motivic chromatic homotopy theory

## Idea

[[chromatic homotopy theory]] in the context of [[motives]].

## Related concepts

* [[motives]]
  * [[mixed motives]]
  * [[motivic homotopy theory]]
    * **chromatic motivic homotopy theory**

## References

* Jens Hornbostel, _Motivic Chromatic Homotopy Theory_ ([web](http://www.math.uiuc.edu/K-theory/0642/))

[[!redirects chromatic motivic homotopy theory]]


motivic coarse space

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## Grothendieck topology on bornological coarse spaces

The category of [[bornological coarse spaces]] admits
the structure of a [[site with an interval]].
See §3 in Bunke and Engel for the relevant definitions.

## Definition

The $\infty$-category of motivic coarse spaces is obtained
from the $\infty$-category of $\infty$-sheaves on the site of [[bornological coarse spaces]]
by performing the reflective localization with respect to the maps $I\times X\to X$
for any representable $X$, where $I=\{0,1\}$ is the interval object in bornological coarse spaces.
(This step is formally identical to the one used to construct [[motivic spaces]].)
Next, we also localize with respect to morphisms of representables $\emptyset\to E$,
where $E$ is a flasque bornological coarse space, as defined in Definition 3.19 of Bunke and Engel.
Finally, we localize with respect to morphisms of presheaves of the form $colim_{U\in C}X_U\to X$,
for any $C\in C$, where $X_U$ is obtained from $X$ using the same bornology
and the coarse structure generated by a single element $U$.

## References

* {#BunkeEngel} [[Ulrich Bunke]], [[Alexander Engel]],
_Homotopy theory with bornological coarse spaces_,
[arXiv:1607.03657](http://arxiv.org/abs/1607.03657).

[[!redirects motivic coarse spaces]]


motivic coarse spectrum

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## Definition

The $\infty$-category of motivic coarse spectra
is the [[stabilization]] of the $\infty$-category
of [[motivic coarse spaces]].

## References

* {#BunkeEngel} [[Ulrich Bunke]], [[Alexander Engel]],
_Homotopy theory with bornological coarse spaces_,
[arXiv:1607.03657v1](http://arxiv.org/abs/1607.03657v1).


motivic cohomology

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#### Cohomology
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## Idea ##

Motivic cohomology is a cohomology theory for schemes which in many ways plays the rôle of [[singular cohomology]] in algebraic geometry. It was first conjectured to exist by [[Alexander Beilinson]] and [[Stephen Lichtenbaum]] in the mid 1980s, and it was then defined by [[Vladimir Voevodsky]] in the mid 1990s.

Motivic cohomology must not be confused with the hypothetical "universal" cohomology theory envisioned by [[Alexander Grothendieck]] in the 1960s as the underlying reason for the [[standard conjectures on algebraic cycles]]. The former is an *absolute cohomology theory* with values in abelian groups, while the latter is a *geometric cohomology theory* with values in the still conjectural abelian category of [[mixed motives]]. They are related in that motivic cohomology with _rational_ coefficents should appear as particular [[Ext-group]]s in the category of motives, an idea which can now be made precise using the various existing constructions of the [derived category of motives](http://ncatlab.org/nlab/show/motive#DerivedMotives).

The motivic cohomology groups of a scheme $X$ form a bigraded family of abelian groups $H^{p,q}(X,\mathbb{Z})$. Several competing definitions of these groups exist but they are known to all agree when $X$ is [[smooth scheme|smooth]] over a [[field]]. With rational coefficients, the motivic cohomology groups of $X$ are the associated graded of the $\gamma$-filtration on the rational [[algebraic K-theory]] groups of $X$ (at least if $X$ is [[regular scheme|regular]]). With coefficients in $\mathbb{Z}/p$, they are closely related to the [[étale cohomology]] of $X$ with coefficients in the sheaf $\mu_p$ of $p$th [[roots of unity]] (if $p$ is invertible on $X$) and to the logarithmic de Rham-Witt cohomology of $X$ (if $p$ equals the characteristic of $X$).

The motivic cohomology of a sufficiently nice scheme $X$ is also related to the [[algebraic K-theory]] of $X$ via the _motivic spectral sequence_
$$ H^{*,*}(X,\mathbb{Z}) \Rightarrow K_*(X). $$
which degenerates rationally. The search for this spectral sequence was one of the motivating factor in the development of motivic cohomology. More generally, spectral sequences whose first page consists of motivic cohomology groups exist for any cohomology theory represented by a [[motivic spectrum]]; they are analogous to the [[Atiyah-Hirzebruch spectral sequence]]s in topology.

## Definitions

We give three definitions of motivic cohomology with integral coefficients, in historical order: the first, due to Bloch and later generalized by Levine, only works for [[smooth schemes]] over [[Dedekind domain]]s. The other two, due to Voevodsky, work for arbitrary schemes. All definitions are known to agree for [[smooth scheme]]s over [[field]]s, but the equivalence of any pair of them is an open question for more general schemes. It is generally accepted that the Bloch–Levine definition produces the desired motivic cohomology groups as far as it applies, but there is no consensus beyond that.

Note that in each definition motivic cohomology is *absolute*: the groups $H^{p,q}(X,\mathbb{Z})$ depend only of the scheme $X$ and not on any base scheme.

### As Bloch's higher Chow groups {#HigherChowGroups}

The first and most elementary definition of motivic cohomology groups was Bloch's definition of higher Chow groups ([Bloch](#Bloch86)), although they were only recognized as such later by Voevodsky.

Let $X$ be a [[smooth scheme]] over a [[field]] $k$. The group $z^*(X)$ of _algebraic cycles_ on $X$ is the free abelian group generated by the irreducible closed subschemes of $X$, graded by codimension.

The **algebraic $n$-simplex** $\Delta^n$ is the $k$-scheme

$$\Delta^n = Spec(k[t_0, \dots, t_n]/(\sum_i t_i - 1)). $$

Note that $\Delta^n$ is isomorphic to affine $n$-space $\mathbb{A}^n$. There are obvious coface and codegeneracy maps that turn $\Delta^\bullet$ into a [[cosimplicial object|cosimplicial]] $k$-scheme. The graded [[simplicial abelian group]] $z^*(X,\bullet)$ is the subgroup of $z^*(X\times\Delta^\bullet)$ generated in simplicial degree $n$ by the cycles which intersect all faces $X\times\Delta^m \subset X\times\Delta^n$ properly. One then defines the **higher Chow groups** $CH^*(X,n)$ by

$$ CH^*(X,n) = \pi_n(z^*(X, \bullet)). $$

The groups $CH^*(X,0)$ are the ordinary [[Chow groups]] of algebraic cycles modulo rational equivalence.

Voevodsky proved that these groups agree with [his definition](#VoevodskysDef) of motivic cohomology under the re-indexing

$$ H^{p,q}(X,\mathbb{Z}) = CH^{q}(X,2q-p) .$$

Levine extended Bloch's definition to smooth schemes over [[Dedekind domains]] in such a way that motivic cohomology supported at a prime fits in the expected long exact sequence ([Levine](#Levine01)).

#### References

* [[Spencer Bloch]], _Algebraic Cycles and Higher $K$-Theory_, Adv. Math. 61 (1986), pp. 267–304
{#Bloch86}

* [[Marc Levine]], _Techniques of localization in the theory of algebraic cycles_, J. Algebraic Geom. 10 (2001), pp. 299–363, [pdf](http://www.uni-due.de/~bm0032/publ/TechLoc.pdf)
{#Levine01}

### As Zariski cohomology of certain complexes of sheaves {#VoevodskysDef}

In the mid 1990s [[Vladimir Voevodsky]] gave the first "official" definition of the motivic cohomology of a scheme $X$ as the [[hypercohomology]] of certain [[complexes]] of [[sheaves]] $\mathbb{Z}(q)$ on the [[Zariski site|Zariski]] [[site]] of $X$ (an analog of the [[category of open subsets]] of a [[topological space]]). The complexes $\mathbb{Z}(q)$, $q\geq 0$, are called the **motivic complexes**; the existence of such complexes was predicted as part of the so-called Beilinson dream.

+-- {: .num_defn}
###### Definition
The motivic cohomology of a [[scheme]] $X$ is the [[hypercohomology]] of the complexes of sheaves $\mathbb{Z}(q)$ on the Zariski site:

$$ H^{p,q}(X,\mathbb{Z}) := H^p_{Zar}(X,\mathbb{Z}(q)) $$
=--

This is [MaVoWe, Definition 3.4](#MaVoWe).

Voevodsky's definition, for smooth schemes over fields, has been shown to have most properties that Beilinson and Lichtenbaum had demanded of the hypothetical cohomology theory, except that to date it hasn't been shown that the cohomology groups vanish in negative degree, as they should. This open question is known as the _Beilinson vanishing conjecture_.

#### As hom-sets in the derived category of motives

Voevodsky also gave an accompanying definition of an integral version of the [[derived category]] of the hypothetical category of [[motive|mixed motives]] (see there for the definition) and showed that the motivic cohomology appears as derived hom-complexes in this derived category (see [MaVoWe, rop. 14.16](#MaVoWe) for a precise statement).

#### References ##

* [[Carlo Mazza]], [[Vladimir Voevodsky]] and [[Charles Weibel]], _Lectures in motivic cohomology_ ([web](http://math.rutgers.edu/~weibel/motiviclectures.html) [pdf](http://www.claymath.org/library/monographs/cmim02.pdf))
{#MaVoWe}

### As cohomology with coefficients in Eilenberg-Mac Lane objects {#ChowGroupsAsInfStackCohomology}

From the point of view of the [[motivic homotopy theory]] of Morel and Voevodsky, one would like the motivic cohomology of $X$ to be representable in the [[motivic homotopy theory|stable motivic homotopy category]] $SH(X)$ over $X$. Voevodsky gave a definition of motivic cohomology in this setting as the [bigraded cohomology theory](http://ncatlab.org/nlab/show/motivic+homotopy+theory#CohomologyTheories) represented by the _motivic Eilenberg–Mac Lane spectrum_ $H(\mathbb{Z})\in SH(X)$.

The motivic spectrum $H(\mathbb{Z})$ is built out of _motivic Eilenberg–Mac Lane spaces_ $K(\mathbb{Z}(n),2n)$. Below we only discuss the definition of these spaces over a [[field]] $k$. The definition in general is essentially the same, but it relies on the notion of finite correspondence over more general bases which is technical.

To define motivic Eilenberg-Mac Lane spaces, a first guess might be to apply the general definition of an [[Eilenberg-Mac Lane object]] in the [[Nisnevich (∞,1)-topos]] and then take its $\mathbb{A}^1$-localization. While this is an interesting construction, these spaces can only be assembled into an $S^1$-spectrum and we want a $\mathbb{P}^1$-spectrum. This is not easy: Voevodsky states in his [ICM-talk article](#ICM) (on p. 596) that every morphism $\mathbb{P}^1 \wedge K(\mathbb{Z},n) \rightarrow K(\mathbb{Z},n+1)$ is trivial in the $\mathbb{A}^1$-homotopy category.

Instead one applies a recipe which, when applied to the usual topological spheres produces the (topological) [[Eilenberg-Mac Lane spaces]], to the algebro-geometric sphere $\mathbb{P}^1$:

The [[Dold-Thom theorem]] says that in topology the reduced singular homology of a space $X$ can be produced as

$$H_i(X) = \pi_i (\mathrm{colim}_N\, \mathrm{Sym}^N X)^+$$

where $\mathrm{Sym}^N X = (X\times X\times ...\times X)/\Sigma_N$ is the free strictly commutative monoid on $X$ and $(-)^+$ denotes group completion. Inserting the topological n-sphere $S^n$ yields that $(\mathrm{Sym}^N X)^+$ is an Eilenberg-Mac Lane space.

The symmetric powers $\mathrm{Sym}^N$ also make sense for quasi-projective $k$-schemes, and they can be formally extended to pointed [[presheaves]] on such schemes. If $X$ is a pointed presheaf, we have maps $\mathrm{Sym}^N (X) \rightarrow \mathrm{Sym}^{N+1} (X)$ (lengthening an $N$-letter word by one, attaching the base point) and the colimit over these maps, followed by group completion, gives a functor $L(-):=(\mathrm{Sym}^{\infty} (-))^+$. Over a field of characteristic zero, one defines the *motivic Eilenberg-Mac Lane spaces* by

$$K(\mathbb{Z}(n),2n):=L((\mathbb{P}^1)^{\wedge n}).$$

These assemble to give the *motivic Eilenberg-Mac Lane spectrum* $H(\mathbb{Z}):=(\mathbb{Z},K(\mathbb{Z}(1),2),K(\mathbb{Z}(2),4),\ldots)$ with bonding maps induced by $\mathbb{P}^1 \wedge \mathrm{Sym}^N ({\mathbb{P}^1}^{\wedge m}) \rightarrow \mathrm{Sym}^N ({\mathbb{P}^1}^{\wedge m+1}), (x, \sum x_i) \mapsto \sum (x,x_i)$ (i.e. take the extra $\mathbb{P}^1$-point as new coordinate in the bigger $\wedge$-product of $\mathbb{P}^1$s).

This definition does not quite work over fields of positive characteristic. In general one has to take cycles as described by Denis-Charles Cisinski below. Intuitively the points of $(\mathrm{Sym}^{\infty} (X))^+$ are finite formal sums of points of X, i.e. zero-cycles, which links this story to the functor $L$ described below. In characteristic zero both coincide. The link to higher Chow groups however only becomes apparent in the cycle description.

> the following paragraphs are due to [[Denis-Charles Cisinski]], taken from [this MathOverflow thread](http://mathoverflow.net/questions/2520/homotopy-theory-of-schemes-examples).

To keep things simple, let us assume we work over a [[perfect field]] $k$. The easiest part of motivic cohomology which we can get is the [[Picard group]] (i.e. the [[Chow group]] in degree 1). This works essentially like in [[Top]]: in the [[model structure on simplicial presheaves|(model) category of simplicial Nisnevich sheaves]] (over smooth $k$-schemes), the classifying space of the multiplicative group $\mathbb{G}_m := \mathbb{A}^1 - \{0\}$  has the $\mathbb{A}^1$-[[homotopy n-type|homotopy type]] of the infinite dimensional projective space.

Moreover, as the Picard group is homotopy invariant for regular schemes (semi-normal is even enough), the fact that $H^1(X,\mathbb{G}_m) = Pic(X)$ reads as

$$
  [X,\mathbf{B} \mathbb{G}_m]
  =
  Pic(X)
  =
  CH^1(X)
  \,
$$

where $[-,-]$ are the hom-sets in the [[motivic homotopy category]] $\mathrm{H}(k)$.

In general, we denote by $K(\mathbb{Z}(n),2n)$ the $n$-th motivic
[[Eilenberg-Mac Lane object]], i.e. the object of $\mathrm{H}(k)$ which represents the $n$-th [[Chow group]]: for any smooth $k$-scheme $X$, one has

$$
  [\Sigma^i X , K(\mathbb{Z}(n), 2n)]
  =
  H^{2n -i}(X, \mathbb{Z}(n))
  \,.
$$

There are several models for $K(\mathbb{Z}(n),2n)$, one of the smallest being constructed as follows. What is explained above is that $K(\mathbb{Z}(1),2)$ is the infinite projective space. $K(\mathbb{Z}(0),0)$ is simply the constant sheaf. For higher $n$, here is the following construction due to Voevodsky.

Given a $k$-scheme $X$, denote by $L(X)$ the [[presheaf with transfers]] associated to $X$, that is the presheaf of abellian groups whose sections over a smooth $k$-scheme $V$ are the finite [[correspondence]]s from $V$ to $X$ (i.e. the finite linear combinations of cycles $\sum n_i Z_i$  in $V \times X$ such that $Z_i$ is finite and surjective over $V$). This is a presheaf, where the pullbacks are defined using the pullbacks of cycles (the condition that the $Z_i$; are finite and surjective over a smooth (hence normal) scheme $V$ makes that this is well defined without working up to rational equivalences, and as we consider only pullbacks along maps $U \to V$ with $U$ and $V$ smooth (hence regular) ensures that the multiplicities which will appear from these pullbacks will always be integers). The [[presheaf]] $L(X)$ is a [[sheaf]] for the [[Nisnevich topology]]. This construction is functorial in $X$ (we will need this functoriality only for closed immersions).

Let $X$ (resp. $Y$) be the cartesian product of $n$ (resp. $n-1$) copies of the projective line. The point at infinity gives a family of $n$ maps $u_i : Y \to X$. Then a model of the [[Eilenberg-Mac Lane object]] $K(\mathbb{Z}(n), 2n)$ is the sheaf of sets obtained as the quotient (in the category of Nisnevich sheaves of abelian groups) of $L(X)$ by the subsheaf generated by the images of the maps $L(u_i) : L(Y) \to L(X)$.

#### References

The original definition of motivic Eilenberg–Mac Lane spaces and spectra is in

* [[Vladimir Voevodsky]], _$A^1$-homotopy Theory_, Documenta Mathematica, Extra Volume ICM 1998, I, 579-604 [ps](http://www.emis.de/journals/DMJDMV/xvol-icm/00/Voevodsky.MAN.ps.gz)
{#ICM}

More details are in §4 of

* [[Marc Hoyois]], _From algebraic cobordism to motivic cohomology_ ([pdf](http://www.mathematik.ur.de/hoyois/papers/hopkinsmorel.pdf))

## Applications

Motivic cohomology is used to construct examples of [[Euler systems]], which in turn have applications to [[special values of L-functions]]. See also chapter 3 of [#LoefflerZerbes18](#LoefflerZerbes18).

## Related concepts

* [[Beilinson conjectures]]

* [[algebraic K-theory]]
* [[algebraic cobordism]]
* [[étale cohomology]]

* [[motivic homotopy theory]]
* [[motivic chromatic homotopy theory|chromatic motivic homotopy theory]]

* [[motivic spectrum]]

* [[absolute cohomology]]

* [[motives in physics]]

## References

* [[Eric Friedlander]], _[Algebraic Cycles and algebraic K-theory, II](http://www.math.northwestern.edu/~eric/lectures/)_ ([lecture 6 (pdf)](http://www.math.northwestern.edu/~eric/lectures/ihp/ihplec6.pdf))

* J. F. Jardine, _Motivic symmetric spectra_, Doc. Math. 5 (2000), 445--553

* [[Alexander Beilinson|A. Be?linson]], R. MacPherson, V. Schechtman, _Notes on motivic cohomology_,  Duke Math. J.  54  (1987),  no. 2, 679--710; [doi](http://dx.doi.org/10.1215/S0012-7094-87-05430-5).

* [[Vladimir Voevodsky]], [[Pierre Deligne]], _Lectures on motivic cohomology 2000/2001_ ([web](http://www.math.uiuc.edu/K-theory/0527/))

* [[Marc Levine]], _Algebraic cycle complexes_, talk notes, 2008, [pdf](https://www.uni-due.de/~bm0032/publ/CycleComplexes.pdf).

A discussion of an [[equivariant cohomology|equivariant]] version of motivic cohomology is in

* Ben Williams, _Equivariant Motivic Cohomology_ ([pdf](http://math.stanford.edu/~williams/MotivicRSS.pdf))

For a discussion of the relation betwen [[motive]]s and motivic cohomology, see for instance section 0.1.8 of

* [[Spencer Bloch]], _Lectures on Mixed Motives_ ([ps](http://www.math.uchicago.edu/~bloch/motive.ps))

The application of motivic cohomology to constructing [[Euler systems]] is discussed in chapter 3 of

* {#LoefflerZerbes18} David Loeffler and Sarah Zerbes, _Euler Systems_, Arizona Winter School 2018 Notes ([pdf](https://swc-math.github.io/aws/2018/2018LoefflerZerbesNotes.pdf))

On ([[stable cohomotopy|stable]]) [[motivic cohomology|motivic]] [[Cohomotopy]] of [[schemes]] (as  [[motivic homotopy theory|motivic homotopy classes]] of maps into [[motivic sphere|motivic]] [[Tate spheres]]):

* [[Aravind Asok]], [[Jean Fasel]], [[Mrinal Kanti Das]], *Euler class groups and motivic stable cohomotopy*, Journal of the EMS **24** 8 (2022) 2775–2822 [[arXiv:1601.05723](https://arxiv.org/abs/1601.05723), [doi:10.4171/jems/1156](https://doi.org/10.4171/jems/1156)]

* [[Samuel Lerbet]], *Motivic stable cohomotopy and unimodular rows*, Advances in Mathematics **436** 109415 (2024) [[arXiv:2206.11688](https://arxiv.org/abs/2206.11688), [doi:10.1016/j.aim.2023.109415](https://doi.org/10.1016/j.aim.2023.109415)]

category: algebraic geometry

[[!redirects motivic complex]]
[[!redirects motivic complexes]]
[[!redirects higher Chow group]]
[[!redirects higher Chow groups]]


motivic cohomology - contents

**[[motive]]**, **[[motivic cohomology]]**

## Ingredients

* [[cohomology]]

* [[stable homotopy theory]]

## Definitions

* [[pure motive]]

  * [[Chow motive]]

  * [[numerical motive]]

    * [[periods]], [[motivic Galois group]], [[cosmic Galois group]]

* [[mixed motive]]

  * [[Nori motive]]

  * [[Voevodsky motive]]

    * [[A1-homotopy theory]]

* [[noncommutative motive]]

  * [[bivariant algebraic K-theory]], [[KK-theory]]

## Related

* [[motivic function]]

* [[motivic integration]]

* [[motivic Donaldson-Thomas invariant]]

[Edit this sidebar](/nlab/edit/motivic+cohomology+-+contents)


motivic function

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## Definition

Given a [[small category]] $\mathcal{C}$ with [[coproducts]] and given an [[object]] $X \in \mathcal{C}$, the [[abelian group]] $Mot(X)$ of **motivic functions** on $X$ is defined by [[generators and relations]] as follows: it is the [[quotient]] of the [[free abelian group]] on the [[morphisms]] $S \to X$ by the relations

$$
  [S_1 \coprod S_2 \stackrel{(f_1,f_2)}{\to} X]
  =
  [S_1 \stackrel{f_1}{\to} X]
  +
  [S_1 \stackrel{f_2}{\to} X]
  \,.
$$

## Related concepts

The construction of motivic functions has some similarity with

* the construction of [[Baum-Douglas geometric cycles]]

* the construction that Baez-Dolan called [[groupoidification|de-groupoidification]].

## References

Section 2.2 of

* [[Maxim Kontsevich]], [[Yan Soibelman]], _Motivic Donaldson-Thomas invariants: summary of results_ ([arXiv:0910.4315](http://arxiv.org/abs/0910.4315))

[[!redirects motivic functions]]


motivic Galois group

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## Idea

Simply by changing the symmetry in the [[symmetric monoidal category|symmetric monoidal]] [[category of pure motives|category of pure]] [[numerical motives]] $Mot_{num}(k,\mathbb{Q})$, we can obtain a [[Tannakian category]]
$\tilde Mot_{num}(k,\mathbb{Q})$, which is thus [[equivalence of categories|equivalent]] to a [[category of representations]] of some [[algebraic group]] $GMot_k$:

$$
  \tilde Mot_{num}(k,\mathbb{Q}) \simeq Rep(GMot_k)
  \,.
$$

In the sense of [[Galois theory]], this [[algebraic group]] is called the _motivic Galois group_ for pure motives. There is also a motivic Galois group of mixed motives.

This group is, or is closely related to, the group of algebraic _[[periods]]_, and as such is related to expressions appearing in [[deformation quantization]] and in [[renormalization]] in [[quantum field theory]], whence it is also sometimes referred to as the _[[cosmic Galois group]]_. See there for more on this.

##(Mixed) Motivic Galois groups

###Nori's Motivic Galois group

[[M. Levine]], _Mixed motives._ (3.3 Motives by Tannakian formalism) In, Handbook of K-theory, Vol. 1, Friedlander and Grayson, eds., p. 429–521, Springer Verlag (2005).([pdf](https://www.uni-due.de/~bm0032/publ/MixMotKHB.pdf)).

###Ayoub's Motivic Galois group

* [[Joseph Ayoub]], _L'algèbre de Hopf et le groupe de Galois motiviques d'un corps de caractéristique nulle I_,
([pdf] (http://user.math.uzh.ch/ayoub/PDF-Files/GaloisMotivic-1.pdf)).{#Ayoub}

* [[Joseph Ayoub]], _L'algèbre de Hopf et le groupe de Galois motiviques d'un corps de caractéristique nulle II_, ([pdf](http://user.math.uzh.ch/ayoub/PDF-Files/GaloisMotivic-2.pdf)).

* [[Joseph Ayoub]], _From motives to comodules over the motivic Hopf algebra_, ([pdf](http://user.math.uzh.ch/ayoub/PDF-Files/Motivic-Hopf-3.pdf)).

[[Joseph Ayoub]], [Lecture 1](https://www.youtube.com/watch?v=f4vgBfPkiWg),[Lecture 2](https://www.youtube.com/watch?v=JHSdJ28iED0),[Lecture 3](https://www.youtube.com/watch?v=j967ozw9nXU), The lectures were held within the framework of the (Junior) Hausdorff Trimester Program Topology and the Workshop: Interactions between operads and motives.

[Mathoverflow, Derived version of equivalence between motives and representations of Motivic galois groups?](http://mathoverflow.net/questions/254341/derived-version-of-equivalence-between-motives-and-representations-of-motivic-ga).

###Comparison

_Ayoub's weak Tannakian formalism applied to the Betti realization for Voevodsky motives yields a pro-algebraic group, a candidate for the motivic Galois group.  In particular, each Voevodsky motive gives rise to a representation of this group. On the other hand Nori motives are just representations of Nori's motivic Galois group._ These groups are isomorphic.

* [[Utsav Choudhury]], [[Martin Gallauer Alves de Souza]], _An isomorphism of motivic galois groups_, ([arXiv:1410.6104](https://arxiv.org/pdf/1410.6104v2.pdf)).

_the existence of a motivic t-structure (which renders the Betti realization
t-exact) would imply the isomorphism of motivic Galois groups._

* [[J.P.Pridham]], _Tannaka duality for enhanced triangulated categories_, ([arXiv:1309.0637](https://arxiv.org/pdf/1309.0637v4.pdf)).

##Noncommutative Motivic Galois groups

* [[Matilde Marcolli]], [[Gonçalo Tabuada]] _Unconditional noncommutative motivic galois groups_,([arXiv:1112.5422](https://arxiv.org/pdf/1112.5422.pdf)).

* [[Matilde Marcolli]], [[Gonçalo Tabuada]] _Noncommutative numerical motives, Tannakian structures, and motivic Galois groups._ J. Eur. Math. Soc. 18 (2016), 623-655. doi: 10.4171/JEMS/598

## References

### General

A quick survey is in

* [[Bruno Kahn]], _Motivic Galois groups_ ([pdf](http://www.math.jussieu.fr/~kahn/preprints/PaloAlto2.pdf))
 {#Kahn}

More is in

* [[Jean-Pierre Serre]], _Propriétés conjecturales des groups de Galois motiviques et des représentations l-adiques_, in U. Jannsen, S. Kleiman, J.-P. Serre (eds)  _Motives_, Proceedings of Symposia in Pure Mathematics, AMS, vol. 55, part 1, 377 - 400 (1994).

* [[Yves André]], _Groupes de Galois motiviques et périodes_, [pdf](http://www.bourbaki.ens.fr/TEXTES/1104.pdf), Séminaire Bourbaki, 68ème année, 2015-2016 n°1104.

* [[Yves André]], _What is...a motivic Galois group_, writeup of a seminar, [pdf](https://www.him.uni-bonn.de/fileadmin/him/Lecture_Notes/motivic_Galois_group.pdf)

* AB Goncharov, _Polylogarithms and Motivic galois groups_, ([pdf](http://users.math.yale.edu/users/ag727/polylog.pdf)).

* R Sujatha, _Motives from a categorical point of view_, ([pdf](http://www.math.tifr.res.in/~sujatha/ihes.pdf)).

* [[Minhyong Kim]], _(Videos) Motivic fundamental groups and Diophantine geometry_, [I](https://www.youtube.com/watch?v=qiR8un1mwIA), [II](https://www.youtube.com/watch?v=SZLE_dqf5lk).

* Annette Huber, Stefan Müller-Stach, _On the relation between Nori motives and Kontsevich periods_, [arxiv/1105.0865](http://arxiv.org/abs/1105.0865)

> We show that the spectrum of Kontsevich's algebra of formal periods is a torsor under the motivic Galois group for mixed motives over the rational numbers. This assertion is stated without proof by Kontsevich and originally due to Nori. In a series of appendices, we also provide the necessary details on Nori's category of motives.

[Mathoverflow, What are the possible motivic Galois groups over Q
?](http://mathoverflow.net/questions/127633/what-are-the-possible-motivic-galois-groups-over-mathbb-q?rq=1).

[Mathoverflow, Why would the category of Motives be Tannakian?](http://mathoverflow.net/questions/71660/why-would-the-category-of-motives-be-tannakian).

### Relation to Grothendieck-Teichmüller group

The [[Grothendieck-Teichmüller group]] is supposed to be a [[quotient]] of the motivic Galois group. This is a conjecture due to ([Drinfeld 91](#Drinfeld91)).

* [[Vladimir Drinfel'd]], _On quasi-triangular Quasi-Hopf algebras and a group closely related with $Gal(\overline{\mathbb{Q}/\mathbb{Q}})$, Leningrad Math. J., 2 (1991), 829 - 860.
 {#Drinfeld91}

### In quantum field theory

A conjecture that the motivic Galois group naturally acts on the space of [[deformation quantizations]] of [[free field theories]] is in section 5 of

* [[Maxim Kontsevich]], _Operads and motives in deformation quantization_, Lett. Math. Phys.48:35-72, 1999 ([arXiv:math/9904055](http://arxiv.org/abs/math/9904055))

Regarding that this approach is related to Hodge theory periods, and that QFT can be attacked using the Batalin-Vilkovsky formalism, Barannikov has developed
further the Batalin-Vilkovisky geometry and related Hodge theory in a direction which is probably a ground for better understanding of motivic Galois group as well.

An analogous statement, that a motivic Galois group naturally acts on structures in [[renormalization]] is in

* [[Alain Connes]], [[Matilde Marcolli]], _Renormalization and motivic Galois theory_, [arXiv:math/0409306](http://arxiv.org/abs/math/0409306)
 {#ConnesMarcolli04}

* [[Alain Connes]], _Symetries Galoisiennes et Renormalisation_, [arXiv:math/0211199](https://arxiv.org/abs/math/0211199)

For more along these lines see at _[[cosmic Galois group]]_.

###Grothendieck's Yoga Remarks

## Related concepts

* [[Galois theory]]

* [[motive]]

* [[cosmic Galois group]], [[Grothendieck-Teichmüller group]]

[[!redirects motivic Galois groups]]


motivic homotopy theory

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
#### Motivic cohomology
+--{: .hide}
[[!include motivic cohomology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

_Motivic homotopy theory_ or _$\mathbf{A}^1$-homotopy theory_ is the [[homotopy theory]] of smooth [[schemes]], where the [[affine line]] $\mathbf{A}^1$ plays the role of the [[interval]].  Hence what is called the _motivic homotopy category_ or the _$\mathbb{A}^1$-homotopy category_ bears the same relation to [[smooth varieties]] that the ordinary [[homotopy category]] $Ho(Top)$ bears to [[smooth manifolds]].

Both are special cases of a [[homotopy theory]] induced by any sufficiently well-behaved [[interval object]] $I$ in a [[site]] $C$ via [[homotopy localization at an object|localization at that object]]. Ordinary homotopy theory is obtained by taking $C$ to be the [[Diff|site of smooth manifolds]] and $I$ to be the [[real line]] $\mathbb{R}$, and $\mathbb{A}^1$-homotopy theory over a [[Noetherian scheme]] $S$ is obtained when $C$ is the [[Nisnevich site]] of [[smooth scheme]]s of finite type over $S$ and

\[
  \label{AffineLine}
  I \coloneqq \mathbb{A}^1
\]

is the standard [[affine line]] in $C$.

As for the standard homotopy theory, one can furthermore pass to [[spectrum objects]] and consider the [[stable homotopy category]]. In the following we first discuss

* _[The unstable motivic homotopy category](#UnstableCategory)_

and then

* _[The stable motivic homotopy category](#StableCategory)_.

## The unstable motivic homotopy category
 {#UnstableCategory}

Let $S$ be a fixed [[Noetherian scheme|Noetherian base scheme]], and let $Sm/S$ be the category of [[smooth schemes]] of finite type over $S$.

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

The _motivic homotopy category_ $\mathrm{H}(S)$ over $S$ is the [[homotopy localization]] at the [[affine line]] $\mathbb{A}^1$ (eq:AffineLine) of the [[(∞,1)-topos of (∞,1)-sheaves]] on the [[Nisnevich site]] $Sm/S$. Objects of $\mathrm{H}(S)$ are called _motivic spaces_.
=--

Thus, a motivic space over $S$ is an [[(∞,1)-presheaf]] $F$ on $Sm/S$ such that

* $F$ is an [[(∞,1)-sheaf]] for the [[Nisnevich topology]]
* $F$ is _$\mathbb{A}^1$-homotopy invariant_: for every $X\in Sm/S$, the projection $X\times\mathbb{A}^1\to X$ induces an [[equivalence]] $F(X)\simeq F(X\times\mathbb{A}^1$).

As for any [[homotopy localization]], the inclusion $\mathrm{H}(S)\subset PSh(Sm/S)$ admits a [[left adjoint]] [[localization functor]], and one can show that it preserves finite (∞,1)-products.

The (∞,1)-category $\mathrm{H}(S)$ is a [[locally presentable (∞,1)-category|locally presentable]] and [[locally cartesian closed (∞,1)-category|locally cartesian closed]] (∞,1)-category. However, it is not an [[(∞,1)-topos]] (see Remark 3.5 in [Spitzweck-Østvær, _Motivic twisted K-theory_, pdf](http://arxiv.org/pdf/1008.4915v1.pdf))

### Motivic spheres {#MotivicSpheres}

The _[[Tate sphere]]_ is the [[pointed object]] of $\mathrm{H}(S)$ defined by

$$ T {\coloneqq} \mathbb{A}^1/\mathbb{G}_m, $$

that is, $T$ is the [[homotopy cofiber]] of the inclusion $\mathbb{G}_m\hookrightarrow \mathbb{A}^1$.
More generally, any algebraic [[vector bundle]] $V$ on $S$ (or, equivalently, any [[quasicoherent sheaf|locally free sheaf]] of finite rank on $S$) has an associated _motivic sphere_ given by its _[[Thom space]]_:

$$ S^V {\coloneqq} V/V^\times. $$

Thus, $T=S^{\mathbb{A}^1}$.

A crucial observation is that $T$ is the [[suspension]] of $\mathbb{G}_m$ (pointed at $1$):

$$ T \simeq S^1\wedge\mathbb{G}_m. $$

Indeed, this follows from the definition of $T$ and the fact that $(\mathbb{A}^1,1)$ is [[contractible]] as a pointed motivic space. It is common to write, for $p\geq q\geq 0$,

$$ S^{p,q} {:=} S^{p-q}\wedge \mathbb{G}_m^{\wedge q}. $$

+-- {: .num_prop}
###### Proposition
There is a canonical equivalence of pointed motivic spaces $T\simeq (\mathbb{P}^1,1)$.

=--

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

The cartesian square

$$
  \array{
    \mathbb{G}_m &\stackrel{}{\to}& \mathbb{A}^1
    \\
    \downarrow && \downarrow
    \\
    \mathbb{A}^1 &\stackrel{}{\to}& \mathbb{P}^1
  }
$$

becomes [[(∞,1)-pushout|homotopy cocartesian]] in the [[Nisnevich site|Nisnevich]] (even [[Zariski site|Zariski]]) (∞,1)-topos. By pointing all schemes at $1$ and using that $(\mathbb{A}^1,1)$ is contractible, we deduce that $(\mathbb{P}^1,1)\simeq S^1\wedge \mathbb{G}_m\simeq T.$
=--

## The stable motivic homotopy category
 {#StableCategory}

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

The _stable motivic homotopy category_ $SH(S)$ over $S$ is the [[inverse limit]] of the tower of (∞,1)-categories

$$ \dots \stackrel{\Omega_T}{\to} \mathrm{H}_*(S) \stackrel{\Omega_T}{\to} \mathrm{H}_*(S) \stackrel{\Omega_T}{\to} \mathrm{H}_*(S),$$

where $H(S)$ is the ordinary motivic homotopy category from def. \ref{MotivicHomotopyCategory}, and where $\Omega_T {:=}Hom(T, -)$. An object of the stable motivic homotopy category is called a _[[motivic spectrum]]_ (or _$T$-spectrum_).
=--

Thus, a motivic spectrum $E$ is a sequence of pointed motivic spaces $(E_0,E_1,E_2\dots)$ together with [[equivalences]]

$$ \Omega_T E_{i+1}\simeq E_i. $$

Since $T\simeq \mathbb{P}^1$, we could equivalently use $\mathbb{P}^1$ instead of $T$ in the above definition.

Since $T\simeq S^1\wedge \mathbb{G}_m$, $SH(S)$ is indeed a [[stable (∞,1)-category]].

### Symmetric monoidal structure and universal property
 {#SymmetricMonoidalStructureAndUniversalProperty}

The functor $\Omega^\infty_T\colon SH(S)\to \mathrm{H}_*(S)$ sending a motivic spectrum $E$ to its first component $E_0$ admits a [[left adjoint]] $\Sigma_T^\infty$. One can then equip the category $SH(S)$ with the structure of a [[symmetric monoidal (∞,1)-category]] in such a way that the (∞,1)-functor $\Sigma_T^\infty$ can be promoted to a symmetric monoidal (∞,1)-functor. As such, $SH(S)$ is characterized by a [[universal property]]:

+-- {: .num_prop}
###### Proposition
Let $\mathcal{C}$ be a [[locally presentable (∞,1)-category|locally presentable]] [[symmetric monoidal (∞,1)-category|symmetric monoidal]] [[(∞,1)-category]].
The (∞,1)-functor

$$ Fun^{\otimes, L}(SH(S),\mathcal{C}) \to Fun^{\otimes} (Sm/S, \mathcal{C}), \quad F\mapsto F\circ \Sigma_T^\infty(-)_+ $$

(where $\otimes$ means "symmetric monoidal" and $L$ means "colimit-preserving")
is [[fully faithful]] and its essential image consists of the symmetric monoidal (∞,1)-functors $F\colon Sm/S\to \mathcal{C}$ satisfying:

* [[Nisnevich site|Nisnevich excision]]
* $\mathbb{A}^1$-homotopy invariance
* _$T$-stability_: the [[homotopy cofiber]] of $F(\mathbb{G}_m)\to F(\mathbb{A}^1)$ is $\otimes$-invertible.

=--

This is ([Robalo, Corollary 5.11](#Robalo)).

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

Similar characterizations exist for [[noncommutative motives]], see at _[Noncommutative motive -- As the universal additive invariant](noncommutative+motive#AsUniversalAditiveInvariant)_.

=--

### Stable motivic spheres {#StableMotivicSpheres}

Because $T\simeq S^{2,1}$, the stable motivic spheres $S^{p,q}$ are defined for all $p,q\in\mathbb{Z}$.

All the other motivic spheres $S^V$, for $V$ a vector bundle on $S$, also become invertible in $SH(S)$. In fact, the [[Picard ∞-groupoid]] of $SH(S)$ receives a map from the [[algebraic K-theory]] of $S$. These invertible objects are all *exotic* in the sense that they are not equivalent to any of the "categorical" spheres $S^n = \Sigma^n\Sigma^\infty_T S_+$. These exotic spheres play an important rôle in the formalism of [[six operations]] in stable motivic homotopy theory (see [Ayoub](#Ayoub)).

### Cohomology theories {#CohomologyTheories}

Any motivic spectrum $E\in SH(S)$ gives rise to a bigraded [[cohomology|cohomology theory]] for smooth $S$-schemes and more generally for motivic spaces:

$$ E^{p,q}(X) {:=} [\Sigma^\infty_T X_+, \Sigma^{p,q} E], $$

as well as a bigraded homology theory:

$$ E_{p,q}(X) {:=} [S^{p,q}, \Sigma^\infty_T X_+ \wedge E]. $$

## Main features

### The six operations {#SixOperations}

The categories $SH(S)$ for varying base scheme $S$ support a formalism of [[six operations]]. This means that to every morphism of schemes $f: X\to Y$ is associated an ([[inverse image]] $\dashv$ [[direct image]])-[[adjunction]]

$$ f^* : SH(Y) \to SH(X) : f_* $$

and, if $f$ is [[separated morphism|separated]] of finite type, a ([[direct image with compact support]] $\dashv$ [[exceptional inverse image]])-adjunction

$$ f_! : SH(X) \to SH(Y): f^!,$$

satisfying the properties listed at [[six operations]]. For more details see [Ayoub](#Ayoub).

**Stable homotopy functors.** To construct the six operations for $SH$, Voevodsky introduced an axiomatic setting which also subsumes the classical case of [[étale cohomology]]. A _stable homotopy functor_ is a contravariant (∞,1)-functor

$$ D\colon Schemes^{op} \to PrStab (\infty,1) Cat, $$

from some category of schemes to the (∞,1)-category $PrStab (\infty,1) Cat$ of [[locally presentable (∞,1)-category|locally presentable]] [[stable (∞,1)-categories]] and colimit-preserving exact functors, satisfying the following axioms (for $f$ a morphism of schemes, we denote $D(f)$ by $f^*$ and its right adjoint by $f_*$):

1. $D(\emptyset)=0$.

2. If $i: X\to Y$ is an immersion of schemes, then $i_*\colon D(X)\to D(Y)$ is [[fully faithful (∞,1)-functor|fully faithful]].

3. (Smooth base change/[[Beck-Chevalley condition]]) If $f$ is [[smooth scheme|smooth]], then $f^*$ admits a left adjoint $f_\sharp$. Moreover, given a cartesian square
$$
  \array{
    Y' &\stackrel{k}{\to}& X'
    \\
    _h\downarrow && \downarrow \;_f
    \\
    Y &\stackrel{g}{\to}& X
  }
$$
with $f$ smooth, there is a canonical equivalence $h_\sharp k^*\simeq g^* f_\sharp$.

4. (Locality) If $i: Z\hookrightarrow X$ is a [[closed immersion]] with open complement $j: U\hookrightarrow X$, then the pair $(i^*,j^*)$ is [[conservative functor|conservative]].

5. (Homotopy invariance) If $p: X\times\mathbb{A}^1\to X$ is the projection, then $p^*$ is [[fully faithful (∞,1)-functor|fully faithful]].

6. ($T$-stability) If $p$ is as above and $s$ is the zero section of $p$, then $p_\sharp s_*: D(X)\to D(X)$ is an [[equivalence of (∞,1)-categories]].

+-- {: .num_theorem}

###### Theorem
(Ayoub, Voevodsky) Every stable homotopy functor admits a formalism of four operations $f^\ast$, $f_\ast$, $f_!$, and $f^!$.
=--

For a more precise statement, see [Ayoub, Scholie 1.4.2](#Ayoub).

+-- {: .num_theorem}
###### Theorem
$SH$ is a stable homotopy functor.
=--

This is essentially proved in [Morel-Voevodsky 99](#MorelVoevodsky99). In fact, something stronger is expected to be true:

+-- {: .num_theorem}
###### Expected Theorem
$SH$ is the [[initial object]] in the [[(∞,2)-category]] of stable homotopy functors.
=--

This theorem has not been proved yet; however, [Ayoub's thesis](#Ayoub) shows that every stable homotopy functor factors through $SH$.

### Realization functors {#RealizationFunctors}

**Complex realization.** The functor

$$ Sm/\mathbb{C}\to SmoothMfd,\quad X\mapsto X(\mathbb{C}) $$

associating to a smooth $\mathbb{C}$-scheme $X$ its set of $\mathbb{C}$-points with its structure of [[smooth manifold]] induces a functor from $\mathrm{H}(\mathbb{C})$ to the [[homotopy localization]] of the [[smooth ∞-groupoid|smooth (∞,1)-topos]] at the [[interval object]] $\mathbb{A}^1(\mathbb{C})\simeq \mathbb{R}^2$. As this localization is equivalent to the [[(∞,1)-topos]] $\infty Grpd$ of [[discrete ∞-groupoid]]s, we obtain the _complex realization functor_

$$ \mathrm{H}(\mathbb{C}) \to \infty Grpd. $$

After $T$-stabilization, we obtain a functor from $SH(\mathbb{C})$ to the [[(∞,1)-category of spectra]].

**Real realization.** The functor

$$ Sm/\mathbb{R}\to \mathbb{Z}/2-SmoothMfd,\quad X\mapsto X(\mathbb{C}) $$

associating to a smooth $\mathbb{R}$-scheme $X$ its set of $\mathbb{C}$-points with its structure of [[smooth manifold]] together with the action of $\mathbb{Z}/2$ by [[complex conjugation]] induces as in the complex case the _Real realization functor_

$$ \mathrm{H}(\mathbb{R}) \to PSh_\infty(\mathcal{O}_{\mathbb{Z}/2}), $$

where $\mathcal{O}_{\mathbb{Z}/2}$ is the [[orbit category]] of $\mathbb{Z}/2$. After $T$-stabilization, we obtain a functor from $SH(\mathbb{R})$ to the (∞,1)-category of [[equivariant stable homotopy theory|genuine]] $\mathbb{Z}/2$-spectra.

**Étale realization.** Over a separably closed field $k$, we can consider the [[étale homotopy|étale homotopy type]] functor

$$ Sh_{\infty}((Sm/k)_{Nis})\to Sh_{\infty}((Sm/k)_{et}) \stackrel{\Pi}{\to} Pro(\infty Grpd) $$

(see [[shape of an (∞,1)-topos]]). However, it does not descend to $\mathrm{H}(k)$ because the étale homotopy type is not $\mathbb{A}^1$-homotopy invariant. To rectify this, we choose a prime $l\neq \operatorname{char}(k)$ and consider the reflexive [[localization]] $Pro(\infty Grpd)^\wedge_{l}$ of $Pro(\infty Grpd)$ at the class of maps inducing isomorphisms on pro-homology groups with coefficients in $\mathbb{Z}/l$. We then obtain an _étale realization functor_

$$ \mathrm{H}(k) \to Pro(\infty Grpd)^\wedge_{l}. $$

### The slice filtration {#SliceFiltration}

_See [[motivic slice filtration]]._

The slice filtration is a filtration of $\mathrm{H}(S)$ and of $SH(S)$ which is analogous to the [[Postnikov tower in an (∞,1)-category|Postnikov filtration]] for [[(∞,1)-topoi]]. It generalizes the _coniveau filtration_ in [[algebraic K-theory]], the _fundamental filtration_ on [[Witt groups]], and the _weight filtration_ on [[mixed Tate motives]].

If $S$ is [[smooth scheme|smooth]] over a [[field]], the layers of the slice filtration of a motivic spectrum (called its _slices_) are modules over the [motivic Eilenberg–Mac Lane spectrum](#MotivicCohomology) $H(\mathbb{Z})$. At least if $S$ is a [[field]] of characteristic zero, this is the same thing as an integral [[motive]]. The [[spectral sequences]] associated to the slice filtration are analogous to the [[Atiyah-Hirzebruch spectral sequence]]s in that their first page consists of [[motivic cohomology]] groups.

### The $\mathbb{A}^1$-Postnikov filtration
{#A1PostnikovFiltration}

One can also consider the filtration on $\mathrm{H}(S)$ induced by the [[Postnikov tower in an (∞,1)-category|Postnikov filtration]] in the containing [[Nisnevich (∞,1)-topos]]. A motivic space is _$\mathbb{A}^1$-n-connected_ if it is [[n-connected]] as a Nisnevich (∞,1)-sheaf, and the _$\mathbb{A}^1$-homotopy groups_ $\pi_n^{\mathbb{A}^1}(X,x)$ of a motivic space are its [[homotopy groups in an (∞,1)-topos|homotopy groups]] as a Nisnevich (∞,1)-sheaf.

If $S$ has finite [[Krull dimension]], $\mathbb{A}^1$-homotopy groups detect [[equivalences]] because the [[Nisnevich (∞,1)-topos]] is [[hypercomplete]].

+-- {: .num_remark}
###### Remark
 The usage of the $\mathbb{A}^1$- prefix in the above definitions may seem strange since all these notions are simply inherited from the Nisnevich (∞,1)-topos. The point is that, when a smooth scheme $X$ is viewed as a motivic space, a localization functor is implicitly applied. The underlying Nisnevich (∞,1)-sheaf of the motivic space "$X$" can thus be very different from the Nisnevich (∞,1)-sheaf represented by $X$ (for which these definitions would not be interesting at all!).
=--

Intuitively, $\mathbb{A}^1$-connectedness corresponds to the topological connectedness of the _real points_ rather than of the complex points. For example, $\mathbb{G}_m$ is not $\mathbb{A}^1$-connected, and $\mathbb{P}^1$ is $\mathbb{A}^1$-connected but not $\mathbb{A}^1$-simply connected.

+-- {: .num_theorem}
###### Theorem
Let $k$ be a [[perfect field]] (resp. a [[field]]) and $X$ a [[Nisnevich site|Nisnevich]] (∞,1)-sheaf of spaces (resp. of spectra) on $Sm/k$. If $X$ is [[n-connected]] for some n, then its $\mathbb{A}^1$-localization is also n-connected.
=--

This is Morel's _connectivity theorem_ ([Morel, Theorem 5.38](#Morel)). It follows that the $\mathbb{A}^1$-Postnikov filtration on $\mathrm{H}(k)$ "extends" to a [[t-structure]] on the stable (∞,1)-category $SH(k)$, called the _homotopy t-structure_.

### Relation to the theory of motives

The stable motivic homotopy category $SH(S)$ is the basis for several definitions of the [[motive|derived category of mixed motives]] over $S$. See there for more details.

### Relation to the theory of symmetric bilinear forms

Motivic homotopy theory is also related to the classical theory of symmetric [[bilinear forms]] (or [[quadratic forms]] in characteristic $\neq 2$). Invariants such as [[Witt groups]], oriented [[Chow groups]], and [[Hermitian K-theory]] are representable in the motivic homotopy category.

A central theorem of [[Fabien Morel]] states that, if $k$ is a [[field]], the ring of endomorphisms of the motivic sphere spectrum $S^0\in SH(k)$ is canonically isomorphic to the _Grothendieck–Witt ring_ $GW(k)$: this is the [[group completion]] of the [[semiring]] of isomorphism classes of nondegenerate symmetric [[bilinear form]]s over $k$ ([Morel, Corollary 5.43](#Morel)). The case $k=\mathbb{R}$ is especially enlightening: there the stable homotopy class of a pointed endomorphism of $\mathbb{P}^1$ corresponds to the nondegenerate symmetric bilinear form over $\mathbb{R}$ whose dimension is the degree of the induced endomorphism of $\mathbb{P}^1(\mathbb{C})\simeq S^2$ and whose signature is the degree of the induced endomorphism of $\mathbb{P}^1(\mathbb{R})\simeq S^1$.

## Equivariant motivic homotopy theory
 {#EquivariantMotivicHomotopyTheory}

A general theory of [[equivariant motivic homotopy theory|equivariant]] (unstable and stable) motivic homotopy theory was introduced in ([Carlsson-Joshua 2014](#CJ2014)) and further developed in ([Hoyois 15](#Hoyois15)).

## Applications and examples

### $\mathbb{A}^1$-coverings and the $\mathbb{A}^1$-fundamental groupoid
{#A1coverings}

Like the $\mathbb{A}^1$-[Postnikov filtration](#A1PostnikovFiltration), $\mathbb{A}^1$-coverings and the $\mathbb{A}^1$-fundamental groupoid $\Pi_1^{\mathbb{A}^1}$ are defined in the containing [[Nisnevich (∞,1)-topos]]. A morphism of motivic spaces $f: Y\to X$ is an _$\mathbb{A}^1$-covering_ if it is [[n-truncated object of an (∞,1)-category|0-truncated]] as a morphism between Nisnevich (∞,1)-sheaves. Such a morphism is determined by its 1-truncation $\tau_{\leq 1}f$, and hence there is an equivalence between the category of $\mathbb{A}^1$-coverings of $X$ and that of $\mathbb{A}^1$-invariant objects in the [[classifying topos]] of the Nisnevich sheaf of groupoids
$$ \Pi_1^{\mathbb{A}^1}(X):=\tau_{\leq 1}X. $$

+-- {: .num_theorem}
###### Theorem
Let $k$ be a [[field]] and $X\in\mathrm{H}_\ast(k)$ a pointed $\mathbb{A}^1$-connected motivic space. Let $\tilde X$ be the $\mathbb{A}^1$-localization of the 1-connected cover of $X$ (as a pointed Nisnevich (∞,1)-sheaf). Then:

1. $\tilde X$ is the initial object in the category of pointed $\mathbb{A}^1$-coverings of $X$.

2. $\tilde X$ is the unique pointed $\mathbb{A}^1$-covering of $X$ which is $\mathbb{A}^1$-simply connected.

3. The Nisnevich sheaf of (unpointed) automorphisms $Aut_X(\tilde X)$ is canonically isomorphic to $\pi_1^{\mathbb{A}^1}(X)$.
=--

This is [Morel, Theorem 6.8](#Morel). The key input for this theorem is the fact that $\Pi_1^{\mathbb{A}^1}(X)$ is $\mathbb{A}^1$-invariant; this is only known for $\mathbb{A}^1$-connected motivic spaces over fields, which explains the hypotheses of the theorem.

When $X$ is a smooth $S$-scheme, examples of $\mathbb{A}^1$-coverings of $X$ include $\mathbb{G}_m$-torsors and finite Galois coverings of degree prime to the characteristics of $X$ ([Morel, Lemma 6.5](#Morel)). This can be used to compute some $\mathbb{A}^1$-fundamental groups, for example:

+-- {: .num_prop}
###### Proposition
If $n\geq 2$, $\pi_1^{\mathbb{A}^1}(\mathbb{P}^n)=\mathbb{G}_m$.
=--

+-- {: .proof}
###### Proof
The projection $\mathbb{A}^{n+1}-0\to\mathbb{P}^n$ is a $\mathbb{G}_m$-torsor and hence an $\mathbb{A}^1$-covering. If $n\geq 2$, $\mathbb{A}^{n+1}-0$ is moreover $\mathbb{A}^1$-simply connected and hence is the universal $\mathbb{A}^1$-covering of $\mathbb{P}^n$. Thus, $\pi_1^{\mathbb{A}^1}(\mathbb{P}^n)=Aut_{\mathbb{P}^n}(\mathbb{A}^{n+1}-0)=\mathbb{G}_m$.
=--

### $\mathbb{A}^1$-h-cobordisms and the classification of surfaces

An _$\mathbb{A}^1$-h-cobordism_ is a surjective [[proper map]] $X\to\mathbb{A}^1$ in $Sm/S$ such that, for $i=0,1$, the fiber $X_i$ is smooth and the inclusion $X_i\hookrightarrow X$ becomes an equivalence in $\mathrm{H}(S)$.

Asok and Morel used $\mathbb{A}^1$-h-cobordisms to classify rational smooth proper surfaces over [[algebraically closed field]]s up to $\mathbb{A}^1$-homotopy. See [Asok-Morel](#AsokMorel).

### Euler classes and splittings of algebraic vector bundles

Let $k$ be a [[perfect field]]. If $X$ is a smooth affine $k$-scheme, Morel proved that

$$ Vect_n(X) \cong [X,BGL_n], $$

where $[-,-]$ denote homotopy classes of maps in the motivic homotopy category $\mathrm{H}(k)$, def. \ref{MotivicHomotopyCategory}. The classical problem of determining whether a [[rank]] $n$ [[vector bundle]] splits off a trivial [[line bundle]] is thus equivalent to determining whether the classifying map $X\to BGL_n$ lifts to $BGL_{n-1}$ in $\mathrm{H}(k)$. If the Nisnevich [[cohomological dimension]] of $X$ is at most $n$, we can use [[obstruction theory]] together with the [[fiber sequence]]

$$ \mathbb{A}^n-0 \to BGL_{n-1} \to BGL_n $$

to obtain the following criterion:

+-- {: .num_theorem}
###### Theorem
(Morel)
Suppose that $n\geq 2$ and that $X$ has dimension $\leq n$. Let $\xi$ be a vector bundle of rank $n$ over $X$.
Then there exists a canonical class
$$ e(\xi)\in H^n_{Nis}(X,\pi_{n-1}^{\mathbb{A}^1}(\mathbb{A}^n-0)(\mathrm{det} \xi)) $$
which vanishes if and only if $\xi$ splits off a trivial line bundle.
=--

The twist by the determinant $\mathrm{det} \xi$ comes from the nontrivial $\mathbb{A}^1$-fundamental group $\pi_1^{\mathbb{A}^1}(BGL_n)=\mathbb{G}_m$.

The Nisnevich sheaf $\pi_{n-1}^{\mathbb{A}^1}(\mathbb{A}^n-0)$ is the n-th Milnor–Witt K-theory sheaf $\mathbf{K}^{MW}_n$, so that

$$ H^n_{Nis}(X,\pi_{n-1}^{\mathbb{A}^1}(\mathbb{A}^n-0)(\mathrm{det} \xi)) \cong \widetilde{CH}^n(X; \mathrm{det} \xi) $$

is the n-th oriented [[Chow group]] of $X$. When $k$ is [[algebraically closed field|algebraically closed]], this is just the usual Chow group of $X$ and $e(\xi)$ can be identified with the top Chern class of $\xi$.

### Motivic cohomology {#MotivicCohomology}

Suppose that $S$ is a [[smooth scheme]] over a [[field]]. Then the [[motivic cohomology]] of smooth $S$-schemes is representable in $\mathrm{H}(S)$ and in $SH(S)$. If $A$ is an [[abelian group]] and $p\geq q\geq 0$, there exist a pointed motivic spaces $K(A(q),p)$, called a _motivic Eilenberg–MacLane space_, such that, for every $X\in Sm/S$,

$$ H^{p-r,q-s}(X,A) = [\Sigma^{r,s}X_+, K(A(q),p)]. $$

Voevodsky's _cancellation theorem_ implies that $\Omega_T K(A(q+1), p+2)\simeq K(A(q), p)$. It follows that the sequence of motivic spaces $K(A(n),2n)$ form a $T$-spectrum $H(A)$, called a _motivic Eilenberg–MacLane spectrum_, such that

$$ H^{p,q}(X,A) = [\Sigma_T^\infty X_+, \Sigma^{p,q} H(A)]. $$

If $R$ is a [[commutative ring]], the motivic spectrum $H(R)$ has a canonical structure of $E_\infty$-algebra in $SH(S)$ which induces the ring structure in motivic cohomology.

Unlike in topology, $H(\mathbb{Q})$ is not always equivalent to the rational motivic sphere spectrum $S^0_{\mathbb{Q}}$: this is only the case if $-1$ is a sum of squares in the base field. In general, $H(\mathbb{Q})$ is a direct summand of $S^0_{\mathbb{Q}}$.

The stable (∞,1)-category of $H(\mathbb{Q})$-[[modules]] is equivalent to the [[derived category of mixed motives]]. See there for more details.

### Algebraic K-theory

See [[algebraic K-theory spectrum]].

### Algebraic cobordism
 {#AlgebraicCobordism}

See [[algebraic cobordism]].

## Related concepts

* [[equivariant homotopy theory]]

* [[motive]]
* [[motivic cohomology]]

* [[slice spectral sequence]]

* [[B1-homotopy theory]]

* There is an analog of $\mathbb{A}^1$-homotopy theory for other geometries. The extra left adjoint on a [[cohesive (infinity,1)-topos]] may realize the localization at an abstract [[continuum]] [[line object]]. See at _[[cohesion]]_ for more details.

## References

### General

The original references:

* [[Vladimir Voevodsky]], $\mathbf{A}^1$-Homotopy Theory. Proceedings of the International Congress of Mathematicians, Vol. I (Berlin, 1998). Doc. Math. 1998, Extra Vol. I, 579–604 (electronic).  [web](http://www.math.uiuc.edu/documenta/xvol-icm/00/Voevodsky.MAN.html)

* {#MorelVoevodsky99} [[Fabien Morel]], [[Vladimir Voevodsky]], _$\mathbb{A}^1$-homotopy theory of schemes_, Publications Mathématiques de l'IHÉS, Volume 90 (1999), p. 45-143  ([Numdam:PMIHES_1999__90__45_0](http://www.numdam.org/item/?id=PMIHES_1999__90__45_0) [K-Theory:0305](https://faculty.math.illinois.edu/K-theory/0305/) )

* {#Morel} [[Fabien Morel]], _$\mathbb{A}^1$-algebraic topology over a field_, LNM 2052, 2012, ([pdf](https://www.mathematik.uni-muenchen.de/~morel/Prepublications/A1TopologyLNM.pdf))

* {#AsokMorel} [[Aravind Asok]], [[Fabien Morel]], _Smooth varieties up to $\mathbb{A}^1$-homotopy and algebraic h-cobordisms_, Adv. Math. **227** (5) (2011) 1990-2058 [[arXiv:0810.0324](http://arxiv.org/abs/0810.0324), [doi:10.1016/j.aim.2011.04.009](https://doi.org/10.1016/j.aim.2011.04.009)]

Readable introductions to the subject are:

* [[Bjørn Ian Dundas]], [[Marc Levine]], [[Paul Arne Østvær]], [[Oliver Röndigs]], [[Vladimir Voevodsky]], _Motivic Homotopy Theory: Lectures at a Summer School in Nordfjordeid, Norway, August 2002_, Springer, Universitext, (2006) ([doi:10.1007/978-3-540-45897-5](https://www.springer.com/de/book/9783540458951))

* [[Marc Levine]], _Motivic Homotopy Theory_, Milan j. math (2008), ([pdf](http://www.math.unam.mx/~javier/levine.pdf))

* [[Fabien Morel]], _An introduction to $\mathbb{A}^1$ homotopy theory_, ICTP Trieste July 2002 ([directory](http://publications.ictp.it/lns/vol15/vol15toc.html), [pdf](http://www.ictp.it/%7Epub_off/lectures/lns015/Morel/Morel.pdf), [ps](https://www.mathematik.uni-muenchen.de/~morel/Prepublications/lectureTrieste.ps))

* [[Fabien Morel]], _On the motivic stable π₀ of the sphere spectrum_ ([ps](https://www.mathematik.uni-muenchen.de/~morel/Prepublications/Newton.ps))

* {#Arndt17} [[Peter Arndt]], _Abstract motivic homotopy theory_, thesis 2017 ([web](https://repositorium.ub.uni-osnabrueck.de/handle/urn:nbn:de:gbv:700-2017021015476?mode=full),  [pdf](https://repositorium.ub.uni-osnabrueck.de/bitstream/urn:nbn:de:gbv:700-2017021015476/6/thesis_arndt.pdf), [[ArndtAbstractMotivic.pdf:file]])

   {#Arndth19} exposition: lecture at _[Geometry in Modal HoTT](https://felix-cherubini.de/abstracts.html)_, 2019 ([recording I](https://www.youtube.com/watch?v=f0wpcNs8hQo), [recording II](https://www.youtube.com/watch?v=sTl8637a2Zo))

Detailed discussion of the [[model structure on simplicial presheaves]] on the [[Nisnevich site]] and its [[homotopy localization]] to [[A1-homotopy theory]]:

* [[John F. Jardine]], *[[Local homotopy theory]]*, Springer Monographs in Mathematics (2015) [[doi:10.1007/978-1-4939-2300-7](https://doi.org/10.1007/978-1-4939-2300-7)]

with brief exposition in:

* [[John F. Jardine]], *Motivic spaces and the motivic stable category* [[pdf](http://www.aimath.org/WWN/motivesdessins/motivic.pdf), [[Jardine-MotivicSpaces.pdf:file]]]

For more on the general procedure see [[homotopy localization]].

The universal property of $SH(S)$ is proved in

* {#Robalo} [[Marco Robalo]], _Noncommutative Motives I: A Universal Characterization of the Motivic Stable Homotopy Theory of Schemes_, 2013 ([pdf](http://arxiv.org/pdf/1206.3645v3.pdf))

For the formalism of [[six operations]] see

* {#Ayoub} [[Joseph Ayoub]], _Les six opérations de Grothendieck et le formalisme des cycles évanescants dans le monde motivique_ PhD thesis, Paris [[pdf](http://www.math.uiuc.edu/K-theory/0761/THESE.pdf), [K:0761](https://faculty.math.illinois.edu/K-theory/0761)]

* [[Joseph Ayoub]], *Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique (I)*, Astérisque **314** (2007) [[numdam:AST_2007__314__R1_0](http://www.numdam.org/issues/AST_2007__314__R1_0)]

* [[Joseph Ayoub]], *Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique (II)*, Astérisque **315** (2007) [[numdam:AST_2007__315__1_0](http://www.numdam.org/issues/AST_2007__315__1_0)]

* [[Vladimir Voevodsky]], [[Pierre Deligne]], _Voevodsky's lectures on cross functors_ ([pdf](https://www.math.ias.edu/vladimir/sites/math.ias.edu.vladimir/files/2015_transfer_from_ps_delnotes01.pdf))

The slice filtration was defined in

* [[Vladimir Voevodsky]], _Open problems in the stable motivic homotopy theory_ K-theory, 0392 ([web](https://faculty.math.illinois.edu/K-theory/0392/) [pdf](https://faculty.math.illinois.edu/K-theory/0392/nowmovo.pdf))

Representability results:

* [[Aravind Asok]], [[Marc Hoyois]], [[Matthias Wendt]], *Affine representability results in $\mathbb{A}^1$-homotopy theory I: Vector bundles*, Duke Math. J. **166** 10 (2017) 1923-1953 [[arXiv:1506.07093](http://arxiv.org/abs/1506.07093), [doi:10.1215/00127094-0000014X](https://doi.org/10.1215/00127094-0000014X)]

* [[Aravind Asok]], [[Marc Hoyois]], [[Matthias Wendt]], _Affine representability results in $\mathbb{A} ^1$-homotopy theory II: principal  bundles and homogeneous spaces_, Geom. Topol. **22** (2018) 1181-1225 [[arXiv:1507.08020](http://arxiv.org/abs/1507.08020), [doi:10.2140/gt.2018.22.1181](https://doi.org/10.2140/gt.2018.22.1181)]

Discussion related to [[étale homotopy]]:

* [[Daniel Isaksen]], _Étale realization of the $\mathbb{A} ^1$-homotopy theory of schemes_, Advances in Mathematics 184 (2004)

Discussion about [[thick ideals]]:

* [[Ruth Joachimi]], _Thick ideals in equivariant and motivic stable homotopy categories_, [arXiv:1503.08456](http://arxiv.org/abs/1503.08456).

On ([[stable cohomotopy|stable]]) [[motivic cohomology|motivic]] [[Cohomotopy]] of [[schemes]] (as  [[motivic homotopy theory|motivic homotopy classes]] of maps into [[motivic sphere|motivic]] [[Tate spheres]]):

* [[Aravind Asok]], [[Jean Fasel]], [[Mrinal Kanti Das]], *Euler class groups and motivic stable cohomotopy*, Journal of the EMS **24** 8 (2022) 2775–2822 [[arXiv:1601.05723](https://arxiv.org/abs/1601.05723), [doi:10.4171/jems/1156](https://doi.org/10.4171/jems/1156)]

* [[Samuel Lerbet]], *Motivic stable cohomotopy and unimodular rows*, Advances in Mathematics **436** 109415 (2024) [[arXiv:2206.11688](https://arxiv.org/abs/2206.11688), [doi:10.1016/j.aim.2023.109415](https://doi.org/10.1016/j.aim.2023.109415)]

### Motivic homotopy theory in other contexts

[[equivariant motivic homotopy theory]] is developed in

* {#CJ2014} [[Gunnar Carlsson]], [[Roy Joshua]], _Equivariant motivic homotopy theory_, [arXiv](http://arxiv.org/abs/1404.1597v1).

This was vastly generalized and studied more thoroughly in

* {#Hoyois15} [[Marc Hoyois]], *The six operations in equivariant motivic homotopy theory*, Adv. Math. **305** (2017) 197-279 [[arXiv:1509.02145](https://arxiv.org/abs/1509.02145), [doi:10.1016/j.aim.2016.09.031](https://doi.org/10.1016/j.aim.2016.09.031)]

  > (cf. [[Grothendieck's six operations]])

Motivic homotopy theory of [[derived noncommutative geometry|noncommutative spaces]] (associative dg-algebras) is studied in

* [[Marco Robalo]], _Noncommutative Motives I: A Universal Characterization of the Motivic Stable Homotopy Theory of Schemes_, 2013 ([pdf](http://arxiv.org/pdf/1206.3645v3.pdf))

Motivic homotopy theory of [[associative ring|associative]] [[nonunital rings]] is studied in

* [[Grigory Garkusha]], _Homotopy theory of associative rings_, [arXiv:math/0608482](http://arxiv.org/abs/math/0608482v2).

* [[Grigory Garkusha]], _Algebraic Kasparov K-theory. II_, [arXiv:1206.0178](http://arxiv.org/abs/1206.0178v2).

See also

* [[analytic motivic homotopy theory]]
* [[logarithmic motivic homotopy theory]]

[[!redirects A1-homotopy]]
[[!redirects A1 homotopy]]
[[!redirects A1-homotopy theory]]
[[!redirects A1 homotopy theory]]
[[!redirects A1-homotopy category]]
[[!redirects A1 homotopy category]]

[[!redirects stable motivic homotopy theory]]
[[!redirects motivic stable homotopy theory]]

[[!redirects motivic homotopy category]]
[[!redirects stable motivic homotopy category]]
[[!redirects motivic space]]
[[!redirects motivic spaces]]
[[!redirects motivic spectrum]]
[[!redirects motivic spectra]]

[[!redirects homotopy theory of schemes]]


motivic integration

+-- {: .rightHandSide}
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###Context###
#### Integration theory
+--{: .hide}
[[!include integration theory - contents]]
=--
#### Motivic cohomology
+--{: .hide}
[[!include motivic cohomology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

__Motivic integration__ attaches to certain subsets of [[arc scheme]]s of varieties an element in a [[Grothendieck ring of varieties]] in a manner close in spirit to [[Euler characteristics]]. Technically it may be viewed as one of generalizations of [[p-adic integration]] to a geometric integration theory obtained by replacing the [[p-adic integers]] by a more broad class of rings. Some examples include [[formal power series ring]] over the [[complex numbers]], and more generally any henselian discretely valued fields of residual characteristic zero. [Recently](#References), these cases have been generalized to the case of any complete [[discrete valuation ring]] (whose [[residue field]] is [[perfect field|perfect]] in the [[mixed characteristic]] case).

In all of the described cases, the idea is to construct a suitable [[scheme]] over the residue field, called the [[arc space|arc scheme]] which serves as a suitable [[measure space]] on which functions are integrated; however, the value of integrals are not [[real number|real numbers]], but instead they are valued in the [[Grothendieck ring]] of varieties. In this sense, the value of a "motivic" integral is often regarded as a _geometric object_, rather than a numerical value.

The notion of arc scheme has a further generalization, known as a [[Greenberg scheme]].

## History

Motivic integration was introduced in the talk of [[Maxim Kontsevich]] at Orsay in 1995 in order to prove that [[Hodge numbers]] of [[Calabi-Yau manifolds]] are [[birational map|birational]] invariants. This talk also dealt with [[orbifold cohomology]] as well as 2 related papers of  [[Lev Borisov]]. The orbifold cohomology has been continued by Weimin Chen, [[Yongbin Ruan]] and collaborators, and later also by algebraic geometers Abramovich, Vistoli, and others. Batyrev is another contributor to both subjects in the context of physics.

Later, a more general framework of motivic integration in model theory has been put forward by Denef and Loeser, partly based on Denef's work on $p$-adic integration. More recent work using a model-theoretic approach is by Hrushovski and Kazhdan.

## Related concepts

* [[p-adic integration]]

* [[motivic cohomology]]

* [[arc space]]

* [[Greenberg scheme]]

## References
 {#References}

A textbook account is in

*  {#CNS18} Antoine Chambert-Loir, Johannes Nicaise, Julien Sebag, _Motivic integration_, Progress in Mathematics 325 Birkhaeuser 2018 ([doi:10.1007/978-1-4939-7887-8](https://doi.org/10.1007/978-1-4939-7887-8))

See also

* Wikipedia _[Motivic integration](https://en.wikipedia.org/wiki/Motivic_integration)_

Lectures include

* Karen Smith, _Motivic integration_, ([pdf](http://www.msri.org/ext/shortermotivic2.pdf))
* Manuel Blickle, _A short course on geometric motivic integration_, [math.AG/0507404](http://arxiv.org/abs/math/0507404)
* [[Raf Cluckers]], _Motivic integration for dummies_, [pdf](http://wis.kuleuven.be/algebra/Raf/MForD.pdf), A course on motivic integration [I](http://wis.kuleuven.be/algebra/Raf/LaRoche3Cluckers.pdf), [II](http://wis.kuleuven.be/algebra/Raf/LaRoche3Cluckers.pdf), [III](http://wis.kuleuven.be/algebra/Raf/LaRoche3Cluckers.pdf)
* Julia Gordon, Yoav Yaffe, _An overview of arithmetic motivic integration_, [arxiv/0811.2160](http://arxiv.org/abs/0811.2160)
* [[Lou van den Dries]], _Lectures on motivic integration_ , Ms. University of Illinois at Urbana-Champaign. ([dvi](http://www.math.uiuc.edu/~vddries/mo.dvi))
* Thomas C. Hales, _What is motivic measure?_, [math.LO/0312229](https://arxiv.org/abs/math/0312229)

The concept originates from

* [[M. Kontsevich]], lecture on motivic integration, Orsay, December 7, 1995.

Original articles include the following:

* Jan Denef, [[François  Loeser]], _Definable sets, motives and $p$-adic integrals_,  J. Amer. Math. Soc. __14__ (2001),  no. 2, 429--469, [doi](http://dx.doi.org/10.1090/S0894-0347-00-00360-X)

* Jan Denef, François  Loeser, _Motivic integration and the Grothendieck group of pseudo-finite fields_ Proc. ICM, Vol. II (Beijing, 2002), 13--23, Higher Ed. Press, Beijing, 2002.

* R. Cluckers, F. Loeser, _Constructible motivic functions and motivic integration_, Invent. Math. __173__ (2008), 23--121 [math.AG/0410203](http://arxiv.org/abs/math/0410203)

* Jan Denef, François  Loeser, _Germs of arcs on singular algebraic varieties and motivic integration_,  Invent. Math. __135__  (1999),  no. 1, 201--232.

* D. Abramovich, M. Mariño, M. Thaddeus, R. Vakil, _Enumerative invariants in algebraic geometry and string theory_, Lectures from the C.I.M.E. Summer School, Cetraro, June 6--11, 2005. Edited by Kai Behrend and Marco Manetti. LNIM 1947, Springer 2008. x+201 pp.

Hrushovski and Kazhdan studied motivic integration from [[model theory]] point of view:

* [[Ehud Hrushovski]], [[David Kazhdan]], _Motivic Poisson summation_, [arxiv/0902.0845](http://arxiv.org/abs/0902.0845)

* [[Ehud Hrushovski]], [[David Kazhdan]], _The value ring of geometric motivic integration and the Iwahori Hecke algebra of $SL_2$_, [math.LO/0609115](http://arxiv.org/abs/math/0609115); _Integration in valued fields_, in Algebraic geometry and number theory, 261–405, Progress. Math. __253__, Birkhäuser Boston, [pdf](http://math.huji.ac.il/~ehud/papers/intv-060428.pd

* [[David Kazhdan]], _Lecture notes in motivic integration_, with intro to logic and model theory, [pdf](http://www.ma.huji.ac.il/~kazhdan/Notes/motivic/b.pdf)
* R. Cluckers, J. Nicaise, J. Sebag (Editors), _Motivic Integration and its Interactions with Model Theory and Non-Archimedean Geometry_, 2 vols.
London Mathematical Society Lecture Note Series __383__, __384__
* Raf Cluckers, Julia Gordon, Immanuel Halupczok, _Motivic functions, integrability, and uniform in p bounds for orbital integrals_, [arxiv:1309.0594](https://arxiv.org/abs/1309.0594)

* Julien Sebag, _Intégration motivique sur les schémas formels_,  Bull. Soc. Math. France __132__:1 (2004) 1--54, [MR2005e:14017](https://mathscinet.ams.org/mathscinet-getitem?mr=2075915)

* Takehiko Yasuda, _Motivic integration over Deligne-Mumford stacks_ (2004) arXiv:[math/0312115](https://arxiv.org/abs/math/0312115)

* [[Michael Larsen]], [[Valery Lunts]], _Motivic measures and stable birational geometry_, Mosc. Math. J. 3 (2003), no. 1, 85--95, 259, [math.AG/0110255](http://arxiv.org/abs/math/0110255), [MR2005a:14026](http://www.ams.org/mathscinet-getitem?mr=1996804), [journal](http://www.ams.org/distribution/mmj/vol3-1-2003/abst3-1-2003.html#larsen-lunts_abstract); _Rationality criteria for motivic zeta functions_, Compos. Math. __140__ (2004), no. 6, 1537--1560, [math.AG/0212158](https://arxiv.org/abs/math/0212158)
* [[Alexei Bondal]], [[Michael Larsen]], [[Valery Lunts]], _Grothendieck ring of pretriangulated categories_, Int. Math. Res. Not. 2004, no. 29, 1461--1495, [math.AG/0401009](https://arxiv.org/abs/math/0401009)

* Emmanuel Bultot, _Motivic integration and logarithmic geometry_, PhD thesis [arxiv/1505.05688](https://arxiv.org/abs/1505.05688)

[[!redirects motivic measure]]


motivic L-function

#Contents#
* table of contents
{:toc}

## Idea

A generalization of the [[Hasse-Weil L-functions]] to more general [[motives]].

## Related concepts

* [[motivic multiple zeta function]]

## References

### General

* Wikipedia, _[Motivic L-functions](http://en.wikipedia.org/wiki/Motivic_L-function)_

### In physics

* [[Rolf Schimmrigk]], _Motivic L-Function Identities from CFT and Arithmetic Mirror Symmetry_ ([arXiv:1301.2225](http://arxiv.org/abs/1301.2225))

See also _[[motives in physics]]_

[[!redirects motivic L-functions]]


motivic multiple zeta values

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

#Contents#
* table of contents
{:toc}

## Idea

Where a [[zeta function]] and [[multiple zeta function]] may be assigned to a suitable [[variety]], so a _motivic multiple zeta function_ is attached to the corresponding [[motive]], like a [[motivic L-function]] is.

Where [[zeta functions]] appear in [[physics]] as expressions for [[vacuum amplitudes]], so [[multiple zeta functions]] appear in expressions for more general [[scattering amplitudes]]. The intricate combinatorics of these becomes often more tractable when re-expressing them as motivic multiple zeta values ([e.g. Schlotterer-Stieberger 12](#SchottererStieberger12)).

## Related concepts

* [[zeta function]]

* [[multiple zeta values]]

* [[motive]], [[motives in physics]]

* [[motivic L-function]]

* [[mixed Tate motive]]

## References

### General

* Francis Brown, _On the decomposition of motivic multiple zeta values_ ([arXiv:1102.1310v2](http://arxiv.org/abs/1102.1310v2))

* A. B. Goncharov, _Galois symmetries of fundamental groupoids and noncommutative geometry_ ([arXiv:math/0208144](http://arxiv.org/abs/math/0208144))

### In physics (scattering amplitudes)
 {#ReferencesScattering}

In [[scattering amplitudes]].

Of the [[superstring]]:

* {#SchottererStieberger12} O. Schlotterer, [[Stephan Stieberger]], _Motivic Multiple Zeta Values and Superstring Amplitudes_ ([arXiv:1205.1516](http://arxiv.org/abs/1205.1516))

* [[Stephan Stieberger]] _Motivic superstring amplitudes_, 2013 ([pdf](http://www.maths.dur.ac.uk/lms/2013/PNTPP13/talks/0163stieberger.pdf))

In [[N=4 D=4 super Yang-Mills theory]]:

* Marcus Spradlin, Anastasia Volovich, _Symbols of One-Loop Integrals From Mixed Tate Motives_ ([arXiv:1105.2024](http://arxiv.org/abs/1105.2024))

See also at _[[motives in physics]]_.

[[!redirects motivic multiple zeta function]]
[[!redirects motivic multiple zeta functions]]