Goursat theorem

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### Context
#### Analysis
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[[!include analysis - contents]]
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#### Complex geometry
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[[!include complex geometry - contents]]
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# The Goursat theorem
* table of contents
{: toc}

## Idea

In [[complex analysis]], the Goursat theorem is the extension (due to [[Édouard Goursat]]) of the [[Cauchy integral theorem]] from [[continuously differentiable functions]] (for which [[Augustin Cauchy]] had proved it) to [[differentiable functions]] (which requires a harder and more technical argument).  Given Cauchy's other work, the immediate corollary is that every differentiable function is in fact [[analytic function|analytic]]; see [[holomorphic function]].  This corollary may also be viewed as the Goursat theorem; everything else in the basic theory of holomorphic functions is due to Cauchy, while Goursat\'s contribution was to remove the hypothesis that the derivative of such a function must be continuous.  On the other hand, the Goursat theorem may be seen as not a new theorem at all; from this perspective, Cauchy had always intended to prove his integral theorem for all complex-differentiable functions, and Goursat simply filled in a gap in Cauchy\'s proof.  But as Goursat was born the year after Cauchy died, that was a long-standing gap!

## Statements

Let $U$ be an [[open subset]] of the [[complex plane]] $\mathbb{C}$ and let $f\colon U \to \mathbb{C}$ be a complex-[[differentiable function]]; that is,
$$ f'(\zeta) \coloneqq \lim_{z \to \zeta} \frac{f(z) - f(\zeta)}{z - \zeta} $$
exists for every point $\zeta$ in $U$.  Then any of the following conclusions may be thought of as the Goursat theorem:

+-- {: .num_theorem #CD}
###### Theorem
The function $f'\colon U \to \mathbb{C}$ is [[continuous function|continuous]].  That is, $f$ is [[continuously differentiable function|continuously differentiable]].
=--

+-- {: .num_theorem #ITR}
###### Theorem
Given any point $\zeta$ in $U$ and any rectangle (or triangle) $C$ in $U$ whose inside contains $\zeta$ and is contained in $U$, the [[contour integral]]
$$ \oint_C f(z) \,\mathrm{d}z = 0 .$$
That is, the [[Cauchy integral theorem]] holds for $f$ on rectangles (or triangles).
=--

+-- {: .num_theorem #IT}
###### Theorem
Given any point $\zeta$ in $U$ and any [[Jordan curve]] $C$ in $U$ whose inside contains $\zeta$ and is contained in $U$, the [[contour integral]]
$$ \oint_C f(z) \,\mathrm{d}z = 0 .$$
That is, the [[Cauchy integral theorem]] holds for $f$.
=--

+-- {: .num_theorem #IF}
###### Theorem
Given any point $\zeta$ in $U$ and any [[Jordan curve]] $C$ in $U$ whose inside contains $\zeta$ and is contained in $U$, the [[contour integral]]
$$ \oint_C \frac{f(z)}{z - \zeta} \,\mathrm{d}z = 2\pi\mathrm{i}f(\zeta) .$$
That is, the [[Cauchy integral formula]] holds for $f$.
=--

+-- {: .num_theorem #AF}
###### Theorem
Given any point $\zeta$ in $U$, there is an [[infinite sequence]] $(c_0, c_1, \ldots)$ such that
$$ f(z) = \sum_i c_i (z - \zeta)^i $$
on some neighbourhood of $\zeta$ contained in $U$.  That is, the function $f$ is [[analytic function|analytic]].
=--

Cauchy had established that the conclusion of each of these theorems implied the conclusion of the next (except that he did not bother with Theorem \ref{ITR}), and it\'s immediate that the last implies the first, so from the perspective of Goursat\'s time, these theorems are all equivalent.  Goursat actually introduced his proof for Theorem \ref{ITR}, which is of interest only because Goursat\'s proof naturally gives it.  Since Goursat was the first to call attention to it, one might call Theorem \ref{ITR} specifically Goursat\'s Theorem, even though it is more of a lemma (for Theorem \ref{IT}) than a theorem in its own right.  (One might call Theorem \ref{ITR} 'Goursat\'s Lemma', but that is already used for an unrelated result about [[pullbacks]] in [[group theory]].)

## Proof

Here\'s a version for Theorem \ref{ITR}: <https://planetmath.org/proofofgoursatstheorem>

[[!redirects Goursat theorem]]
[[!redirects Goursat Theorem]]
[[!redirects Goursat's theorem]]
[[!redirects Goursat's Theorem]]
[[!redirects Goursat's theorem]]
[[!redirects Goursat's Theorem]]
[[!redirects Goursat\'s theorem]]
[[!redirects Goursat\'s Theorem]]


Goutam Mukherjee

* [webpage](https://www.isical.ac.in/~goutam/)

## Selected writings

On [[twisted Bredon cohomology]] ([[twisted cohomology|twisted]] ordinary [[equivariant cohomology]]):

* Amiya Mukherjee, [[Goutam Mukherjee]], _Bredon-Illman cohomology with local coefficients_, The Quarterly Journal of Mathematics, Volume 47, Issue 2, June 1996, Pages 199–219 ([doi:10.1093/qmath/47.2.199](https://doi.org/10.1093/qmath/47.2.199))

* [[Goutam Mukherjee]], Neeta Pandey, _Equivariant cohomology with local coefficients_, Proc. Amer. Math. Soc. 130 (2002), 227-232 ([doi:10.1090/S0002-9939-01-06377-8](https://doi.org/10.1090/S0002-9939-01-06377-8))

* [[Goutam Mukherjee]], Debasis Sen, _Equivariant simplicial cohomology with local coefficients and its classification_, Topology and its Applications, Volume 157, Issue 6, 15 April 2010, Pages 1015-1032 ([doi:10.1016/j.topol.2010.01.004](https://doi.org/10.1016/j.topol.2010.01.004))

category: people


graded abelian group

[[!redirects graded abelian groups]]

## Idea

A [[graded object]] in [[abelian groups]].

## Examples

A [[generalized homology]]/[[generalized cohomology theory]] is a suitable [[covariant functor]]/[[contravariant functor]] from suitable [[topological spaces]] to $\mathbb{Z}$-graded abelian groups ("[[cohomology groups]]").

## Related concept

* [[super abelian group]]

* [[graded module]]

* [[graded vector space]]


graded algebra

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#### Homological algebra
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#Contents#
* table of contents
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## Idea

A _graded algebra_ is an [[associative algebra]] which is with a labelling on its elements by elements of some [[monoid]] or [[group]], and such that the multiplication in the algebra is reflected in the multiplication in the labelling group.

## Definition

Let $G$ be a [[group]].  (Often $G$ will be abelian, and, in fact, one usually takes by default $G = \mathbb{Z}$ the additive group of [[integers]], in which case the actual group being used is omitted from the terminology and notation.)

A **graded ring** is a [[ring]] $R$ equipped with a decomposition of the underlying [[abelian group]] as a [[direct sum]] $R = \oplus_{g \in G} R_g$ such that the product takes $R_{g} \times R_{g'} \to R_{g g'}$.

Analogously there is the notion of graded $k$-[[associative algebra]] over any commutative ring $k$.

Specifically for $k$ a [[field]] a **graded algebra** is a [[monoid in a monoidal category|monoid in]] [[graded vector spaces]] over $k$.

An $\mathbb{N}$-graded algebra is called _connected_ if in degree-0 it is just the ground ring.

A *[[differential graded algebra]]* is a graded algebra $A$ equipped with a [[derivation]] $d : A\to A$ of degree +1 (or -1, depending on conventions) and such that $d \circ d = 0$. This is the same as a [[monoid]] in the category of [[chain complexes]].

\linebreak

{#StronglyGraded} A $\mathbb{N}$-graded algebra is called **strongly $\mathbb{N}$-graded** (in [Ardizzoni & Menini (2007), Def. 3.2](#ArdizzoniMenini07)) if for every $n,p \ge 0$, the [[multiplication]] $A_{n} \otimes A_{p} \rightarrow A_{n+p}$ is an [[epimorphism]].

## Properties

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

For $R$ a commutative ring write $Spec R \in Ring^{op}$ for the corresponding object in the [[opposite category]]. Write $\mathbb{G}_m$ for the multiplicative group underlying the [[affine line]].

There is a [[natural isomorphism]] between

* $\mathbb{Z}$-gradings on $R$;

* $\mathbb{G}_m$-[[action]]s on $Spec R$.

=--

The proof is spelled out at [[affine line]] in the section *[Properties](affine+line#Properties)*.

## Examples

### Group ring

Let $G$ be any (discrete) group and $k[G]$, its [[group algebra]]. This has a direct sum decomposition as a $k$-module,

$$k[G] = \bigoplus_{g\in G}L_g$$

where each $L_g$ is a one dimensional free $k$-module, for which it is convenient, here, to give a basis $\{\ell_g\}$.  The graded algebra structure is obtained by extending the multiplication rule,

$$\ell_{g_1}\cdot \ell_{g_2} = \ell_{g_1g_2},$$

given on basis elements, by $k$-linearity.

### Lazard ring

The [[Lazard ring]], carrying the universal (1-dimensional, commutative) [[formal group law]] is naturally an $\mathbb{N}$-graded ring.

## References

Textbook account:

* {#Karpilovsky87} [[Gregory Karpilovsky]], Chapter 2 of: *The Algebraic Structure of Crossed Products*, Mathematics Studies **142**, North Holland 1987 ([ISBN:9780080872537](https://www.elsevier.com/books/the-algebraic-structure-of-crossed-products/karpilovsky/978-0-444-70239-5))

For [[Hopf algebras]]:

* Ken Brown, Paul Gilmartin, James J. Zhang, _Connected (graded) Hopf algebras_ ([arXiv:1601.06687](http://arxiv.org/abs/1601.06687))

The notion of strongly $\mathbb{N}$-graded algebra is defined in:

* {#ArdizzoniMenini07} Alessandro Ardizzoni, Claudia Menini, _Associated graded algebras and coalgebras_ ([arXiv:0704.2106](https://arxiv.org/abs/0704.2106))

[[!redirects graded algebras]]

[[!redirects graded ring]]
[[!redirects graded rings]]

[[!redirects connected graded algebra]]
[[!redirects connected graded algebras]]


graded bimonoid

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#Contents#
* table of contents
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## Idea

A graded bimonoid is like a [[bimonoid]] but graded, as a [[graded monoid]], with [[underlying]] [[graded object]] $(A_{n})_{n \ge 0}$. We suppose in this entry, that the graduations are taken with indexes in $\mathbb{N}$. The multiplication, comultiplication, unit and counit are defined as morphisms which respect the grading.

Whereas we're not sure if the notion of graded bimonoid under this form is new or not, the notion of special graded bimonoid below is certainly.

Under the conjecture of the page [[symmetric powers in a symmetric monoidal (Q plus)-linear category]], bicommutative special graded bimonoids with trivial units are necessarily given by the [[symmetric powers]] of the [[object]] $A_{1}$.

## Definition

A graded bimonoid in a [[CMon-enriched symmetric monoidal category]] is given by a family $(A_{n})_{n \ge 0}$ of objects, a family $(\nabla_{n,p}:A_{n} \otimes A_{p} \rightarrow A_{n+p})_{n,p \ge 0}$ of morphisms, a family $(\Delta_{n,p}:A_{n+p} \rightarrow A_{n} \otimes A_{p})_{n,p \ge 0}$ of morphisms, a morphism $\eta:I \rightarrow A_{0}$ and a morphism $\epsilon:A_{0} \rightarrow I$ which verify some coherences (often, we don't write the indexes under $\Delta$ or $\nabla$):

* $((A_{n})_{n \ge 0},(\nabla_{n,p})_{n,p \ge 0}, \eta)$ is a [[graded monoid]]
* $((A_{n})_{n \ge 0},(\Delta_{n,p})_{n,p \ge 0}), \epsilon)$ is a [[graded comonoid]]
* Compatibility multiplication/comultiplication:

If $n,p,q,r \ge 0$ are such that $n+p=q+r$ ($n$ must be read as $A_{n}$ in the string diagrams):
\begin{imagefromfile}
        "file_name": "mult-comult.jpg",
        "width": 350
    \end{imagefromfile}

* Compatibility counit/multiplication:

\begin{tikzcd}
A_{0} \otimes A_{0} \arrow[rr, "\nabla"] \arrow[rd, "\epsilon \otimes \epsilon"'] &                     & A_{0} \arrow[ld, "\epsilon"] \\
                                                                                & I \otimes I \cong I &
\end{tikzcd}

* Compatibility unit/comultiplication:

\begin{tikzcd}
I \cong I \otimes I \arrow[rd, "\eta \otimes \eta"] \arrow[d, "\eta"'] &                     \\
A_{0} \arrow[r, "\Delta"']                                             & A_{0} \otimes A_{0}
\end{tikzcd}

* Compatibility unit/counit:

\begin{tikzcd}
I \arrow[d, "\eta"'] \arrow[rd, "Id"] &   \\
A_{0} \arrow[r, "\epsilon"']          & I
\end{tikzcd}

## Special graded bimonoid

* We say that a graded bimonoid is special if it verifies moreover that:

\begin{tikzcd}
A_{n+p} \arrow[d, "\Delta"'] \arrow[rd, "{n+p \choose n}Id"] &         \\
A_{n} \otimes A_{p} \arrow[r, "\nabla"']                     & A_{n+p}
\end{tikzcd}

* We say that a graded bimonoid is bicommutative if the underlying graded monoid is commutative and the underlying graded comonoid is cocommutative.

* We say that a graded bimonoid has trivial units if:
  - $\eta$ is an isomorphism (equivalently, $\epsilon$ is an isomorphism)
  - $\nabla_{n,p}$ is an isomorphism if either $n = 0$ or $p = 0$ (we have these equivalences with further properties assumed: $\nabla_{n,0}$ is an isomorphism iff $\nabla_{0,n}$ is an isomorphism in case of commutativity, $\Delta_{n,0}$ is an isomorphism iff $\Delta_{0,n}$ is an isomorphism in case of cocommutativity, $\nabla_{n,0}$ is an isomorphism iff $\Delta_{n,0}$ is an isomorphism in case of specialty and $\nabla_{0,n}$ is an isomorphism iff $\Delta_{0,n}$ is an isomorphism in case of specialty).

Under the conjecture of the page [[symmetric powers in a symmetric monoidal (Q plus)-linear category]], we have:

\begin{conjecture}
Suppose that $\mathcal{C}$ is a symmetric monoidal $\mathbb{Q}^{+}$-linear category. If $(A_{n})_{n \ge 0}$ is the underlying graded object of a special bicommutative graded bimonoid with trivial units, then each $A_{n}$ is a $n^{th}$ symmetric power of $A_{1}$.
\end{conjecture}

Note that for a special bicommutative graded bimonoids, the condition of trivial units is equivalent to:

  - $\eta$ is an isomorphism
  - $\nabla_{n,0}$ is an isomorphism for every $n \ge 0$

Of course, if our category possesses the symmetric powers of an object, we also generate a special bicommutative graded bimonoid with trivial units by taking $A_{n} = A^{\otimes_{s}n}$. Let's see how it works in a concrete case. Suppose that $\mathbb{K}$ is a field of characteristic $0$. Take a vector space $A$ of basis $X_{1},...,X_{q}$. Then $A^{\otimes_{s}m} \cong \mathbb{K}_{m}[X_{1},...,X_{q}]$, the space of [[homogeneous polynomials]] of degree $m$ in the variables $X_{1},...,X_{q}$. The unit $\mathbb{K} \cong \mathbb{K}_{0}[X_{1},...,X_{q}]$ and the multiplication $\mathbb{K}_{m}[X_{1},...,X_{q}] \otimes \mathbb{K}_{n}[X_{1},...,X_{q}] \rightarrow \mathbb{K}_{m+n}[X_{1},...,X_{q}]$ are straightforward as well as the counit $\mathbb{K}_{0}[X_{1},...,X_{n}] \cong \mathbb{K}$.
The comultiplication
$$\mathbb{K}_{m+n}[X_{1},...,X_{q}] \rightarrow \mathbb{K}_{m}[X_{1},...,X_{q}] \otimes \mathbb{K}_{n}[X_{1},...,X_{q}]$$
is defined by:
$$
X_{i_{1}}...X_{i_{m+n}} \mapsto \frac{1}{n!p!}\underset{\sigma \in \mathfrak{S}_{n+p}}{\sum} X_{\sigma(i_{1})}...X_{\sigma(i_{n})} \otimes X_{\sigma(i_{n+1})}...X_{\sigma(i_{n+p})}
$$
The intuition is that it takes a monomial of degree $n+p$, decomposes it in all the possible decompositions of a monomial of degree $n$ and a monomial of degree $p$ and then sum all the possibilites (however, the result is obtained by considering the different instances of a variable in a monomial as different entities). For example:
$$
\Delta_{2,2}(XYZW) = XY \otimes ZW + XZ \otimes YW + XW \otimes YZ + YZ \otimes ZW + YW \otimes XZ + ZW \otimes XY
$$
and
$$
\Delta_{2,2}(XYZ^{2}) = XY \otimes Z^{2} + XZ \otimes YZ + XZ \otimes YZ + YZ \otimes Z^{2} + YZ \otimes XZ + Z^{2} \otimes XY
$$
$$
= XY \otimes Z^{2} + 2XZ \otimes YZ + 2YZ \otimes Z^{2} + Z^{2} \otimes XY
$$
The number of decomposition of a monomial of degree $n+p$ in one part of degree $n$ and one part of degree $p$ is equal to the number of choice of $n$ of the variables in the monomial amongst the $n+p$ which appear, thus $\binom{n+p}{n}=\frac{(n+p)!}{n!p!}$. When we recompose the two part of the decomposition together, we find each time the same compositie that we had at the beginning. Thus, the result of the decomposition followed by the recomposition is nothing more than a multiplication by the scalar $\binom{n+p}{n}$.
$$
\nabla_{2,2}(\Delta_{2,2}(XYZW)) = \binom{2+2}{2}XYZW = 6XYZW
$$
$$
\nabla_{2,2}(\Delta_{2,2}(XYZ^{2})) = \binom{2+2}{2}XYZ^{2} = 6XYZ^{2}
$$

The comultiplication can be expressed using the [[Hasse-Schmidt derivative]]. As explained in the referred entry, for any commutative rig $R$, given a monomial $P \in R_{m+n}[X_{1},...,X_{q}]$ and a monic monomial $Y_{1}...Y_{p} \in R_{p}[X_{1},...,X_{q}]$, equivalently a multiset of $p$ variables in $\{X_{1},...,X_{q}\}$, the Hasse-Schmidt derivative $D_{Y_{1}...Y_{p}}(P)$ is the number of way of extracting $Y_{1}...Y_{p}$ from $P$, multiplied by $\frac{P}{Y_{1}...Y_{p}}$. If there is no way of extracting $Y_{1}...Y_{p}$, ie. if $Y_{1}...Y_{p}$ isn't a divisor of $P$, then $D_{Y_{1}...Y_{p}}(P)=0$. The comultiplication $\Delta_{m,n}:R_{m+n}[X_{1},...,X_{q}] \rightarrow R_{m}[X_{1},...,X_{q}] \otimes R_{n}[X_{1},...,X_{q}]$ can thus be expressed without using any rational coefficient, ie. without assuming that $R$ is a field of characteristic $0$, or more generally a $\mathbb{Q}^{+}$-algebra. It is then expressed in this way, for any $P \in R_{m+n}[X_{1},...,X_{q}]$ (writing $\mathcal{M}_{p}(X_{1},...,X_{q})$ for the set of all multisets of $p$ elements in $\{X_{1},...,X_{q}\}$):
$$
\Delta_{n,p}(P) = \underset{Y_{1}...Y_{p} \in \mathcal{M}_{p}(X_{1},...,X_{q})}{\sum} D_{Y_{1}...Y_{p}}(P) \otimes Y_{1}...Y_{p}
$$
Unpacking the definition of the Hasse-Schmidt derivative, it is defined on a monomial $P=\lambda.W_{1}...W_{n+p}$ by:
$$
\Delta_{n,p}(P) = \underset{\substack{Y_{1}...Y_{p} \in \mathcal{M}_{p}(X_{1},...,X_{q}) \\ Y_{1}...Y_{p} | P}}{\sum}\left[\underset{\substack{1 \le i \le q \\ X_{i} | Y_{1}...Y_{n}}}{\prod}\binom{nbr\;of\;time\;X_{i}\;in\;P}{nbr\;of\;time\;X_{i}\;in\;Y_{1}...Y_{n}}\right]\frac{P}{Y_{1}...Y_{n}} \otimes Y_{1}...Y_{n}
$$
and then prolongated by linearity:
$$
\Delta_{n,p}(P+Q) = \Delta_{n,p}(P)+\Delta_{n,p}(Q)
$$
in order to define it for every homogeneous polynomial. If this formula looks more intimidating that the one defined by a sum on $\mathfrak{S}_{n+p}$ in characteristic $0$, it is in fact nothing more than a precise formula to express the fact that $\Delta_{n,p}(P)$ for a monomial $P$ of degree $n+p$, is the sum of all the decompositions of $P$ in one part of degree $n$ and one part of degree $p$. Such a [[combinatorial]] description have the virtue of not requiring some divisions which are forbidden in [[positive characteristic]].

## Properties

* With the above notations, if we have a graded bimonoid, then $(A_{0},\nabla_{0,0}, \Delta_{0,0}, \eta, \epsilon)$ is a bimonoid.

* Suppose that our [[CMon-enriched symmetric monoidal category]] possesses a [[zero object]] such that $A \otimes 0 \cong 0$ for every object $A$. We can then associate a graded bimonoid to every bimonoid by putting:
  - $A_{0} = A$
  - $\nabla_{0,0} = \nabla$
  - $\Delta_{0,0} = \Delta$
  - $A_{n} = 0$ if $n \ge 1$

## Related concepts

[[bimonoid]]

[[symmetric powers in a symmetric monoidal (Q plus)-linear category]]

[[graded codifferential category]]

[[Hasse-Schmidt derivative]]


graded C*-algebra

#Contents#
* table of contents
{:toc}

## Idea

A [[graded algebra|graded]] [[C*-algebra]].

Notably $\mathbb{Z}_2$-graded $C^\ast$-algebras appear in [[KK-theory]].

## References

Section 14 in

* [[Bruce Blackadar]], _[[K-Theory for Operator Algebras]]_, 2nd ed. Cambridge University Press, Cambridge (1999)
 {#Blackadar99}

[[!redirects graded C*-algebras]]

[[!redirects graded C-star algebra]]
[[!redirects graded C-star algebras]]


graded codifferential category

#Contents#
* table of contents
{:toc}

## Idea

A graded codifferential category is like a [[codifferential category]] but where the algebra modality is replaced by a graded algebra modality. Whereas $FVec_{\mathbb{K}}$ is only a trivial codifferential category, with all the structure equal to zero, $FVec_{\mathbb{K}}$ is a non-trivial codifferential category.

## Definition

We will use [[graded monads]] with grading in the multiplicative monoid of a commutative [[rig]]. Also, the term rig must be read as "commutative rig" below.
\begin{definition}
Let $R$ be a rig. A $R$-graded monad in a category $\mathcal{C}$ is given by a family $(S_{r}:\mathcal{C} \rightarrow \mathcal{C})_{r \in R}$ of endofunctors, a family $(m_{r,s}A:S_{r}(S_{s}(A)) \rightarrow S_{rs}(A))_{r,s \in R}$ of natural transformations and a natural transformation $u:A \rightarrow S_{1}(A)$ such that:

\begin{tikzcd}
S_{r}(S_{s}(S_{t}(A))) \arrow[dd, "{S(m)}"'] \arrow[rr, "{m}"] &  & S_{rs}(S_{t}(A)) \arrow[dd, "{m}"] \\
                                                                                &  &                                           \\
S_{r}(S_{st}(A)) \arrow[rr, "{m}"']                                            &  & S_{rst}(A)
\end{tikzcd}

and

\begin{tikzcd}
S_{r}(A) \arrow[d, "S(u)"'] \arrow[rrd, "="] &  &          & S_{r}(A) \arrow[rrd, "="] \arrow[d, "u"'] &  &          \\
S_{r}(S_{1}(A)) \arrow[rr, "{m}"']         &  & S_{r}(A) & S_{1}(S_{r}(A)) \arrow[rr, "{m}"']  &  & S_{r}(A)
\end{tikzcd}

\end{definition}

\begin{definition}
Let $R$ be a rig. A $R$-graded algebra modality is a $R$-graded monad with a family of natural transformations $\nabla_{r,s}A:S_{r}(A) \otimes S_{s}(A) \rightarrow S_{r+s}(A)$ and a natural transformation $\eta_{A}:I \rightarrow S_{0}(A)$ such that for every object $A$, we have that $((S_{r}(A))_{r \in R},(\nabla_{r,s})_{r,s \in R}, \eta)$ is a $(R,+,0)$-commutative [[graded monoid]] and this diagram commutes:

\begin{tikzcd}
S_{r}(S_{t}(A)) \otimes S_{s}(S_{t}(A)) \arrow[dd, "\nabla"'] \arrow[rr, "m \otimes m"] &  & S_{rt}(A) \otimes S_{st}(A) \arrow[dd, "\nabla"] \\
                                                                                &  &                                                  \\
S_{r+s}(S_{t}(A)) \arrow[rr, "m"']                                                      &  & S_{(r+s)t}(A)
\end{tikzcd}
\end{definition}

\begin{definition}
A $R$-graded codifferential category is a [[CMon-enriched symmetric monoidal category]] with a $R$-graded algebra modality and a $R$-graded deriving transformation ie. a family $(d_{r}A:S_{r+1}(A) \rightarrow S_{r}(A) \otimes A)$ of natural transformation such that:

* [[Leibniz rule]]:
\begin{tikzcd}
                                                                                &  &  & S_{n+p}(A) \arrow[rrrdd, "d"] &  &  &                         \\
                                                                                &  &  &                                &  &  &                         \\
S_{n}(A) \otimes S_{p}(A) \arrow[rrruu, "\nabla"] \arrow[rrrrrr, "(d \otimes 1)( 1 \otimes \gamma)(\nabla \otimes 1)+ (1 \otimes d)(\nabla \otimes 1)"'] &  &  &                                &  &  & S_{n+p-1}(A) \otimes A
\end{tikzcd}

* [[chain rule|Chain rule]]:

\begin{tikzcd}
S_{n-1}(S_{p}(A)) \otimes S_{p}(A) \arrow[rr, "m \otimes d"] &  & S_{(n-1)p}(A) \otimes S_{p-1}(A) \otimes A \arrow[ddd, "\nabla \otimes 1"'] \\
                                                                &  &                                                                               \\
                                                                &  &                                                                               \\
                                                                &  & S_{np-1}(A) \otimes A                                                        \\
S_{n}(S_{p}(A)) \arrow[uuuu, "d"] \arrow[rr, "m"']            &  & S_{np}(A) \arrow[u, "d"]
\end{tikzcd}

* Linear rule:

\begin{tikzcd}
A \arrow[dd, "u"'] \arrow[rrdd, "\eta \otimes 1"] &  &                     \\
                                                          &  &                     \\
S_{1}(A) \arrow[rr, "d"']                             &  & S_{0}(A) \otimes A
\end{tikzcd}

* Schwarz rule:

\begin{tikzcd}
S_{n-1}(A) \otimes A \arrow[rr, "d \otimes A"] &  & S_{n-2}(A) \otimes A \otimes A \arrow[d, "1 \otimes \gamma"] \\
                                                                    &  & S_{n-2}(A) \otimes A \otimes A                               \\
                                                                    &  &                                                               \\
S_{n}(A) \arrow[uuu, "d"] \arrow[rr, "d"']                 &  & S_{n-1}(A) \otimes A \arrow[uu, "d \otimes 1"']
\end{tikzcd}

\end{definition}

## Related concepts

[[symmetric powers in a symmetric monoidal (Q plus)-linear category]]

[[differential category]]

[[Homogeneous polynomial codifferential category]]

[[Anticommutative graded codifferential category]]

[[!redirects graded codifferential categories]]


graded commutative algebra > history

> see _[[supercommutative algebra]]_


graded commutator

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In [[superalgebra]] the the _graded commutator_ or _supercommutator_ of two elements of homogeneous degree is

$$
  [a,b] \coloneqq  \left( a b - (-1)^{deg(a) deg(b)} b a\right)
$$

and extended from there to the whole superalgebra as a [[graded derivation]] in both arguments.

So when at least one of $a$ or $b$ is even graded, then this is the [[commutator]]. When both are odd graded then this is the [[anti-commutator]].

The graded commutator is just the plain commutator [[internalization|internal]] to the [[symmetric monoidal category]] of [[super vector spaces]].

[[!redirects graded commutators]]

[[!redirects super-commutator]]
[[!redirects super-commutators]]
[[!redirects super commutator]]
[[!redirects super commutators]]
[[!redirects supercommutator]]
[[!redirects supercommutators]]


graded comonoid

A **graded comonoid** in a [[symmetric monoidal category]] $\mathcal{V}$ is a [[graded monoid]] in the [[opposite category]] $\mathcal{V}^{op}$.

## See also

[[comonoid]]
[[graded object]]
[[graded monoid]]
[[graded bimonoid]]


graded derivation

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## Idea

The term *graded derivation* would a priori refer in general to suitably compatible [[derivations]] of [[graded algebras]], but is in practice used mostly for those operations $d \,\colon\, A \to A$ on [[superalgebras]] $A$ that satisfy the condition that for $a_1, a_2 \,\in\, A$ a [[pair]] of [[elements]] each of homogeneous degree $deg(a_i) \in \mathbb{Z} \,mod\, 2$ the derivation of their product is

$$
  d(a_1 \cdot a_2)
  \;=\;
  (d a_1) \cdot a_2
  +
  (-1)^{deg(a_1) \cdot deg(d)}
  a_1 \cdot d a_2
  \,,
$$

where $deg(d) \,\in\, \mathbb{Z} \,mod\,2$ is the degree of the graded derivation.

For $d = 0$ this is the law satisfied by an ordinary (un-graded) [[derivation]], also known as the *[[Leibniz rule]]* satisfied by ordinary [[differentiation]]. The archetypical example of a non-trivially graded derivation is the [[de Rham differential]] acting on the [[de Rham algebra]].

[[!redirects graded derivations]]


graded field

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## Definition

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

An even-[[graded ring]] $R$ is a **graded field** if every non-zero homogenous element is multiplicatively invertible.

=--

For instance ([Lurie, lecture 24, def. 1](#LurieLecture)).

## Related concepts

* [[infinity-field]]

## References

Lecture 24 of

* [[Jacob Lurie]], _Morava E-theory and Morava K-theory_, Lecture notes ([web](http://www.math.harvard.edu/~lurie/252x.html))
 {#LurieLecture}

[[!redirects graded fields]]


graded fusion category

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## Definition

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

Let $\mathcal{C}$ be a [[semisimple category|semisimple]] [[linear category|linear]]([[Vect]]-[[enriched category|enriched]]) (over a field $k$) category and $G$ a finite group.
A **$G$-grading** on $\mathcal{C}$ is a direct sum decomposition of $\mathcal{C}$ into _homogeneous components_ $\mathcal{C}_g$, which are again $k$-linear semisimple categories:

$$\mathcal{C} \simeq \bigoplus_{g \in G} \mathcal{C}_g$$

=--

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

A **graded fusion category** is a fusion category with a chosen $G$-grading for some group, such that the monoidal product maps $\mathcal{C}_g \times \mathcal{C}_h$ into $\mathcal{C}_{g h}$.

=--

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

Let $\Lambda_\mathcal{C}$ be the set of equivalence classes of simple objects in $\mathcal{C}$.
A $G$-grading is a map $\operatorname{deg}\colon \Lambda_\mathcal{C} \to G$ such that for any two simples $X$ and $Y$ and any simple subobject $Z \hookrightarrow X \otimes Y$, we have $\operatorname{deg}([Z]) = \operatorname{deg}([X]) \operatorname{deg}([Y])$.

=--

## Terminology

* The trivial component $\mathcal{C}_1$ of the grading is a full fusion subcategory.
* A $G$-extension of a fusion category $\mathcal{D}$ is a $G$-graded fusion category $\mathcal{C}$ such that $\mathcal{D} \simeq \mathcal{C}_1$.
* A grading is faithful if $\mathcal{C}_g \neq 0$ for all $g \in G$. This is a standard assumption.
* The __adjoint category__ $\mathcal{C}_{\text{ad}}$ is the full fusion subcategory of $\mathcal{C}$ spanned by objects of the form $X \otimes X^*$.

## Examples

* Every fusion category has the trivial grading from the trivial group.
* $G$-graded vector spaces for a finite group $G$ are naturally a graded fusion category.
* The universal grading, see below.
* [[$G$-crossed braided fusion categories]] are a (kind of) [[categorification]] of [[crossed module]]s, and therefore carry a grading as part of the data.

## Universal grading

From the definition, it is clear that gradings are covariant in the group.
A group homomorphism $\phi\colon G \to H$ gives an obvious map of the set of gradings:

$$ \mathcal{C} \simeq \bigoplus_{h \in H} \bigoplus_{g \in \phi^{-1}(h)} \mathcal{C}_g $$

or

$$ \operatorname{deg}_H = \phi \circ \operatorname{deg}_G $$

Morphisms of gradings are therefore simply group homomorphisms.

For every fusion category $\mathcal{C}$, there exists a universal grading by a group $U_\mathcal{C}$.
It has the following properties:

* It is faithful.
* The trivial component is the full fusion subcategory spanned by objects of the form $X \otimes X^*$.
* Every full fusion subcategory $\mathcal{D} \subset \mathcal{C}$ containing the adjoint category $\mathcal{C}_{\text{ad}}$ is of the form $\mathcal{D} \simeq \bigoplus_{h \in H} \mathcal{C}_h$ for some subgroup $H \subset U_\mathcal{C}$.
* The group of monoidal automorphisms of the identity functor is canonically isomorphic to $\operatorname{Hom}(U_\mathcal{C}, k^\times)$.

## References ##

* Shlomo Gelaki, Dmitri Nikshych, [Nilpotent fusion categories](https://arxiv.org/abs/math/0610726)
* [[Pavel Etingof]], Dmitri Nikshych, Victor Ostrik, with an appendix by Ehud Meir, [Fusion categories and homotopy theory](https://arxiv.org/abs/0909.3140v2)
* Vladimir Drinfeld, Shlomo Gelaki, Dmitri Nikshych, Victor Ostrik, [On braided fusion categories I](https://arxiv.org/abs/0906.0620)

[[!redirects graded fusion categories]]


graded manifold

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## Idea

Under the [[embedding of smooth manifolds into formal duals of R-algebras]] a [[smooth manifold]] $X$ is identified with the [[formal dual]] of its $\mathbb{R}$-[[algebra of functions|algebra of]] [[smooth functions]]. Given then a $\mathbb{Z}$-[[graded module]] $\mathfrak{a}$ over $C^\infty(X)$ concentrated away from zero, the [[symmetric algebra|graded symmetric]] [[graded algebra]]

$$
  Sym_{C^\infty(X)}\left(
    \mathfrak{a}
  \right)
$$

may naturally be thought of as the [[formal dual]] of a _$\mathbb{Z}$-graded manifold_.

If moreover $Sym_{C^\infty(X)}\left(\mathfrak{a}\right)$ is equipped with a [[differential]] that makes it a [[differential graded-commutative algebra]], then the formal dual is called a _[[differential graded manifold]]_.

## Related concepts

* [[higher differential geometry]], [[derived geometry]]

* [[Lie infinity-algebroid]]

* [[BRST-complex]], [[BV-complex]]

[[!redirects graded manifolds]]


graded modality

#Contents#
* table of contents
{:toc}

##Idea

Classic [[modalities]] are defined in absolute terms, so that, for instance, [[necessity and possibility]] are understood in terms of a state of affairs obtaining in all or some world. Some [philosophers](#philosophy) in the early 1970s looked to generalize these to graded versions so that a proposition may be said to be, say, quite likely, highly likely, and so on. Since then, linguists and computer scientists have developed these ideas, the latter expanding the topic to include those modalities relating to resources and [[side effects]].

A **graded modality** is an indexed family of [[modalities]] with some additional [[structure]] on the indices which mirrors the structure of the axioms/proof rules. A graded modality is a [[graded monad]] with idempotent components.

For example, the [[exponential modality]] of [[linear logic]] $!$ has a graded counterpart in [[bounded linear logic]], where $!$ is replaced with a family of modalities $!_n$ indexed by the natural numbers (the reuse bound).

More recent variations of [[linear logic]] which use graded modalities, where $!$ is replaced with a family of modalities $!_r$ indexed by a [[semi-ring]] could be designated by the name [[graded linear logic]].

[[Differential linear logic]] and [[graded linear logic]] seem to be mixable under the form of a [[graded differential linear logic]]. Such a logic would be a refinement of [[differential linear logic]] able to deal with differentiable functions which have a grade such as the degree of a polynomial or the number of times the function is differentiable. A more specific system would be a [[symmetric linear logic]] which deal specifically with [[symmetric powers]] and thus [[homogeneous polynomials]]. It seems possible to adapt the idea to an [[anticommutative]] setting giving an [[homological linear logic]]. Such a logic would be probably able to give a syntax for at least [[Koszul complexes]].

## References

* Brunel, Aloïs and Gaboardi, Marco and Mazza, Damiano and Zdancewic, Steve, _A Core Quantitative Coeffect Calculus_, ([doi](https://doi.org/10.1007/978-3-642-54833-8_19)) ([pdf](https://lipn.univ-paris13.fr/~mazza/papers/CoreQuantCoeff.pdf))

* Dan R. Ghica and Alex I. Smith, _Bounded Linear Types in a Resource Semiring
_, ([doi](https://doi.org/10.1007/978-3-642-54833-8_18)) ([pdf](https://www.cs.bham.ac.uk/~drg/papers/esop14.pdf))

* [[Dominic Orchard]], Vilem-Benjamin Liepelt, Harley Eades III, _Quantitative Program Reasoning with Graded Modal Types_, ([pdf](https://www.cs.kent.ac.uk/people/staff/dao7/publ/granule-icfp19.pdf))

* [[Dominic Orchard]], Vilem-Benjamin Liepelt, _Gram:  A linear functional language with graded modal types_, ([extended abstract](https://www.cs.ox.ac.uk/conferences/fscd2017/preproceedings_unprotected/TLLA_Orchard.pdf))

* Marco Gaboardi, [[Shin-ya Katsumata]], [[Dominic Orchard]], Flavien Breuvart, [[Tarmo Uustalu]], *Combining effects and coeffects via grading*, ICFP 2016: Proceedings of the 21st ACM SIGPLAN International Conference on Functional Programming (2016) 476–489 [[doi:10.1145/2951913.2951939](https://doi.org/10.1145/2951913.2951939), [talk abstract](https://icfp16.sigplan.org/details/icfp-2016-papers/31/Combining-Effects-and-Coeffects-via-Grading), [video rec](https://www.youtube.com/watch?v=l1ZNMT3fQCo)]

* Flavien Breuvart, Michele Pagani, _Modelling Coeffects in the Relational Semantics
of Linear Logic_, ([pdf](https://drops.dagstuhl.de/opus/volltexte/2015/5438/pdf/35.pdf))

* Shin-ya Katsumata, _A Double Category Theoretic Analysis of Graded Linear Exponential Comonads_, ([doi](https://doi.org/10.1007/978-3-319-89366-2_6))

* Soichiro Fujii, Shin-ya Katsumata, Paul-André Melliès, _Towards a Formal Theory of Graded Monads_, ([doi](https://doi.org/10.1007/978-3-662-49630-5_30)) ([pdf](https://www.irif.fr/~mellies/papers/fossacs2016-final-paper.pdf))

* [[J-B Vienney|Jean-Baptiste Vienney]], [[Jean-Simon Lemay|Jean-Simon Pacaud Lemay]], _Graded Differential Categories and Graded Differential Linear Logic_, 2023, [arXiv](https://arxiv.org/abs/2303.10586)

A formal framework for type theories with a collection of modalities is in

* [[Daniel Licata]], [[Mike Shulman]], and [[Mitchell Riley]], _A Fibrational Framework for Substructural and Modal Logics (extended version)_, in Proceedings of 2nd International Conference on Formal Structures for Computation and Deduction (FSCD 2017) ([doi: 10.4230/LIPIcs.FSCD.2017.25](http://drops.dagstuhl.de/opus/volltexte/2017/7740/), [pdf](http://dlicata.web.wesleyan.edu/pubs/lsr17multi/lsr17multi-ex.pdf))

On [[dependent linear types]] and [[graded modalities]]:

* [[Benjamin Moon]], [[Harley Eades III]], [[Dominic Orchard]], *Graded Modal Dependent Type Theory*. In: N. Yoshida (ed.) *Programming Languages and Systems ESOP 2021*, Lecture Notes in Computer Science **12648**, Springer (2021) 462-490 [[doi:10.1007/978-3-030-72019-3_17](https://doi.org/10.1007/978-3-030-72019-3_17), [arxiv:2010.13163](https://arxiv.org/abs/2010.13163)]

For discussion in [[philosophy]] and linguistics, see

* Patrick Grosz, _Grading Modality: A New Approach to Modal Concord and its Relatives_, ([paper](https://www.univie.ac.at/sub14/proc/grosz.pdf))

* Daniel Lassiter, _Graded modality_, ([paper](https://web.stanford.edu/~danlass/Lassiter-graded-modality-draft.pdf))

* Daniel Lassiter, _Graded Modality: Qualitative and Quantitative Perspectives_, OUP, ([website](https://global.oup.com/academic/product/graded-modality-9780198701354?cc=us&lang=en&))

 {#philosophy} Earlier work in philosophy includes

* Lou Goble, 1970. _Grades Of Modality_. Logique Et Analyse 13, pp. 323-334, ([paper](http://virthost.vub.ac.be/lnaweb/ojs/index.php/LogiqueEtAnalyse/article/view/405)).
*  David Kaplan, _S5 with multiple possibility_, The Journal of Symbolic Logic, 35:355–356, 1970.

* Kit Fine, 1972. _In so many possible worlds_. Notre Dame Journal of Formal Logics 13: 516–520.

* David Lewis, 1973. _Counterfactuals_. Oxford: Blackwell

* M. Fattorosi-Barnaba  and  F. de Caro.  _Graded  modalities I_. Studia  Logica, 44:197–221,1985

[[!redirects graded modalities]]


graded monad

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## Idea
 {#Idea}

By analogy with [[graded algebra]], an **$\mathcal{M}$-graded monad** in a category $\mathcal{C}$ for a [[monoidal category]], $(\mathcal{M}, \otimes, I)$, is a [[lax monoidal functor]] from $\mathcal{M}$ to the [[endofunctor|endo]]-[[functor category]] of $\mathcal{C}$ (that whose [[monoid objects]] are [[monads]] on $\mathcal{C}$):

$$
  (\mathcal{M}, \otimes, I)
  \longrightarrow
  ([\mathcal{C}, \mathcal{C}], \circ, id_{\mathcal{C}})
  \,.
$$

This generalizes the concept of a plain *[[monad]]* on $\mathcal{C}$, which is recovered as the special case of grading by the the [[terminal category|terminal]] monoidal category $1$.

Equivalently, an $\mathcal{M}$-grading is a [[action of a monoidal category|lax action]] of $\mathcal{M}$ on $\mathcal{C}$, which in turn is equivalently a lax [[2-functor]] to [[Cat]] from the [[delooping#deloopings_of_higher_categorical_structures|delooping]] of $\mathcal{M}$, i.e.: $\mathbf{B} \mathcal{M} \to Cat$.

Just as [[monads]] may be defined in any [[2-category]] $K$ (besides the case $K = $[[Cat]]), this suggests that we may generalize graded monads to lax 2-functors $\mathbf{B} \mathcal{M} \to K$.

A further generalization is to category-valued monads, lax functors from any category to $Cat$ ([OrchWadEad](#OrchWadEad)), and then to 2-category-valued monads from any 2-category.

Graded monads are also known as **parametric** monads. The grading idea may also be applied to [[comonads]].

## Examples

1. The grading may arise from a monoid $(M, \otimes, e)$. Then for some given category, $C$, we have a family of endofunctors, $T_m$, indexed by elements of $M$, with maps $\mu_{m, n, X}: T_m(T_n X) \to T_{m \otimes n} X$ and $\eta_{X}: X \to T_{e} X$, for $m, n$ in $M$ and $X$ in $C$. For instance, there is a $(\mathbb{N}, \times, 1)$-graded monad on sets where $T_n$ returns lists of length $n$ of elements of a set.

1. In computer science, monads model [[side effects|effects]] and comonads coeffects. Grading can therefore allow further annotation of these. For instance, there are graded versions of the [[reader monad]], [[state monad]] and [[writer comonad]] ([OrchPet](#OrchPet)).

1. Any [[graded modality]], such as found in [[bounded linear logic]].

1. For graded monads relevant for probability theory see ([Perrone](#Perrone)).

1. Given the strict action of a monoidal category, $\mathcal{M}$ on a category $\mathcal{B}$, and an adjunction {#FromStrictAction}

\begin{center}
  \begin{tikzcd}
    \mathcal{A}
     \arrow[r, shift right=6pt, "R"', "\bot"]
    &
    \mathcal{B},
     \arrow[l, shift right=6pt, "L"']
  \end{tikzcd}
\end{center}
then $\mathcal{A}$ inherits a lax action of $\mathcal{M}$ and is hence a graded monad. Every lax action can be generated from a strict action in this way. Initial and terminal such resolutions of a lax action then generalize the $\mathcal{M} \cong 1$ situation in which is a monad is resolved into adjunctions with the [[Kleisli category|Kleisli]] and [[Eilenberg-Moore category|Eilenberg-Moore]] categories ([FKM 16](#FKM)).

## Uses

Graded monads can be used to construct ordinary monads by left Kan extension in the 2-category of [[monoidal category|monoidal categories]], [[lax monoidal functor|lax monoidal functors]], and [[monoidal natural transformation|monoidal transformations]]. There are known criteria for when this Kan extension exists; see ([Fritz & Perrone 18, Theorem 2.1](#FPKan)), as well as the references in there on algebraic Kan extensions.

For example, taking the left Kan extension of the graded list monad $\mathbf{B} \mathbb{N} \to [Set,Set]$ described above results in the usual list monad on $Set$, given by a lax monoidal functor $1 \to [Set,Set]$. Based on a similar construction on the category of complete metric spaces, ([Fritz & Perrone 17](#FPKant)) have contructed a monad of Radon probability measures without any appeal to measure theory; the intuitive idea being that a probability measure can be thought of as an idealized version of a finite sample, and spaces of finite samples make up a graded monad. Forgetting the grading by taking the above Kan extension then produces the _Kantorovich monad_, containing all Radon probability measures of finite first moment. This construction reduces certain problems in measure-theoretic probability to purely combinatorial problems.

A *useful feature* of such constructions is that the multiplication of the graded monad is often a _strong_ monoidal functor in practice. For the graded list monad, this is because a list of length $m n$ can be decomposed uniquely into a list of length $m$ of lists of length $n$, so that the multiplication $T_m T_n X \longrightarrow T_{m n} X$ is an isomorphism. For the probability monad mentioned in the previous paragraph, the analogous phenomenon occurs as well, and this can be exploited e.g. in order to prove a disintegration theorem for finite first moment Radon probability measures on complete metric spaces ([Perrone, Theorem 2.6.9](#Perrone)).

## Related concepts

* [[graded modality]]
* [[monad]]
* [[actegory]]

## References

* {#FKM} Soichiro Fujii, Shin-ya Katsumata, and [[Paul-André Melliès]], _Towards a Formal Theory of Graded Monads_, ([pdf](https://www.irif.fr/~mellies/papers/fossacs2016-final-paper.pdf))

* Soichiro Fujii, _A 2-Categorical Study of Graded and Indexed Monads_, ([arXiv:1904.08083](https://arxiv.org/abs/1904.08083))

* Ulrich Dorsch, Stefan Milius and Lutz Schröder, _Graded Monads and Graded Logics for the Linear Time – Branching Time Spectrum_, ([arXiv:1812.01317](https://arxiv.org/abs/1812.01317))

* {#Perrone} [[Paolo Perrone]], _Categorical Probability and Stochastic Dominance in Metric Spaces_, ([thesis](http://www.paoloperrone.org/phdthesis.pdf))

* {#FPKan} [[Tobias Fritz]], [[Paolo Perrone]], _A Criterion for Kan Extensions of Lax Monoidal Functors_, ([arXiv:1809.10481](https://arxiv.org/abs/1809.10481)).

* {#FPKant} [[Tobias Fritz]], [[Paolo Perrone]], _A Probability Monad as the Colimit of Spaces of Finite Samples_, ([arXiv:1712.05363](https://arxiv.org/abs/1712.05363)).

* {#OrchPet} [[Dominic Orchard]], Tomas Petricek, _Embedding effect systems in Haskell_, ([pdf](https://www.doc.ic.ac.uk/~dorchard/publ/haskell14-effects.pdf))

* {#OrchWadEad} Dominic Orchard, Philip Wadler, Harley Eades, _Unifying graded and parameterised monads_, ([arXiv:2001.10274](https://arxiv.org/abs/2001.10274))

Graded monads are shown to be examples of [[enriched]] [[relative monads]] in:

* [[Dylan McDermott]], [[Tarmo Uustalu]]. _Flexibly graded monads and graded algebras_, in: *Mathematics of Program Construction MPC 2022*, Lecture Notes in Computer Science **13544**, Springer (2022) [[doi:10.1007/978-3-031-16912-0_4](https://doi.org/10.1007/978-3-031-16912-0_4)]

[[!redirects graded monads]]


graded monoid

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## Idea

A graded monoid is like a monoid but where elements of the monoid possess some degree and such that when you multiply, you sum the degrees. It can be defined in any monoidal category.

## Definition and properties

Let $M$ be a [[commutative monoid]] (in the category $\mathbf{Sets}$). A **$M$-graded monoid** $\Phi$ in a [[monoidal category]] $\mathcal{V}$ with [[unit object]] $I$ is the data of

* for each $m \in M$, an [[object]] $\Phi_m$,
* for each $m,n \in M$, a [[morphism]]
  $$\nabla_{m,n}: \Phi_m \otimes \Phi_n \to \Phi_{m+n} $$
* a morphism
  $$ \eta: I \to \Phi_0 $$

such that the obvious [[associativity]] and [[unit]] [[axioms]] hold.

Thus, a $M$-graded monoid is in particular a $M$-[[graded object]]. In fact, a $M$-graded monoid is just a [[monoid in a monoidal category|monoid]] in the monoidal category of $M$-graded objects of $\mathcal{V}$, hence a [[lax monoidal functor]] $M \to \mathcal{V}$ by [[Day convolution#DayMonoidsAreLaxMonoidalFunctorsOnTheSite|this proposition]] (where $M$ is viewed as a [[discrete category|discrete]] monoidal category).

We say that a $M$-graded monoid is **connected** if $\eta$ is an isomorphism.

If $\Phi$ is a $M$-graded monoid in $\mathcal{V}$, then $(\Phi_{0},\nabla_{0,0},\eta)$ is a (non-graded) monoid in $\mathcal{V}$.

## Examples

The following are all examples of connected $\mathbb{N}$-graded monoids.

* In the [[symmetric monoidal category]] of [[groups]] with the [[cartesian product]], an example of $\mathbb{N}$-graded monoid is the trivial one $1 = (1)_n$

* In the same category, another example is the **graded monoid of symmetric groups** $\Sigma = (\Sigma_n)_n$.

* In any monoidal category, defining $\Phi_n := A^{\otimes n}$, $\nabla_{m,n}:A^{\otimes m} \otimes A^{\otimes n} \rightarrow A^{\otimes (m+n)}$ given by [[associators]] and $\eta = 1_{I}$, we obtain the connected $\mathbb{N}$-graded monoid of tensor powers.

* In any symmetric monoidal category, defining $S^n(A)$ equal to the [[coequalizer]] of the $n!$ permutations $A^{\otimes n} \rightarrow A^{\otimes n}$ and defining the multiplication $S^n(A) \otimes S^p(A) \rightarrow S^{n+p}(A)$ by using the universal property of the coequalizer, we obtain the connected commutative $\mathbb{N}$-graded monoid of [[symmetric powers]].

* In any symmetric monoidal category, defining $\Gamma^n(A)$ equal to the [[equalizer]] of the $n!$ permutations $A^{\otimes n} \rightarrow A^{\otimes n}$ and defining the multiplication $\Gamma^n(A) \otimes \Gamma^p(A) \rightarrow \Gamma^{n+p}(A)$ by using the universal property of the equalizer, we obtain the connected commutative $\mathbb{N}$-graded monoid of [[divided powers]].

* In any symmetric monoidal category enriched over the category of abelian groups, defining $\Lambda^n(A)$ equal to the coequalizer of the $n!$ signed permutations $sgn(\sigma).\sigma:A^{\otimes n} \rightarrow A^{\otimes}$ where $sgn(\sigma)$ is the [[signature]] of the permutation $\sigma$ and defining the multiplication $\Lambda^n(A) \otimes \Lambda^p(A) \rightarrow \Lambda^{n+p}(A)$ by using the universal property of the equalizer, we obtain the connected graded-commutative  (ie. such that $\sigma;\nabla_{p,n} = sgn(\sigma).\sigma:\Phi_{n} \otimes \Phi_{p} \rightarrow \Phi_{n+p}$) $\mathbb{N}$-graded monoid of [[exterior powers]]. Replacing the coequalizer by an equalizer always provides an isomorphic $\mathbb{N}$-graded monoid.

The  connected commutative $\mathbb{N}$-graded monoid of symmetric powers and of divided powers are not isomorphic in general, but they are if the symmetric monoidal category is enriched over $\mathbb{Q}^{+}$-modules, for instance in the category of [[vector spaces]] over a [[field]] of [[characteristic]] $0$ or in the category of sets and [[relations]] (where $S^{n}(X) \cong \Gamma^{n}(X)$ is equal to the set of all multisets of $n$ elements of $X$).

## See also

* [[monoid object]]
* [[graded object]]
* [[graded comonoid]]
* [[graded bimonoid]]

[[!redirects graded unitary monoid]]
[[!redirects graded unital monoid]]


graded Seely isomorphism

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# Contents
* table of contents
{:toc}

## Idea

This is a generalization of the [[Seely isomorphism]] to [[graded modalities]]. A graded exponential modality $(!_{r}:\mathcal{C} \rightarrow \mathcal{C})_{r \in R}$ graded by some [[rig]] $R$ and in a [[CMon-enriched symmetric monoidal category]] $\mathcal{C}$ verifies the graded Seely isomorphism iff we have:
$$
!_{r}(A \oplus B) \cong \underset{s+t=r}{\bigoplus} !_{s}A \otimes !_{t}A
$$
One needs to combine [[graded linear logic]] and [[differential linear logic]] to go from the usual Seely isomorphism of linear logic to the more general graded Seely isomorphism. The usual Seely isomorphism is obtained as a graded Seely isomorphism when one takes $R$ equal to the zero rig.
## In sequent calculus

... to come ...

## In category theory

... to come ...

## Related concepts

- [[Seely isomorphism]]
- [[graded linear logic]]

## References

It will be discussed in a paper "Graded Differential Categories and Graded Differential Linear Logic" by [[JS Pacaud Lemay]] and [[J-B Vienney]].


Graded Seely isomorphism > history

< [[graded Seely isomorphism]]


graded set

# Graded sets
* table of contents
{: toc}

## Definition

Let $S$ be a [[set]].  Frequently, $S$ is a [[group]] or [[monoid]] (usually [[commutative monoid|commutative]]).

An **$S$-graded set** is an $S$-indexed [[family of sets]] $\{X_s\}_{s\in S}$. This can equivalently be described as a [[function]] $S \to Set$, or as a function $X \to S$ (a [[bundle]] with $X_s$ the [[fiber]] over $s\in S$).

The elements of $X_s$ are often said to have **degree** $s$.

Given a [[pair]] of $S$-graded sets $\{X_s\}_{s\in S}$ and $\{Y_s\}_{s\in S}$, a [[homomorphism]] between them is an $S$-indexed family of functions $\{f_s \colon X_s \to Y_s \}_{s\in S}$.  This can equivalently be described as a [[natural transformation]] between the two associated functions $S \to Set$, or as a function from $X$ to $Y$ that make the [[diagram]] with the associated functions $X \to S$ and $Y \to S$ [[commuting diagram|commute]].

## Examples

The most common choices of $S$ are probably:

* the [[natural numbers]] $\mathbb{N}$.

* the [[integers]] $\mathbb{Z}$.

* the [[cyclic group of order 2|2-element set]] $\mathbb{Z}/2$.  In this case, the elements of degree $0$ are often called **even**, and those of degree $1$ **odd**.

## Monoidal structures and enrichment

Suppose $(S,0,+)$ is a monoid, written additively.  Then the [[category]] $Set^S$ of $S$-graded sets has a [[closed monoidal category|closed]] [[monoidal category|monoidal]] [[mathematical structure|structure]], where

$$ (X \otimes Y)_s = \coprod_{u+v = s} (X_u \times Y_v) $$

This is a special case of [[Day convolution]].

Furthermore, this monoidal structure laxly interchanges with the pointwise product of graded sets:

$$ (X \times Y)_s := X_s \times Y_s $$

where the lax interchange maps $(X \times Y) \otimes (Z \times W) \to (X \otimes Z) \times (Y \otimes W)$ are given by inclusions. These interacting monoidal structures make $Set^S$ into a [[duoidal category]].

A $Set^S$-[[enriched category]] is a [[category]] whose [[morphisms]] all have degrees in $S$, and such that [[identity morphisms]] have degree $0$ and $deg(g f) = deg(g) + deg(f)$.  Note that its underlying ordinary category, in the usual sense of [[enriched category theory]], is the category of degree-$0$ morphisms.

More generally, we may grade by a monoidal category. This leads to the notion of [[locally graded category]].

## Graded objects

Given any set $S$ and any [[category]] $C$, the [[category]] of __$S$-graded objects of $C$__ is simply the [[functor category]] $C^S$ (identifying $S$ with its [[discrete category]]).  This includes graded sets as above, as well as [[graded abelian groups]], [[graded modules]], [[graded vector spaces]].  However, [[graded rings]] and [[graded algebras]] are not the same (and in particular require $S$ to be a [[monoid]]).

## Related concepts

* [[bigraded object]]

* [[graded module]]

* [[graded ring]]

* [[graded manifold]]

* [[differential-graded algebra]]

[[!redirects graded set]]
[[!redirects graded sets]]

[[!redirects graded object]]
[[!redirects graded objects]]


graded vector bundle

## Definition

A __graded vector bundle__ is simply a [[graded object]] in the category of [[vector bundles]],
or, equivalently, a bundle of graded [[vector spaces]].

## Related concepts

* [[derived smooth manifold]]

* [[dg-manifold]]

* [[NQ-supermanifold]]

* [[L-∞-algebroid]]


graded vector space

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Differential-graded objects
+-- {: .hide}
[[!include differential graded objects - contents]]
=--
#### Homological algebra
+-- {: .hide}
[[!include homological algebra - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Definition

Given a [[set]] $G$, an __$G$-graded vector space__ is a map $V$
assigning to each element $g \in G$ a [[vector space]] $V_g$.  Given $G$-graded vector spaces $V$ and $W$, a morphism $f\colon V \to W$ assigns to each element $g \in G$ a linear operator $f_g\colon V_g \to W_g$.  That is, the [[category]] of $G$-graded vector spaces is the [[functor category]] $Vect^G$.

We can just as easily talk about a __$G$-graded [[module]]__ or a __$G$-[[graded object]]__ in any [[category]].  However, a [[graded algebra]] has additional requirements (using a [[monoid]] structure on $G$, as below).

In other words, a $G$-graded vector space is a functor $V\colon G \to Vect$, where the set $G$ is treated as a [[discrete category]], and [[Vect]] is the category of vector spaces. Similarly, a morphism of $G$-graded vector spaces is a natural transformation between such functors.  In short, the category of $G$-graded vector spaces is the [[functor category]] $Vect^G$.

People are usually interested in $G$-graded vector spaces when the set $G$ is equipped with extra structure.  If the set $G$ is a [[monoid]], $Vect^G$ is a [[monoidal category]], with the [[tensor product]] given by
$$
  (V \otimes W)_g = \bigoplus_{h \cdot k = g} V_h \otimes W_k.
$$
The unit is given by having the underlying field $k$ as $V_e$ and the trivial vector space for other gradings. If $G$ is a commutative monoid, then $Vect^G$ is a [[symmetric monoidal category]].

If $G$ is a group, every finite-dimensional $G$-graded vector space has a [[left dual object|left dual]] and a [[right dual object|right dual]].  And if $G$ is an [[abelian group]], these duals coincide.

By far the most widely-used examples are $G = \mathbb{Z}$ and $G = \mathbb{N}$.  Indeed, the term _graded vector space_ is often used to mean a $G$-graded vector space with one of these choices of $G$.  The case $G = \mathbb{Z}/2$ is also important: a $\mathbb{Z}/2$-graded vector space is also called a [[supervector space]]. However, in this case one often uses a different braiding on $Vect^G$, one which uses the [[ring]] structure of $\mathbb{N}$; see [Wikipedia](https://secure.wikimedia.org/wikipedia/en/wiki/Super_vector_space#The_category_of_super_vector_spaces).

### Explicit definition

Let $G$ be a [[set]] with [[decidable equality]]. Then, given a [[commutative ring]] $R$ (usually a [[field]] $F$), a **$G$-graded $R$-module** is an $R$-[[module]] $V$ (usually a [[vector space]] $V$) with a binary function $\langle - \rangle_{(-)}: V \times G \to V$ called the **grade projection operator** such that

* for all $v:V$, $v = \sum_{g:G} \langle v \rangle_g$

* for all $a:R$, $b:R$, $v:V$, $w:V$, and $g:G$, $\langle a v + b w \rangle_g = a \langle v \rangle_g + b \langle w \rangle_g$

* for all $v:V$ and $g:G$, $\langle \langle v \rangle_g \rangle_g = \langle v \rangle_g$

* for all $v:V$, $g:G$, and $h:G$, $(g \neq h) \Rightarrow (\langle \langle v \rangle_g \rangle_h = 0)$

For an element $g:G$, the [[image]] of $\langle - \rangle_g$ under $V$ is denoted as $\langle V \rangle_g$.

$$\langle V \rangle_g \coloneqq \mathrm{im}(\langle - \rangle_g)$$

## Remarks

For the case that $G$ is a group,
this means that the category of $G$-graded vector spaces is a [[vertical categorification|categorification]] of the [[category algebra|group algebra]] of $G$, where numbers are replaced by vector spaces. Recalling from the remark in [[category algebra]] that the group algebra of a group can be identified with the monoid of spans of the form
$$
  \array{
     && G
     \\
     & \swarrow && \searrow
     \\
     pt &&\Rightarrow&& pt
     \\
     & \searrow && \swarrow
     \\
     &&
     Vect
  }
\,,
$$
where $pt \to Vect$ goes to the [[ground field]] $k$,
the monoidal category of $G$-graded vector spaces can be identified with the monoid of spans of the form
$$
  \array{
     && G
     \\
     & \swarrow && \searrow
     \\
     pt &&\Rightarrow&& pt
     \\
     & \searrow && \swarrow
     \\
     &&
     2Vect
  }
  \,.
$$
Here $2Vect$ denotes some version of the category of 2-vector spaces with the property that the category $Vect$ is one of its objects and such that $End_{2Vect}(Vect) \simeq Vect$ (in analogy to how $End_{Vect}(k) \simeq k$) and $pt \to 2Vect$ maps to $Vect$. Possible choices for $2Vect$ is the 2-category of [[Kapranov-Voevodsky 2-vector space]]s or the bigger bicategory [[Bimod]] of algebras and [[bimodule]]s.

More details on this perspective on graded vector spaces are in [[schreiber:Nonabelian cocycles and their quantum symmetries]].

In general, adding algebraic structure onto $G$ will add categorical structure onto $\text{Vect}^G$. A table keeping track of these structures is below:

|  Structure on $G$   | Structure on $\text{Vect}^G$ |
|:--------------------|:-----------------------------|
|         Set         |           Category           |
|        Monoid       |      Monoidal category       |
|     Finite group    |        Fusion category       |
| Finite abelian group|     Braided fusion category  |

## Special case of $\mathbb{Z}$-graded vector spaces
 {#SpecialCaseOfZGradedVectorSpaces}

The case  $G = \mathbb{Z}$ serves as a base for many other applications of the same basic idea. It has some of its own 'traditional' terminology and structure that links it to [[differential object|differential objects]], so that a 'differential graded vector space' is a [[chain complex]] of vector spaces. We will use 'gvs' as an abbreviation for this sort of graded vector space and 'dgvs' for the differential form. (Of course, the theory easily adapts to handle graded modules over a ring, and with some restriction, to graded groups.) Basing algebras on dgvs gives differential graded algebras ([[dg-algebra]]) and so on.

## Concepts

The entry  here will be  a sort of lexicon of some terms which are taken from a source on [[rational homotopy theory]]. This will be more or less 'as-is' from the source (except translating it from  the original French that is!), i.e. without too much editing. This means that there may be conflicts with other entries, which will need resolving later. Some links to other entries have been given but more could be made. There WILL initially be some duplication but that will be eliminated later on.

The lexicon will be spread over a number of entries with links given in the table of contents ion the right hand side at the top.

**Note** With $\mathbb{Z}$-graded vector spaces (and sometimes with other examples as well), some authors work with a direct sum of the various vector spaces instead of using an indexed family.

### (Pre-)graded vector spaces

 A _pre-$\mathbb{Z}$-graded vector space_ (pre-gvs) is a direct sum $V = \bigoplus_{p\in \mathbb{Z}} V_p$.  The elements of $V_p$ are said to be _homogeneous of degree $p$_.  If $x \in V_p$, write $|x| = p$.

Sometimes it may be convenient to write $\bar{x} = (-1)^{|x|}x$ and $V_+ = \bigoplus_{p\gt 0}V_p$.  Another very useful piece of notation is $V^p = V_{-p}$.  In this case we will refer to an 'upper grading' with, in contrast, the other notation being a 'lower grading'.  These are merely for convenience and have little or no mathematical significance.

For the purposes of this lexicon:

A _graded vector space_ (gvs) is a positively or negatively graded pre-gvs that is either $V = \sum_{p\geq 0}V_p$ or $V = \sum_{p\leq 0}V^p$. (This effectively restricts from a $\mathbb{Z}$-grading to one over $\mathbb{N}$

We consider the field $k$ to be a pre-gvs with $(k)_0 = k$, and $(k)_p =0$ if $p\neq 0$.  We say $V$ is of _finite type_ if $dim(V_p) \lt \infty$ for all $p$.

A linear map $f :V\to W$ between pre-gvs is _of degree $p$_ if $f(V_q) \subseteq W_{p+q}$ for all $q$. (Note this may also occur as $f(V^q) \subseteq W^{q-p}$.)

A _morphism_ $f : V\to W$ is a linear map of degree zero. Pregraded vector spaces and the morphisms between them define the category ${pre GVS}$.

The set of all linear maps of degree $p$ from $V$ to $W$ will be denoted $Hom_p(V,W)$ and we set

$$Hom(V,W) = \bigoplus_p Hom_p(V,W).$$

Of course, we now have two notations for the same object, ${pre GVS}(V,W) = Hom_0(V,W)$.

### Suspension

If $r\in \mathbb{Z}$, the $r$-suspension of $V$ is given by $(s^r V)_n = V_{n-r}.$

We will need $s$, the 1-suspension, and $s^{-1}$ in particular.  Of course, $(s^{-1}V)_n = V_{n+1}$.  It is also useful to note $(s^r V)^p = V^{p+r}$. Again, of course, $s^r: V\to s^r V$ is an isomorphism of degree $-r$ having $s^{-r}$ as its inverse.

(This is the basic example of the **suspension functor** discussed in [[triangulated category]].)

### Duals

The dual of a (pre-)gvs $V$ is $\#V$ defined by

$$
  \begin{aligned}
     (\#V)_p &: = Hom_p(V,k)\\
             &\cong{Vect}(V_{-p},k)\\
	     &\cong\#(V_{-p})\\
	     & =\#(V^p).
\end{aligned}
$$

If $f : V\to W$ is of degree $|f|$, then its _transpose_

$$^t f : \#W \to \#V$$

is given by

$$(^t f)(\psi)(x) = (-1)^{|f||\psi|}\psi f(x),$$

for $\psi \in \#W$ and $x\in V$.  Thus if
$V\stackrel{f}{\to}W\stackrel{g}{\to}X$, then

$^t (g\circ f)   = (-1)^{|f||g|}(^t f\circ {}^t g).$}
In particular, for $f$ an isomorphism
$$( ^t f)^{-1} = (-1)^{|f|} ^t (f^{-1}).$$

### Duality

Let $V$ be a gvs, by convention in the duality

$$\langle \quad ; \quad \rangle: V \leftrightarrow \#V,$$

we will usually assume $V$ is non-negatively graded (so $V = \bigoplus_{p\geq 0}V_p$), whilst the right hand side is non-positively graded.

If $V$ is of finite type then $\#\#V \cong V$, of course. The suspension of the dual $s(\#V)$ can be identified with $\#(s^{-1}V)$ and similarly
$s^{-1}(\#V) = \#s(V)$.  These identifications are via the rules:

$$\begin{aligned}
\langle s^{-1}z;su\rangle &= (-1)^{|z|} \langle z;u \rangle,\\
\langle s z;s^{-1}u  \rangle &= (-1)^{|z|+1}\langle z;u \rangle.\end{aligned}$$

This sign convention is needed to ensure that $s s^{-1} = id$.

### Tensor products

The _[[tensor product]]_ of two pre-gvs, $V$ and $W$, is $V\otimes W$, where

$$(V\otimes W)_n = \bigoplus_{p + q = n}V_p\otimes W_q.$$

On morphisms

$$(f\otimes g) (v\otimes w) = (-1)^{|g||f|}(f(v) \otimes g(w))$$

and is of degree $|f| + |g|$.

In particular there is a natural injection $(\#V)\otimes (\#W) \to \#(V\otimes W)$, and this is an isomorphism if either $V$ or $W$ is of finite type.

This tensor product makes the category of graded vector spaces into a [[monoidal category]]. This becomes a [[symmetric monoidal category]] with [[braiding]] the evident braiding inherited from the underlying vector spaces.

Notice that given suitable [[cocycles]] on $G$, then $G$-graded vector spaces may be equiped with a non-trivial braiding. For instance on the monoidal category of $\mathbb{Z}/2$-graded vector spaces there is one non-trivial symmetric braiding, up to equivalence. Imposing this yields [[super vector spaces]].

## Related concepts

* [[graded set]], [[graded object]]

* [[super vector space]]

* [[chain complex]]

* [[differential graded algebra]]

## References

* {#AragonEtAl97} G. Aragón, J.L. Aragón, M.A. Rodríguez (1997), Clifford Algebras and Geometric Algebra, _Advances in Applied Clifford Algebras_ Vol. 7 No. 2, pg 91–102, doi:10.1007/BF03041220, S2CID:120860757

[[!redirects graded vector space]]
[[!redirects graded vector spaces]]
[[!redirects graded module]]
[[!redirects graded modules]]


gradient

+-- {: .rightHandSide}
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### Context
#### Riemannian geometry
+-- {: .hide}
[[!include Riemannian geometry - contents]]
=--
#### Differential geometry
+-- {: .hide}
[[!include synthetic differential geometry - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Definition

Let $(X,g)$ be a [[Riemannian manifold]] and $f \in C^\infty(X)$ a [[function]].

The **gradient** of $f$ is the [[vector field]]

$$
  \nabla f := g^{-1} d_{dR} f \in \Gamma(T X)
  \,,
$$

where $d_{dR} : C^\infty(X) \to \Omega^1(X)$ is the [[de Rham differential]].

This is the unique vector field $\nabla f$ such that

$$
  d_{dR} f = g(-,\nabla f)
$$

or equivalently, if the manifold is [[oriented]],  this is the unique vector field such that

$$
  d_{dR} f = \star_g \iota_{\nabla f} vol_g
  \,,
$$

where $vol_g$ is the [[volume form]] and $\star_g$ is the [[Hodge star operator]] induced by $g$.  (The result is independent of [[orientation]], which can be made explicit by interpreting both $vol$ and $\star$ as valued in [[pseudoforms]].)

Alternatively, the gradient of a scalar field $A$ in some point $x\in M$ is calculated (or alternatively defined) by the integral formula

$$
grad A = lim_{vol D\to 0} \frac{1}{vol D} \oint_{\partial D} \vec{n} A d S
$$

where $D$ runs over the domains with smooth boundary $\partial D$ containing point $x$ and $\vec{n}$ is the unit vector of outer normal to the surface $S$. The formula does not depend on the shape of boundaries taken in limiting process, so one can typically take a coordinate chart and balls with decreasing radius in this particular coordinate chart.

## Example

If $(M,g)$ is the [[Cartesian space]] $\mathbb{R}^n$ endowed with the standard Euclidean metric, then

$$
\nabla f= \sum_{i=1}^n\frac{\partial f}{\partial x^i}\partial_i
.$$

This is the classical gradient from [[vector analysis]].

## Remark

In many classical applications of the gradient in [[vector analysis]], the Riemannian structure is actually irrelevant, and the gradient can be replaced with the [[differential form|differential 1-form]].

## Related concepts

* [[nabla]]

* **gradient**

  * [[gradient flow]]

  * [[symplectic gradient]]

* [[slope of a line]]

* [[Hamiltonian flow]]

* [[curl]]

* [[divergence]]

[[!redirects gradient]]
[[!redirects gradients]]
[[!redirects gradient vector field]]
[[!redirects gradient vector fields]]


gradient flow

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

#Contents#
* table of contents
{:toc}

## Idea

For $(X,g)$ a [[Riemannian manifold]] and $f : X \to \mathbb{R}$ a [[smooth function]], let

$$
  \nabla f := g^{-1}(d_{dR} f) \in \Gamma(T X)
$$

be the [[gradient]] [[vector field]] of $X$. The [[flow]] induced by this on $X$ is the **gradient flow** of $f$.

## Examples

* [[Yang-Mills instanton]]s are the gradient flow trajectories of the [[Chern-Simons action functional]].

* [[Ricci flow]] is the gradient flow of the [[action functional]] of [[dilaton gravity]].

  (This is a key part of Perelman's proof of the [[Poincare conjecture]].)

* the learning algorithm of a [[neural network]] is a gradient descent for the loss function ([Thierry-Mieg 18](neural+network#ThierryMieg18))

## References

* Philippe Clément, _Introduction to Gradient Flows in Metric Spaces (II)_ ([pdf](https://igk.math.uni-bielefeld.de/sites/default/files/study_materials/notes2.pdf))

[[!redirects gradient flows]]

[[!redirects gradient descent]]


grading

* [[graded set]]

* [[graded vector space]]

* [[graded algebra]]

[[!redirects gradings]]


Graeme Segal

[Wikimedia Commons image](http://commons.wikimedia.org/wiki/File:Graeme_Segal.jpeg), taken by [[George Bergman]] in 1982

* [website](https://www.maths.ox.ac.uk/people/graeme.segal)

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

## Selected writings

On [[classifying spaces]] and [[spectral sequences]] (and introducing, following [Grothendicek 61](Segal+condition#Grothendieck61), the "[[Segal conditions]]", see also at _[[complete Segal space|complete]] [[Segal space]]_):

* {#Segal} [[Graeme Segal]], _Classifying spaces and spectral sequences_, Inst. Hautes Études Sci. Publ. Math., vol. 34, pp. 105–112 (1968)

On the [[representation rings]] of [[compact Lie groups]]:

* {#Segal68} [[Graeme Segal]], _The representation ring of a compact Lie group_, Publications Mathématiques de l'Institut des Hautes Études Scientifiques, January 1968, Volume 34, Issue 1, pp 113-128 ([numdam:PMIHES_1968__34__113_0](http://www.numdam.org/item/?id=PMIHES_1968__34__113_0))

On the [[group completion theorem]]:

* {#McDuffSegal76} [[Dusa McDuff]], [[Graeme Segal]], _Homology fibrations and the “group-completion” theorem_, Invent. Math. __31__:3 (1976) 279--284 ([doi:10.1007/BF01403148](https://doi.org/10.1007/BF01403148))

On [[equivariant K-theory]]:

* {#Segal68} [[Graeme Segal]], _Equivariant K-theory_, Inst. Hautes Etudes Sci. Publ. Math.  No. 34 (1968) p. 129-151

On the [[Atiyah-Segal completion theorem]]:

* [[Michael Atiyah]], [[Graeme Segal]], _Equivariant $K$-theory and completion_,  J. Differential Geom. Volume 3, Number 1-2 (1969), 1-18. ([Euclid](http://projecteuclid.org/euclid.jdg/1214428815))

On [[equivariant stable homotopy theory]] and the isomorphism between the [[Burnside ring]] and the [[equivariant stable Cohomotopy]] of the point:

* {#Segal71} [[Graeme Segal]], _Equivariant stable homotopy theory_, Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 2, pp. 59--63. Gauthier-Villars, Paris, 1971 ([[SegalEquivariantStableHomotopyTheory.pdf:file]])

On [[configuration spaces of points]] from [[iterated loop spaces]]:

* {#Segal73} [[Graeme Segal]], _Configuration-spaces and iterated loop-spaces_, Invent. Math. __21__ (1973), 213–221. MR 0331377 ([pdf](http://dodo.pdmi.ras.ru/~topology/books/segal.pdf))

On the [[Kahn-Priddy theorem]] (and a pre-cursor of [[Snaith's theorem]]):

* {#Segal73} [[Graeme Segal]], _The stable homotopy of complex of projective space_, The quarterly journal of mathematics (1973) 24 (1): 1-5. ([[Segal72.pdf:file]], [doi:10.1093/qmath/24.1.1]( https://doi.org/10.1093/qmath/24.1.1))

On [[K-theory of permutative categories]], [[Gamma spaces]] as models for [[connective spectra]], and the identification of [[stable Cohomotopy]] with the [[K-theory of a permutative category|K-theory]] of [[finite sets]]:

* {#Segal74} [[Graeme Segal]], _Categories and cohomology theories_, Topology vol 13, pp.  293-312,  1974  (doi:10.1016/0040-9383(74)90022-6, [pdf](http://ncatlab.org/nlab/files/SegalCategoriesAndCohomologyTheories.pdf))

Proving the [[equivariant Whitehead theorem]]:

* {#JamesSegal78}  [[Ioan James]], [[Graeme Segal]], Theorem 1.1 in: _On equivariant homotopy type_, Topology __17__:3, 1978, 267--272 (doi:10.1016/0040-9383(78)90030-7)

On the [[homotopy type of spaces of rational functions]] from the [[Riemann sphere]] to itself (related to the [[moduli space of monopoles]] in $\mathbb{R}^3$ and to the [[configuration space of points]] in $\mathbb{R}^2$):

* [[Graeme Segal]], _The topology of spaces of rational functions_, Acta Math. __143__ (1979), 39-72 ([euclid:1485890033](https://projecteuclid.org/euclid.acta/1485890033))

* [[Ralph L. Cohen]], [[John D. S. Jones]], [[Graeme B. Segal]], *Stability for holomorphic spheres and Morse theory*, in: K. Grove, I. H. Madsen, E. K. Pedersen (eds.) *Geometry and Topology: Aarhus*, Contemporary Mathematics
Volume: 258;  ([arXiv:math/9904185](https://arxiv.org/abs/math/9904185), [ ISBN:978-0-8218-2158-9](https://bookstore.ams.org/conm-258))

On the [[ordinary cohomology]] of the [[moduli space of Yang-Mills monopoles]]:

* [[Graeme Segal]], Alex Selby, *The cohomology of the space of magnetic monopoles*, Commun. Math. Phys. 177, 775–787 (1996) ([doi:10.1007/BF02099547](https://doi.org/10.1007/BF02099547))

On [[equivariant bundles]] with [[abelian group|abelian]] [[structure group]]:

* {#LashofMaySegal83} [[Richard Lashof]], [[Peter May]], [[Graeme Segal]], _Equivariant bundles with abelian structural group_, Contemporary Mathematics __19__, 1983 ([doi:10.1090/conm/019](https://doi.org/10.1090/conm/019), [pdf](http://www.math.uchicago.edu/~may/PAPERS/45.pdf), [[LashofMaySegalEquivariantBundles.pdf:file]])

On [[loop groups]] of [[compact Lie groups]] and their [[Kac-Moody group]] [[central extension]]:

* {#PressleySegal} Andrew Pressley, [[Graeme Segal]], _Loop groups_ Oxford University Press (1988)

* [[Graeme Segal]], _Loop groups_ ([pdf](http://people.mpim-bonn.mpg.de/zagier/files/doi/10.1007/BFb0084581/chapter08.pdf))

On integrable functions in terms of infinitedimensional [[Sato-Segal-Wilson Grassmannian]]

* G. Segal, [[George Wilson]], _Loop groups and equations of KdV type_, Publ. Math. IHÉS __61__, 5--65 (1985) [numdam](http://www.numdam.org/item/PMIHES_1985__61__5_0) [MR87b:58039](https://mathscinet.ams.org/mathscinet-getitem?mr=87b:58039) [Zbl0592.35112](https://zbmath.org/?q=an:0592.35112)

On [[2d conformal field theory]] and [[modular functors]]:

* [[Graeme Segal]], *Two-dimensional conformal field theories and modular functors*, in: *IXth International Congress on Mathematical Physics*, Swansea 1988, Adam Hilger, Bristol and New York (1989) 22–37 [[pdf](https://dmitripavlov.org/scans/segal-two-dimensional-conformal-field-theories-and-modular-functors.pdf), [[Segal-2dCFTandModularFunctors.pdf:file]]]

Further on the [[FQFT|functorial]]-definition of [[2d conformal field theory]]:

* [[Graeme Segal]], *The definition of conformal field theory*, in: K. Bleuler, M. Werner (eds.), *Differential geometrical methods in theoretical physics* (Proceedings of Research Workshop, Como 1987), NATO Adv. Sci. Inst., Ser. C: Math. Phys. Sci. **250** Kluwer Acad. Publ., Dordrecht (1988) 165-171 $[$[doi:10.1007/978-94-015-7809-7](https://doi.org/10.1007/978-94-015-7809-7)$]$

* [[Graeme Segal]], *The definition of conformal field theory*, in: [[Ulrike Tillmann]] (ed.), *Topology, geometry and quantum field theory* , London Math. Soc. Lect. Note Ser. **308**, Cambridge University Press (2004) 421-577 $[$[doi:10.1017/CBO9780511526398.019](https://doi.org/10.1017/CBO9780511526398.019), [pdf](https://people.maths.ox.ac.uk/segalg/0521540496txt.pdf), [[SegalDefinitionCFT.pdf:file]]$]$

On [[twisted K-theory]] and [[twisted equivariant K-theory]]:

* {#AtiyahSegal04} [[Michael Atiyah]], [[Graeme Segal]], _Twisted K-theory_, Ukrainian Math. Bull. **1**, 3 (2004) ([arXiv:math/0407054](http://arxiv.org/abs/math/0407054), [journal page](http://iamm.su/en/journals/j879/?VID=10), [published pdf](http://iamm.su/upload/iblock/45e/t1-n3-287-330.pdf))

* {#AtiyahSegal05} [[Michael Atiyah]], [[Graeme Segal]], _Twisted K-theory and cohomology_ ([arXiv:math/0510674](http://arxiv.org/abs/math/0510674))

On [[quantization]] of the [[electromagnetic field]] in view of [[Dirac charge quantization]] and [[higher U(1)-gauge theory]]:

* [[Daniel S. Freed]], [[Gregory W. Moore]], [[Graeme Segal]], p. 7 of: *The Uncertainty of Fluxes*, Commun. Math. Phys. 271:247-274, 2007 ([arXiv:hep-th/0605198](https://arxiv.org/abs/hep-th/0605198), [doi:10.1007/s00220-006-0181-3](https://doi.org/10.1007/s00220-006-0181-3))

* [[Daniel Freed]], [[Gregory Moore]], [[Graeme Segal]], _Heisenberg Groups and Noncommutative Fluxes_, Annals Phys. 322:236-285 (2007) ([arXiv:hep-th/0605200](http://arxiv.org/abs/hep-th/0605200))

On [[Wick rotation]] in [[quantum field theory]] in terms of complex metrics:

* [[Maxim Kontsevich]], [[Graeme Segal]], *Wick rotation and the positivity of energy in quantum field theory*, The Quarterly Journal of Mathematics **72** 1-2, (2021) 673–699 [[arXiv:2105.10161](https://arxiv.org/abs/2105.10161), [doi:10.1093/qmath/haab027](https://doi.org/10.1093/qmath/haab027)]

review talks:

* [[Graeme Segal]], _Wick rotation and the positivity of energy in quantum field theory_, talk at Institut des Hautes Études Scientifiques (IHÉS), June 2014 ([video recording](https://www.youtube.com/watch?feature=player_embedded&v=vTvXHL6ZJik))

* [[Graeme Segal]], *Wick Rotation and the Positivity of Energy in Quantum Field Theory*, talk at IAS Physics Group Meeting, December 2021 ([video recording](https://www.ias.edu/video/wick-rotation-and-positivity-energy-quantum-field-theory))

[[!redirects G. Segal]]

[[!redirects Graeme B. Segal]]

[[!redirects Segal]]

category: people


Graham Ellis

Graham Ellis works at the National University of Ireland Galway. His recent work is on [[computational homological algebra]], homology and related areas of [[topological data analysis]].  He is also expert in non-abelian homotopy theoretic topics such as [[crossed modules]] and [[Lie crossed modules]]

* [Home Page](http://hamilton.nuigalway.ie/)

## Selected writings

Comoutational  classification of [[strict 2-groups|strict]] [[2-groups]] ([[crossed modules]]):

* [[Graham Ellis]], Luyen Van Le, *Homotopy 2-types of Low order*, Experimental Mathematics **23** 4 (2014)  [[doi:10.1080/10586458.2014.912059](https://doi.org/10.1080/10586458.2014.912059), [pdf](http://hamilton.nuigalway.ie/preprints/2t.pdf)]

On [[computational topology]]:

* [[Graham Ellis]], *An Invitation to Computational Homotopy*, Oxford University Press (2019) [[ISBN:9780198832973](https://global.oup.com/academic/product/an-invitation-to-computational-homotopy-9780198832973?cc=de&lang=en&), [MAA review](https://www.maa.org/press/maa-reviews/an-invitation-to-computational-homotopy)]

category:people

[[!redirects G.J. Ellis]]
[[!redirects G. J. Ellis]]
[[!redirects G. Ellis]]
[[!redirects G.J.Ellis]]


Gram-Schmidt process

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

#Contents#
* table of contents
{:toc}

## Idea

The Gram--Schmidt process is an [[algorithm]] which takes as input an ordered [[basis]] of an [[inner product space]] and produces as output an ordered [[orthonormal basis]].

In terms of [[matrix|matrices]], the Gram--Schmidt process is a procedure of factorization of a invertible matrix $M$ in the [[general linear group]] $GL_n(\mathbb{R})$ (or $GL_n(\mathbb{C})$) as a [[matrix product|product]] $M = U T$ where $T$ is an upper triangular matrix and $U$ is an orthonormal (or unitary) matrix; as such it is a special case of the more general [[Iwasawa decomposition]] for a (connected) semisimple [[Lie group]].

Since the factorization depends smoothly on the parameters, the Gram--Schmidt procedure enables the reduction of the structure group of an inner product [[vector bundle]] (e.g., the [[tangent bundle]] of a [[Riemannian manifold]] or a [[Kähler manifold]]) from $GL_n$ to the [[orthogonal group]] $O_n$ (or the [[unitary group]] $U_n$).

## Gram--Schmidt process on Hilbert spaces
  {#GramSchmidtOnHilbertSpaces}

In this section, "basis" is understood to signify an ordered independent set whose [[linear span]] is [[dense subspace|dense]] in a [[Hilbert space]] $H$ seen as a [[metric space]]. We will describe the Gram--Schmidt process as applied to a $d$-dimensional Hilbert space for some [[cardinal]] $d$ with a basis $v_0, v_1, \ldots$ consisting of $d$ vectors.

The orthonormal basis $u_0, u_1, \ldots$ produced as output is defined recursively by a) subtracting the orthogonal projection to the closed subspace generated by all previous vectors and b) normalizing. We denote the orthogonal projection onto a closed subspace $A$ by $\pi_A\colon H\to A$ and the normalization $v/\|v\|$ of a vector $v \in H$ by $N(v)$. For [[ordinal]]s $\alpha \lt d$ define

$$u_\alpha \coloneqq N\left(v_\alpha - \pi_{S_\alpha}(v_\alpha)\right)$$

where $S_\alpha$ is the closure of the span of $\{v_\beta: \beta \lt \alpha\}$, noting that the projection $\pi_{S_\alpha}$ is known to exist, since $H$ is complete. This can be rewritten more explicitly using transfinite recursion as

$$u_\alpha =  N\left(v_\alpha - \sum_{\beta \lt \alpha} \langle v_\alpha, u_\beta\rangle u_\beta\right)$$

where the sum on the right is well defined by the Bessel inequality, i.e. only countably many coefficients are non-zero and they are square-summable. A simple (transfinite) inductive argument shows that the $u_\alpha$ are unit vectors orthogonal to each other, and that the span of $\left\{u_\beta \colon \beta \lt \alpha\right\}$ is equal to the span of $\left\{v_\beta \colon \beta \lt \alpha\right\}$ for $\alpha \leq d$. Therefore $u_0, u_1, \ldots$ is an orthonormal basis of $H$.

+-- {: .num_remark}
###### Remark
**(Application to non-bases)**

If we apply the Gram--Schmidt process to a well-ordered independent set whose closed linear span $S$ is *not* all of $H$, we still get an orthonormal basis of the subspace $S$.  If we apply the Gram--Schmidt process to a dependent set, then we will eventually run into a vector $v$ whose norm is zero, so we will not be able to take $N(v)$.  In that case, however, we can simply remove $v$ from the set and continue; then we will still get an orthonormal basis of the closed linear span.  (This conclusion is not generally valid in [[constructive mathematics]], since it relies on [[excluded middle]] applied to the statement that $\|v\| \neq 0$.  However, it does work to [[discrete fields]], such as the algebraic closure of the rationals, as seen in elementary undergraduate linear algebra.)

=--

## Via Gaussian elimination
 {#ViaGaussianElimination}

There is an alternative algorithm via [[Gaussian elimination]] which for a set of vectors produces an orthogonal basis for its linear span ([Pursell-Trimble 91](#PursellTrimble91)).

In the special case that the original vectors are linearly independent and after normalizing the resulting orthogonal basis to an orthonormal basis, the output of this algorithm is an orthonormal basis as produced also by the Gram-Schmidt process.

+-- {: .num_lemma #LUDecompositionOfPositiveSemidefiniteMatrices}
###### Lemma
**(LU-decomposition of [[positive semidefinite matrices]])**

Every [[positive semidefinite matrix]] $M$ has a [[matrix product]]-decomposition

$$
  M \;=\; L \cdot U
$$

where

1. $L$ is

   1. a [[lower triangular matrix]]

   1. an [[invertible matrix]]

1. $U$ is an [[upper triangular matrix]].

=--

([Pursell-Trimble 91, p. 4](#PursellTrimble91))

+-- {: .num_prop #OrthogonalBasisOfLinearSpanByLUDecomposition}
###### Proposition
**(orthogonal basis of linear span via LU-decomposition)**

Let $(a_j)$ be a [[tuple]] of [[vectors]] of the same length, hence let

$$
  A = (A_{i j}) \coloneqq ((a_j)_i)
$$

be the [[matrix]] whose $j$th column is $a_j$.

Since the [[matrix product]] $M \coloneqq A^T \cdot A$ of $A$ with its [[transpose matrix]] is necessarily a [[positive semidefinite symmetric matrix]] it has an LU-decomposition according to Lemma \ref{LUDecompositionOfPositiveSemidefiniteMatrices}:

$$
  A^T \cdot A \;=\; L \cdot U
  \,.
$$

Then the column vectors $(\hat q_j)$ of the matrix

$$
  \widehat Q \;\coloneqq\; A \cdot (L^{-1})^T
$$

constitute an orthogonalization of the original tuple of vectors $(a_i)$ in that the non-zero $\hat q_j$ form an orthogonal [[linear basis]] of the [[linear span]] of the $(a_j)$.

=--

([Pursell-Trimble 91, top of p. 5](#PursellTrimble91))

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

That the matrix $L$ in Prop. \ref{OrthogonalBasisOfLinearSpanByLUDecomposition} is [[lower triangular matrix|lower triangular]] and [[invertible matrix|invertible]] (by Lemma \ref{LUDecompositionOfPositiveSemidefiniteMatrices}) means that its [[inverse matrix]] $L^{-1}$ is also a [[lower triangular matrix]] which reflects a process of [[row reduction]] from $A^T \cdot A$ to $U$.

Accordingly, the orthogonalization in Prop. \ref{OrthogonalBasisOfLinearSpanByLUDecomposition} may be understood as applying [[Gauss elimination]] to

$$
  \big[A^T \cdot A \vert A^T \big]
  \;\mapsto\;
  \big[ \widehat Q^T\big]
$$

=--

([Pursell-Trimble 91, p. 3](#PursellTrimble91))

## Examples

### Legendre polynomials

A classic illustration of Gram--Schmidt is the production of the [[Legendre polynomials]].

Let $H$ be the [[Hilbert space]] $H = L^2([-1, 1])$ of [[square integrable functions]] in the [[closed interval]], equipped with the standard [[inner product]] defined by

$$\langle f, g\rangle = \int_{-1}^1 \widebar{f(x)} g(x) d x$$

By the [[Stone-Weierstrass theorem]], the space of [[polynomials]] $\mathbb{C}[x]$ is dense in $H$ according to its standard inclusion, and so the polynomials $1, x, x^2, \ldots$ form an ordered basis of $H$.

Applying the Gram--Schmidt process as [above](#GramSchmidtOnHilbertSpaces), one readily computes the first few orthonormal functions:

$$u_1(x) = N(1) = 1/2$$

$$u_2(x) = N(x - 0) = \sqrt{3/2} x$$

$$u_3(x) = N(x^2 - \langle x^2, 1/2\rangle 1/2 - 0) = N(x^2 - 1/3) = 3\sqrt{5/2}/2(x^2 - 1/3)$$

The classical Legendre polynomials $P_n(x)$ are scalar multiplies of the functions $u_n$, adjusted so that $P_n(1) = 1$; they satisfy the orthogonality relations

$$\langle P_n, P_m\rangle = \frac{2}{2n + 1}\delta_{m, n}$$

where $\delta_{m, n}$ is the [[Kronecker delta]].

## Categorified Gram--Schmidt process
 {#CategorifiedGramSchmidtProcess}

Many aspects of the Gram--Schmidt process can be [[categorification|categorified]] so as to apply to [[2-Hilbert spaces]] as indicated at _[[Schur's lemma]]_ in the section _[In terms of categorical algebra](Schur's+lemma#InterpretationInCategoricalAlgebra)_;

We will illustrate the basic idea with an example that was suggested to us by [[James Dolan]].  (See also at _[[permutation representation]]_ the sections _[Examples -- Virtual permutation representations](permutation+representation#ExamplesVirtualPermutationRepresentations)_.)

Consider the [[category of representations]] $S_n Rep$ over the [[complex numbers]] of the [[symmetric group]] $G \coloneqq S_n$. (As a running example, we consider $S_4$; up to [[isomorphism]], there are five [[irreducible representations]]

$$U_{(4)}, \, U_{(3 1)}, \, U_{(2 2)}, \, U_{(2 1 1)}, \, U_{(1 1 1 1)}$$

classified by the five [[Young diagrams]] of size 4. To save space, we denote these as $U_1$, $U_2$, $U_3$, $U_4$, $U_5$.)

The [[irreducible representations]] $U_i$ of $S_n$ form a _$2$-orthonormal basis_ in the sense that any two of them $U_i, U_j$ satisfy the relation

$$hom_G(U_i, U_j) \cong \delta_{i j} \cdot \mathbb{C}$$

(where $hom_G$ denotes the [[hom-object|hom vector space]] in $S_n Rep$, and $n \cdot \mathbb{C}$ indicates a [[direct sum]] of $n$ copies of $\mathbb{C}$).
In fact, the irreducible representations are uniquely determined up to isomorphism by these relations.

There is however another way of associating representations to [[partitions]] or [[Young diagrams]]. Namely, consider the [[subgroup]] of permutations which take each row of a Young diagram or Young tableau of size $n$ to itself; this forms a [[parabolic subgroup]] of $S_n$, [[conjugation|conjugate]] to one of type $P_{(n_1 \ldots n_k)} = S_{n_1} \times \ldots \times S_{n_k}$ where $n_i$ is the length of the $i^{th}$ row of the Young diagram. The group $S_n$ [[transitive action|acts transitively]] on the [[orbit space]] of [[cosets]]

$$S_n/P_{(n_1 \ldots n_k)}$$

and these actions give linear [[permutation representations]] $\mathbb{C}\big[S_n/P_{(n_1 \ldots n_k)}\big]$ of $S_n$. Equivalently, these are [[linear representations]] $V_i$ which are [[induced representation|induced]] from the [[trivial representation]] along inclusions of parabolic subgroups.

We claim that

1. these representations form a $\mathbb{Z}$-[[linear basis]] of the [[representation ring]] $R(S_n)$,

1. we may calculate their characters using a categorified Gram--Schmidt process.

We indicate the proof:

Given two such [[parabolic subgroups]] $P$, $Q$ in $G = S_n$, the _$2$-inner product_

$$hom_G(\mathbb{C}[G/P], \mathbb{C}[G/Q])$$

may be identified with the [[linear span|free vector space]] on the set of [[double cosets]] $P\backslash G/Q$.

One may count the number of double cosets by hand in a simple case like $G = S_4$. That is, for the 5 representations $V_1, \ldots, V_5$ induced from the 5 parabolic subgroups $P_i$ corresponding to the 5 Young diagrams listed above, the dimensions of the 2-inner products $hom(V_i, V_j)$ are the sizes of the corresponding double coset spaces $P_i\backslash S_4 /P_j$. These numbers form a [[matrix]] as follows (following the order of the $5$ [[partitions]] listed above):

$$\left( \array
{1 & 1 & 1 & 1 & 1 \\
1 & 2 & 2 & 3 & 4 \\
1 & 2 & 3 & 4 & 6 \\
1 & 3 & 4 & 7 & 12 \\
1 & 4 & 6 & 12 & 24
}\right)
$$

To reiterate: this matrix is the [[decategorification]] (a matrix of [[dimensions]]) of a matrix of $2$-inner products where the $(i j)$-entry is of the form

$$hom_G(V_i, V_j) \cong V_i^* \otimes_G V_j$$

where the $V_i$ are [[induced representation|induced]] from inclusions of [[parabolic subgroups]]. The $V_i$ are $\mathbb{N}$-linear combinations of irreducible representations $U_i$ which form a $2$-orthonormal basis, and we may perform a series of elementary row operations which convert this matrix into an [[upper triangular matrix]], and which will turn out to be the [[decategorification|decategorified]] form of the 2-matrix with entries

$$hom_G(U_i, V_j) \cong U_i^* \otimes_G V_j$$

where $U_i$ is the [[irrep]] corresponding to the $i$th Young diagram (as listed above). The [[upper triangular matrix]] is

$$\left( \array
{1 & 1 & 1 & 1 & 1 \\
0 & 1 & 1 & 2 & 3 \\
0 & 0 & 1 & 1 & 2 \\
0 & 0 & 0 & 1 & 3 \\
0 & 0 & 0 & 0 & 1} \right)
$$

and we read off from the columns the following decompositions into irreducible components:

$$V_1 \cong U_1$$

$$V_2 \cong U_1 + U_2$$

$$V_3 \cong U_1 + U_2 + U_3$$

$$V_4 \cong U_1 + 2 U_2 + U_3 + U_4$$

$$V_5 \cong U_1 + 3 U_2 + 2 U_3 + 3 U_4 + U_5$$

The last representation $V_5$ is the [[regular representation]] of $S_4$ (because the [[parabolic subgroup]] is [[trivial group|trivial]]). Since we know from general theory that the multiplicity of the irreducible $U_i$ in the [[regular representation]] is its [[dimension]], we get as a by-product the dimensions of the $U_i$ from the expression for $V_5$:

$$dim(U_1) = 1, \, dim(U_2) = 3, \, dim(U_3) = 2, \, dim(U_4) = 3, \, dim(U_5) = 1$$

(the first of the $U_i$ is the [[trivial representation]], and the last $U_5$ is the [[alternating representation]]).

The row operations themselves can be assembled as the lower triangular matrix

$$ \left( \array
{1 & 0 & 0 & 0 & 0 \\
-1 & 1 & 0 & 0 & 0 \\
0 & -1 & 1 & 0 & 0 \\
1 & -1 & -1 & 1 & 0 \\
-1 & 2 & 1 & -3 & 1 }
\right)$$

and from the rows we read off the irreducible representations as "virtual" (i.e., $\mathbb{Z}$-linear) combinations of the parabolically induced representations $V_i$:

$$U_1 \cong V_1$$

$$U_2 \cong -V_1 + V_2$$

$$U_3 \cong -V_2 + V_3$$

$$U_4 \cong V_1 - V_2 - V_3 + V_4$$

$$U_5 \cong -V_1 + 2V_2 + V_3 - 3 V_4 + V_5$$

which can be considered the result of the categorified Gram--Schmidt process.

It follows from these representations that the $V_i$ form a $\mathbb{Z}$-[[linear basis]] of the representation ring $Rep(S_4)$.

Analogous statements hold for each symmetric group $S_n$.

## Related entries

* [[Hermite normal form]]

* [[Smith normal form]]

* [[matrix analysis]]

## References

See also

* Wikipedia, _[Gram-Schmidt process](https://en.wikipedia.org/wiki/Gram%E2%80%93Schmidt_process)_

The formulation via Gaussian elimination is due to

* {#PursellTrimble91} Lyle Pursell, S. Y. Trimble, _Gram-Schmidt orthogonalization by Gauss Elimination_, The American Mathematical Monthly
Vol. 98, No. 6 (1991), pp. 544-549 ([jstor:2324877](https://www.jstor.org/stable/2324877))

[[!redirects Gram–Schmidt process]]
[[!redirects Gram--Schmidt process]]
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[[!redirects Gram-Schmidt procedure]]
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[[!redirects Gram-Schmidt factorization]]
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[[!redirects Gram-Schmidt factorisation]]
[[!redirects Gram–Schmidt factorisation]]
[[!redirects Gram--Schmidt factorisation]]

[[!redirects Gram–Schmidt algorithm]]

[[!redirects Gram-Schmidt orthogonalization]]
[[!redirects Gram-Schmidt orthogonalizations]]


grammar

\tableofcontents

## Idea

A __grammar__ is an explicit or implicit set/system of organizational rules governing some system of signs (see *[[semiotics]]*), for example a cultural code, a language, or a [[formal logic|logical system]].
In particular, a grammar should help distinguish well-formed or meaningful expressions. Regarding that the meaning of a sign is part of the system, grammars in general may contain rules concerning or depending on [[semantics]], though sometimes one restricts grammar to form.

It is widely believed that advanced _cognition_ itself includes intermediate representations which is symbolic and uses intermediate signs and their grammar.
Patterns like analogies which contain placeholders to fill can also be viewed as part of grammar.

Vocabulary (the list of signs) itself is usually, from the point of view of a grammar, considered as an open class  (that is, they can in principle be extended) hence it is not viewed as a part of a grammar, except for a special reserved signs (for example auxiliary verbs in a language grammar) which are typically viewed within closed classes (the lists which are considered exhaustive).

## In linguistics

In [[linguistics]], a _descriptive grammar_ is an explicit grammar of some language/idiom. That is not the natural grammar itself but the scientific account of it (for example, Panini's grammar of Sanskrit which is a particular record writen in verse). A _prescriptive grammar_ is an explicit set of rules of advices how to attain to some norm of a language.
Linguists consider prescriptive grammars political or social construct rather than linguistical (scientific) though they may include the research into tradition and norm and some experience with socially popular practice. In ideal, a descriptive grammar should include information on variation, but often does not and is received often as prescriptive. A scientifically sound descriptive ground should never hesitate to include phrases/idioms which look  as lacking some logics, if they are not etymologically sound or not confirming to some general rules (or its original usage in history) if such idioms indeed appear often in practice (for example usage of double negation meaning single negation etc.).

Notable modern theories of grammar in linguistics include generative grammar, transformational grammar, cognitive grammar, functional grammar, generative semantics etc.

## Related entries

* [[formal grammar]]

* [[linguistics]]

* [[syntax]], [[syntax (linguistics)]],

* [[semantics]]

* [[semiotics]], [[tree]]

## Literature

* [[David Mumford]], _Grammar isn't merely part of language_, 2016 [blog](https://www.dam.brown.edu/people/mumford/blog/2016/grammar.html)

* wikipedia [generative grammar](https://en.wikipedia.org/wiki/Generative_grammar), [dependency grammar](https://en.wikipedia.org/wiki/Dependency_grammar), [transformational grammar](https://en.wikipedia.org/wiki/Transformational_grammar), [cognitive grammar](https://en.wikipedia.org/wiki/Cognitive_grammar), [functional discourse grammar](https://en.wikipedia.org/wiki/Functional_discourse_grammar)

Generative semantics in linguistics has the basic thesis that the computation of syntactic structures is driven by the semantic meaning of constituents

* [[George Lakoff]], _Toward a generative semantics_, 1963, 1976 [pdf](https://escholarship.org/content/qt64m2z2b1/qt64m2z2b1.pdf)
* wikipedia: [generative semantics](https://en.wikipedia.org/wiki/Generative_semantics)