Contents

# Contents

## 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 localization at that object. Ordinary homotopy theory is obtained by taking $C$ to be the 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 schemes of finite type over $S$ and

(1)$I \coloneqq \mathbb{A}^1$

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

and then

## The unstable motivic homotopy category

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

###### Definition

The motivic homotopy category $\mathrm{H}(S)$ over $S$ is the homotopy localization at the affine line $\mathbb{A}^1$ (1) 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 and locally cartesian closed (∞,1)-category. However, it is not an (∞,1)-topos (see Remark 3.5 in Spitzweck-Østvær, Motivic twisted K-theory, pdf)

### Motivic spheres

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 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 $p\geq q\geq 0$,

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

There is a canonical equivalence of pointed motivic spaces $T\simeq (\mathbb{P}^1,1)$.

###### Proof

The cartesian square

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

becomes homotopy cocartesian in the Nisnevich (even 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

###### 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. , 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

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:

###### Proposition

Let $\mathcal{C}$ be a locally presentable 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 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).

###### Remark

Similar characterizations exist for noncommutative motives, see at Noncommutative motive – As the universal additive invariant.

### Stable motivic spheres

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).

### Cohomology theories

Any motivic spectrum $E\in SH(S)$ gives rise to a bigraded 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

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 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.

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 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.

3. (Smooth base change/Beck-Chevalley condition) If $f$ is 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.

5. (Homotopy invariance) If $p: X\times\mathbb{A}^1\to X$ is the projection, then $p^*$ is 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.

###### 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.

###### Theorem

$SH$ is a stable homotopy functor.

This is essentially proved in Morel-Voevodsky 99. In fact, something stronger is expected to be true:

###### 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 shows that every stable homotopy functor factors through $SH$.

### Realization functors

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 (∞,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 ∞-groupoids, 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, called the complex Betti realization

$\mathrm{Be}:\mathrm{Sp}(\mathbb{C}) \rightarrow \mathrm{Sp}$

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 genuine $\mathbb{Z}/2$-spectra, called the real Betti realization

$\mathrm{Be}:\mathrm{Sp}(\mathbb{R}) \rightarrow \mathrm{Sp}_{C_2}$

(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

See motivic slice filtration.

The slice filtration is a filtration of $\mathrm{H}(S)$ and of $SH(S)$ which is analogous to the 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 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 $H(\mathbb{Z})$. 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 sequences in that their first page consists of motivic cohomology groups.

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

One can also consider the filtration on $\mathrm{H}(S)$ induced by the 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 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.

###### 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.

###### Theorem

Let $k$ be a perfect field (resp. a field) and $X$ a 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). 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 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 forms over $k$ (Morel, Corollary 5.43). 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

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

## Applications and examples

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

Like the $\mathbb{A}^1$-Postnikov filtration, $\mathbb{A}^1$-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 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

$\Pi_1^{\mathbb{A}^1}(X):=\tau_{\leq 1}X.$
###### 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. 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). This can be used to compute some $\mathbb{A}^1$-fundamental groups, for example:

###### Proposition

If $n\geq 2$, $\pi_1^{\mathbb{A}^1}(\mathbb{P}^n)=\mathbb{G}_m$.

###### 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 fields up to $\mathbb{A}^1$-homotopy. See Asok-Morel.

### 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. . 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:

###### 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, this is just the usual Chow group of $X$ and $e(\xi)$ can be identified with the top Chern class of $\xi$.

### Motivic cohomology

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.

## References

### General

The original references:

Readable introductions to the subject are:

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

with brief exposition in:

For more on the general procedure see homotopy localization.

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

• Marco Robalo, Noncommutative Motives I: A Universal Characterization of the Motivic Stable Homotopy Theory of Schemes, 2013 (pdf)

For the formalism of six operations see

The slice filtration was defined in

Representability results:

Discussion related to étale homotopy:

• Daniel Isaksen, Étale realization of the $\mathbb{A} ^1$-homotopy theory of schemes, Advances in Mathematics 184 (2004)

• Ruth Joachimi?, Thick ideals in equivariant and motivic stable homotopy categories, arXiv:1503.08456.

On (stable) motivic Cohomotopy of schemes (as motivic homotopy classes of maps into motivic Tate spheres):

### Motivic homotopy theory in other contexts

equivariant motivic homotopy theory is developed in

This was vastly generalized and studied more thoroughly in

Motivic homotopy theory of 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)

Motivic homotopy theory of associative nonunital rings is studied in