May spectral sequence

> under construction

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###Context###
#### Homological algebra
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[[!include homological algebra - contents]]
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#### Stable Homotopy theory
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[[!include stable homotopy theory - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

A _May spectral sequence_ ([May 64](#May64), [May 74](#May74), [Ravenel 86, 3.2](#Ravenel86)) is a certain type of [[spectral sequence]] that computes the second page of the [[classical Adams spectral sequence]]. More generally, it computes [[Ext]]-groups (or [[Cotor]]-groups, via [this lemma](cotensor+product#ComoduleHomInTermsOfCotensorProduct)) of [[comodules]] over [[commutative Hopf algebras]] (the [[dual Steenrod algebra]] in the classical case) starting from the corresponding $Ext$-groups over just an [[associated graded]] Hopf algebra.

The May spectral sequence is the [[spectral sequence of a filtered complex]] induced by suitably filtering the canonical (but intractable) [[bar complex]] model for [[Ext]]/[[Cotor]] (see [Ravenel 86, chapter 3, section 2](#Ravenel86), [Kochman 96, section 5.3](#Kochman96)).

## Preliminaries

### Cobar complex

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

Given $(\Gamma,A)$ a [[graded commutative Hopf algebroid]], then $A$ becomes a left comodule over $\Gamma$ with coaction given by the right unit

$$
  A \overset{\eta_R}{\longrightarrow} \Gamma \simeq \Gamma \otimes_A A
$$

and it becomes a right comodule with coaction given by the left unit

$$
  A \overset{\eta_L}{\longrightarrow} \Gamma \simeq A \otimes_A \Gamma
  \,.
$$

=--

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

Dually this is the action of morphisms on objects given by evaluation at the source or target, respectively.

=--

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

Let $(\Gamma,A)$ be a [[commutative Hopf algebra]], hence a [[commutative Hopf algebroid]] for which the left and right units coincide $\eta \;\colon\; A \longrightarrow \Gamma$.

Then the **unit coideal** of $\Gamma$ is the [[cokernel]]

$$
  \overline{\Gamma}
    \coloneqq
  coker( A \overset{\eta}{\longrightarrow} \Gamma)
  \,.
$$

=--

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

Let $(\Gamma,A)$ be a [[commutative Hopf algebra]], hence a [[commutative Hopf algebroid]] for which the left and right units coincide $\eta \;\colon\;
A \longrightarrow \Gamma$.

Then the unit coideal $\overline{\Gamma}$ (def. \ref{UnitCoidealInCommutativeHopfAlgebra}) carries the structure of an $A$-[[bimodule]] such that the [[projection]] morphism

$$
  \Gamma \longrightarrow \overline{\Gamma}
$$

is an $A$-bimodule homomorphism. Moreover, the coproduct $\Psi \;\colon\; \Gamma \longrightarrow \Gamma \otimes_A \Gamma$ descends to a coproduct $\overline{\Gamma} \;\colon\; \overline{\Gamma} \longrightarrow \overline{\Gamma} \otimes_A \overline{\Gamma}$ such that the projection intertwines the two coproducts.

=--

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

For the first statement, consider the [[commuting diagram]]

$$
  \array{
    A \otimes A
      &\overset{A \otimes \eta}{\longrightarrow}&
    A \otimes \Gamma
      &\longrightarrow&
    A \otimes \overline{\Gamma}
    \\
    \downarrow && \downarrow && \downarrow^{\mathrlap{\exists}}
    \\
    A
      &\underset{\eta}{\longrightarrow}&
    \Gamma
      &\longrightarrow&
    \overline{\Gamma}
  }
  \,,
$$

where the left [[commuting square]] exhibits the fact that $\eta$ is a homomorphism of left $A$-modules.

Since the [[tensor product of abelian groups]] $\otimes$ is a [[right exact functor]] it preserves cokernels, hence $A \otimes \overline{\Gamma}$ is the cokernel of $A \otimes A \to A\otimes \Gamma$ and hence the right vertical morphisms exists by the [[universal property]] of cokernels. This is the compatible left module structure on $\overline{\Gamma}$. Similarly the right $A$-module structure is obtained.

For the second statement, consider the [[commuting diagram]]

$$
  \array{
    A
      &\overset{\eta}{\longrightarrow}&
    \Gamma
      &\longrightarrow&
    \overline{\Gamma}
    \\
    {}^{\mathllap{\eta}}\downarrow
      &&
    \downarrow^{\mathrlap{\Psi}}
      &&
    \downarrow^{\mathrlap{\exists}}
    \\
    \Gamma \simeq \Gamma \otimes_A A
      &\underset{id \otimes_A \eta}{\longrightarrow}&
    \Gamma \otimes_A \Gamma
      &\longrightarrow&
    \overline{\Gamma} \otimes_A \overline{\Gamma}
  }
  \,.
$$

Here the left square commutes by one of the co-unitality conditions on $(\Gamma,A)$, equivalently this is the co-action property of $A$ regarded canonically as a $\Gamma$-comodule.

Since also the bottom morphism factors through zero, the [[universal property]] of the cokernel $\overline{\Gamma}$ implies the existence of the right vertical morphism as shown.

=--

+-- {: .num_defn #CobarComplex}
###### Definition
**(cobar complex)**

Let $(\Gamma,A)$ be a [[commutative Hopf algebra]], hence a [[commutative Hopf algebroid]] for which the left and right units coincide $A \overset{\eta}{\longrightarrow} \Gamma$. Let $N$ be a left $\Gamma$-comodule.

The **cobar complex** $C^\bullet_\Gamma(N)$ is the [[cochain complex]] of abelian groups with terms

$$
  C^s_\Gamma(N)
   \coloneqq
    \underset{s\; factors}{
    \underbrace{
    \overline{\Gamma}
      \otimes_A
      \cdots
      \otimes_A
    \overline{\Gamma}
    }
    }
  \otimes_A N
$$

(for $\overline{\Gamma}$ the unit coideal of def. \ref{UnitCoidealInCommutativeHopfAlgebra}, with its $A$-bimodule structure via lemma \ref{PropertiesOfUnitCoideal})

and with [[differentials]] $d_s \colon C^s_\Gamma(N) \longrightarrow C^{s+1}_\Gamma(N)$ given by the alternating sum of the coproducts via lemma \ref{PropertiesOfUnitCoideal}.

=--

([Ravenel 86, def. A1.2.11](#Ravenel86))

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

Let $(\Gamma,A)$ be a [[commutative Hopf algebra]], hence a [[commutative Hopf algebroid]] for which the left and right units coincide $A \overset{\eta}{\longrightarrow} \Gamma$. Let $N$ be a left $\Gamma$-comodule.

Then the [[cochain cohomology]] of the cobar complex $C^\bullet_\Gamma(N)$ (def. \ref{CobarComplex}) is the [[Ext]]-groups of comodules from $A$ (regarded as a left comodule via def. \ref{ComoduleStructureOnGroundRing}) into $N$

$$
  H^\bullet(C^\bullet_\Gamma(N))
    \;\simeq\;
  Ext^\bullet_\Gamma(A,N)
  \,.
$$

=--

([Ravenel 86, cor. A1.2.12](#Ravenel86), [Kochman 96, prop. 5.2.1](#Kochman96))

### Self-Ext of free graded commutative coalgebras

Throughout, let $A$ be a [[commutative ring]] and let $\Gamma$ be a graded [[commutative Hopf algebroid|commutative Hopf algebra]] over $A$. We write $(\Gamma,A)$ for this data.

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

Let $(\Gamma,A)$ be a graded [[commutative Hopf algebra]] such that

1. the underlying algebra is free graded commutative;

1. $\eta \colon A \to \Gamma$ is a [[flat morphism]];

1. $\Gamma$ is generated by [[primitive elements]] $\{x_i\}_{i\in I}$

then the [[Ext]] of $\Gamma$-comodules from $A$ and itself is the (graded) polynomial algbra on these generators:

$$
  Ext_\Gamma(A,A) \simeq A[\{x_i\}_{i \in I}]
$$

=--

([Ravenel 86, lemma 3.1.9](#Ravenel86), [Kochman 96, prop. 3.7.5](#Kochman96))

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

Consider the co-free left $\Gamma$-comodule ([prop.](commutative+Hopf+algebroid#CoFreeComodules))

$$
  \Gamma \otimes_A A[\{y_i\}_{i \in I}]
$$

and regard it as a [[chain complex]] of left comodules by defining a [[differential]] via

$$
  d \colon x_i \mapsto y_i
$$

$$
  d \colon y_i \mapsto 0
$$

and extending as a graded [[derivation]].

We claim that $d$ is a homomorphism of left comodules: Due to the assumption that all the $x_i$ are primitive we have on generators that

$$
  \begin{aligned}
    (id,d) ( \Psi(x_i)  )
    & =
    (id,d) ( x_i \otimes 1 + 1 \otimes x_i )
    \\
    & =
    \underset{= 0}{x_i  \otimes \underbrace{(d 1)} }
     +
    \underset{= y_i}{ 1 \otimes \underbrace{(d x_i)} }
    \\
    & =
    \Psi( d x_i )
  \end{aligned}
$$

and

$$
  \begin{aligned}
    (id,d)( \Psi(y_i) )
    & =
    (id,d) ( 1, y_i )
    \\
    & =
    (1, d y_i)
    \\
    & =
    0
    \\
    & =
    \Psi( 0 )
    \\
    & =
    \Psi(d y_i)
  \end{aligned}
  \,.
$$

Since $d$ is a graded derivation on a free graded commutative algbra, and $\Psi$ is an algebra homomorphism, this implies the statement for all other elements.

Now observe that the canonical [[chain map]]

$$
  (\Gamma \otimes_A A[\{y_i\}_{i \in I}]  ,\; d)
    \overset{\simeq_{qi}}{\longrightarrow}
  A
$$

(which projects out the generators $x_i$ and $y_i$ and is the identity on $A$), is a [[quasi-isomorphism]], by construction. Therefore it constitutes a co-free resolution of $A$ in left $\Gamma$-comodules.

Since the counit $\eta$ is assumed to be flat, and since $A$ is trivially a [[projective module]] over itself, [this prop.](commutative+Hopf+algebroid#CoFreeHopfComodulesAreHomNAcyclicForProjectiveN) now implies that the above is an [[acyclic resolution]] with respect to the functor $Hom_{\Gamma}(A,-) \colon \Gamma CoMod \longrightarrow A Mod$. Therefore it computes the [[Ext]]-functor. Using that forming co-free comodules is [[right adjoint]] to forgetting $\Gamma$-comodule structure over $A$ ([prop.](commutative+Hopf+algebroid#CoFreeComodules)), this yields:

$$
  \begin{aligned}
    Ext^\bullet_\Gamma(A,A)
    & \simeq
    H_\bullet(Hom_\Gamma(A,  \Gamma \otimes_A A[\{y_i\}_{i \in I}] ), d)
    \\
    & \simeq
    H_\bullet(Hom_A(A,  A[\{y_i\}_{i \in I}] ), d= 0 )
    \\
    & \simeq
    Hom_A(A,  A[\{y_i\}_{i \in I}] )
    \\
    & \simeq
    A[\{y_i\}_{i \in I}]
  \end{aligned}
  \,.
$$

=--

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

If $(\Gamma,A)$ is equipped with a [[filtering]] right/left $\Gamma$-comodules $N_1$ and $N_2$ are compatibly filtered, then there is a [[spectral sequence]]

$$
  \mathcal{E}_1
  =
  Ext_{gr_\bullet \Gamma}(gr_\bullet N_1, gr_\bullet N_2)
  \;\Rightarrow\;
  Ext_{\Gamma}(N_1, N_2)
$$

converging to the [[Ext]] over $\Gamma$ between $N_1$ and $N_2$, whose first page is the Ext over the [[associated graded]] Hopf algebra $gr_\bullet \Gamma$ between the associated graded modules $gr_\bullet N_1$ and $gr_\bullet N_2$.

=--

([Ravenel 86, lemma 3.1.9](#Ravenel86), [Kochman 96, prop. 3.7.5](#Kochman96))

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

The filtering induces a filtering on the standard cobar complex (def. \ref{CobarComplex}) which computes $Cotor(A,A)$. The spectral sequence in question is the corresponding [[spectral sequence of a filtered complex]]. Its first page is the homology of the associated graded complex (by this [prop.](spectral+sequence+of+a+filtered+complex#FirstPages)).

=--

## Construction of the May spectral sequence

Let now $A \coloneqq \mathbb{F}_2$,  $\Gamma \coloneqq \mathcal{A}^\bullet_{\mathbb{F}_2}$ be the mod 2 [[dual Steenrod algebra]]. By [Milnor's theorem](Steenrod+algebra#MilnorTheoremOnDualSteenrodAlgebra), as an $\mathbb{F}_2$-algebra this is

$$
  \mathcal{A}^\bullet_{\mathbb{F}_2}
  =
  \mathbb{F}_2[\xi_1, \xi_2, \cdots]
  \,.
$$

and the coproduct is given by

$$
  \Psi(\xi_n)
  =
  \underoverset{k = 0}{i}{\sum} \xi_{i - k}^{2^k} \otimes \xi_k
  \,,
$$

where we set $\xi_0 \coloneqq 1$.

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

Set

$$
  h_{i,n} \coloneqq \xi_i^{2^n}
  \,.
$$

Also one abbreviates

$$
  h_n \coloneqq h_{1,n} = \xi_1^{2^n}
  \,.
$$

=--

By binary expansion of powers, there is a unique way to express every monomial in $\mathbb{F}_2[\xi_1, \xi_2, \cdots]$ as a product of the elements $h_{i,n}$ from def. \ref{hGeneratorsInClassicalAdamsSpectralSequence}, such that each such element appears at most once in the product. E.g.

$$
  \begin{aligned}
    \xi_i^5 \xi_j^7
      & =
    \xi_i^{2^0 + 2^2} \xi_j^{2^0 + 2^1 + 2^2}
     \\
     & =
     h_{i,0} h_{i,2} h_{j,0} h_{j,1} h_{j,2}
  \end{aligned}
  \,.
$$

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

In terms of the generators $\{h_{i,n}\}$, the coproduct on $\mathcal{A}^\ast_{\mathbb{F}_2}$ takes the following simple form

$$
  \Psi(h_{i,n})
  \;=\;
  \underoverset{k = 0}{i}{\sum} h_{i-k,n+k}\otimes h_{k,n}
  \,.
$$

=--

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

Using that the coproduct of a [[bialgebra]] is a [[homomorphism]] for the algebra structure and using [[freshman's dream]] arithmetic over $\mathbb{F}_2$, one computes:

$$
  \begin{aligned}
    \Psi(h_{i,n})
    & =
    \Psi\left(\xi_i^{2^n}\right)
    \\
    & = (\Psi(\xi_i))^{2n}
    \\
    & =
    \left(\underoverset{k = 0}{i}{\sum} \xi_{i-k}^{2^k} \otimes \xi_k\right)^{2^n}
    \\
    & = \underoverset{k = 0}{i}{\sum} \left(\xi_{i-k}^{2^k}\right)^{2^n} \otimes \xi^{2^n}_k
    \\
    & = \underoverset{k = 0}{i}{\sum} \xi_{i-k}^{2^k \cdot 2^n} \otimes \xi^{2^n}_k
    \\
    & = \underoverset{k = 0}{i}{\sum} \xi_{i-k}^{2^{(k+n)}} \otimes \xi^{2^n}_k
    \\
    & = \underoverset{k = 0}{i}{\sum} h_{i-k,n+k}\otimes h_{k,n}
  \end{aligned}
  \,.
$$

=--

Define a [[graded algebra|grading]] on $\mathcal{A}^\bullet_{\mathbb{F}_2}$ by setting (this is due to ([Ravenel 86, p.69](#Ravenel86)))

$$
  {\vert h_{i,n} \vert}
    \coloneqq
  2i-1
$$

and extending this additively to these unique representative.

For instance

$$
  \begin{aligned}
    \vert \xi_i^5 \xi_j^7\vert
    & =
    {\vert h_{i,0} h_{i,2} h_{j,0} h_{j,1} h_{j,2} \vert}
    \\
    & =
    2(2i-1) + 3(2j-1)
  \end{aligned}
  \,.
$$

Consider the corresponding [[filtered object|filtering]]

$$
  \cdots
   \subset
  F_p \mathcal{A}^\ast_{\mathbb{F}_2}
   \subset
  F_{p+1} \mathcal{A}^\ast_{\mathbb{F}_2}
    \subset
  \cdots
    \subset
  \mathcal{A}^\ast_{\mathbb{F}_2}
$$

with filtering stage $p$ containing all elements of total degree $\leq p$.

Observe that

$$
  \begin{aligned}
    \Psi(\xi_i)
    &
   =
   \underset{deg = 2i-1}{\underbrace{\xi_{i} \otimes 1}}
   +
    \underoverset{0 \lt k \lt i}{}{\sum}
     \underset{deg = 2i-2}{\underbrace{\xi_{i-k}^{p^k} \otimes \xi_k}}
   +
    \underset{deg = 2i-1}{\underbrace{1 \otimes \xi_i}}
  \end{aligned}
  \,.
$$

This means that after projection to the [[associated graded]] Hopf algebra

$$
  F_\bullet \mathcal{A}^\ast_{\mathbb{F}_2}
   \longrightarrow
  gr_\bullet \mathcal{A}^\ast_{\mathbb{F}_2}
$$

all the generators $h_{i,n}$ become [[primitive elements]]:

$$
  \begin{aligned}
    \Psi(h_{i,n})
    &
   =
   h_{i,n}\otimes 1
   +
   1 \otimes h_{i,n}
   \;\;\;\;\;
   \in
   gr_\bullet \mathcal{A}^\ast_{\mathbb{F}_2}
   \otimes
   gr_\bullet \mathcal{A}^\ast_{\mathbb{F}_2}
  \end{aligned}
  \,.
$$

Hence lemma \ref{CoModuleExtFromAToAForPrimitivelyGeneratedExteriorAlgebra} applies and says that thet $Ext$ from $\mathbb{F}_2$ to itself over the [[associated graded]] Hopf algebra is the polynomial algebra in these generators:

$$
  Ext_{gr \mathcal{A}^\ast_{\mathbb{F}_2}}(\mathbb{F}_2,\mathbb{F}_2)
  =
  \mathbb{F}_2[\{h_{i,n}\}_{{i \geq 1,} \atop {n \geq 0}}]
  \,.
$$

Moreover, lemma \ref{SpectralSequenceConvergingToExtForFilteredHopfAlgebra} says that this is the first page of a spectral sequence that converges to the $Ext$ over the original Hopf algebra:

$$
  \mathcal{E}_1
  =
  \mathbb{F}_2[\{h_{i,n}\}_{{i \geq 1} \atop {n \geq 0}}]
  \;\Rightarrow\;
  Ext_{\mathcal{A}^\ast_{\mathbb{F}_2}}(\mathbb{F}_2,\mathbb{F}_2)
  \,.
$$

This is the _May spectral sequence_ for the computation of $Ext_{\mathcal{A}^\ast_{\mathbb{F}_2}}(\mathbb{F}_2,\mathbb{F}_2)$. Notice that since everything is $\mathbb{F}_2$-linear, its [extension problem](spectral+sequence#ExtensionProblem) is trivial.

Moreover, again by lemma \ref{SpectralSequenceConvergingToExtForFilteredHopfAlgebra}, the differentials on any $r$-page are the restriction of the differentials of the bar complex to the $r$-almost cycles ([prop.](Introduction+to+Stable+homotopy+theory+--+I#DifferentialsOnAlmostChains)). The differential of the bar complex is the alternating sum of the coproduct on $\mathcal{A}^\ast_{\mathbb{F}_2}$, hence by prop.  \ref{CoproductOnDualSteenrodInTermsOfAdaptedGenerators} this is:

$$
  d_1 (h_{i,n})
  =
  \underoverset{k = 0}{i}{\sum} h_{i-k,n+k}\otimes h_{k,n}  \,.
$$

## The second page of the classical Adams spectral sequence
 {#TheSecondPageOfTheClassicalAdamsSpectralSequence}

Now we use the above formula to explicitly compute the cohomology of the second page of the [[classical Adams spectral sequence]].

In doing so it is now crucial that the differential in the standard cobar complex (def. \ref{CobarComplex}) for $Cotor$ lands in $\overline{\Gamma} \coloneqq coker(\eta)$ where the generator $h_{0,n} = \xi_0 = 1$ disappears.

Recall the further abbreviation

$$
  h_n \coloneqq h_{1,n}
  \,.
$$

Hence we find using the formula from prop. \ref{CoproductOnDualSteenrodInTermsOfAdaptedGenerators}, that

$$
  \begin{aligned}
    d_1(h_n)
    & \coloneqq
    d_1(h_{1,n})
    \\
    & =
    \overline{\Psi}(h_{1,n})
    \\
    & =
    h_{1,n} \otimes \underset{= 0}{\underbrace{ h_{0,n} }}
    +
    \underset{= 0}{\underbrace{ h_{0,n+1} }} \otimes h_{1,n}
    \\
    & =
    0
  \end{aligned}
$$

and hence all the elements $h_n$ are cocycles.

$$
  d_1(h_{2,0})
    =
  h_{2,0} \otimes \underset{= 0}{\underbrace{h_{0,0}}}
  +
  h_{1,1} \otimes h_{1,0}
  +
  \underset{= 0}{\underbrace{h_{0,2}}} h_{2,0}
$$

$$
  d_1( h_{2,1} )
    =
  h_{2,1} \otimes \underset{= 0}{\underbrace{h_{0,n}}}
    +
  h_{1,2} \otimes h_{1,1}
    +
  \underset{ = 0}{\underbrace{h_{0,3}}} \otimes h_{2,1}
$$

$$
  d_1( h_{2,2} )
    =
  h_{1,3} \otimes h_{1,2}
$$

$$
  d_1( h_{2,3} )
    =
  h_{1,4} \otimes h_{1,3}
$$

$$
  d_1( h_{3,0} )
    =
  h_{2,1} \otimes h_{1,0}#
    +
  h_{1,2} \otimes h_{2,0}
$$

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

In the range $t - s \leq 13$, the second page of the May spectral sequence for $Ext_{\mathbb{A}^\ast_{\mathbb{F}_2}}(\mathbb{F}_2,\mathbb{F}_2)$ has as generators all the

* $h_n$

* $b_{i,n} \coloneqq (h_{i,n})^2$

as well as the element

*  $x_7 \coloneqq h_{2,0} h_{2,1} + h_{1,1} h_{3,0}$

subject to the relations

* $h_n h_{n+1} = 0$

* $h_2 b_{2,0} = h_0 x_7$

* $h_2 x_7 = h_0 b_{2,1}$.

The differentials in this range are

1. $d_r(h_{n}) = 0$

1. $d_2(b_{2,n}) = h_n^2 h_{n+2} + h_{n+1}^3$

1. $d_2(x_7) = h_0 h_2^2$

1. $d_2(b_{3,0}) = h_1 b_{2,1} + h_3 b_{2,0}$

1. $d_4(b_{2,0}^2) = h_0^4 h_3$.

=--

e.g. ([Ravenel 86, lemma 3.2.8 and lemma 3.2.10](#Ravenel86), [Kochman 96, lemma 5.3.2 and lemma 5.3.3](#Kochman96))

Hence this solves the May spectral sequence, hence gives the second page of the classical Adams spectral sequence. Inspection just of the degrees then shows that in this range there is no non-trivial differential in the Adams spectral sequece in this range. This way one arrives at:

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

In low $t-s$ the group $Ext_{\mathcal{A}^\ast_{\mathbb{F}_2}}(\mathbb{F}_2,\mathbb{F}_2)$ is spanned by the items in the following table

<img src="http://ncatlab.org/nlab/files/ClassicalAdamsSpectralSequence.jpg" width="600" >

(graphics taken from ([[Symmetric spectra|Schwede 12]]))

=--

([Ravenel 86, theorem 3.2.11](#Ravenel86), [Kochman 96, prop. 5.3.6](#Kochman96))

Hence the first dozen [[stable homotopy groups of spheres]] 2-locally are

| $k =$ | 0 | 1 | 2 | 3 | 4  | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 |
|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|
| $\pi_k(\mathbb{S}\otimes \mathbb{Z}_{(2)}) = $ | $\mathbb{Z}_{(2)}$  | $\mathbb{Z}/2$  |  $\mathbb{Z}/2$ |  $\mathbb{Z}/8$ |  $0$ |  $0$ | $\mathbb{Z}/2$ |   $\mathbb{Z}/16$ | $(\mathbb{Z}/2)^2$ |  $(\mathbb{Z}/2)^3$ | $\mathbb{Z}/2$ | $\mathbb{Z}/8$ | $0$ | $0$ |

Remark: The full answer turns out to be this:

| $k =$ | 0 | 1 | 2 | 3 | 4  | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | $\cdots$ |
|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|
| $\pi_k(\mathbb{S}) = $ | $\mathbb{Z}$  | $\mathbb{Z}_2$  |  $\mathbb{Z}_2$ |  $\mathbb{Z}_{24}$ |  $0$ |  $0$ | $\mathbb{Z}_2$ |   $\mathbb{Z}_{240}$ | $(\mathbb{Z}_2)^2$ |  $(\mathbb{Z}_2)^3$ | $\mathbb{Z}_6$ | $\mathbb{Z}_{504}$ | $0$ | $\mathbb{Z}_3$ | $(\mathbb{Z}_2)^2$ | $\mathbb{Z}_{480} \oplus \mathbb{Z}_2$ | $\cdots$ |

## Related concepts

* [[Curtis algorithm]]

## References

The origin is in

* {#May64} [[Peter May]],  _The cohomology of restricted Lie algebras and of Hopf algebras; application to the Steenrod algebra_, Thesis,  Princeton 1964

* {#May66} [[Peter May]],  _The cohomology of restricted Lie algebras and of Hopf algebras_, Journal of Algebra 3, 123-146 (1966) ([pdf](http://math.uchicago.edu/~may/PAPERS/3.pdf))

* [[Peter May]], _Some remarks on the structure of Hopf algebras_, Proceedings of the AMS, vol 23, No. 3 (1969) ([pdf](http://www.math.uchicago.edu/~may/PAPERS/9.pdf))

Further computational improvements and a computation of the first 70 differentials (for the case of the mod 2 [[Steenrod algebra]]) was given in

* {#Tangora17} [[Martin Tangora]], _On the cohomology of the Steenrod algebra_, Math. Z. 116, 18-64 (1970)

More on the $E_2$-term is in

* {#May74} [[Peter May]], _The Steenrod algebra and its associated graded algebra_, University of Chicago preprint, 1974.

Review includes

* {#Ravenel86} [[Doug Ravenel]], chapter 3, section 2 of _[[Complex cobordism and stable homotopy groups of spheres]]_, 1986

* {#Kochman96} [[Stanley Kochman]], section 5.3 of _[[Bordism, Stable Homotopy and Adams Spectral Sequences]]_, AMS 1996

* Wikipedia, _[May spectral sequence](http://en.wikipedia.org/wiki/May_spectral_sequence)_

[[!redirects May spectral sequences]]


maybe monad

+-- {: .rightHandSide}
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### Context
#### Computation
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[[!include constructivism - contents]]
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#### Categorical algebra
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[[!include categorical algebra -- contents]]
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#### Type theory
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[[!include type theory - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

What is called the _maybe monad_ is a simple [[monad in computer science]] which is used to implement the most basic kind of "[[exceptions]]" indicating the failure of a computation in terms of [[functional programming]]: The maybe monad models the exception which witnesses a failure without however producing any further information.

On the type system the [[maybe monad]] is the operation $X \mapsto X \sqcup \ast$ of forming the [[coproduct type]] with the [[unit type]]. (The [[exception monad]] for the [[unit type]].)

The idea here is that a function $X \longrightarrow Y$ in its [[Kleisli category]] is in the original category a function of the form $X \longrightarrow Y \coprod \ast $ so either returns indeed a value in $Y$ or else returns the unique element of the [[unit type]]/[[terminal object]] $\ast$ -- it is a _[[partial function]]_. The latter case is naturally interpreted as "no value returned", hence as indicating a "failure in computation".

## Properties

> under construction

### Kleisli category

The [[Kleisli category]] of the maybe monad on $Set$ is the category whose objects are sets, and whose morphisms are [[partial functions]].

This observation generalizes as follows: if $\mathbf{C}$ is an [[extensive category]] and has a [[terminal object]] $\ast$, then a morphism $g: X \to Y \coprod \ast$ as an object of the [[overcategory]] $\mathbf(C)/(Y \coprod \ast)$ determines (uniquely up to canonical isomorphism) an object

$$(f: X_1 \to Y, !: X_2 \to \ast) \in \mathbf{C}/Y \times \mathbf{C}/\ast$$

such that $g = f \coprod !$, in other words a partial morphism $f: X \rightharpoonup Y$ whose domain of definition is a subobject $X_1 \hookrightarrow X$ with complement $X_2 \hookrightarrow X$. In brief, maps in the Kleisli category are partial maps with complemented domain.

In particular, in the case of a [[Boolean topos]], the Kleisli category is the category of objects and partial maps; see also [[partial map classifier]].

### EM-category and Relation to pointed objects
 {#EMCategoryAndRelationToPointedObjects}

The [[algebra over a monad|algebras]] over the maybe monad are  [[pointed objects]].

Moreover, assuming $\mathbf{C}$ has [[finite products]] and an appropriate form of [[distributive category|distributivity]] (which obtains if for example $\mathbf{C}$ is [[extensive category|lextensive]]), the maybe monad on $\mathbf{C}$ is a [[monoidal monad]] on the [[cartesian monoidal category]] $\mathbf{C}$. It follows (by the discussion at _[[commutative monad]]_, see also ([Seal 12](#Seal12))) that its [[Eilenberg-Moore category]] of algebras canonically inherits the structure of a [[monoidal category]], at least under the mild assumption that it has [[reflexive coequalizers]]. Note that the maybe monad $T$ preserves reflexive coequalizers, so the monadic functor [[created limit|creates]] reflexive coequalizers if the base category has them; in this abstract setting the monoidal product on algebras $(X, \alpha: T X \to X)$, $(Y, \beta: T Y \to Y)$ is given explicitly as the [[coequalizer]] of $T(\alpha \times \beta): T(T X \times T Y) \to T(X \times Y)$ and

$$T(T X \times T Y) \stackrel{T(\phi_{X, Y})}{\to} T T(X \times Y) \stackrel{\mu}{\to} T(X \times Y)$$

where $\phi$ is one of the structural constraints on the monoidal monad $T$ and $\mu$ is the multiplication on $T$. One finds that this coequalizer yields the usual [[smash product]] of pointed objects.

+-- {: .num_remark}
###### Remark
The smash product as the correct monoidal product can also be deduced in a perhaps more perspicuous manner if we assume more of the base category: that it is [[cartesian closed category|cartesian closed]], [[finite limit|finitely complete]], and finitely cocomplete. In that case we construct the [[internal hom]] of $T$-algebras, i.e., the internal hom of pointed objects $(Y, \beta: T Y \to Y)$ and $(Z, \gamma: T Z \to Z)$ directly as an [[equalizer]] of maps

$$\array{
Z^Y & \to & T Z^{T Y} \\
 & \mathllap{Z^\beta} \searrow & \downarrow \mathrlap{\gamma^{T Y}} \\
 & & Z^{T Y}
}$$

where the top arrow expresses [[enriched functor|enriched functoriality]] of $T$ (which in turn is closely related to the [[strong functor|strength]] on $T$). The success of this is guaranteed by the [[commutative monad|commutativity]] of the monad (which here takes a particularly simple form, being given by the commutative _[[monoid]]_ $\ast$ with respect to coproduct $\coprod$). Then, by taking the monoidal product that is [[adjunct|adjoint]] to the [[internal hom]], one is led to the [[smash product]] $(X \wedge Y)_\ast$ all the same: that is, one can read off the smash product from the fact that pointed maps $X_\ast \to \hom_\ast(Y_\ast, Z_\ast)$ should correspond to pointed maps $(X \wedge Y)_\ast \to Z_\ast$.
=--

### Relation to natural number objects

Regarding just the underlying [[endofunctor]] of the maybe monad, its [[initial algebra over an endofunctor]] is a [[natural numbers object]].

### On the augmented simplex category

We may view the [[simplex category|augmented simplex category]] as the subcategory of $\Set$ whose objects are the finite [[von Neumann ordinals]] and whose morphisms are the monotone functions between them. Then the maybe monad on $\Set$ restricts to $\Delta_a$ to give the monad that sends the object $\mathbf{n}$ to $\mathbf{n+1}$ and the morphism $f:\mathbf{n}\to\mathbf{m}$ to the morphism $T(f):\mathbf{n+1}\to\mathbf{m+1}$ defined by

$$T(f)(k) = \begin{cases}
f(k) & k \lt n\\
m & k = n
\end{cases}$$

In fact, $\Delta_a$ is freely generated by this structure and $\mathbf{0}$ in the sense that its objects are given by $\mathbf{n}=T^n\mathbf{0}$, the face maps are given by $\delta_i^n=T^{n-i-1}\eta_\mathbf{i}$, the degeneracy maps are given by $\sigma_i^n=T^{n-i-1}\mu_\mathbf{i}$, and the [[simplicial identities]] are precisely the monad axioms. Another way to put this is that $(\Delta_a,T,\mathbf{0})$ is the initial object of the $2$-category whose objects are categories equipped with a monad and an object.

This means that if C is some other category equipped with a **co**monad and an object then we get a canonical functor $\Delta_a^\mathrm{op}\to C$ and hence an augmented [[simplicial object]] in $C$. In particular when $C$ is the [[Eilenberg-Moore category| category of algebras]] of a monad on $D$ we get a simplicial object for each algebra, whose underlying simplicial object in $D$ is the [[bar construction]].

The comonad $T^\mathrm{op}$ on $\Delta_a^\mathrm{op}$ induces the [[decalage#DecalageComonad|Décalage comonad]].

## Related concepts

* [[exception monad]]

* [[state monad]]

* [[continuation monad]]

* [[successor monad]]

## References

* {#Seal12} Gavin J. Seal, _Tensors, monads and actions_ [[arXiv:1205.0101](http://arxiv.org/abs/1205.0101)]

Around (0.4.24.2) in

* {#Durov07} [[Nikolai Durov]], _New Approach to Arakelov Geometry_ ([arXiv:0704.2030](http://arxiv.org/abs/0704.2030))

the algebraic structure of the would be "[[field with one element]]" is regarded as being the maybe monad, hence [[modules]] over $\mathbb{F}_1$ are defined to be [[algebra over a monad|monad-algebras]] over the maybe monad, hence [[pointed sets]].

[[!redirects maybe monads]]


Mayer-Vietoris sequence

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###Context###
#### Homotopy theory
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[[!include homotopy - contents]]
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#### Limits and colimits
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[[!include infinity-limits - contents]]
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#### Cohomology
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[[!include cohomology - contents]]
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#Contents#
* table of contents
{:toc}

## Ideas

_Mayer-Vietoris sequence_ is the term for the [[fiber sequence]] -- or often for the corresponding [[long exact sequence of homotopy groups]] -- induced from an [[(∞,1)-pullback]] (or for a [[homotopy pullback]] presenting it).

## Definition

Let $\mathcal{C}$ be an [[(∞,1)-category]] with [[finite (∞,1)-limits]] and let $X, Y, B$ be [[pointed objects]] and

$$
  f : X \to B
$$

and

$$
  g : Y \to B
$$

be any two [[morphisms]] with common [[codomain]] preserving the base points. Let $X \times_B Y$ be the [[(∞,1)-pullback]]

$$

  \array{
    X \times_B Y &\to& Y
    \\
    \downarrow &\swArrow_\simeq& \downarrow^{\mathrlap{g}}
    \\
    X &\stackrel{f}{\to}& B
  }
  \,.
$$

The corresponding **Mayer-Vietoris sequence** is the [[fiber sequence]] of the induced morphism $X \times_B Y \to X \times Y$. Often the term is used (only) for the corresponding [[long exact sequence of homotopy groups]].

## Properties

### General
 {#GeneralProperties}

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

Let $\mathcal{C}$ be a [[presentable (∞,1)-category]].

Then $X \times_B Y$, which by definition sits in

$$
  \array{
    X \times_B Y &\to& Y
    \\
    \downarrow &\swArrow_{\simeq}& \downarrow^{\mathrlap{g}}
    \\
    X  &\stackrel{f}{\to}& B
  }
$$

is equivalently also the following [[(∞,1)-pullback]]

$$
  \array{
    X \times_B Y &\to& B
    \\
    \downarrow &\swArrow_{\simeq}& \downarrow^{\mathrlap{\Delta_B}}
    \\
    X \times Y &\stackrel{(f,g)}{\to}& B \times B
  }
  \,,
$$

where the right vertical morphism is the [[diagonal]].

Moreover, the [[homotopy fiber]] of $X \times_B Y \to X \times Y$ is the [[loop space object]] $\Omega B$.

=--

See also at _[[homotopy pullback]]_ [this corollary](homotopy+pullback#HomotopyPullbackByFactorizationLemma).

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

The first statement one checks for instance by choosing a presentation by a [[combinatorial model category]] and then proceeding as below in the discussion _[Presentation by fibrant objects](#PresentationByFibrantObjects)_.
Then by the [[pasting law]] for $(\infty,1)$-pullbacks it follows that with the left square in

$$
  \array{
    \Omega B &\to& X \times_B Y &\to & B
    \\
    \downarrow &\swArrow_{\simeq}& \downarrow &\swArrow_{\simeq}& \downarrow
    \\
    * &\to& X \times Y &\stackrel{(f,g)}{\to}& B \times B
  }
$$

an $(\infty,1)$-pullback, so is the total outer rectangle. But again by the first statement, this is equivalent to the $(\infty,1)$-pullback

$$
  \array{
    \Omega B &\to& *
    \\
    \downarrow &\swArrow_{\simeq}& \downarrow
    \\
    * &\to& B
  }
  \,,
$$

which is the defining pullback for the [[loop space object]].

=--

Therefore the Mayer-Vietoris [[homotopy fiber sequence]] is of the form

$$
  \Omega B \to X \times_B Y \to X \times Y
  \,.
$$

For $\mathcal{C} = $ [[∞Grpd]] $\simeq L_{whe}$ [[Top]], this point of view is amplified in ([Dyer-Roitberg 80](#DyerRoitberg80)).

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

The corresponding [[long exact sequence of homotopy groups]] is of the form

$$
  \cdots \to \pi_{n+1} B \to \pi_n X \times_B Y \stackrel{(f_*, g_*)}{\to} \pi_n X \oplus \pi_n Y \stackrel{f_* - g_*}{\to} \pi_n B \to \cdots
$$

$$
  \cdots \to \pi_2 B \to \pi_1 X \times_B Y \stackrel{(f_+, g_*)}{\to}  \pi_1 X \times \pi_1 Y \stackrel{f_* \cdot g_*^{-1}}{\to} \pi_1 B \to \pi_0 (X \times_B Y) \to \pi_0 (X \times Y)
  \,.
$$

=--

This is what has historically been the definition of Mayer-Vietories sequences ([Eckmann-Hilton 64](#EckmannHilton)).

### Presentation by fibrant objects
 {#PresentationByFibrantObjects}

Suppose that the [[(∞,1)-category]] $\mathcal{C}$ is [[simplicial localization|presented]] by a [[category of fibrant objects]] $C$ (for instance the [[subcategory]] on the fibrant objects of a [[model category]]).

Then the $(\infty,1)$-pullback $X \times_B Y$ is presented by a [[homotopy pullback]], and by the [[factorization lemma]], this is given by the ordinary [[limit]]

$$
  \array{
    X \times^h_B Y &\to& &\to& Y
    \\
    \downarrow && && \downarrow^{\mathrlap{g}}
    \\
    && B^I &\to& B
    \\
    \downarrow && \downarrow
    \\
    X &\stackrel{f}{\to}& B
  }
  \,,
$$

where $B \stackrel{\simeq}{\to} B^I \to B \times B$ is a [[path object]] for $B$. This limit coincides, up to [[isomorphism]], with the [[pullback]]

$$
  \array{
    X \times_B^h Y &\to& B^I
    \\
    \downarrow && \downarrow
    \\
    X \times Y &\stackrel{(f,g)}{\to}& B \times B
  }
  \,.
$$

This implies in particular that the [[homotopy fiber]] of $X \times_B^h Y \to X \times Y$ is the [[loop space object]] $\Omega B$, being the [[fiber]] of the [[path space object]] projection.

### Over an $\infty$-group
 {#OverAGroupObject}

We consider now the case where $B$ carries the structure of an  [[∞-group]] (or just a grouplike [[H-space]] object) in a [[presentable (∞,1)-category]] or [[locally Cartesian closed (∞,1)-category]] $\mathcal{C}$.

In this case (as discussed in a moment), we have an [[(∞,1)-pullback]]

$$
  \array{
    B &\to& *
    \\
    \downarrow^{\mathrlap{\Delta_B}}
    &\swArrow_{\simeq}&
    \downarrow^{\mathrlap{e}}
    \\
    B \times B
    &\stackrel{(-)\cdot (-)^{-1}}{\to}&
    B
  }
  \,,
$$

where the bottom horizontal morphism is the composite

$$
  (-)\cdot (-)^{-1} : B \times B \stackrel{(id, (-)^{-1})}{\to} B \times B \stackrel{\cdot}{\to} B
$$

of a morphism that sends the second argument to its inverse with the group composition operation.

It then follows by the [[pasting law]] and prop. \ref{SequenceFromDiagonal} that in this case the morphism $X \times_B Y \to X \times Y$ in the Mayer-Vietoris sequence is itself the homotopy fiber of $X \times Y \stackrel{f \cdot g^{-1}}{\longrightarrow} B$, hence that we have a long homotopy fiber sequence of the form

$$
  \Omega B \longrightarrow X \times_B Y \longrightarrow X \times Y
  \stackrel{f \cdot g^{-1}}{\longrightarrow} B
  \,.
$$

First consider two more concrete special cases.

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

Let $S$ be a small [[site]] and let $\mathcal{C} = Sh_{(\infty,1)}(S)$ be the [[(∞,1)-category of (∞,1)-sheaves]] on $S$.

This is [[presentable (∞,1)-category|presented]] by the projective [[model structure on simplicial presheaves]]

$$
  \mathcal{C} \simeq ([S^{op}, sSet]_{proj, loc})^\circ
  \,.
$$

As discussed there, the [[Dold-Kan correspondence]] prolongs to a [[Quillen adjunction]] on [[presheaves]] whose [[right adjoint]] is

$$
  \Xi :
  [S^{op}, Ch_{\bullet \leq 0}(Ab)]_{proj}
  \to
  [S^{op}, sAb]_{proj}
  \to
  [S^{op}, sSet]_{proj}
  \,.
$$

Let then $B \in \mathcal{C}$ be an object with a presentation in $[S^{op}, sSet]$ in the image of this $\Xi$. We write $B$ also for this presentation, and hence $B = \Xi(\tilde B)$ for some presheaf of chain complexes $\tilde B$.

We claim now that such $B$ satisfies the above assumption.

To see this, first notice that the evident morphism $- : \tilde B \times \tilde B \to \tilde B$ is degreewise an [[epimorphism]], hence it is a fibration in $[S^{op}, Ch_{\bullet \geq 0}(Ab)]_{proj}$, and since $\Xi$ is [[Quillen adjunction|right Quillen]], so is the corresponding morphism $- : B \times B \to B$ in $[S^{op}, sSet]_{proj}$.

Therefore the ordinary pullback of presheaves of chain complexes

$$
  \array{
    \tilde B &\to& *
    \\
    \downarrow^{\mathrlap{\Delta_{\tilde B}}} && \downarrow^{\mathrlap{0}}
    \\
    \tilde B \times \tilde B &\stackrel{-}{\to}& \tilde B
  }
$$

is a [[homotopy pullback]] in $[S^{op}, Ch_{\bullet \geq 0}(Ab)]_{proj}$, as is the ordinary pullback of simplicial presheaves

$$
  \array{
    B &\to& *
    \\
    \downarrow^{\mathrlap{\Delta_B}} && \downarrow^{\mathrlap{0}}
    \\
    B \times B &\stackrel{-}{\to}& B
  }
$$

in $[S^{op}, sSet]_{proj}$.

Since [[∞-stackification]] preserves [[finite (∞,1)-limits]], this presents an [[(∞,1)-pullback]] also in $\mathcal{C}$.

=--

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

Let $\mathcal{C}$ be an [[(∞,1)-topos]] with a 1-[[site]] $S$ of definition (a [[n-localic (∞,1)-topos|1-localic (∞,1)-topos]]).

Then (as discussed there) every [[∞-group]] object in $\mathcal{C}$
has a presentation by a presheaf of [[simplicial groups]]

$$
  B \in [S^{op}, sGrp]_{proj} \to [S^{op}, sSet]_{proj}
  \,.
$$

We claim that the canonical morphism $- : B \times B \to B$ is
objectwise a [[Kan fibration]] and hence a fibration in the
projective [[model structure on simplicial presheaves]].

Let $U \in S$ be any test object. A diagram

$$
  \array{
    \Lambda[k]^i &\stackrel{(ha, hb)}{\to}& B(U) \times B(U)
    \\
    \downarrow^{\mathrlap{j}} && \downarrow
    \\
    \Delta[k] &\stackrel{\sigma}{\to}& B(U)
  }
$$

corresponds to a $k$-cell $\sigma \in B(U)$ together with a
choice of decomposition of the $i$th [[horn]] $j^* \sigma$
as a difference

$$
  (j^* \sigma)_l = ha_l \cdot hb_l^{-1}
  \,.
$$

Since $B(U)$ itself is a [[Kan complex]] (being a [[simplicial group]], as discussed there) there is a filler $b \colon \Delta[k] \to B(U)$
of the [[horn]] $hb \colon \Lambda[k]^i \to B(U)$.
Define then

$$
  a \coloneqq \sigma \cdot b
  \,.
$$

Since all the face maps are group homomorphisms, this is indeed a filler of $ha$:

$$
  \begin{aligned}
    \delta_l(a)
      & = \delta_l(\sigma \cdot b)
    \\
      & = \delta_l(\sigma) \cdot \delta_l(b)
    \\
      & = \delta_l(\sigma) \cdot hb_l
    \\
      & = ha_l
  \end{aligned}
  \,.
$$

Moreover, by construction, $(a,b)$ is a filler in

$$
  \array{
    \Lambda[k]^i &\stackrel{(ha, hb)}{\to}& B(U) \times B(U)
    \\
    \downarrow^{\mathrlap{i}}
    &{}^{(a,b)}\nearrow&
    \downarrow
    \\
    \Delta[k] &\stackrel{\sigma}{\to}& B(U)
  }
  \,.
$$

Since therefore $- \colon B \times B \to B$ is a projective fibration, it follows
as before that the ordinary pullback

$$
  \array{
    B &\to& *
    \\
    \downarrow^{\mathrlap{\Delta_B}} && \downarrow^{e}
    \\
    B \times B &\stackrel{-}{\to}& B
  }
$$

is a [[homotopy pullback]].

=--

+-- {: .num_prop #SequenceOverGroupObjectIn1LocalicSituation}
###### Proposition

For $B$ an [[∞-group]] object as above, the [[(∞,1)-pullback]] $X \times_B Y$ is equivalently given by the $(\infty,1)$-pullback

$$
  \array{
   X \times_B Y &\to& *
   \\
   \downarrow &\swArrow_{\simeq}& \downarrow^{\mathrlap{0}}
   \\
   X \times Y &\stackrel{f \cdot g^{-1}}{\to}& B
  }
  \,.
$$

=--

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

By prop. \ref{SequenceFromDiagonal} the object $X \times_B Y$ is the $(\infty,1)$-pullback in

$$
  \array{
    X \times_B Y &\to& B
    \\
    \downarrow &\swArrow_{\simeq}& \downarrow^{\mathrlap{\Delta_B}}
    \\
    X \times Y &\stackrel{(f,g)}{\to}& B \times B
  }
  \,.
$$

By the [[pasting law]] this is equivalently given by the  composite pullback of

$$
  \array{
    X \times_B Y &\to& B &\to& *
    \\
    \downarrow
    &\swArrow_{\simeq}&
    \downarrow^{\mathrlap{\Delta_B}}
    &\swArrow_{\simeq}&
    \downarrow^{\mathrlap{0}}
    \\
    X \times Y &\stackrel{(f,g)}{\to}& B \times B
    &\stackrel{-}{\to}& B
  }
  \,.
$$

Here the composite bottom morphism is $(f - g)$.

=--

Summing this up:

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

For $\mathbf{H}$ an [[(∞,1)-sheaf]] [[(∞,1)-topos]], $B$ an [[∞-group]]-object in $\mathbf{H}$ and $f\colon X \to B$ and $g \colon Y\to B$ two morphisms, then there is a long [[homotopy fiber sequence]] of the form

$$
  \Omega B \longrightarrow X \times_B Y \longrightarrow X \times Y
  \stackrel{f \cdot g^{-1}}{\longrightarrow} B
  \,.
$$

=--

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

For $\mathcal{C}$ an [[(∞,1)-site]] of definition, there is a [[reflective sub-(infinity,1)-category|reflection]]

$$
  \mathbf{H} \stackrel{\longleftarrow}{\hookrightarrow}
  [C^{op},\infty Grpd]
$$

of $\mathbf{H}$ into an [[(∞,1)-category of (∞,1)-presheaves]].

By prop. \ref{SequenceOverGroupObjectIn1LocalicSituation} the statement
holds in $[C^{op},\infty Grpd]$. Since embedding and reflection both preserve
[[finite (∞,1)-limits]], it hence also holds in $\mathbf{H}$.

=--

Still more generally and more simply:

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

Let $\mathcal{C}$ be a [[locally Cartesian closed (∞,1)-category]].
Let $G$ be an [[∞-group]] object (or just a grouplike [[H-space]]-object). Then for
$\phi \colon D \longrightarrow G$ any morphism we have a [[homotopy pullback]] square of the form

$$
  \array{
     G \times D &\longrightarrow& D
     \\
     \downarrow && \downarrow^{\mathrlap{\phi}}
     \\
     G \times G &\stackrel{(-)\cdot (-)^{-1}}{\longrightarrow}& G
  }
  \,.
$$

=--

([nForum discussion](http://nforum.mathforge.org/discussion/6403/generality-of-mayervietoris-in-an-infinitytopos-/?Focus=51417#Comment_51417))

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

By [this discussion](locally+cartesian+closed+infinity,1-category#InternalLogic) we may use [[homotopy type theory]] reasoning. Starting out with the discussion at _[homotopy pullback -- In homotopy type theory](homotopy+pullback#InHomotopyTypeTheory)_ we obtain

$$
\begin{aligned}
  D\times_G  (G\times G)
  &= \sum_{d:D} \sum_{g_1:G} \sum_{g_2:G} (g_1\cdot g_2^{-1} = \phi(d)) \\
  &= \sum_{d:D} \sum_{g_1:G} \sum_{g_2:G} (g_1 = \phi(d)\cdot g_2) \\
  &= \sum_{d:D} \sum_{g_2:G} \sum_{g_1:G} (g_1 = \phi(d)\cdot g_2) \\
  &= \sum_{d:D} \sum_{g_2:G} \mathbf{1}\\
  &= D\times G
\end{aligned}
  \,,
$$

where the second but last step consists of observing a contractible based [[path space object]] (see the discussion at [[factorization lemma]]).

=--

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

Let $\mathcal{C}$ be a [[locally Cartesian closed (∞,1)-category]].
Let $G$ be an [[∞-group]] object (or just a grouplike [[H-space]]-object).

Then for $f \colon X \to G$ and $g \colon Y \to G$ two morphisms, there is a Mayer-Vietoris-type [[homotopy fiber sequence]]

$$
  \cdots \to \Omega G
  \longrightarrow
  X \times_G Y
  \longrightarrow
  X \times Y
  \stackrel{f \cdot (g^{-1})}{\longrightarrow}
  G
  \,.
$$

=--

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

Use prop. \ref{HTTArgumentForPullback} with $\phi$ being the canonical point, i.e. the inclusion $e \colon \ast \to G$ of [[generalized the|the]] neutral element to find the homotopy pullback

$$
  \array{
     G  &\longrightarrow& \ast
     \\
     \downarrow^{\mathrlap{\Delta}} && \downarrow^{\mathrlap{e}}
     \\
     G \times G &\stackrel{(-)\cdot (-)^{-1}}{\longrightarrow}& G
  }
  \,.
$$

Then use the [[pasting law]] as above.

=--

## Examples

### (Co)Homology of a cover
 {#CohomologyOfACover}

A special case of the general Mayer-Vietoris sequence, corollary \ref{LongExactSequenceOfHomotopyGroups} -- which historically was the first case considered -- applies to the [[cohomology]]/[[homology]] of a [[topological space]] $X$ equipped with an [[open cover]] $\{U_1, U_2 \to X\}$.

Being a [[cover]] means (see [[effective epimorphism in an (∞,1)-category]]) that there is a [[homotopy pushout]] [[diagram]] of the form

$$
  \array{
    U \cap V &\hookrightarrow& U
    \\
    \downarrow && \downarrow
    \\
    V &\to& X
  }
$$

in the [[(∞,1)-topos]] $\mathbf{H} = $ [[∞Grpd]]/[[Top]].

When this is [[presentable (∞,1)-category|presented]] by the standard [[model structure on simplicial sets]] or in terms of [[CW-complex]]es by the [[model structure on topological spaces]], it is given by an ordinary [[pushout]].

Let then $A \in \infty Grpd \simeq Top$ be some coefficient object, for instance an [[Eilenberg-MacLane object]] $\mathbf{B}^n G$ ([[Eilenberg-MacLane space]] $\cdots \simeq K(G,n)$) for the definition of ordinary [[singular cohomology]] with coefficients in an [[abelian group]] $G$.

Then applying the [[derived hom space]] functor $\mathbf{H}(-, A) : \mathbf{H}^{op} \to \infty Grpd$ yields the [[(∞,1)-pullback]] [[diagram]]

$$
  \array{
    \mathbf{H}(X, A) &\to& \mathbf{H}(U,A)
    \\
    \downarrow && \downarrow
    \\
    \mathbf{H}(V,A) &\to& \mathbf{H}(U \cap V, A)
  }
$$

to which we can apply the homotopical Mayer-Vietoris sequence.

Notice that (as discussed in detail at [[cohomology]]) the [[homotopy groups]] of the [[∞-groupoid]] $\mathbf{H}(X,\mathbf{B}^n G)$ are the [[cohomology groups]] of $X$ with coefficients in $G$

$$
  \pi_k \mathbf{H}(X, \mathbf{B}^n G)
  \simeq
  H^{n-k}(X, G)
  \,.
$$

By the above [general properties](GeneralProperties) the above homotopy pullback is equivalent to

$$
  \mathbf{H}(X,A) \to \mathbf{H}(U,A) \times \mathbf{H}(V,A)
  \to \mathbf{H}(U \cap V, A)
$$

being a [[fiber sequence]]. The corresponding long exact sequence in cohomology (as discussed above) is what is traditionally called the Mayer-Vietoris sequence of the cover of $X$ by $U$ and $V$ in $A$-cohomology.

By duality (see [[universal coefficient theorem]]) an analogous statement holds for the [[homology]] of $X$, $U$ and $V$.

## Related concepts

* [[diamond principle]]

* [[Brown-Gersten property]]

## References

An original reference is

* {#EckmannHilton} [[Beno Eckmann]] and [[Peter Hilton]], _Unions and intersections in homotopy theory_, Comment. Math. Helv. 3 (1964),2 93-307, [doi](http://dx.doi.org/10.1007/BF02566918)

A more modern review that emphasizes the role of [[homotopy fiber sequences]] is in

* {#DyerRoitberg80} [[Eldon Dyer]], Joseph Roitberg, _Note on sequence of Mayer-Vietoris type_, Proceedings of the AMS, volume 80, number 4 (1980) ([pdf](http://www.ams.org/journals/proc/1980-080-04/S0002-9939-1980-0587950-8/S0002-9939-1980-0587950-8.pdf))

Discussion in the context of [[stable model categories]] includes

* [[Peter May]], lemma 5.7 of _The additivity of traces in triangulated categories_, Adv. Math., 163(1):34-73, 2001 ([pdf](http://www.math.uchicago.edu/~may/PAPERS/AddJan01.pdf))

Discussion in the context of [[homotopy type theory]] includes

* E Cavallo et al, _Exactness of the Mayer-Vietoris Sequence
in Homotopy Type Theory_ ([pdf](http://www.contrib.andrew.cmu.edu/~ecavallo/works/mayer-vietoris.pdf))

[[!redirects Mayer-Vietoris sequences]]
[[!redirects Mayer-Vietoris long exact sequence]]


Mayuko Yamashita

* [personal page](https://www.kurims.kyoto-u.ac.jp/~mayuko/english.html)

## Selected writings

On lifting the [[Witten genus]] of the [[heterotic string]] to [[topological modular forms]]:

* [[Yuji Tachikawa]], _Topological modular forms and the absence of a heterotic global anomaly_ ([arXiv:2103.12211](https://arxiv.org/abs/2103.12211))

* [[Yuji Tachikawa]], [[Mayuko Yamashita]], *Topological modular forms and the absence of all heterotic global anomalies*, Comm. Math. Phys. **402** (2023) 1585-1620 [[arXiv:2108.13542](https://arxiv.org/abs/2108.13542), [doi:10.1007/s00220-023-04761-2](https://doi.org/10.1007/s00220-023-04761-2)]

and in relation to [[Anderson duality]] of [[tmf]]:

* [[Yuji Tachikawa]], [[Mayuko Yamashita]], *Anderson self-duality of topological modular forms, its differential-geometric manifestations, and vertex operator algebras* [[arXiv:2305.06196](https://arxiv.org/abs/2305.06196)]

On [[differential K-theory|differential]] [[KO-theory]] via [[mass terms]]:

* [[Kiyonori Gomi]], [[Mayuko Yamashita]], *Differential KO-theory via gradations and mass terms* [[arXiv:2111.01377](https://arxiv.org/abs/2111.01377)]

On the mod-2 [[elliptic genus]], both as a [[partition function]] of a [[superstring]] as well as via the [[string-orientation of tmf]]:

* [[Yuji Tachikawa]], [[Mayuko Yamashita]], [[Kazuya Yonekura]], *Remarks on mod-2 elliptic genus* [[arXiv:2302.07548](https://arxiv.org/abs/2302.07548)&rbrack

category: people


Mazur-Ulam theorem

The Mazur--Ulam theorem (after [[Stanislaw Mazur]] and [[Stanislaw Ulam]]) states that if $E$ and $F$ are [[normed vector space]]s over $\mathbb{R}$ and $f\colon E \to F$ is an [[isometry|isometric]] isomorphism, then $f$ is an [[affine space|affine]] map. For a proof, see either of the references below by Akhil Mathew and Jussi Väisälä. It is based on the idea that a midpoint between two points can be characterized in purely norm-theoretic terms. This is more or less trivial if the norm is strictly convex (compare the discussion at [[isometry]]); in the general case, one cleverly exploits properties of the reflection map $x \mapsto 2 z - x$ where $z$ is a midpoint in question, to arrive at the conclusion.

## References

A. Mathew, Post on his blog Climbing Mount Bourbaki, November 19, 2009 ([link](http://amathew.wordpress.com/2009/11/19/a-theorem-of-mazur-ulam-on-isometries-of-vector-spaces/))

J. Väisälä, A proof of the Mazur--Ulam theorem ([link](http://www.helsinki.fi/~jvaisala/mazurulam.pdf))

[[!redirects Mazur-Ulam theorem]]
[[!redirects Mazur–Ulam theorem]]
[[!redirects Mazur--Ulam theorem]]


Mazurkiewicz trace

## Idea

Mazurkiewicz trace theory is one of the oldest forms of [[trace semantics]]. Its models are based on labelled [[posets]], the labelling being by a [[trace alphabet]].

[[!redirects Mazurkiewicz trace]]
[[!redirects Mazurkiewicz traces]]
[[!redirects Marzurkiewicz trace]]
[[!redirects Marzurkiewicz traces]]


McKay correspondence

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
=--
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea
 {#Idea}

Generally, the _McKay correspondence_ is a subtle correspondence between the theory of [[finite subgroups of SU(2)]], the corresponding [[orbifold]] [[singularities]] ([[du Val singularities]]) and that of [[simple Lie groups]] falling into the [[ADE classification]].

[[!include ADE -- table]]

The correspondence may be understood at different levels:

1. **McKay quivers.** The original correspondence due to [McKay 80](#McKay80) is the observation that the [[McKay quiver]] associated to the [[orbifold]] [[singularity]] $\mathbb{C}^2 \sslash G$ of a [[finite subgroup of SU(2)]] $G \subset SU(2)$ happens to be an extended [[Dynkin quiver]], hence happens to be an extended [[Dynkin diagram]] of the kind that also arises in the [[ADE-classification]] of [[simple Lie groups]].

   More on this in _[McKay quivers](#McKayQuivers)_, below.

1. **Equivariant K-theory.** A more conceptual explanation for this correspondence comes from the classical fact that the [[blow up]] $\widehat{\mathbb{C}^2\sslash G}$ of the [[ADE singularity]] $\mathbb{C}^2\sslash G$ is given by a sequence of [[spheres]] that touch along the vertices of the corresponding [[Dynkin quiver]]. In [Gonzalez-Sprinberg & Verdier 83](#GSV83) it was shown that the $G$-[[equivariant K-theory]] $K_G(\mathbb{C}^2)$ of the singularity is isomorphic to the [[K-theory]] of the desingularization $\widehat {\mathbb{C}^2\sslash G}$ such that under this isomorphism the generators of $K_G(\mathbb{C}^2) \simeq R_\mathbb{C}(G)$, hence of the [[representation ring]], hence the [[irreducible representations]], map to K-theory classes supported on the [[exceptional divisors]] of this blowup. This gives a conceptual geometric/cohomological explanation for the identifications observed by [McKay 80](#McKay80).

   More on this in _[As an equivalence of K-theories](#AsAnEquivalenceOfKtheories)_

1. **Seiberg-Witten theory.** Under the interpretation of [[K-theory]]-classes as [[D-brane]] charges, the K-theoretic McKay correspondence of [Gonzalez-Sprinberg & Verdier 83](#GSV83) identifies [[wrapped brane]] charges of the desingularized space with [[fractional brane]] charges at the singularity.

   This leads to a further (currently non-rigorous) explanation of the McKay correspondence in terms of the [[AdS-CFT duality|dual]] [[worldvolume]] [[quantum field theories]] on these branes, which are [[N=2 D=4 super Yang-Mills theory|N=2]] [[quiver gauge theories]]: The [[moduli space]] of [[scalar fields]] of these theories has two "branches", called the _[[Higgs branch]]_ and the _[[Coulomb branch]]_, and the idea is that depending on which of the branches the [[vacuum state]] of the theory is in, it describes the brane as being either on the [[ADE singularity]] or on its [[resolution of singularities|resolution]], but since it's the same quantum field theory in both cases, these two situations are actually suitably equivalent.

  More on this in _[Via N=2 SYM theory](#ViaSuperYangMillsTheory)_ below.

### Via McKay quivers
 {#McKayQuivers}

Generally, for $G$ a [[finite group]] and $V$ a [[linear representation]] of $G$ on a [[finite dimensional vector space|finite dimensional]] [[complex vector space]], the _[[McKay quiver]]_ or _[[McKay graph]]_ associated with $V$ is the [[quiver]] whose [[vertices]] correspond to the [[irreducible representations]] $\rho_i$ of $G$ and which has $a_{i j} \in \mathbb{N}$ [[edges]] between the $i$th and the $j$th vertex, for $a_{i j}$ the [[coefficients]] in the expansion into [[irreps]] of the [[tensor product of representations]] of $V$ with these irreps:

$$
  V \otimes \rho_i
  \;\simeq\;
  \underset{j}{\bigoplus}
  a_{i j} \cdot \rho_j
  \,.
$$

Specifically this applies to the special case where $G \subset $ [[SU(2)]] a [[finite subgroup of SU(2)]] and $V$ its defining representation on $\mathbb{C}^2$.

The original _McKay correspondence_ ([McKay 80](#McKay80)) states that in this case the corresponding [[McKay quivers]] are [[Dynkin quivers]]/[[Dynkin diagrams]] in the same [[ADE classification]] as the [[ADE singularity]] $\mathbb{C}^2 \sslash G$.

More precisely: If one uses _all_ [[irreducible representations]] including the 1-dimensiona [[trivial representation]] $\rho_0$ then one gets the "extended Dynkin diagram", where the extra node corresponds to $\rho_0$. This is the vertex indicated by a cross in the following diagrams:

\begin{center}
<img src="https://ncatlab.org/nlab/files/ExtendedADEQuivers.jpg" width="400">
\end{center}

> graphics grabbed from [GSV 83, p. 4](#GSV83)

In particular, for $G =\mathbb{Z}_N \subset SU(2)$ a [[cyclic group]] of [[order of a group|order]] $N$, there are $N$ complex [[irreps]] and the McKay quiver, i.e. the extended Dynkin diagram, has $N$-vertices, connected by edges to form a circle.

### As an equivalence of K-theories
 {#AsAnEquivalenceOfKtheories}

A strong version of the McKay correspondence is obtained when the [[cohomology theory]] is taken be ([[equivariant K-theory|equivariant]]) _[[K-theory]]_ ([Gonzalez-Sprinberg & Verdier 83](#GSV83)):

Here the McKay correspondence becomes an [[isomorphism]] between the [[equivariant K-theory]] $K_{G_{ADE}}(\ast)$ of an [[ADE-singularity]] (equivalently the [[representation ring]] $R(G_{ADE})$) and the plain [[K-theory]] $K(\tilde X)$ of its [[blow-up]] [resolution](ADE+singularity#ResolutionBySpheresTouchingAlongADynkinDiagram) $\tilde X$. Much like in [[topological T-duality]], this isomorphism is given by [[integral transform]] ([[Fourier-Mukai transform]]) through the canonical [[correspondence]] that these spaces constitute:

\[
  \label{KMcKay}
  \array{
     &&
     && K( X \times_{X/G} \tilde X )
     \\
     &&
     & {}^{\mathllap{ p_1^\ast }}\nearrow && \searrow^{\mathrlap{Inv \circ (p_2)_\ast}}
     \\
     R(G) \simeq K_G(\ast) &\simeq& K_G(X) && \overset{\simeq}{\longrightarrow}  && K(\tilde X)
  }
\]

In terms of [[physics]] ([[string theory]]), the [[K-theory]] classes appearing here may be interpreted as [[groups]] of [[fractional D-brane|fractional]] [[D-brane charges]] (see [there](fractional+D-brane#DBranesAtOrbifoldSingularities)).
In terms of the [[worldvolume]] [[Chan-Paton gauge field|Chan-Paton]]-[[Yang-Mills theory]] on the D-branes the McKay correspondence is then seen as passage from the [[Higgs branch]] to the [[Coulomb branch]] (see [there](fractional+D-brane#InTermsOfTheWorldvolumeGaugeTheories)).

The proof of these statements generally proceeds by relating both sides of the equivalence to [[Dynkin diagrams]] of [[ADE-classification|ADE-type]].

The classical McKay correspondence, named after [[John McKay]], is a one-to-one correspondence between the [[McKay graphs]] of [[finite group|finite]] [[subgroups]] $G \subset \text{SL}_2(\mathbb{C})$ and the extended Dynkin diagrams of ADE type.

### Via $N=2 $ super Yang-Mills theory
 {#ViaSuperYangMillsTheory}

Various seemingly unrelated structures in mathematics fall into an [[ADE classification]]. Notably [[finite group|finite]] [[subgroups]] of [[special unitary group|SU(2)]] and [[compact Lie group|compact]] [[simple Lie groups]] do. The way this works usually is that one tries to classify these structures somehow, and ends up finding that the classification is governed by the combinatorics of [[Dynkin diagrams]].

While that does explain a bit, it seems the statement that both the [[icosahedral group]] and the Lie group [[E8]] are related to the same [[Dynkin diagram]] somehow is still more a question than an answer. Why is that so?

The first key insight is due to [Kronheimer 89](ADE+singularity#Kronheimer89a). He showed that the (resolutions of) the [[orbifold]] quotients $\mathbb{C}^2/\Gamma$ for finite subgroups $\Gamma$ of $SU(2)$ are precisely the generic form of the [[gauge group|gauge]] [[orbits]] of the [[direct product group]] of $U(n_i)$s acting in the evident way on the [[direct sum]] of $Hom(\mathbb{C}^{n_i}, \mathbb{C}^{n_j})$-s, where $i$ and $j$ range over the vertices of the [[Dynkin diagram]], and $(i,j)$ over its edges.

This becomes more illuminating when interpreted in terms of [[gauge theory]]: in a [[quiver gauge theory]] the [[gauge group]] is a [[direct product group]] of  $U(n_i)$ factors associated with vertices of a [[quiver]], and the [[particles]] which are [[charged particle|charged]] under this gauge group arrange, as a [[linear representation]], into a [[direct sum]] of $Hom(\mathbb{C}^{n_i}, \mathbb{C}^{n_j})$-s, for each edge of the quiver.

Pick one such particle, and follow it around as the gauge group transforms it. The space swept out is its gauge [[orbit]], and [Kronheimer 89](#Kronheimer89) says that if the quiver is a Dynkin diagram, then this gauge orbit looks like $\mathbb{C}^2/\Gamma$.

On the other extreme, gauge theories are of interest whose gauge group is not a big direct product, but is a [[simple Lie group]], such as [[special unitary group|SU(N)]] or [[E8]]. The mechanism that relates the two classes of examples is [[spontaneous symmetry breaking]] ("[[Higgs field|Higgsing]]"): the ground state energy of the field theory may happen to be achieved by putting the fields at any one point in a higher dimensional space of field configurations, acted on by the gauge group, and fixing any one such point "spontaneously" singles out the corresponding [[stabilizer subgroup]].

Now here is the final ingredient: [[N=2 D=4 super Yang-Mills theory]] ("[[Seiberg-Witten theory]]"). These theories have a [[potential energy|potential]] such that its [[vacua]] break a simple gauge group, such as $SU(N)$, down to a [[Dynkin diagram]] [[quiver gauge theory]]. One place where this is reviewed, physics style, is in [Albertsson 03, section 2.3.4](N=2+D=4+super+Yang-Mills+theory#Albertsson03).

More precisely, these theories have two different kinds of vacua, those on the "[[Coulomb branch]]" and those on the "[[Higgs branch]]" depending on whether the scalars of the "[[vector multiplets]]" (the gauge field sector) or of the "[[hypermultiplet]]" (the matter field sector) vanish. The statement above is for the Higgs branch, but the Coulomb branch is supposed to behave "dually".

So that then finally is the relation, in the ADE classification, between the simple Lie groups and the finite subgroups of SU(2): start with an N=2 super Yang Mills theory with gauge group a simple Lie group. Let it spontaneously find its vacuum and consider the orbit space of the remaining spontaneously broken symmetry group. That is (a resolution of) the orbifold quotient of $\mathbb{C}^2$ by a discrete subgroup of $SU(2)$.

## Related concepts

* [[McKay quiver]]

* [[ADE classification]]

* [[quiver gauge theory]]

* [[wrapped brane]]

* [[AGT correspondence]]

* [[3d-3d correspondence]]

* [[knots-quivers correspondence]]

## References

The original article is

* {#McKay80} [[John McKay]], _Graphs, singularities, and finite groups_ Proc. Symp. Pure Math. Vol. 37. No. 183. 1980

As an isomorphism between the [[equivariant K-theory]] of [[ADE-singularity]] and the plain [[topological K-theory]] of its [resolution](ADE+singularity#ResolutionBySpheresTouchingAlongADynkinDiagram), the McKay correspondence is proven in:

* {#GSV83} [[Gérard Gonzalez-Sprinberg]], [[Jean-Louis Verdier]], _Construction géométrique de la correspondance de McKay_, Ann. Sci. ́École Norm. Sup. **16** 3 (1983) 409–449 ([numdam:ASENS_1983_4_16_3_409_0](http://www.numdam.org/item?id=ASENS_1983_4_16_3_409_0))

An analogous discussion for [[derived categories]] of coherent sheaves is in

* [[Tom Bridgeland]], [[Alastair King]], [[Miles Reid]], _The McKay correspondence as an equivalence of derived categories_, J. Amer. Math. Soc. 14 (2001) 535--554  ([pdf](https://www.ams.org/journals/jams/2001-14-03/S0894-0347-01-00368-X/S0894-0347-01-00368-X.pdf) [doi](https://doi.org/10.1090/S0894-0347-01-00368-X))

See also:

* [[Duiliu-Emanuel Diaconescu]], Mauro Porta, Francesco Sala, _McKay correspondence, cohomological Hall algebras and categorification_, Represent. Theory **27** (2023) 933-972 [[arXiv:2004.13685](https://arxiv.org/abs/2004.13685), [doi:10.1090/ert/649](https://doi.org/10.1090/ert/649)]

* J. Denef, F. Loeser, _Motivic integration, quotient singularities and the McKay correspondence_, Compositio Math. 131 (2002) 267--290

* V. Batyrev, D. Dais, _Strong McKay correspondence, string-
theoretic Hodge numbers and mirror symmetry_, Topology __35__ (1996) 901--929 alg-geom/9410001

Introductions and surveys include

* Graham Leuschke, _The McKay correspondence_ ([pdf](http://www.leuschke.org/uploads/McKay-total.pdf))

* Bockland, _Character tables and McKay quivers_ ([pdf](https://staff.fnwi.uva.nl/r.r.j.bocklandt/notes/kleinian.pdf))

* Drew Armstrong, _Lectures on the McKay correspondence_, 2015 ([pdf scan of notes](http://www.math.miami.edu/~armstrong/Talks/McKay_Talca.pdf))

* Max Lindh, _An Introduction to the McKay Correspondence
Master Thesis in Physics_ ([pdf](http://www.diva-portal.org/smash/get/diva2:1184051/FULLTEXT01.pdf))

* [[John Baez]], _The Geometric McKay Correspondence_, ([Part I](https://golem.ph.utexas.edu/category/2017/06/the_geometric_mckay_correspond.html), [Part II](https://golem.ph.utexas.edu/category/2017/07/the_geometric_mckay_correspond_1.html))

Literature collection:

* [[Miles Reid]], _[Links to papers on McKay correspondence](http://homepages.warwick.ac.uk/staff/Miles.Reid/McKay/)_


McKay quiver

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

#Contents#
* table of contents
{:toc}

## Idea

Generally, for $G$ a [[finite group]] and $V$ a [[linear representation]] of $G$ on a [[finite dimensional vector space|finite dimensional]] [[complex vector space]], the _McKay quiver_ or _McKay graph_ associated with $V$ is the [[quiver]] whose [[vertices]] correspond to the [[irreducible representations]] $\rho_i$ of $G$ and which has $a_{i j} \in \mathbb{N}$ [[edges]] between the $i$th and the $j$th vertex, for $a_{i j}$ the [[coefficients]] in the expansion into [[irreps]] of the [[tensor product of representations]] of $V$ with these irreps:

$$
  V \otimes \rho_i
  \;\simeq\;
  \underset{j}{\bigoplus}
  a_{i j} \cdot \rho_j
  \,.
$$

Specifically this applies to the special case where $G \subset $ [[SU(2)]] a [[finite subgroup of SU(2)]] and $V$ its defining representation on $\mathbb{C}^2$. The [[McKay correspondence]] states that in this case the corresponding McKay quivers are [[Dynkin quivers]]/[[Dynkin diagrams]] in the same [[ADE classification]] as the [[ADE singularity]] $\mathbb{C}^2 \sslash G$.

More precisely: If one uses _all_ [[irreducible representations]] including the 1-dimensiona [[trivial representation]] $\rho_0$ then one gets the "extended Dynkin diagram", where the extra node corresponds to $\rho_0$. This is the vertex indicated by a cross in the following diagrams:

\begin{center}
<img src="https://ncatlab.org/nlab/files/ExtendedADEQuivers.jpg" width="400">
\end{center}

> graphics grabbed from [GSV 83, p. 4](#GSV83)

In particular, for $G =\mathbb{Z}_N \subset SU(2)$ a [[cyclic group]] of [[order of a group|order]] $N$, there are $N$ complex [[irreps]] and the McKay quiver, i.e. the extended Dynkin diagram, has $N$-vertices, connected by edges to form a circle.

## Related concepts

* [[Dynkin quiver]]

* [[McKay correspondence]]

* [[quiver gauge theory]]

## References

The construction is due to

* {#McKay80} [[John McKay]], _Graphs, singularities, and finite groups_ Proc. Symp. Pure Math. Vol. 37. No. 183. 1980

Interpretation in terms of [[equivariant K-theory]] of the corresponding [[ADE-singularity]] and plain [[K-theory]] of its [[blowup]] is due to

* {#GSV83} [[Gérard Gonzalez-Sprinberg]], [[Jean-Louis Verdier]], p. 4 fo _Construction géométrique de la correspondance de McKay_, Ann. Sci. ́École Norm. Sup.16 (1983) 409–449. ([numdam](http://www.numdam.org/item?id=ASENS_1983_4_16_3_409_0))

See also

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

[[!redirects McKay quivers]]

[[!redirects McKay graph]]
[[!redirects McKay graphs]]


mdata

I'm Matias Data. I'm one of [Eduardo](Eduardo+Dubuc)'s students.


meager set

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

# Meager set
* table of contents
{: toc}

## Definition

A [[topological subspace|subset]] $A\subseteq X$ of a [[topological space]] $X$ is called **meager** (alias **first category**, often **meagre** outside the USA) if it is a [[countable set|countable]] [[union]] of [[nowhere dense set|nowhere dense]] [[topological subspace|subsets]] of $X$.

## Related pages

* [[nowhere dense subspace]]
* [[Baire category theorem]]

[[!redirects meager set]]
[[!redirects meager sets]]
[[!redirects meagre set]]
[[!redirects meagre sets]]
[[!redirects set of first category]]
[[!redirects sets of first category]]
[[!redirects set of the first category]]
[[!redirects sets of the first category]]

[[!redirects meager subset]]
[[!redirects meager subsets]]
[[!redirects meagre subset]]
[[!redirects meagre subsets]]
[[!redirects subset of first category]]
[[!redirects subsets of first category]]
[[!redirects subset of the first category]]
[[!redirects subsets of the first category]]

[[!redirects meager subspace]]
[[!redirects meager subspaces]]
[[!redirects meagre subspace]]
[[!redirects meagre subspaces]]
[[!redirects subspace of first category]]
[[!redirects subspaces of first category]]
[[!redirects subspace of the first category]]
[[!redirects subspaces of the first category]]


Mealy machine

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

#Contents#
* table of contents
{:toc}

## Idea

A _Mealy machine_ (named after [Mealy 1955](#Mealy55)) -- traditionally understood as a particular type of [[finite state automaton]] -- is a [[map]] (traditionally thought of as a [[pair]] of maps, see Def. \ref{DefinitionInComponents} below) which takes input data *and* a given "state" to output data *and* an update of that "state". In modern language of [[monadic computational effects]] this just means that a Mealy machine is a *[[state monad|stateful]]-map*, namely a [[Kleisli morphisms]] for a [[state monad]] (see Rem. \ref{MealyMachinesAsEffectfulMaps}  below).

## Definition

### Original definition

\begin{definition}
\label{DefinitionInComponents}
**(traditional component definition)**
\linebreak
A (finite) _Mealy machine_ consists of ([[finite sets|finite]]) [[sets]]

* $S$, of _states_;

* $I$, the _input alphabet_;

* $O$, the _output alphabet_;

and [[functions]]

* $trans \colon S \times I \to S$, a _transition function_;

* $out \colon S \times I \to O$, an _output function_

and often an [[element]]

* $s_0 \in S$, the _initial state_.

\end{definition}

\begin{remark}\label{MealyMachinesAsEffectfulMaps}
**(Mealy machines as effectful maps)**
\linebreak
More concisely, a Mealy machine as in Def. \ref{DefinitionInComponents} is (except for the specification of the initial state) just a [[map]] between [[Cartesian products]] of this form:

\[
  \label{MealyMachineAsMapBetweenProducts}
  (out, trans)
  \;\colon\;
  I \times S \longrightarrow O \times S
\]

As such this makes sense [[internalization|in]] any [[cartesian monoidal category]] other than [[Sets]].

In a *[[cartesian closed monoidal category|closed]]* cartesian monoidal category (such as [[Sets]]) with [[internal hom]] $[\text{-},\text{-}]$, such a map (eq:MealyMachineAsMapBetweenProducts) is equivalent to (namely: is the "[[currying|curried]]" [[internal hom]]-[[adjunct]] of) a map of the form

$$
  S \longrightarrow \big[I,\, O \times S\big]
  \,.
$$

In this form Mealy machines are discussed as [[coalgebras of an endofunctor]], for instance in [[Pattinson 2003, 1.1.3](#Pattinson03)], [[Bonsangue, Rutten & Silva 2008, p. 1](BonsangueRuttenSilva08)], [[Ghica, Kaye & Sprunger 2022, Def. 25](#GhicaKayeSprunger22)].

But of course, by the same token  (eq:MealyMachineAsMapBetweenProducts) is also [[adjunct]] to a map of the form

\[
  \label{MealyMachineAsStateEffectufulFunction}
  I \longrightarrow \big[S ,\, S \times O \big]
  \,
\]

which makes more manifest how a Mealy machine is a map sending input $i \in I$ to output $o \in O$ after also reading out a stated $s \in S$ and then also updating that state.

In fact, in this form (eq:MealyMachineAsStateEffectufulFunction) Mealy machines are exactly the *$S$-effectful maps* in these sense of [[monadic computational effects]], namely the [[Kleisli morphisms]] for the $S$-[[state monad]] (mentioned as such for instance in [Oliveira & Miraldo 2016, p. 462](#OliveiraMiraldo16)).
For discussion of this perspective in [[Haskell]]: [[github.com/orakaro/MonadicMealyMachine](https://github.com/orakaro/MonadicMealyMachine)], [[Perone & Karachalias 2023, p. 3](#PeroneKarachalias23)].
\end{remark}

### As spans in monoidal categories
 {#AsSpans}

Instead of regarding the pair given by the transition- and output function of a Mealy machine (Def. \ref{DefinitionInComponents}) as a single map from $I \times S$ into the [[Cartesian product]] $O \times S$ (eq:MealyMachineAsMapBetweenProducts), one may, of course, regard them as a [[span]]:

\begin{tikzcd}[sep=20pt]
  & I \times S
  \ar[dl, "{ \mathrm{out} }"{sloped}]
  \ar[dr, "{ \mathrm{trans} }"{sloped}]
  \\
  O && S
  \mathrlap{\,.}
\end{tikzcd}

In this form, a [[Cartesian product]] is not needed to pair these two operations, and hence it is tempting to consider a generalized notion of Mealy machines in (not-necessarily [[cartesian monoidal category|cartesian]]) [[symmetric monoidal categories]] $(\mathcal{C}, \otimes, \mathbb{1})$, given by [[spans]] of the above form, but using the given [[monoidal product]] to pair what are now $\mathcal{C}$-objects $I$ and $S$:

\begin{tikzcd}[sep=20pt]
  & I \otimes S
  \ar[dl, "{ \mathrm{out} }"{sloped}]
  \ar[dr, "{ \mathrm{trans} }"{sloped}]
  \\
  O && S
  \mathrlap{\,.}
\end{tikzcd}

This definition is considered for instance in [BLLL23a, Def. 2.1](#BLLL23a).

An evident notion of [[homomorphisms]] between such spans are maps $f \colon S \to S'$ such that the following [[commuting diagram|diagram commutes]] ([BLLL23a, Rem. 2.2](#BLLL23a)):

\begin{tikzcd}[sep=20pt]
  & I \otimes S
  \ar[dl, "{ \mathrm{out} }"{sloped}]
  \ar[dr, "{ \mathrm{trans} }"{sloped}]
  \ar[dd, "{ I \otimes f }"{description}]
  \\
  O
  \ar[dd, equals]
  &&
  S
  \ar[dd, "{f}"]
  \\
  &
  I \otimes S'
  \ar[dl, "{ \mathrm{out}' }"{sloped}]
  \ar[dr, "{ \mathrm{trans}' }"{sloped}]
  \\
  O && S'
\end{tikzcd}

The authors of [[BLLL23a, Prop. 3.5](#BLLL23a)] [[BLLL23b, Def. 2.1](#BLLL23b)] find it useful to rephrase this as follows:

\begin{definition}
\label{FoscoDefinition}
A Mealy machine internal to a [[symmetric monoidal category]]
$(\mathcal{C},\,\otimes,\,\mathbb{1})$ for

* input alphabet object $I \,\in\, Ob(\mathcal{C})$

* output alphabet object $O \,\in\, Ob(\mathcal{C})$

is an object of the following [[pullback]] $Mly(I,O)$ in [[Cat]]:

\begin{centre}
\begin{tikzcd}
  \mathrm{Mly}(I,O)
  \ar[r]
  \ar[d]
   &
  (-\otimes I)/O
  \ar[d, "V"]
  \\
  \mathrm{Alg}(-\otimes I)
  \ar[r, "U"']
  &
  \mathcal{C}
  \mathrlap{\,,}
\end{tikzcd}
\end{centre}

where:

* $Alg(-\otimes I)$ is the category of [[algebra over an endofunctor|algebras over]] the [[endofunctor]]

  $$
    \array{
      \mathllap{
        (\text{-})\otimes I \,\colon\;
      }
      \mathcal{C}
        &\longrightarrow&
      \mathcal{C}
      \\
      S &\mapsto& S\otimes I
     }
   $$

* $U$ is the forgetful functor,

* $(-\otimes I)/O$ is the [[comma category]] of morphisms $out \colon  X\otimes I\to O$,

* $V$ is the forgetful functor $(X,out)\mapsto X$.

\end{definition}

\begin{remark}

If $\mathcal{C}$ has [[countable set|countable]] [[coproducts]], a semiautomaton $trans \colon S\otimes I \to S$ consists of an [[action#actions_of_a_monoid|action]] of the [[free monoid]] on $I$, $\sum_{n=0}^\infty I^{\otimes n}$; this is well-explained [[action#actions_of_a_set|_ibi_]].

Concisely, there is an [[equivalence of categories]]
\[
  Alg(-\otimes I) \simeq EM(I^*)
\]
between $I$-semiautomata and the [[Eilenberg-Moore category]] of the monad [[action monad|induced]] by the monoid $I^*$.
\end{remark}

Then, a Mealy machine as in Def. \ref{FoscoDefinition} consists indeed of a [[span]]
\[
  S \leftarrow S\otimes I \to O
  \,.
\]

## Related entries

* [[state monad]]

* [[Moore machine]]

* [[automata]]

* [[Mealy morphism]]

## References

The notion is due to:

* {#Mealy55} [[George H. Mealy]], *A Method for Synthesizing Sequential Circuits*, Bell System Technical Journal **34** 5 (1955) 1045-1079 [[doi:10.1002/j.1538-7305.1955.tb03788.x](https://doi.org/10.1002/j.1538-7305.1955.tb03788.x)]

For discussion in the context of [[finite state automata]] see:

* [[Mark V. Lawson]], *Finite automata*, CRC Press (2003) [[webpage](http://www.ma.hw.ac.uk/~markl/books.html), course notes:[pdf](http://www.ma.hw.ac.uk/~markl/teaching/AUTOMATA/kleene.pdf)]

Discussion via [[category theory]] and as [[coalgebras of an endofunctor]]

* {#Pattinson03} [[Dirk Pattinson]], Section 1.1.3 in: _An Introduction to the Theory of Coalgebras_ (2003) [[pdf](https://nasslli.sitehost.iu.edu/2003/datas/DirkPattinson.pdf), [[Pattinson-Coalgebras.pdf:file]]]

* {#BonsangueRuttenSilva08} M. M. Bonsangue, Jan Rutten & Alexandra Silva , *Coalgebraic Logic and Synthesis of Mealy Machines*,  in: *Foundations of Software Science and Computational Structures. FoSSaCS 2008*, Lecture Notes in Computer Science, **4962** (2008) [[doi:10.1007/978-3-540-78499-9_17](https://doi.org/10.1007/978-3-540-78499-9_17)]

* H. Hansen, Jan Rutten, *Symbolic synthesis of Mealy machines from arithmetic bitstream functions*, Scientific Annals of Computer Science **20** (2010) 97–130 [[pub:16639](https://ir.cwi.nl/pub/16639), [pdf](https://core.ac.uk/download/pdf/301659512.pdf)]

* {#GhicaKayeSprunger22} [[Dan R. Ghica]], George Kaye, David Sprunger, Def. 25 in: *A compositional theory of digital circuits* [[arXiv:2201.10456](https://arxiv.org/abs/2201.10456)]

Discussion as [[state monad|stateful]]-maps:

* {#OliveiraMiraldo16} José Nuno Oliveira, Victor Cacciari Miraldo, *A practical approach to state-based system calculi*, Journal of Logical and Algebraic Methods in Programming **85** 4 (2016) 449-474 [[doi:10.1016/j.jlamp.2015.11.007](https://doi.org/10.1016/j.jlamp.2015.11.007)]

* {#PeroneKarachalias23} Marco Perone, Georgios Karachalias, *Composable Representable Executable Machines* [[arXiv:2307.09090](https://arxiv.org/abs/2307.09090)]

Discussion in symmetric monoidal categories:

* {#BLLL23a} Guido Boccali, Andrea Laretto, [[Fosco Loregian]], Stefano Luneia, *Completeness for categories of generalized automata* [[arXiv:2303.03867](https://arxiv.org/abs/2303.03867)]

* {#BLLL23b} Guido Boccali, Bojana Femić, Andrea Laretto, [[Fosco Loregian]], Stefano Luneia, *The semibicategory of Moore automata* [[arXiv:2305.00272](https://arxiv.org/abs/2305.00272)]

[[!redirects Mealy machines]]


Mealy morphism

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#### Enriched category theory
+--{: .hide}
[[!include enriched category theory contents]]
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#### Categorical algebra
+-- {: .hide}
[[!include categorical algebra -- contents]]
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# Contents
* table of contents
{: toc}

## Idea

When we define [[internal categories]] of some category $\mathcal{C}$ as [[monads]] in the [[category of spans]] $\mathrm{Span}(\mathcal{C})$, there is a subtlety concerning the definition of [[internal functors]]. It is not the case that the morphisms of monads (namely [[colax morphism|colax]] monad morphisms) give us [[internal functors]] directly: we need to ask that the 1-cell of the morphism is a [[left adjoint]] in $\mathrm{Span}(\mathcal{C})$ (which corresponds to asking that the left leg of the span be the identity, or an isomorphism). (This is related to the [[profunctor#Properties|behaviour of profunctors]] in that they correspond to functors (via the [[Yoneda embedding]]) exactly when they are [[left adjoints]].)  If in considering morphisms of monads in $Span(\mathcal{C})$ we _don't_ require this left-adjoint condition then we end up constructing **Mealy morphisms** between internal categories.

'Mealy morphisms' are  named after [[Mealy machines]], which, in turn, were named after  [[George H. Mealy]].

## Definition

Let $\mathbf{A}$ and $\mathbf{B}$ be [[categories]] [[enriched category|enriched]] in a [[monoidal category]] $(\mathbf{V}, I, \otimes)$.

A *Mealy morphism* $(F, \phi) \colon \mathbf{A} \rightarrow \mathbf{B}$
consists of a [[pair]] of [[functions]] $F \colon Ob\mathbf{A} \rightarrow Ob\mathbf{B}$ and $\phi \colon \mathbf{A} \rightarrow \mathbf{V}$,
and for each [[pair]] of [[objects]] $A, A'$ in $\mathbf{A}$ a function
\[
\phi(A, A') \colon \mathbf{A}(A, A') \otimes \phi(A) \longrightarrow \phi(A') \otimes \mathbf{B}(FA, FA')
\]
satisfying the following commutative diagrams.
\begin{tikzcd}
I \otimes \phi(A)
\arrow[d, "\cong"']
\arrow[r, "{j_{A} \otimes 1}"]
& \mathbf{A}(A, A) \otimes \phi(A)
\arrow[d, "{\phi(A, A)}"]
\\
\phi(A) \otimes I
\arrow[r, "{1 \otimes j_{FA}}"']
& \phi(A) \otimes \mathbf{B}(FA, FA)
\end{tikzcd}
\begin{tikzcd}
\mathbf{A}(A', A'') \otimes \mathbf{A}(A, A') \otimes \phi(A)
\arrow[r, "{\circ_{A, A', A''} \otimes \phi(A)}"]
\arrow[d, "1 \otimes {\phi(A, A')}"']
&[+3em] \mathbf{A}(A, A'') \otimes \phi(A)
\arrow[dd, "{\phi(A, A'')}"]
\\
\mathbf{A}(A', A'') \otimes \phi(A') \otimes \mathbf{B}(FA, FA')
\arrow[d, "{\phi(A', A'') \otimes 1}"']
&
\\
\phi(A'') \otimes \mathbf{B}(FA', FA'') \otimes  \mathbf{B}(FA, FA')
\arrow[r, "1 \otimes \circ_{FA, FA', FA''}"']
& \phi(A'') \otimes \mathbf{B}(FA, FA'')
\end{tikzcd}

## Properties

- There is a [[bicategory]] of small $V$-enriched categories, Mealy morphisms, and Mealy cells. It is the [[Kleisli bicategory]] for the [[free cocompletion]] of a small $V$-category under [[copowers]]. This is analogous to the universal property of the bicategory [[Prof]] of [[profunctors]] as the Kleisli bicategory for the free cocompletion of a small category under [[small colimits]]. This is proven in [Paré 2012](#Pare12).

## References

The notion of Mealy morphism between [[enriched categories]] was introduced in the paper:

* {#Pare12} [[Robert Paré]], _Mealy Morphisms of Enriched Categories_, Applied Categorical Structures **20** (2012) pp 251--273, doi:[10.1007/s10485-010-9238-8](https://doi.org/10.1007/s10485-010-9238-8).

Mealy morphisms are mentioned in Example 16.8 and Example 16.27 in the paper:

* {#GarnerShulman16} [[Richard Garner]], [[Michael Shulman]], _Enriched categories as a free cocompletion_, Advances in Mathematics, 289, 2016 ([arXiv:1301.3191](https://arxiv.org/abs/1301.3191), [doi:10.1016/j.aim.2015.11.012](https://doi.org/10.1016/j.aim.2015.11.012))

Mealy morphisms are also considered under the name "two-dimensional partial map" (a notion attributed to [[Lawvere]]) in the Appendix of the paper:

* {#LackStreet02} [[Stephen Lack]], [[Ross Street]], _The formal theory of monads II_, Journal of Pure and Applied Algebra, 175, 2002 ( [doi:10.1016/S0022-4049(02)00137-8](https://doi.org/10.1016/S0022-4049%2802%2900137-8))

Some talk slides which mention Mealy morphisms:

* [[Robert Paré]], [Mealy Morphisms for Tensor Categories](https://www.mscs.dal.ca/~pare/June2010.pdf)

[[!redirects Mealy morphisms]]


mean

## Related concepts

* [[mean value theorem]]

* [[expectation value]]

## References

' Wikipedia, _[Mean](https://en.wikipedia.org/wiki/Mean)_

[[!redirects means]]


mean value theorem

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#### Differential geometry
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# The Mean Value Theorem
* table of contents
{: toc}

## Idea {#idea}

The motivating idea of the [[derivative]] in [[differential calculus]] is that it is approximated by a [[ratio]] of differences.  We may also say that an instantaneous rate of change is approximated by an average rate of change.  The Mean Value Theorem (MVT) reverses this, and says that any average rate of change is equal to some instantaneous rate of change, if certain differentiability conditions are met.

The name comes from the fact that, due to the [[fundamental theorem of calculus]], an average rate of change over an [[interval]] may be viewed as an average (or [[mean]]) of the instantaneous rates of change along the interval.  Thus, the theorem states that the mean value of the derivative on an interval is attained somewhere in that interval.

## Statements {#statement}

There are traditionally three versions of increasing generality, although even the most general version is implicit in the most specific version (requiring only a linear coordinate transformation).

+-- {: .num_theorem #rolle}
###### Rolle\'s Theorem

Suppose that $a \lt b$ are [[real numbers]] and $f$ is a [[continuous map|continuous]] [[real number|real]]-valued [[function]] on $[a,b]$.  If $f$ is [[differentiable function|differentiable]] on the [[interior]] $]{a,b}[$, and if $f(a) = f(b)$, then for some $c \in {]{a,b}[}$,
$$ f'(c) = 0 .$$
=--

+-- {: .num_theorem #lagrange}
###### Lagrange\'s Theorem

Suppose that $a \lt b$ are [[real numbers]] and $f$ is a [[continuous map|continuous]] [[real number|real]]-valued [[function]] on $[a,b]$.  If $f$ is [[differentiable function|differentiable]] on the [[interior]] $]{a,b}[$, then for some $c \in {]{a,b}[}$,
$$ f'(c) = \frac {f(b) - f(a)} {b - a} ,$$
or equivalently
$$ f'(c) (b - a) = f(b) - f(a) .$$
=--

+-- {: .num_theorem #cauchy}
###### Cauchy\'s Theorem

Suppose that $a \lt b$ are [[real numbers]] and $f$ and $g$ are [[continuous map|continuous]] real-valued [[functions]] on $[a,b]$.  If $f$ and $g$ are [[differentiable function|differentiable]] on the interior $]{a,b}[$, then for some $c \in {]{a,b}[}$,
$$ f'(c) (g(b) - g(a)) = g'(c) (f(b) - f(a)) ;$$
assuming that $f'$ and $g'$ are never simultaneously zero in $]{a,b}[$ and that $(f(a),g(a)) \neq (f(b),g(b))$, then for some $c \in {]{a,b}[}$,
$$ \frac {f'(c)} {g'(c)} = \frac {f(b) - f(a)} {g(b) - g(a)} ,$$
where either side of this equation is allowed to be interpreted as $\infty$ in case it is division by zero (necessarily with a nonzero dividend under these conditions); or perhaps better, an equality of [[ratio]]s:
$$ f'(c) : g'(c) :: f(b) - f(a) : g(b) - g(a) .$$
=--

If we write $u$ for $f(x)$ and $v$ for $g(x)$, then this last version states that
$$ \left.{\frac{\mathrm{d}u}{\mathrm{d}v}}\right|_{x=c} = \left.{\frac{\Delta{u}}{\Delta{v}}}\right|_{x=a}^b .$$
Compare this to the definition
$$ \left.{\frac{\mathrm{d}u}{\mathrm{d}v}}\right|_{x=a} \coloneqq \lim_{b\to{a}} \left.{\frac{\Delta{u}}{\Delta{v}}}\right|_{x=a}^b $$
(although this is really only a *definition* when $v$ is $x$, which reduces Cauchy\'s theorem to Lagrange\'s).

## Naming

Rolle\'s theorem is usually called just 'Rolle\'s' theorem, being the only result attributed today to [[Michel Rolle]]; but Lagrange\'s and Cauchy\'s theorems must be called 'mean value' theorems, as [[Joseph-Louis Lagrange]] and [[Augustin-Louis Cauchy]] did far more.  By default, the term 'Mean Value Theorem' usually refers to Lagrange\'s theorem.  (But neither Rolle nor Lagrange proved their theorem in the general case; the first proofs of all of them are due to Cauchy in 1823, a decade after Lagrange\'s death and more than a century after Rolle\'s death.)

## Mean-value inequalities

One consequence of these mean-value theorems if that if the relevant derivatives (or ratios of derivatives) are bounded, then the corresponding differences (or ratios of differences) will also be bounded.  We state this for Lagrange\'s theorem, although there are versions that correspond more to Rolle\'s or Cauchy\'s.

+-- {: .num_theorem #inequality}
###### The Mean-Value Inequality aka the Law of Bounded Change

Suppose that $a \lt b$ are [[real numbers]] and $f$ is a [[continuous map|continuous]] [[real number|real]]-valued [[function]] on $[a,b]$.  If $f$ is [[differentiable function|differentiable]] on the [[interior]] $]{a,b}[$, and we have $m \leq f' \leq M$ on $]{a,b}[$ for some constants $m$ and $M$, then
$$ m \leq \frac {f(b) - f(a)} {b - a} \leq M .$$
A slightly weaker statement is
$$ {|{f(b) - f(a)}|} \leq {|{b - a}|} \, \sup_{]{a,b}[} {|{f'}|} ,$$
which is true even if the derivative is unbounded (in which case the right-hand side is [[infinity|infinite]]).
=--

In [[constructive mathematics]], the mean-value theorems generally cannot be proved, since it may be impossible to find the value $c$ (although some variations with stronger hypotheses or weaker conclusions can often be proved, similarly to the [[Intermediate-Value Theorem]]).  However, the mean-value inequality is true in constructive mathematics, as long as $f$ is [[uniformly differentiable map|uniformly differentiable]] on every closed subinterval of $]{a,b}[$ (as is typical).

The second form of the mean-value inequality shows the relationship of differentiability to [[Lipschitz continuity]]: a continuous function on an interval with bounded derivative on the interior of the interval is Lipschitz continuous on that interval (and the [[supremum]] of the [[absolute value]] of the [[derivative]] is the [[Lipschitz constant]]).

## References {#refs}

* Wikipedia, _[Mean value theorem](http://en.wikipedia.org/wiki/Mean_value_theorem)_

* nForum discussion of [constructive versions](https://nforum.ncatlab.org/discussion/5574/constructive-mean-value-theorem/)

[[!redirects mean value theorem]]
[[!redirects mean value theorems]]
[[!redirects Mean Value Theorem]]
[[!redirects Mean Value Theorems]]
[[!redirects mean-value theorem]]
[[!redirects mean-value theorems]]
[[!redirects Mean-value Theorem]]
[[!redirects Mean-value Theorems]]
[[!redirects Mean-Value Theorem]]
[[!redirects Mean-Value Theorems]]
[[!redirects MVT]]

[[!redirects Rolle's theorem]]
[[!redirects Rolle theorem]]
[[!redirects Rolle's Theorem]]
[[!redirects Rolle Theorem]]

[[!redirects mean value inequality]]
[[!redirects mean value inequalities]]
[[!redirects Mean Value Inequality]]
[[!redirects Mean Value Inequalities]]
[[!redirects mean-value inequality]]
[[!redirects mean-value inequalities]]
[[!redirects Mean-value Inequality]]
[[!redirects Mean-value Inequalities]]
[[!redirects Mean-Value Inequality]]
[[!redirects Mean-Value Inequalities]]


Meaning and Necessity

* [[Rudolf Carnap]]

_Meaning and Necessity_

[Wikipedia](http://en.wikipedia.org/wiki/Meaning_and_Necessity)

##Related entries

* [[modal logic]]
* [[Naming and Necessity]]

category: reference