A¹ homotopy theory

Choose a base scheme ${\displaystyle S}$. Classically, ${\displaystyle S}$ is asked to be Noetherian, but many modern authors such as Marc Hoyois work with quasi-compact quasi-separated base schemes. In any case, many important results are only known over a perfect base field, such as the complex numbers, it's perfectly fine to consider only this case.

Step 1a: Nisnevich sheaves. Classically, the construction begins with the category ${\displaystyle Shv(Sm_{S})_{Nis}}$ of Nisnevich sheaves on the category ${\displaystyle Sm_{S}}$ of smooth schemes over ${\displaystyle S}$. Heuristically, this should be considered as (and in a precise technical sense is) the universal enlargement of ${\displaystyle Sm_{S}}$ obtained by adjoining all colimits and forcing Nisnevich descent to be satisfied.

Step 1b: simplicial sheaves. In order to more easily perform standard homotopy theoretic procedures such as homotopy colimits and homotopy limits, ${\displaystyle Shv_{Nis}(Sm_{S})}$ replaced with the following category of simplicial sheaves.

Let Δ be the simplex category, that is, the category whose objects are the sets

{0}, {0, 1}, {0, 1, 2}, ...,

and whose morphisms are order-preserving functions. We let ${\displaystyle \Delta ^{op}Shv(Sm_{S})_{Nis}}$ denote the category of functors ${\displaystyle \Delta ^{op}\to Shv(Sm_{S})_{Nis}}$. That is, ${\displaystyle \Delta ^{op}Shv(Sm_{S})_{Nis}}$ is the category of simplicial objects on ${\displaystyle Shv(Sm_{S})_{Nis}}$. Such an object is also called a simplicial sheaf on ${\displaystyle Sm_{S}}$.

Step 1c: fibre functors. For any smooth ${\displaystyle S}$-scheme ${\displaystyle X}$, any point ${\displaystyle x\in X}$, and any sheaf ${\displaystyle F}$, let's write ${\displaystyle x^{*}F}$ for the stalk of the restriction ${\displaystyle F|_{X_{Nis}}}$ of ${\displaystyle F}$ to the small Nisnevich site of ${\displaystyle X}$. Explicitly, ${\displaystyle x^{*}F=colim_{x\to V\to X}F(V)}$ where the colimit is over factorisations ${\displaystyle x\to V\to X}$ of the canonical inclusion ${\displaystyle x\to X}$ via an étale morphism ${\displaystyle V\to X}$. The collection ${\displaystyle \{x^{*}\}}$ is a conservative family of fibre functors for ${\displaystyle Shv(Sm_{S})_{Nis}}$.

Step 1d: the closed model structure. We will define a closed model structure on ${\displaystyle \Delta ^{op}Shv(Sm_{S})_{Nis}}$ in terms of fibre functors. Let ${\displaystyle f:{\mathcal {X}}\to {\mathcal {Y}}}$ be a morphism of simplicial sheaves. We say that:

• f is a weak equivalence if, for any fibre functor x of T, the morphism of simplicial sets ${\displaystyle x^{*}f:x^{*}{\mathcal {X}}\to x^{*}{\mathcal {Y}}}$ is a weak equivalence.
• f is a cofibration if it is a monomorphism.
• f is a fibration if it has the right lifting property with respect to any cofibration which is a weak equivalence.

The homotopy category of this model structure is denoted ${\displaystyle {\mathcal {H}}_{s}(T)}$.

This model structure has Nisnevich descent, but it does not contract the affine line. A simplicial sheaf ${\displaystyle {\mathcal {X}}}$ is called ${\displaystyle \mathbb {A} ^{1}}$-local if for any simplicial sheaf ${\displaystyle {\mathcal {Y}}}$ the map

${\displaystyle {\text{Hom}}_{{\mathcal {H}}_{s}(T)}({\mathcal {Y}}\times \mathbb {A} ^{1},{\mathcal {X}})\to {\text{Hom}}_{{\mathcal {H}}_{s}(T)}({\mathcal {Y}},{\mathcal {X}})}$

induced by ${\displaystyle i_{0}:\{0\}\to \mathbb {A} ^{1}}$ is a bijection. Here we are considering ${\displaystyle \mathbb {A} ^{1}}$ as a sheaf via the Yoneda embedding, and the constant simplicial object functor ${\displaystyle Shv(Sm_{S})_{Nis}\to \Delta ^{op}Shv(Sm_{S})_{Nis}}$.

A morphism ${\displaystyle f:{\mathcal {X}}\to {\mathcal {Y}}}$ is an ${\displaystyle \mathbb {A} ^{1}}$-weak equivalence if for any ${\displaystyle \mathbb {A} ^{1}}$-local ${\displaystyle {\mathcal {Z}}}$, the induced map

${\displaystyle {\text{Hom}}_{{\mathcal {H}}_{s}(T)}({\mathcal {Y}},{\mathcal {Z}})\to {\text{Hom}}_{{\mathcal {H}}_{s}(T)}({\mathcal {X}},{\mathcal {Z}})}$

is a bijection. The ${\displaystyle \mathbb {A} ^{1}}$-local model structure is the localisation of the above model with respect to ${\displaystyle \mathbb {A} ^{1}}$-weak equivalences.

Finally we may define the A¹ homotopy category.

Definition. Let S be a finite-dimensional Noetherian scheme (for example ${\displaystyle S=Spec(\mathbb {C} )}$ the spectrum of the complex numbers), and let Sm_/S denote the category of smooth schemes over S. Equip Sm_/S with the Nisnevich topology to get the site (Sm_/S)_Nis. The homotopy category (or infinity category) associated to the ${\displaystyle \mathbb {A} ^{1}}$-local model structure on ${\displaystyle \Delta ^{op}Shv_{*}(Sm_{S})_{Nis}}$ is called the A¹-homotopy category. It is denoted ${\displaystyle {\mathcal {H}}_{s}}$. Similarly, for the pointed simplicial sheaves ${\displaystyle \Delta ^{op}Shv_{*}(Sm_{S})_{Nis}}$ there is an associated pointed homotopy category ${\displaystyle {\mathcal {H}}_{s,*}}$.

Note that by construction, for any X in Sm_/S, there is an isomorphism

X ×_S A¹
_S ≅ X,

in the homotopy category.

Because we started with a simplicial model category to construct the ${\displaystyle \mathbf {A} ^{1}}$-homotopy category, there are a number of structures inherited from the abstract theory of simplicial models categories. In particular, for ${\displaystyle {\mathcal {X}},{\mathcal {Y}}}$ pointed simplicial sheaves in ${\displaystyle \Delta ^{op}{\text{Sh}}_{*}({\text{Sm}}/S)_{nis}}$ we can form the wedge product as the colimit

${\displaystyle {\mathcal {X}}\vee {\mathcal {Y}}={\underset {\to }{\text{colim}}}\left\{{\begin{matrix}*&\to &{\mathcal {X}}\\\downarrow &&\\{\mathcal {Y}}\end{matrix}}\right\}}$

and the smash product is defined as

${\displaystyle {\mathcal {X}}\wedge {\mathcal {Y}}={\mathcal {X}}\times {\mathcal {Y}}/{\mathcal {X}}\vee {\mathcal {Y}}}$

recovering some of the classical constructions in homotopy theory. There is in addition a cone of a simplicial (pre)sheaf and a cone of a morphism, but defining these requires the definition of the simplicial spheres.

From the fact we start with a simplicial model category, this means there is a cosimplicial functor

${\displaystyle \Delta ^{\bullet }:\Delta \to \Delta ^{op}{\text{Sh}}_{*}({\text{Sm}}/S)_{nis}}$

defining the simplices in ${\displaystyle \Delta ^{op}{\text{Sh}}_{*}({\text{Sm}}/S)_{nis}}$. Recall the algebraic n-simplex is given by the ${\displaystyle S}$-scheme

${\displaystyle \Delta ^{n}={\text{Spec}}\left({\frac {{\mathcal {O}}_{S}[t_{0},t_{1},\ldots ,t_{n}]}{(t_{0}+t_{1}+\cdots +t_{n}-1)}}\right)}$

Embedding these schemes as constant presheaves and sheafifying gives objects in ${\displaystyle \Delta ^{op}{\text{Sh}}_{*}({\text{Sm}}/S)_{nis}}$, which we denote by ${\displaystyle \Delta ^{n}}$. These are the objects in the image of ${\displaystyle \Delta ^{\bullet }([n])}$, i.e. ${\displaystyle \Delta ^{\bullet }([n])=\Delta ^{n}}$. Then using abstract simplicial homotopy theory, we get the simplicial spheres

${\displaystyle S^{n}=\Delta ^{n}/\partial \Delta ^{n}}$

We can then form the cone of a simplicial (pre)sheaf as

${\displaystyle C({\mathcal {X}})={\mathcal {X}}\wedge \Delta ^{1}}$

and form the cone of a morphism ${\displaystyle f:{\mathcal {X}}\to {\mathcal {Y}}}$ as the colimit of the diagram

${\displaystyle C(f)={\underset {\to }{\text{colim}}}\left\{{\begin{matrix}{\mathcal {X}}&\xrightarrow {f} &{\mathcal {Y}}\\\downarrow &&\\C({\mathcal {X}})\end{matrix}}\right\}}$

In addition, the cofiber of ${\displaystyle {\mathcal {Y}}\to C(f)}$ is simply the suspension ${\displaystyle {\mathcal {X}}\wedge S^{1}=\Sigma {\mathcal {X}}}$. In the pointed homotopy category there is additionally the suspension functor

${\displaystyle \Sigma$ :{\mathcal {H}}_{s,*}(Sm/S)_{Nis}\to {\mathcal {H}}_{s,*}(Sm/S)_{Nis}} given by ${\displaystyle \Sigma ({\mathcal {X}})={\mathcal {X}}\wedge S^{1}}$

and its right adjoint

${\displaystyle \Omega$ :{\mathcal {H}}_{s,*}(Sm/S)_{Nis}\to {\mathcal {H}}_{s,*}(Sm/S)_{Nis}}

called the loop space functor.

The setup, especially the Nisnevich topology, is chosen as to make algebraic K-theory representable by a spectrum, and in some aspects to make a proof of the Bloch-Kato conjecture possible.

After the Morel-Voevodsky construction there have been several different approaches to A¹ homotopy theory by using other model category structures or by using other sheaves than Nisnevich sheaves (for example, Zariski sheaves or just all presheaves). Each of these constructions yields the same homotopy category.

There are two kinds of spheres in the theory: those coming from the multiplicative group playing the role of the 1-sphere in topology, and those coming from the simplicial sphere (considered as constant simplicial sheaf). This leads to a theory of motivic spheres S^p,q with two indices. To compute the homotopy groups of motivic spheres would also yield the classical stable homotopy groups of the spheres, so in this respect A¹ homotopy theory is at least as complicated as classical homotopy theory.

For an abelian group ${\displaystyle A}$ the ${\displaystyle (p,q)}$-motivic cohomology of a smooth scheme ${\displaystyle X}$ is given by the sheaf hypercohomology groups

${\displaystyle H^{p,q}(X,A):=\mathbb {H} ^{p}(X_{nis},A(q))}$

for ${\displaystyle A(q)=\mathbb {Z} (q)\otimes A}$. Representing this cohomology is a simplicial abelian sheaf denoted ${\displaystyle K(p,q,A)}$ corresponding to ${\displaystyle A(q)[+p]}$ which is considered as an object in the pointed motivic homotopy category ${\displaystyle H_{\bullet }(k)}$. Then, for a smooth scheme ${\displaystyle X}$ we have the equivalence

${\displaystyle {\text{Hom}}_{H_{\bullet }(k)}(X_{+},K(p,q,A))=H^{p,q}(X,A)}$

showing these sheaves represent motivic Eilenberg-Maclane spaces[1]^{pg 3}.

• Morel (2002) An Introduction to A¹-homotopy theory
• Antieau, Benjamin; Elmanto, Elden (2016), A primer for unstable motivic homotopy theory, arXiv:1605.00929, Bibcode:2016arXiv160500929A

• Motivic stable homotopy groups
• Morel, Fabien; Voevodsky, Vladimir (1999), "A¹-homotopy theory of schemes" (PDF), Publications Mathématiques de l'IHÉS, 90 (90): 45–143, doi:10.1007/BF02698831, MR 1813224, S2CID 14420180, retrieved 9 May 2008
• Voevodsky, Vladimir (1998), "A¹-homotopy theory" (PDF), Documenta Mathematica, Proceedings of the International Congress of Mathematicians, Vol. I (Berlin, 1998): 579–604, ISSN 1431-0635, MR 1648048
• Voevodsky, Vladimir (2008) "Unstable motivic homotopy categories in Nisnevich and cdh-topologies"

• Voevodsky, Vladimir (2001) "Reduced power operations in motivic cohomology"
• Voevodsky, Vladimir (2008) "Motivic Eilenberg-Maclane spaces"

• The motivic Adams spectral sequence
• Motivic chromatic homotopy theory

• Jardine. (1999) Motivic Symmetric Spectra

• The Gersten conjecture for Milnor K-theory
• Tate twists and cohomology of P¹

• The motivic Steenrod algebra in positive characteristic
• Motivic stable homotopy groups
• On the Motivic ${\displaystyle \pi _{0}}$ of the Sphere Spectrum (Springer)
• The first stable homotopy groups of the motivic sphere
• On the zero slice of the sphere spectrum
• Vanishing in stable motivic homotopy sheaves

• Bachmann, Tom (2018), "Motivic and Real Etale Stable Homotopy Theory", Compositio Mathematica, 154 (5): 883–917, arXiv:1608.08855, doi:10.1112/S0010437X17007710, S2CID 119305101