brane charge to equivariant Cohomotopy theory|Lift of fractional D-brane charge to equivariant Cohomotopy theory]]_ ([arXiv:1812.09679](https://arxiv.org/abs/1812.09679), [Python code](https://arxiv.org/src/1812.09679v1/anc))

### de Sitter vacua

On the question whether [[de Sitter spacetime]]-[[vacua]] belong to the [[swampland]] or not:

* {#DanielssonVanRiet18} [[Ulf Danielsson]], [[Thomas Van Riet]],  _What if string theory has no de Sitter vacua?_, International Journal of Modern Physics D, Vol. 27, No. 12, 1830007 (2018)  ([arXiv:1804.01120](https://arxiv.org/abs/1804.01120), [doi:10.1142/S0218271818300070](https://doi.org/10.1142/S0218271818300070))

* {#vanRiet18} [[Thomas Van Riet]], _Is dS space in the Swampland_, talk at [StringPheno18](http://sp18.fuw.edu.pl/) ([pdf slides](http://sp18.fuw.edu.pl/wp-content/uploads/participants-database/thomasvanriet.pdf))

* {#vanRiet18a} [[Thomas Van Riet]], _Status of KKLT_, talk at _[Simons summer workshop 2018](http://scgp.stonybrook.edu/archives/24870)_ ([recording](http://scgp.stonybrook.edu/video_portal/video.php?id=3730))

* {#ObiedOoguriSpodyneikoVafa18} Georges Obied, [[Hirosi Ooguri]], Lev Spodyneiko, [[Cumrun Vafa]], _De Sitter Space and the Swampland_ ([arXiv:1806.08362](https://arxiv.org/abs/1806.08362))

* {#AgrawalObiedSteinhardtVafa18} Prateek Agrawal, Georges Obied, Paul Steinhardt, [[Cumrun Vafa]], _On the Cosmological Implications of the String Swampland_ ([arXiv:1806.09718](https://arxiv.org/abs/1806.09718))

* {#Vafa18} [[Cumrun Vafa]], _Cosmology and the String Swampland_, talk at _[Strings 2018](https://indico.oist.jp/indico/event/5/)_ ([pdf slides](https://indico.oist.jp/indico/event/5/picture/96.pdf), [recording](https://www.youtube.com/watch?v=fU8sJRCRz24&t=1904s))

* [[Frederik Denef]], Arthur Hebecker, Timm Wrase, _The dS swampland conjecture and the Higgs potential_ ([arXiv:1807.06581](https://arxiv.org/abs/1807.06581))

* {#Danielsson18} [[Ulf Danielsson]], _The quantum swampland_ ([arXiv:1809.04512](https://arxiv.org/abs/1809.04512))

  (argues that the issue with [[string theory|stringy]] de Sitter [[moduli stabilization]] raised in [Sethi 17](#Sethi17) is related to the de Sitter instability seen in QFT, according to the references [above](de+Sitter+spacetime#InPerturbativeQuantumGravity))

* Georges Obied, [[Hirosi Ooguri]], Lev Spodyneiko, [[Cumrun Vafa]], _De Sitter Space and the Swampland_ ([arXiv:1806.08362](https://arxiv.org/abs/1806.08362))

* Jakob Moritz, Ander Retolaza, Alexander Westphal, _On uplifts by warped anti-D3-branes_ ([arXiv:1809.06618](https://arxiv.org/abs/1809.06618))

* Iosif Bena, Emilian Dudas, Mariana Graña, Severin Lüst, _Uplifting Runaways_ ([arXiv:1809.06861](https://arxiv.org/abs/1809.06861))

* [[Clay Cordova]], G. Bruno De Luca, [[Alessandro Tomasiello]], _Classical de Sitter Solutions of Ten-Dimensional Supergravity_ ([arXiv:1812.04147](https://arxiv.org/abs/1812.04147))

On  [[de Sitter spacetime]] [[cosmology]] realized in [[brane world models]] in ambient $\sim$[[anti de Sitter spacetime|AdS]]-[[bulk]] [[spacetime]]:

* {#BDDGS18} Souvik Banerjee, [[Ulf Danielsson]], Giuseppe Dibitetto, Suvendu Giri, Marjorie Schillo, _Emergent de Sitter cosmology from decaying AdS_ ([arXiv:1807.01570](https://arxiv.org/abs/1807.01570))

* Souvik Banerjee, [[Ulf Danielsson]], Giuseppe Dibitetto, Suvendu Giri, Marjorie Schillo, _de Sitter Cosmology on an expanding bubble_ ([arXiv:1907.04268](https://arxiv.org/abs/1907.04268))

Discussion in the context of [[M-theory on G2-manifolds]]:

* Beatriz de Carlos, Andre Lukas, Stephen Morris, _Non-perturbative vacua for M-theory on G2 manifolds_, JHEP 0412:018, 2004 ([arxiv:hep-th/0409255](https://arxiv.org/abs/hep-th/0409255))

  which concludes that with taking [[non-perturbative effects]] from [[membrane instantons]] into account one gets 4d vacua with vanishing and negative [[cosmological constant]] ([[Minkowski spacetime]] and [[anti-de Sitter spacetime]]) but not with positive [[cosmological constant]] ([[de Sitter spacetime]]). They close by speculating that [[M5-brane]] instantons might yield [[de Sitter spacetime]].

* Johan Blåbäck, [[Ulf Danielsson]], Giuseppe Dibitetto, Suvendu Giri, _Constructing stable de Sitter in M-theory from higher curvature corrections_ ([arXiv:1902.04053](https://arxiv.org/abs/1902.04053))

  which suggests that including [[higher curvature corrections]] makes it work

* Iosif Bena, Alex Buchel, Severin Lüst, _Throat destabilization (for profit and for fun)_ ([arxiv:1910.08094](https://arxiv.org/abs/1910.08094))

[[!include swampland cobordism conjecture -- references]]

[[!redirects swampland conjecture]]
[[!redirects swampland conjectures]]

[[!redirects swampland cobordism conjecture]]


swampland cobordism conjecture -- references

### Swampland cobordism conjecture
  {#SwamplandCobordismConjectureReferences}

The _[swampland cobordism conjecture](swampland#SwamplandCobordismConjecture)_ is the hypothesis that consistency of [[string theory vacuum|vacua]] in [[string theory]]/[[M-theory]]/[[F-theory]] (hence their being in the "[[landscape of string theory vacua|landscape]]" instead of the "[[swampland]]") requires (undefined) [[string theory|stringy]]/[[quantum gravity]]-analogs of [[cobordism groups]] to vanish:

* {#McNamaVafa17} [[Jacob McNamara]], [[Cumrun Vafa]], _Cobordism Classes and the Swampland_ ([arXiv:1909.10355](https://arxiv.org/abs/1909.10355), [spire:1755441](https://inspirehep.net/literature/1755441))

* [[Miguel Montero]], [[Cumrun Vafa]], _Cobordism Conjecture, Anomalies, and the String Lamppost Principle_, J. High Energ. Phys. 2021, 63 (2021) ([arXiv:2008.11729](https://arxiv.org/abs/2008.11729))

* [[Markus Dierigl]], [[Jonathan Heckman]], _On the Swampland Cobordism Conjecture and Non-Abelian Duality Groups_, Physical Review D 2021 ([arXiv:2012.00013](https://arxiv.org/abs/2012.00013))

* [[Hirosi Ooguri]], [[Tadashi Takayanagi]], _Cobordism Conjecture in AdS_ ([arXiv:2006.13953](https://arxiv.org/abs/2006.13953))

* David Andriot, [[Nils Carqueville]], Niccolò Cribiori, _Looking for structure in the cobordism conjecture_ ([arXiv:2204.00021](https://arxiv.org/abs/2204.00021))

Relation to [[topological K-theory]]:

* [[Ralph Blumenhagen]], Niccolò Cribiori, Christian Kneissl, Andriana Makridou, *Dimensional Reduction of Cobordism and K-theory*, Journal of High Energy Physics 2023, 181, [[doi:10.1007/JHEP03(2023)181](https://doi.org/10.1007/JHEP03%282023%29181)], [[arXiv:2208.01656](https://arxiv.org/abs/2208.01656)]

* [[Ralph Blumenhagen]], Christian Kneissl, Chuying Wang, *Dynamical Cobordism Conjecture: Solutions for End-of-the-World Branes* $[$[arXiv:2303.03423](https://arxiv.org/abs/2303.03423)$]$

Review:

* [[Ralph Blumenhagen]], *Nullifying Cobordism in Quantum Gravity*, talk at *[QFT and Cobordism](https://nyuad.nyu.edu/en/events/2023/march/quantum-field-theories-and-cobordisms.html)*, [[CQTS]] (Mar 2023) $[$[web](Center+for+Quantum+and+Topological+Systems#BlumenhagenMar2023)$]$

A more concrete consequence of this conjecture is claimed ([McNamara-Vafa 17, Sec. 5.2, 2nd paragr.](https://arxiv.org/pdf/1909.10355.pdf#page=31)) to be the statement that -- paraphrasing/extrapolating somewhat: _[[Dirac charge quantization|brane charges are quantized]] in [[Cobordism cohomology]]_; so that, in particular _[[tadpole cancellation]] of brane charges is to happen in [[Cobordism cohomology]]._ This statement is discussed, as a consequence of [[Hypothesis H]], in (see [p. 83](https://arxiv.org/pdf/2103.01877.pdf#page=83)):

* {#SatiSchreiber21a} [[Hisham Sati]], [[Urs Schreiber]], $\,$_[[schreiber:M-Theory as Mf-Theory|M/F-Theory as Mf-Theory]]_ ([arXiv:2103.01877](https://arxiv.org/abs/2103.01877))


Swapneel Mahajan

Swapneel Mahajan works on categories, higher categorical structures, and their applications to combinatorics.  Together with [[Marcelo Aguiar]] he has written an 836-page book on this subject, which is currently available for free online:

* Marcelo Aguiar and Swapneel Mahajan, _[[Monoidal Functors, Species and Hopf Algebras]]_, to be published by Cambridge U. Press.  ([web](http://www.math.tamu.edu/~maguiar/a.pdf))

See also Mahajan's ([website](http://www.math.iitb.ac.in/~swapneel/)).

category: people


Sweedler coring

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## Idea

A canonical or Sweedler [[coring]] is an algebraic structure that is roughly the [[duality|formal dual]] of the [[Čech nerve]] of a [[cover]]: it is used to describe [[descent]] in algebraic contexts.

See also [[monadic descent]].

## Definition

### In components

Let $f : R \hookrightarrow S$ be the extension of associative unital $k$-[[associative algebra|algebra]]s (where $k$ is a commutative unital [[ring]]).

The corresponding __canonical coring__ or __Sweedler coring__ is the $S$-[[coring]]

$$
  C = S\otimes_R S
$$

with coproduct

$$
  \Delta : C\to C\otimes_S C \cong S\otimes_R S\otimes_R S
$$

given by the bilinear extension of the formula

$$
  \Delta: s_1\otimes s_2 \mapsto s_1\otimes 1 \otimes s_2
$$

and counit

$$
  \epsilon : C\to S,\,\,\,\,\,\,\,
 \epsilon: s_1 \otimes s_2 \mapsto s_1 s_2
 \,.
$$

The element $1\otimes 1$ is a [[grouplike element]] in the Sweedler's coring.

### Geometric interpretation {#GeomInterpretationOfCoring}

We give a dual geometric interpretation of the Sweedler coring.

Suppose a context of spaces and function algebras on spaces that satisfies the basic axioms of [[geometric function theory]], in that the algebra of functions $C(Y_1 \times_X Y_2)$ on a [[fiber product]]

$$
  \array{
    Y_1 \times_X Y_2 &\to& Y_2
    \\
    \downarrow && \downarrow
    \\
    Y_1 &\to& X
  }
$$

is the [[tensor product]] of the functions on the factors:

$$
  C(Y_1 \times_X Y_2) = C(Y_1) \otimes_{C(X)} C(Y_2)
  \,.
$$

Then let $\pi : Y \to X$ be a morphism of spaces and set

$$
  R := C(X)
$$

and

$$
  S = C(Y)
$$

and

$$
  (R \hookrightarrow S) := \pi^* : C(X) \to C(Y)
  \,.
$$

The morphism $\pi$ induces its augmented [[Čech nerve]]

$$
  \left(
     \cdots
     \stackrel{\to}{\stackrel{\to}{\stackrel{\to}{\to}}}
     Y \times_X Y \times_X Y
     \stackrel{\to}{\stackrel{\to}{\to}}
     Y \times_X Y \stackrel{\to}{\to}
     Y
     \stackrel{\pi}{\to}
     X
  \right)
  \,.
$$

Taking function algebras of this yields, by the above,

$$
  \left(
     \cdots
     \stackrel{\leftarrow}{\stackrel{\leftarrow}{\stackrel{\leftarrow}{\leftarrow}}}
     S \otimes_R S \otimes_R S
     \stackrel{\leftarrow}{\stackrel{\leftarrow}{\leftarrow}}
     S \otimes_R S
     \stackrel{\leftarrow}{\leftarrow}
     S
     \stackrel{\pi^*}{\leftarrow}
     R
  \right)
  \,.
$$

Writing again $C = S \otimes_R S$ for the Sweedler coring, this is

$$
  \left(
     \cdots
     \stackrel{\leftarrow}{\stackrel{\leftarrow}{\stackrel{\leftarrow}{\leftarrow}}}
     C \otimes_S C
     \stackrel{\leftarrow}{\stackrel{\leftarrow}{\leftarrow}}
     C
     \stackrel{\leftarrow}{\leftarrow}
     S
     \stackrel{\pi^*}{\leftarrow}
     R
  \right)
  \,.
$$

## Properties

### Relation to ring extensions

Various properties of canonical coring correspond to adequate properties of the ring extension. For example, [[coseparable coring|coseparable]] canonical corings correspond to [[split extension]]s (the $k$-algebra extension $R\to S$ is split if there is an $R$-bimodule map $h: S\to R$ with $h(1_S) = 1_R$).

In the case of the trivial ring extension $R\to R$ the coproduct of the canonical coring is the canonical isomorphism
$R\to R\otimes_R R$ and the counit is the identity $R\to R$. Thus, every right $R$-module is a comodule over the canonical coring. Such a canonical coring is called a __trivial coring__. This example puts the rings (or associative unital algebras) and their categories of modules as a special case of corings and their categories of comodules. In view of the next paragraph, this is a generalization of a coring of a cover for the case of a trivial (identity) cover.

### Descent in terms of coring comodules
 {#DescentIntermsOfCoringModules}

Given a morphism $f : R \to S$ with corresponding Sweedler coring $(C = S \otimes_R S,\Delta,\epsilon)$ as above, the category of [[descent]] data $\mathrm{Desc}(S/R)$ for the categories of right modules along $k$-algebra extension $R\to S$  is precisely the category of right $C$-[[comodule]]s.

In other words, the objects of $\mathrm{Desc}(S/R)$ are the pairs $(N,\alpha)$ where $N$ is a right $S$-[[module]], and $\alpha: N\to N\otimes_R S$ is a right $S$-module morphism and if we write $\alpha(m) = \sum_i m_i \otimes s_i$ then

* $\sum_i \alpha(m_i)\otimes s_i = \sum_i m_i\otimes 1\otimes s_i$,

* $\sum_i m_i s_i = m$.

#### In terms of (co)monadic descent {#ComonadicDescent}

This [[coring]]-formulation of [[descent]] may be understood as special case of [[comonadic descent]] (see also the discussion at [[Bénabou-Roubaud theorem]]). See e.g. ([Hess 10, section 2](#Hess10)) for a review. We spell this out in a bit more detail:

The [[bifibration]] in question is

$p :$ [[Mod]] $\to$ [[CRing]]

that sends an object in the category [[Mod]] of [[module]]s to the ring that it is a module over.

A descent datum for a morphism $f : R \to S$ with respect to this bifibration is a (co)algebra object over the co[[monad]] $f_* f^*$ induced by this morphism. We have that

* the morphism $f_*$ sends an $R$-module $N$ to the $S$-module $N \otimes_R S$;

* the morphism $f^*$ sends an $S$-module $P$ to the $R$-module $P \otimes_S S_R$, where $S_R$ is $S$ regarded as a left $S$- and a right $R$-module. So $P \otimes_S S_R$ is just the $S$-module $P$ with only the right $R$-action remembered.

Accordingly, the comonad with underlying functor $f_* f^*$ sends an $S$-module $P$ to the $S$-module $P \otimes_S S \otimes_R S = P \otimes_S C$.

A (co)algebra object for this comonad is hence a co-action morphism

$$
  P \to P \otimes_S C
$$

compatible with the monad action. This is precisely a comodule over the Sweedler coring, as defined above.

#### Geometric interpretation {#GeomInterpretationOfDescent}

Descent for Sweedler corings is a special case of [[monadic descent|comonadic descent]]. We describe this in detail and relate it by duality to the geometrically more intuitive [monadic descent for codomain fibrations](http://ncatlab.org/nlab/show/monadic+descent#ForCodomainFibs).

Assuming again a suitable geometric context as above, we may identify a [[module]] over $R = C(X)$ with (the collection of [[section]]s of) a [[vector bundle]] (or rather a suitable generalization of that: a  [[coherent sheaf]]) over $X$. Similarly for $Y$. So we write

$$
  Vec(X) := R Mod
$$

and

$$
  Vec(Y) := S Mod
$$

for the corresponding [[category|categories]] of [[module]]s. The assignment of such categories to spaces

$$
  Vec : Z \mapsto Vec(Z)
$$

extends to a contravariant [[pseudofunctor]]

$$
  Vec : Spaces^{op} \to Cat
$$

by assigning to a morphism $f : Y \to X$ of spaces the corresponding [[functor]]

$$
  Vec(X) \simeq C(X) Mod \stackrel{- \otimes_{f} C(Y)}{\to} C(Y) Mod \simeq Vec(Y)
  \,.
$$

This way $Vec$ becomes a [[stack|prestack]] of categories on our category of spaces.

If this prestack satisfies [[descent]] along suitable [[cover]]s, it is a [[stack]].

Geometrically this is the case if for each morphism $\pi : Y\to X$ that is regarded as a [[cover]], the category $Desc(Y,Vec)$ whose objects are tuples consisting of

* an [[object]] $a \in Vec(Y)$

* an [[isomorphism]] $g : \pi_1^* a \to \pi_2^* a$

* such that

  $$
    \array{
       && \pi_2^* a
       \\
       & {}^{\mathllap{\pi_{12}^* g}}\nearrow && \searrow^{\mathrlap{\pi_{23}^* g}}
       \\
       \pi_1^* a &&\stackrel{\pi_{13}^*}{\to}&& \pi_3^*
    }
  $$

  commutes.

Morphism are defined similarly (see [[stack]] and [[descent]] for details).

To get the geometric pucture that underlies, by duality, the above comodule definition of descent, we need to reformulate this just a little bit more:

every ordinary [[vector bundle]] $E \to X$ (of finite rank) is the [[associated bundle]] $E \simeq P \times_{O(n)} V$ of an [[orthogonal group|O(n)]]-[[principal bundle]] $P \to X$, and as such its [[section]]s may be identified with $O(n)$-equivariant functions $P \to V \simeq \mathbb{R}^n$ on the total space of $P$.

Using this we may think of the $C(X)$-module of sections of $E$ as a submodule of the $C(X)$-module of all functions on $P$

$$
  \Gamma(E) \subset C(P)
  \,.
$$

We now reformulate the geometric descent for [[vector bundle]]s in terms of geometric descent for their underlying [[principal bundle]]s, and then take functions on everything in sight to obtain the comodule definition of descent that we want to describe:

A descent datum (transition function) for a [[principal bundle]] $Q \to Y$ may be thought of as the the morphism $g$ in the double [[pullback]] diagram

$$
  \array{
    &&Y\times_X Y \times_Y Q &\to& Q
    \\
    &&\downarrow && \downarrow
    \\
    &{}^{\mathllap{g}}\swarrow&Y \times_X Y &\to& Y
    \\
    &&\downarrow && \downarrow^{\mathrlap{\pi}}
    \\
    Q &\to& Y &\stackrel{\pi}{\to}& X
  }
  \,.
$$

Because here $Y \times_X Y \times_Y Q$ is the space whose points consist of a point in a double overlap of the cover and a point in the fiber of $Q$ over that with respect to one patch, and the morphism identifies this with a point in the fiber of $Q$ regarded as sitting over the other patch. Analogously, there is a cocycle condition on $g$ on triple overlaps.

Now, blindly applying our functor that takes functions of spaces to the above diagram yields the double [[pushout]] diagram

$$
  \array{
    &&C(Y\times_X Y \times_Y Q) &\leftarrow& C(Q)
    \\
    &&\uparrow && \uparrow
    \\
    &{}^{\mathllap{g}^*}\nearrow&C(Y \times_X Y) &\leftarrow& C(Y)
    \\
    &&\uparrow && \uparrow^{\mathrlap{\pi^*}}
    \\
    C(Q) &\leftarrow& C(Y) &\stackrel{\pi^*}{\leftarrow}& C(X)
  }
  \,.
$$

We may restrict to $N := \Gamma(E) \subset C(Q)$ as just discussed and switch to the notation from above to get

$$
  \array{
    &&N \otimes_{S} C &\leftarrow& N
    \\
    &&\uparrow && \uparrow
    \\
    &{}^{\mathllap{g}^*}\nearrow& C &\leftarrow& S
    \\
    &&\uparrow && \uparrow^{\mathrlap{\pi^*}}
    \\
    N &\leftarrow& S &\stackrel{\pi^*}{\leftarrow}& R
  }
  \,.
$$

The morphism

$$
  \alpha := g^* : N \to N \otimes_S C
$$

obtained this way is the co-action morphism from the above algebraic definition.

The further cocycle condition on $g$ similarly translates into the condition that $\alpha$ really satisfies the [[comodule]] property.

### Relation to generalized cohomology and Adams spectral sequence
 {#RelationToGeneralizedCohomology}

Applied to [[E-infinity rings]] the Sweedler coring construction
yields the [[Hopf algebroids]] of dual [[Steenrod algebras]]
and appears in the [[Adams spectral sequence]].

## Related concepts

* [[Amitsur complex]], [[Hopf algebroid]]

* [[descent]]

  * [[cover]]

  * [[cohomological descent]]

  * [[monadic descent]],

    * **Sweedler coring**

    * [[monadic descent]], [[higher monadic descent]]

    * [[descent in noncommutative algebraic geometry]]

* [[sheaf]], [[(2,1)-sheaf]], [[2-sheaf]] [[(∞,1)-sheaf]]

## References

Sweedler corings are named after [[Moss Sweedler]].

A textbook account is in

*  [[Tomasz Brzezinski]], [[Robert Wisbauer]], section 29 of _Corings and Comodules_, Cambridge University Press, London Math. Soc. LN 309 (2003), ([errata pdf](http://www.math.uni-duesseldorf.de/~wisbauer/corinerr.pdf))

Section 29 there discusses the relation to the [[Amitsur complex]] and the [[descent theorem]].

* MO: [how-is-a-descent-datum-the-same-as-a-comodule-structure](http://mathoverflow.net/questions/150547/how-is-a-descent-datum-the-same-as-a-comodule-structure)

Discussion in the context of ([[higher monadic descent|higher]]) [[monadic descent]] is around example 2.24 of

* [[Kathryn Hess]], section 6 of _A general framework for homotopic descent and codescent_ ([arXiv:1001.1556](http://arxiv.org/abs/1001.1556))
 {#Hess10}

[[!redirects Sweedler corings]]
[[!redirects Sweedler's coring]]
[[!redirects Sweedler's corings]]

[[!redirects canonical coring]]
[[!redirects canonical corings]]
[[!redirects canonical co-ring]]
[[!redirects canonical co-rings]]


Sweedler notation

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## Idea

_Sweedler notation_ is a special notation for discussion of operations in [[coalgebra]]s

## Definition

If $C$ is a [[coassociative coalgebra|coassociative]] [[coalgebra]] and then for $c\in C$, the comultiplication $\Delta$ maps $c$ to an element in $C\otimes C$ which is therefore a sum of the form $\sum_{i=1}^n a_i\otimes b_i$. Sweedler suggests that we do not make up new symbols like $a$ and $b$ but rather use composed symbols $c_{(1)}$ and $c_{(2)}$. Therefore

$$\Delta(c) = \sum_{i=1}^n c_{(1)i}\otimes c_{(2)i}.$$

Sweedler notation means that for certain manipulations involving just generic **linear** operations we actually do not need to think of the summation symbol $i$, so we can just write

$$\Delta(c) = \sum c_{(1)}\otimes c_{(2)} $$

with or even without summation sign. Surely in either case we need to remember that we do not have a factorization but we do have a sum of possibly more than one entry. One can formalize in fact which manipulations are allowed with such a reduced notation.

It becomes more useful when we take into account coassociativity to justify extending the notation to write

$$ \array{\sum c_{(1)}\otimes c_{(2)} \otimes c_{(3)} := \sum c_{(1)(1)}\otimes c_{(1)(2)}\otimes c_{(2)} \\
= \sum c_{(1)}\otimes c_{(2)(1)}\otimes c_{(2)(2)}.}$$

Furthermore, we can extend it to [[coactions]], e.g. $\rho:V\to V\otimes C$, by $\rho(v) = \sum v_{(0)}\otimes v_{(1)}$. Then we can use the coaction axiom $(id_V\otimes \Delta)\circ\rho = (\rho\otimes id_C)\circ \rho$ to write

$$ \array{v_{(0)}\otimes v_{(1)} \otimes v_{(2)} := v_{(0)(0)}\otimes v_{(0)(1)} \otimes v_{(1)}\\
= v_{(0)}\otimes v_{(1)(0)}\otimes v_{(1)(1)},}$$

where we used the *sumless* Sweedler notation.

One big use is that the scalars like $\epsilon(a_{(3)})$ can be moved freely along the expression, which is difficult to write without calculating with Sweedler components: one would need lots of brackets and flip operators, and this could be messy and abstract.

## References

The notation is named after [[Moss Sweedler]]. Sometimes (though rarely) it is also called Heyneman-Sweedler notation.


Swiss cheese operad

## Idea

The Swiss cheese operad is an analogue of the [[little disks operad]],
where disks are replaced by half-disks,
which contain both ordinary disks in their interior,
as well as half-disks positioned at the flat boundary.

This structure can be organized into an [[operad]]
in the category of [[modules]] over the [[little disks operad]].

The Swiss cheese operad over rationals is not formal,
unlike the little disks operad.

## Related concepts

* [[little disks operad]]

* [[module over an operad]]

## References

* [[Alexander Voronov]], _The Swiss-Cheese Operad_, [arXiv:math/9807037](https://arxiv.org/abs/math/9807037).

* [[Najib Idrissi]], _Swiss-Cheese operad and Drinfeld center_, [arXiv:1507.06844](https://arxiv.org/abs/1507.06844).

See also

* Wikipedia, [Swiss-cheese operad](https://en.wikipedia.org/wiki/Operad#Swiss-cheese_operad) in Operad.


Sy-David Friedman

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

## Selected publications

* [[Neil Barton]], [[Sy-David Friedman]], *Set Theory and Structures*, in: *[[Reflections on the Foundations of Mathematics]]*, Synthese Library **407** Springer (2019)  [[doi:10.1007/978-3-030-15655-8_10](https://doi.org/10.1007/978-3-030-15655-8_10), [pdf](https://philsci-archive.pitt.edu/15908/7/Set_theory_structures_2.pdf)]

category: people


SYK-model and AdS2-CFT1 -- references

### SYK-model in $AdS_2/CFT_1$

Discussion of the [[SYK-model]] as the [[AdS-CFT|AdS/CFT dual]] of [[JT-gravity]] in [[nearly AdS2/CFT1]] and [[AdS-CFT in condensed matter physics]]:

Original articles:

* {#MaldacenaStanford16} [[Juan Maldacena]], [[Douglas Stanford]], _Comments on the Sachdev-Ye-Kitaev model_, Phys. Rev. D 94, 106002 (2016)([arXiv:1604.07818](https://arxiv.org/abs/1604.07818))

* {#Sachdev10} [[Subir Sachdev]], _Holographic metals and the fractionalized Fermi liquid_, Phys. Rev. Lett. 105:151602, 2010 ([arXiv:1006.3794](https://arxiv.org/abs/1006.3794))

Review:

* Gábor Sárosi, _$AdS_2$ holography and the SYK model_, Proceedings of Science 323 ([arXiv:1711.08482](https://arxiv.org/abs/1711.08482),  [doi:10.22323/1.323.0001](https://doi.org/10.22323/1.323.0001))

* {#Maldacena18} [[Juan Maldacena]], _Toy models for black holes II_, talk at PiTP 2018 _From QBits to spacetime_ ([recording](https://video.ias.edu/PiTP/2018/0726-JuanMaldacena))

* Dmitrii A. Trunin, _Pedagogical introduction to SYK model and 2D Dilaton Gravity_ ([arXiv:2002.12187](https://arxiv.org/abs/2002.12187))

Relation to [[black holes]] in terms of [[Majorana dimer]] states:

* Ioanna Kourkoulou, [[Juan Maldacena]], *Pure states in the SYK model and nearly-$AdS_2$ gravity* ([arXiv:1707.02325](https://arxiv.org/abs/1707.02325))

Relation to [[black holes in string theory]] and [[random matrix theory]]:

* {#CGHSS16} Jordan S. Cotler, Guy Gur-Ari, Masanori Hanada, [[Joseph Polchinski]], Phil Saad, [[Stephen Shenker]], [[Douglas Stanford]], Alexandre Streicher, Masaki Tezuka, _Black Holes and Random Matrices_,  JHEP 1705:118, 2017 ([arXiv:1611.04650](https://arxiv.org/abs/1611.04650))

* Tomoki Nosaka, Tokiro Numasawa, _Quantum Chaos, Thermodynamics and Black Hole Microstates in the mass deformed SYK model_ ([arXiv:1912.12302](https://arxiv.org/abs/1912.12302))

On [[non-perturbative effects]] and [[resurgence]]:

* [[Paolo Gregori]], [[Ricardo Schiappa]], *From Minimal Strings towards Jackiw-Teitelboim Gravity: On their Resurgence, Resonance, and Black Holes* ([arXiv:2108.11409](https://arxiv.org/abs/2108.11409))

See also

* Yuri D. Lensky, Xiao-Liang Qi, Pengfei Zhang, _Size of bulk fermions in the SYK model_ ([arXiv:2002.01961](https://arxiv.org/abs/2002.01961))

* Xiao-Liang Qi, Pengfei Zhang, _The Coupled SYK model at Finite Temperature_ ([arXiv:2003.03916](https://arxiv.org/abs/2003.03916))

* Akash Goel, [[Herman Verlinde]], _Towards a String Dual of SYK_ ([arXiv:2103.03187](https://arxiv.org/abs/2103.03187))

* Tarek Anous, Felix M. Haehl, *The quantum $p$-spin glass model: A user manual for holographers* ([arXiv:2106.03838](https://arxiv.org/abs/2106.03838))


Syksy Räsänen

* [webpage](http://www.helsinki.fi/~syrasane/)

## Selected writings

On [[inhomogeneous cosmology]]:

* {#Rasanen08} [[Syksy Räsänen]], _Evaluating backreaction with the peak model of structure formation_, JCAP 0804:026,2008 ([arXiv:0801.2692](https://arxiv.org/abs/0801.2692))

* {#Rasanen10} [[Syksy Räsänen]], _Backreaction as an alternative to dark energy and modified gravity_ ([arXiv:1012.0784](https://arxiv.org/abs/1012.0784))

* {#Rasanen03} [[Syksy Räsänen]], _Dark energy from backreaction_, JCAP 0402:003, 2004 ([arXiv:astro-ph/0311257](https://arxiv.org/abs/astro-ph/0311257))

* {#BuchertRasanen11} [[Thomas Buchert]], [[Syksy Räsänen]], _Backreaction in late-time cosmology_, Annual Review of Nuclear and Particle Science 62 (2012) 57-79 ([arXiv:1112.5335](https://arxiv.org/abs/1112.5335))

* {#MontanariRasanen17} Francesco Montanari, [[Syksy Räsänen]], _Evaluating backreaction with the ellipsoidal collapse model_, JCAP12(2017)008 ([arXiv:1710.02451](https://arxiv.org/abs/1710.02451))

On [[dark matter]]:

* [[Syksy Räsänen]], _Dark matter_ cosmo 2015 lecture notes ([pdf](http://www.courses.physics.helsinki.fi/teor/cos1/cosmo2015_07.pdf))

## Related $n$Lab entries

* [[inhomogeneous cosmology]]

* [[dark matter]]

category: people


sylleptic 3-group

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## Definition

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

A _sylleptic 3-group_ is a [[3-group]] equipped with the following equivalent [[structure]]:

1. Regarded as a [[monoidal 2-category]], $G$ is a [[sylleptic monoidal 2-category]].

1. The [[A-∞ algebra]]/[[E1-algebra]] structure on $G$ refines to an [[E3-algebra]] structure.

1. $G$ is a [[k-tuply groupal n-groupoid|3-tuply monoidal 2-groupoid]].

1. $G$ is a groupal [[k-tuply monoidal (n,r)-category|3-tuply monoidal (2,0)-category]].

=--

## Examples

* If $(\mathcal{C}, \otimes)$ is a [[sylleptic monoidal 2-category]], then its [[Picard 3-group]] is a sylleptic 3-group.

## Related concepts

* [[sylleptic monoidal 2-category]]

* [[∞-group]], [[k-tuply groupal n-groupoid]]

* [[braided ∞-group]],

  * [[braided 2-group]]

  * [[braided 3-group]]

* [[sylleptic ∞-group]]

  * [[sylleptic 3-group]]

* [[abelian ∞-group]]

  * [[symmetric 2-group]]

  * [[abelian 3-group]]

## References

* [[Nick Gurski]], [[Angélica M. Osorno]], Section 2.2 of: *Infinite loop spaces, and coherence for symmetric monoidal bicategories*, Adv. Math. 246 (2013) 1-32 ([arXiv:1210.1174](https://arxiv.org/abs/1210.1174))

[[!redirects sylleptic 3-groups]]


sylleptic monoidal 2-category

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#Contents#
* table of contents
{:toc}

## Idea

A _sylleptic monoidal (weak) 2-category_ is a [[monoidal 2-category]] with a categorified sort of "commutativity" that lies in between a [[braided monoidal 2-category|braiding]] and a [[symmetric monoidal 2-category|symmetry]].

That is, it is a [[2-category]] $C$ equipped with a [[tensor product]] $\otimes : C \times C \to C$ [[2-functor]] which satisfies the first two in a hierarchy of conditions for being commutative up to equivalence.  In the language of [[k-tuply monoidal n-category|k-tuply monoidal n-categories]], a braided monoidal 2-category is a *triply monoidal 2-category*.  As described there, this may be identified with a pointed 5-category with a single $k$-morphism for $k=0,1,2$.  We can also say that it is a monoidal 2-category whose [[En-algebra|E1-algebra]] structure is refined to an [[En-algebra|E3-algebra]] structure.

## Related concepts

* [[monoidal 2-category]], [[3-group]]

* [[braided monoidal 2-category]], [[braided 3-group]]

* **sylleptic monoidal 2-category**, [[sylleptic 3-group]]

* [[symmetric monoidal 2-category]], [[symmetric 3-group]]

* [[k-tuply monoidal (n,r)-category]]

## References

* [[Brian Day]], [[Ross Street]], Section 5 of: *Monoidal Bicategories and Hopf Algebroids*, Advances in Mathematics Volume 129, Issue 1, 15 July 1997, Pages 99-157 ([doi:10.1006/aima.1997.1649](https://doi.org/10.1006/aima.1997.1649))

* [[Sjoerd E. Crans]], Section 4 of: *Generalized Centers of Braided and Sylleptic Monoidal 2-Categories*, Advances in Mathematics, Volume 136, Issue 2, 25 June 1998, Pages 183-223 ([doi:10.1006/aima.1998.1720](https://doi.org/10.1006/aima.1998.1720))

[[!redirects sylleptic monoidal 2-categories]]

[[!redirects sylleptic 2-category]]
[[!redirects sylleptic 2-categories]]

[[!redirects sylleptic monoidal bicategory]]
[[!redirects sylleptic monoidal bicategories]]


sylleptic ∞-group

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#Contents#
* table of contents
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## Definition

A **sylleptic $\infty$-group** is an [[∞-group]] $G$ equipped with the following equivalent [[structure]]

1. a lift of its [[A-∞ algebra|A-∞]]/[[E-n algebra|E1]]-algebra structure to an [[E-n algebra|E3]] algebra structure;

1. the structure of a [[braided ∞-group]] on its [[delooping]] $\mathbf{B}G$

1. the structure of an [[∞-group]] on its double [[delooping]] $\mathbf{B}^2 G$

1. a triple [[delooping]] $\mathbf{B}^3 G$.

## Related concepts

* [[sylleptic monoidal 2-category]]

* [[∞-group]], [[k-tuply groupal n-groupoid]]

* [[May recognition theorem]]

* [[braided ∞-group]],

  * [[braided 2-group]]

  * [[braided 3-group]]

* [[sylleptic ∞-group]]

  * [[sylleptic 3-group]]

* [[abelian ∞-group]]

  * [[symmetric 2-group]]

## References

On [[sylleptic 3-groups]]:

* [[Nick Gurski]], [[Angélica M. Osorno]], Section 2.2 of *Infinite loop spaces, and coherence for symmetric monoidal bicategories*, Adv. Math. 246 (2013) 1-32 ([arXiv:1210.1174](https://arxiv.org/abs/1210.1174))

[[!redirects sylleptic ∞-groups]]

[[!redirects sylleptic infinity-group]]
[[!redirects sylleptic infinity-groups]]

[[!redirects sylleptic n-group]]
[[!redirects sylleptic n-groups]]


syllogism

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#Contents#
* table of contents
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## Idea

A **syllogism** is a pattern of [[deduction|deductive]] argument of a certain limited range of form. The study of syllogisms was originated by [[Aristotle]] and later developed by his followers as well as by the Stoics. Syllogisms were central objects of study for medieval logicians.

Syllogisms involve two premises and a conclusion, each expressing a statement of the form that either all, none or some members of a class do or do not have a certain [[property]]. For example,

* No fish is a mammal.
* Some sea creatures are fish.
* Therefore, some sea creatures are not mammals.

In the nineteenth century, syllogism became seen to be insufficiently expressive as more complicated relations and [[quantification]] were encountered. [[Predicate logic]] came to take its place.

## Original definition

The definition by Aristotle that has received the most attention (out of several variants) is from _Prior Analytics_ 24b18-22 (here taken from translation by [Striker (2009)](#Striker)):

>A 'syllogismos' is an argument (logos) in which, (i) certain things being posited (tethentôn), (ii) something other than what was laid down (keimenôn) (iii) results by necessity (eks anagkês sumbainei)(iv) because these things are so. By 'because these things are so' I mean that it results through these, and by 'resulting through these' I mean that no term is required from outside for the necessity to come about.

## Related entries

* [[natural deduction]]

* [[co-Heyting negation]]

## References

* {#SEP} Stanford Encyclopedia of Philosophy, _[Aristotle's logic](http://plato.stanford.edu/entries/aristotle-logic/)_

* Gisela Striker, _Aristotle's Prior Analytics Book I: Translated with an Introduction and Commentary_, OUP, Oxford (2009) [PhilPapers entry](http://philpapers.org/rec/STRAPA-2)
{#Striker}

* [[Georg Hegel]], _[[Lectures on the History of Philosophy]] -- [Aristotle -- Logic](Lectures+on+the+History+of+Philosophy#AristotleLogic)_.

* [[Georg Hegel]], _[[Science of Logic]]  -- [Subjective logic -- Subjectivity -- Syllogism](Science+of+Logic#SchlussSyllogism)_

An interpretation of syllogistic term logic in modern [[predicate logic]] is discussed in detail in the textbook on [[mathematical logic]]:

* {#HilbertAckermann59} [[David Hilbert]], [[Wilhelm Ackermann]], _The calculus of classes (monadic predicate calculus)_ ([web](http://books.google.it/books?id=45ZGMjV9vfcC&pg=PA44#v=onepage&q&f=false)), chapter II of _Grundzüge der Theoretischen Logik_ , 4th ed. Springer Heidelberg 1959 [1928]

After J. Lukasiewicz's classic monograph on Aristotelian logic from a modern point of view, the 1970s saw influential interpretations of it as systems of [[natural deduction]]:

* J. Bacon, _Natural-deduction rules for syllogistic_ , JSL **31** (1966) pp.686-7.

* J. Corcoran, _Completeness of an ancient logic_ , JSL **37** (1972) pp.696-702.

* T. Smiley, _What is a syllogism_ , JPL **2** (1973) pp.136-154.

An approach to Aristotelian logic using [[category theory]] is in

* {#PalmeReyesMacnamaraReyes94} [[Marie La Palme Reyes]], John Macnamara, [[Gonzalo Reyes]] , _Functoriality and Grammatical Role in Syllogisms_ , Notre Dame J. Formal Logic **35** no.1 (1994) pp.41-66. ([Euclid](http://projecteuclid.org/euclid.ndjfl/1040609293), [pdf](http://projecteuclid.org/download/pdf_1/euclid.ndjfl/1040609293))

* {#RRZ94} M. La Palme Reyes, J. Macnamara, [[Gonzalo E. Reyes|G. E. Reyes]], H. Zolfaghari, _The non-Boolean logic of natural language negation_ , Phil. Math. **2** no.1 (1994) pp.45-68.

A proposal for formalization of syllogisms within [[linear logic]] is in

* Ruggero Pagnan, _A diagrammatic calculus of syllogisms_, Journal of Logic, Language and Information July 2012, Volume 21, Issue 3, pp 347-364 ([arXiv:1001.1707](http://arxiv.org/abs/1001.1707))

* {#Pagnan13} Ruggero Pagnan, _Syllogisms in Rudimentary Linear Logic, Diagrammatically_ ([arXiv:1302.7111](http://arxiv.org/abs/1302.7111))

[[!redirects syllogisms]]

[[!redirects Aristotle's logic]]
[[!redirects Aristotelian logic]]


Sylow p-subgroup

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## Definition

In a [[finite group]] $G$, and for a [[prime number|prime]] $p$, a maximal [[p-torsion|$p$-torsion]] [[torsion subgroup|subgroup]] of $G$ is also known as a **Sylow $p$-subgroup**.

## Properties

The following statements are known as [[Sylow's theorems]], a partial converse to [[Lagrange's theorem]].

Let the [[order of a group|order]] of $G$ be $ r p^k $, where $ r $ is [[coprime integer|coprime]] to $p$.

Let $H \lt G$ be a [[subgroup]] of rank $p^l$, and consider the left [[group action|action]] of $G$ on right [[cosets]] of $H$:

$$
  (g, x H )
  \;\mapsto\;
  (g x) H
  \,.
$$

This induces a further action of $G$ on $(G / H) \mathbf{C} (p^{k-l})$, the subsets of $G/H$ of size $p^{k-l}$.  The number of these, $\binom{ r p^{k-l} }{ p^{k-l} } $, is congruent to $r$, mod $p$, and hence some [[orbit]] must have size [[coprime integer|coprime]] to $p$, hence necessarily dividing $r$, hence some set of cosets must have [[stabilizer subgroup|stabilizer]] of size at least $ p^k $.  [[exercise|One checks]] that, on the other hand, the stabilizer of a set of cosets is at most the size of their [[union]] for a *very good* reason, and furthermore is a subgroup of $G$.  Lastly, every orbit contains a representative that contains $H$. In consequence:

+-- {: .num_theorem #pSylowModp}
###### Theorem

Every $p$-subgroup $H$ of $G$ is contained in a subgroup of order $p^k$, which is necessarily a maximal $p$-subgroup.  The number of maximal $p$-subgroups including $H$ is congruent to $1$ mod $p$.

=--

One also has:

+--  {: .num_theorem #SylowConj}
###### Theorem

Any two Sylow $p$-subgroups of $G$ are conjugate.

=--

See *[[class equation]]* for a detailed discussion of these matters. (Now updated to take into account the proof below. The discussion above refers to a more involved proof from an earlier page version, which in turn was adapted from the Wikipedia article; it may be found [here](https://nforum.ncatlab.org/discussion/6646/class-equation/), [comment 4](https://nforum.ncatlab.org/discussion/6646/class-equation/?Focus=72005#Comment_72005).) The following slick proof for the existence of Sylow subgroups was suggested to us by [[Benjamin Steinberg]].

+-- {: .proof}
###### Proof that Sylow subgroups exist

First observe that if a group $G$ has a $p$-Sylow subgroup $P$, then so does each of its subgroups $H$. For we let $H$ act on $G/P$ by left translation, and then note that since $G/P$ has cardinality prime to $p$, so must one of its connected components $H/Stab(a_x)$ in the $H$-set decomposition

$$G/P \cong \sum_{orbits\; x} H/Stab(a_x)$$

($a_x \in G/P$ a representative of its orbit $x$), making $Stab(a_x)$ a $p$-Sylow subgroup of $H$.

Then, if $H$ is any group, apply this observation to the embedding

$$
  H
    \stackrel{Cayley}{\hookrightarrow}
  Perm({|H|})
    \hookrightarrow
  GL_{{|H|}}(\mathbb{Z}/(p))
  \,,
$$

where we embed the [[permutation group]] via permutation [[matrices]] into the group $G$ consisting of matrices ${|H|} \times {|H|} \to \mathbb{Z}/(p)$. Letting $n$ be the cardinality of $H$, the order of $G$ is $(p^n - 1)(p^n - p)\ldots (p^n - p^{n-1})$, with maximal $p$-factor $p^{n(n-1)/2}$. This $G$ has a $p$-Sylow subgroup given by unitriangular matrices, i.e., upper-triangular matrices with all $1$'s on the diagonal, and we are done.

=--

## Related pages

* [[Sylow's theorem]]

* [[group actions on n-spheres]]

## References

See also

* Wikipedia, *[Sylow theorems](https://en.wikipedia.org/wiki/Sylow_theorems)*

For a generalisation of Sylow theory to [[homotopy type with finite homotopy groups|$\pi$-finite]] [[∞-groups]], that is, ∞-groups with finitely many non-trivial [[homotopy groups]] which are all [[finite group|finite]], see

* [[Matan Prasma]], [[Tomer Schlank]], _Sylow theorems for ∞-groups_ ([arXiv:1602.04494](https://arxiv.org/abs/1602.04494))

[[!redirects Sylow p-subgroups]]


Sylvain Lavau

* [webpage](https://www.sylvainlavau.com/)

## Selected writings

Relation of [[Leibniz algebras]] to the [[embedding tensor]] and [[tensor hierarchies]] in [[gauged supergravity]]:

* [[Sylvain Lavau]], _Tensor hierarchies and Leibniz algebras_, J. Geom. Phys. 144:147-189 (2019) ([arXiv:1708.07068](https://arxiv.org/abs/1708.07068))

A proposal for understanding [[tensor hierarchies]] as [[dg-Lie algebra]]/[[L-infinity algebra]]-refinenents of [[Leibniz algebras]]:

* [[Sylvain Lavau]], [[Jakob Palmkvist]], _Infinity-enhancing of Leibniz algebras_ ([arXiv:1907.05752](https://arxiv.org/abs/1907.05752))

* [[Sylvain Lavau]], [[Jim Stasheff]], _$L_\infty$-algebra extensions of Leibniz algebras_ ([arXiv:2003.07838](https://arxiv.org/abs/2003.07838))

category: people


Sylvie Paycha

* [webpage](http://users.math.uni-potsdam.de/~paycha/)

category: people


Symanzik effective field theory

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## Idea

In the field of [[quantum chromodynamics]], what is known as _Symanzik's effective field theory_ ([Symanzik 82](#Symanzik82), [Symanzik 83](#Symanzik83)) is a type of [[effective field theory]] used in [[lattice QCD]] in order to deal with lattice-spacing effects.

In [[lattice QCD]], the extraction of [[hadron|hadronic]] [[observables]] involves setting a [[scale]] for a particular lattice [[action functional]] with given sea-quark configurations. This involves the extrapolation of the dimensionless product of various quantities, such as gradient-flow scales and the omega [[baryon]] mass, to the continuum and infinite volume limit. This is in turn, interpolated to quark mass points in the [[isospin]] limit. These extrapolation functions must incorporate:

1. Light and Strange Quark Mass Dependence

2. Finite Volume Corrections

3. Discretization Corrections

Symanzik's effective field theory is constructed by expanding the discretized action about the continuum limit.

## Related concepts

[[!include effective field theories of nuclear physics -- contents]]

## References

Original articles:

* {#Symanzik82} [[Kurt Symanzik]], Section 3 of: _Some topics in quantum field theory_ In: Schrader R., Seiler R., Uhlenbrock D.A. (eds.) _Mathematical Problems in Theoretical Physics_ Lecture Notes in Physics, vol 153, Springer 1982 ([doi:10.1007/3-540-11192-1_11](https://doi.org/10.1007/3-540-11192-1_11))

* {#Symanzik83} [[Kurt Symanzik]], _Continuum limit and improved action in lattice theories: (I). Principles and $\phi^4$ theory_, Nuclear Physics B Volume 226, Issue 1, 26 September 1983, Pages 187-204 Nuclear Physics B (doi:10.1016/0550-3213(83)90468-6)

Review:

* Andreas S. Kronfeld, Section 4 in: _Uses of Effective Field Theory in Lattice QCD_, in: _At the Frontier of Particle Physics_ pp. 2412-2477, World Scientific 2002 ([arXiv:hep-lat/0205021](https://arxiv.org/abs/hep-lat/0205021), [doi:10.1142/9789812777270_0004](https://doi.org/10.1142/9789812777270_0004))

See also:

* Stephen R. Sharpe, Robert L. Singleton,Jr, _Spontaneous  flavor  and  parity  breaking  with  Wilson fermions_ Phys. Rev.D58, 074501 (1998), ([arXiv:hep-lat/9804028])

* _Scale setting the Möbius Domain Wall Fermion on gradient-flowed HISQ action using the omega baryon mass and the gradient-flow scales $t_0$ and $w_0$_ ([arXiv:hep-lat/2011.12166](https://arxiv.org/abs/2011.12166))

[[!redirects Symanzik's effective field theory]]

[[!redirects Symanzik effective theory]]
[[!redirects Symanzik's effective theory]]


symbol map

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## Idea

For $V$ an [[inner product space]], the **symbol map** constitutes an [[isomorphism]] of [[super vector spaces]] between the [[Clifford algebra]] of $V$ and the [[exterior algebra]] on $V$.

## Definition

Let $V$ be an [[inner product space]]. Write $Cl(V)$ for its [[Clifford algebra]] and $\wedge^\bullet V$ for its [[Grassmann algebra]].

For $v \in V$ any vector, write

$$
  v\wedge : \wedge^\bullet V \to \wedge^\bullet V
$$

for the [[linear map]] given by exterior product with $v$.

Let

$$
  \langle -,-\rangle : \wedge^\bullet V \otimes \wedge^\bullet V \to \mathbb{R}
$$

be the [[Hodge inner product]] on the [[exterior algebra]] induced from the inner product. With respect to this inner product the above multiplication operator has an [[adjoint operator]]

$$
  \iota_v : \wedge^\bullet V \to \wedge^\bullet V
$$

called _contraction_ with $V$. These operators satisfy the [[canonical anticommutation relations]]

$$
  [v\wedge, w\wedge ] = 0
$$

$$
 [\iota_v, \iota_w] = 0
$$

$$
  [\iota_v, w\wedge] = \langle v,w\rangle
$$

(where all these are [[supercommutators]], hence in fact [[anticommutators]] in the present case).

There is a canonical [[representation]] of the [[Clifford algebra]] on the exterior algebra induced by this construction

$$
  \rho : Cl(V)\otimes \wedge^\bullet V \to \wedge^\bullet V
$$

given by

$$
  (\gamma_v, \phi) \mapsto (v \wedge + \iota_v) \phi
  \,.
$$

The **symbol map** is the restriction of this action to the identity element $1 \in \wedge^\bullet V$:

$$
  \sigma := \rho(-,1) : Cl(V) \to \wedge^\bullet V
  \,.
$$

This is an [[isomorphism]] of $\mathbb{Z}_2$-[[graded vector space]].

The [[inverse]] maps is on even-graded elements given by sending [[bivector]]s to their Clifford incarnation

$$
  \sigma^{-1} : v \wedge w \mapsto \frac{1}{2}\left(\gamma_v \cdot \gamma_w - \gamma_w \cdot \gamma_v\right)
  \,.
$$

## Related concepts

* The symbol map may be thought of as a special case of a _[[symbol of a differential operator]]_. See there for more.

## References

For instance section 2.5 of

* [[Eckhard Meinrenken]], _Clifford algebras and Lie groups_ ([pdf](https://sites.icmc.usp.br/grossi/other/meinrenken.pdf))


symbol of a differential operator

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## Definition

For $X$ a [[smooth manifold]], $E \to X$ a [[vector bundle]] and $D : \Gamma(E) \to \Gamma(E)$ a [[linear differential operator]] on [[sections]] of $E$, its **symbol** is the bundle morphism

$$
   \sigma(D) \;:\; T^* X \otimes_X E \to E
$$

from the [[tensor product of vector bundles]] with the [[cotangent bundle]] which is given at any point $x \in X$ on a [[cotangent vector]] of the form $(\mathbf{d}f)_x \in \Gamma(T^* X)_x$ by

$$
  \sigma(D)_x \;\colon\; \mathbf{d}f_x \mapsto [D,f]_x
  \,,
$$

where in the [[commutator]] on the right we regard multiplication by $f$ as an [[endomorphism]] of $\Gamma(E)$.

{#InKTheory} The symbol of an [[elliptic differential operator]]  may naturally be thought of as an element in the [[topological K-theory]] of $T^\ast X$ (see [Freed 1987](#Freed87), [p. 13](https://ncatlab.org/nlab/files/FreedGeometryOfDiracOperators.pdf#page=13)).

Generally, the symbol of a possibly non-linear [[differential operator]] is similarly the map on the [[cotangent bundle]] given by "replacing [[partial derivatives]] by [[covectors]]" in the definition of the differential operator. In this case the [[principal symbol]] is the highest degree [[homogeneous function|homogeneous]] component of the symbol. Many authors say total symbol for symbol in the sense of this entry and symbol for a principal symbol.

## Examples

+-- {: .num_defn #SymbolMap}
###### Definition
**([[symbol map]])**

For $X = \mathbb{R}^n$ a [[Cartesian space]] and $D$ the [[Dirac operator]] of the [[flat connection]], the symbol of $D$ reproduces the [[symbol map]] between [[differential forms]] and [[Clifford algebra]] elements.

=--

+-- {: .num_example #BicharacteristicFlow}
###### Example
**([[bicharacteristic flow]])**

For $E$ a [[real vector bundle|real]] [[trivial line bundle]] then the principal symbol is equivalently just a real-valued [[smooth function]] on the [[cotangent bundle]]. Since any cotangent bundle is canonically a [[symplectic manifold]], in this case the symbol may be regarded as a [[Hamiltonian]] funtion. The corresponding [[Hamiltonian flow]] is called the _[[bicharacteristic flow]]_ of the given differential operator.

=--

## Related concepts

* [[symbol order]], [[symbol map]], [[principal symbol]]

* [[elliptic differential operator]], [[elliptic chain complex]]

* [[hyperbolic differential operator]]

* [[propagation of singularities theorem]]

## References

* [[Nigel Higson]], [[John Roe]], chapter 2.5 of  _Lectures on operator K-theory and the Atiyah-Singer Index Theorem_ ([pdf](http://folk.uio.no/rognes/higson/Book.pdf))

* {#Freed87} [[Daniel Freed]], *Geometry of Dirac operators* (1987) [[pdf](http://www.ma.utexas.edu/users/dafr/DiracNotes.pdf), [[FreedGeometryOfDiracOperators.pdf:file]]]

See also

* Wikipedia, _[Symbol of a differential operator](http://en.wikipedia.org/wiki/Symbol_of_a_differential_operator)_


symbol order

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## Definition

+-- {: .num_defn #SymbolOrder}
###### Definition
**([[symbol order]])**

A [[smooth function]] $q$ on a [[cotangent bundle]] (e.g. the [[symbol of a differential operator]]) is of _order $m$_ (and type $1,0$, denoted $q \in S^m = S^m_{1,0}$), for $m \in \mathbb{N}$, if on each [[coordinate chart]] $((x^i), (k_i))$ we have that for every [[compact subset]] $K$ of the base space and all multi-indices $\alpha$ and $\beta$, there is a  [[real number]] $C_{\alpha, \beta,K } \in \mathbb{R}$ such that the [[absolute value]] of the [[partial derivatives]] of $q$ is bounded by

$$
  \left\vert
    \frac{\partial^\alpha}{\partial k_\alpha}
    \frac{\partial^\beta}{\partial x^\beta}
    q(x,k)
  \right\vert
  \;\leq\;
  C_{\alpha,\beta,K}\left( 1+ {\vert k\vert}\right)^{m - {\vert \alpha\vert}}
$$

for all $x \in K$ and all cotangent vectors $k$ to $x$.

A [[Fourier integral]] operator $Q$ is of _[[symbol class]]_ $L^m = L^m_{1,0}$ if

1. it is of the form

   $$
     Q f (x)
     \;=\;
     \int \int e^{i k \cdot (x - y)} \hat f(x,y,k) f(y) \, d y d k
   $$

1. its [[principal symbol]] $q$ is of order $m$, in the above sense.

=--

([Hörmander 71, def. 1.1.1 and first sentence of section 2.1](#Hoermander71))

## Examples

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

The [[wave operator]]/[[Klein-Gordon operator]] on [[Minkowski spacetime]] is of class $L^2$, according to def. \ref{SymbolOrder}.

=--

## Related concepts

* [[propagation of singularities theorem]]

## References

* {#Hoermander71} [[Lars Hörmander]], _Fourier  integral  operators I._ Acta  Mathematica  127, 79-183 (1971) ([Euclid](https://projecteuclid.org/euclid.acta/1485889699))

[[!redirects symbol orders]]

[[!redirects symbol class]]
[[!redirects symbol classes]]


Symbol Sandbox > history

< [[Symbol Sandbox]]


symmetric 2-group

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#### Group Theory
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[[!include group theory - contents]]
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#### $(\infty,1)$-Category theory
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[[!include quasi-category theory contents]]
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#Contents#
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## Definition

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###### Definition

A _symmetric 2-group_ is a [[2-group]] equipped with the following equivalent structure:

1. Regarded as a [[monoidal category]], $G$ is a [[symmetric monoidal category]].

1. Regarded as a [[truncated object in an (infinity,1)-category|1-truncated]] [[∞-group]] it has the structure of an [[abelian ∞-group]].

1. The [[delooping]] [[2-groupoid]] $\mathbf{B}G$ is a [[braided 3-group]].

1. The double [[delooping]] [[3-groupoid]] $\mathbf{B}^2 G$ is a [[4-group]].

1. The triple [[delooping]] [[4-groupoid]] $\mathbf{B}^4 G$ exists.

1. The [[A-∞ algebra]]/[[E1-algebra]] structure on $G$ refines to an [[E3-algebra]] structure.

1. $G$ is a [[k-tuply groupal n-groupoid|3-tuply monoidal groupoid]].

1. $G$ is a groupal [[k-tuply monoidal (n,r)-category|3-tuply monoidal (1,0)-category]].

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## Related concepts

* [[symmetric monoidal groupoid]]

* [[ring groupoid]], [[symmetric ring groupoid]]

* [[∞-group]], [[k-tuply groupal n-groupoid]]

* [[braided ∞-group]],

  * [[braided 2-group]]

  * [[braided 3-group]]

* [[sylleptic ∞-group]]

  * [[sylleptic 3-group]]

* [[abelian ∞-group]]

  * [[abelian group]]

  * **symmetric 2-group**

  * [[abelian 3-group]]

## References

Under the name "symmetric cat-groups" and thought of as a sub-class of [[braided monoidal categories]], the notion of braided 2-groups is considered in:

* [[M. Bullejos]], [[Pilar Carrasco]], [[Antonio M. Cegarra]], *Cohomology with coefficients in symmetric cat-groups. An extension of Eilenberg–MacLane's classification theorem*, Mathematical Proceedings of the Cambridge Philosophical Society , Volume 114 , Issue 1 , July 1993 , pp. 163 - 189 ([doi:10.1017/S0305004100071498](https://doi.org/10.1017/S0305004100071498))

[[!redirects symmetric 2-groups]]

[[!redirects abelian 2-group]]
[[!redirects abelian 2-groups]]


symmetric 3-group

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

#Contents#
* table of contents
{:toc}

## Definition

A **symmetric 3-group** is the following equivalent [[structure]]

1. A [[3-group]] whose [[E1-algebra]] structure is equipped with a lift to to an [[E4-algebra]] structure.

1. A [[3-group]] which regarded as an [[∞-group]] is equipped with the
structure of an [[abelian ∞-group]].

1. A [[3-group]] such that the [[delooping]] $\mathbf{B}G$ is equipped with the structure of a [[braided 4-group]].

1. A [[symmetric monoidal 2-category]] all whose objects are invertible under the tensor product.

## Related concepts

* [[k-tuply groupal n-groupoid]]

[[!redirects symmetric 3-groups]]