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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{motivic homotopy theory} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{homotopy_theory}{}\paragraph*{{Homotopy theory}}\label{homotopy_theory} [[!include homotopy - contents]] \hypertarget{motivic_cohomology}{}\paragraph*{{Motivic cohomology}}\label{motivic_cohomology} [[!include motivic cohomology - contents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{UnstableCategory}{The unstable motivic homotopy category}\dotfill \pageref*{UnstableCategory} \linebreak \noindent\hyperlink{MotivicSpheres}{Motivic spheres}\dotfill \pageref*{MotivicSpheres} \linebreak \noindent\hyperlink{StableCategory}{The stable motivic homotopy category}\dotfill \pageref*{StableCategory} \linebreak \noindent\hyperlink{SymmetricMonoidalStructureAndUniversalProperty}{Symmetric monoidal structure and universal property}\dotfill \pageref*{SymmetricMonoidalStructureAndUniversalProperty} \linebreak \noindent\hyperlink{StableMotivicSpheres}{Stable motivic spheres}\dotfill \pageref*{StableMotivicSpheres} \linebreak \noindent\hyperlink{CohomologyTheories}{Cohomology theories}\dotfill \pageref*{CohomologyTheories} \linebreak \noindent\hyperlink{main_features}{Main features}\dotfill \pageref*{main_features} \linebreak \noindent\hyperlink{SixOperations}{The six operations}\dotfill \pageref*{SixOperations} \linebreak \noindent\hyperlink{RealizationFunctors}{Realization functors}\dotfill \pageref*{RealizationFunctors} \linebreak \noindent\hyperlink{SliceFiltration}{The slice filtration}\dotfill \pageref*{SliceFiltration} \linebreak \noindent\hyperlink{A1PostnikovFiltration}{The $\mathbb{A}^1$-Postnikov filtration}\dotfill \pageref*{A1PostnikovFiltration} \linebreak \noindent\hyperlink{relation_to_the_theory_of_motives}{Relation to the theory of motives}\dotfill \pageref*{relation_to_the_theory_of_motives} \linebreak \noindent\hyperlink{relation_to_the_theory_of_symmetric_bilinear_forms}{Relation to the theory of symmetric bilinear forms}\dotfill \pageref*{relation_to_the_theory_of_symmetric_bilinear_forms} \linebreak \noindent\hyperlink{EquivariantMotivicHomotopyTheory}{Equivariant motivic homotopy theory}\dotfill \pageref*{EquivariantMotivicHomotopyTheory} \linebreak \noindent\hyperlink{applications_and_examples}{Applications and examples}\dotfill \pageref*{applications_and_examples} \linebreak \noindent\hyperlink{A1coverings}{$\mathbb{A}^1$-coverings and the $\mathbb{A}^1$-fundamental groupoid}\dotfill \pageref*{A1coverings} \linebreak \noindent\hyperlink{hcobordisms_and_the_classification_of_surfaces}{$\mathbb{A}^1$-h-cobordisms and the classification of surfaces}\dotfill \pageref*{hcobordisms_and_the_classification_of_surfaces} \linebreak \noindent\hyperlink{euler_classes_and_splittings_of_algebraic_vector_bundles}{Euler classes and splittings of algebraic vector bundles}\dotfill \pageref*{euler_classes_and_splittings_of_algebraic_vector_bundles} \linebreak \noindent\hyperlink{MotivicCohomology}{Motivic cohomology}\dotfill \pageref*{MotivicCohomology} \linebreak \noindent\hyperlink{algebraic_ktheory}{Algebraic K-theory}\dotfill \pageref*{algebraic_ktheory} \linebreak \noindent\hyperlink{AlgebraicCobordism}{Algebraic cobordism}\dotfill \pageref*{AlgebraicCobordism} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak \noindent\hyperlink{references_2}{References}\dotfill \pageref*{references_2} \linebreak \noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak \noindent\hyperlink{motivic_homotopy_theory_in_other_contexts}{Motivic homotopy theory in other contexts}\dotfill \pageref*{motivic_homotopy_theory_in_other_contexts} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} \emph{Motivic homotopy theory} or \emph{$\mathbf{A}^1$-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 \emph{motivic homotopy category} or the \emph{$\mathbb{A}^1$-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 \begin{equation} I \coloneqq \mathbb{A}^1 \label{AffineLine}\end{equation} 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 \begin{itemize}% \item \emph{\hyperlink{UnstableCategory}{The unstable motivic homotopy category}} \end{itemize} and then \begin{itemize}% \item \emph{\hyperlink{StableCategory}{The stable motivic homotopy category}}. \end{itemize} \hypertarget{UnstableCategory}{}\subsection*{{The unstable motivic homotopy category}}\label{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 $S$. \begin{defn} \label{MotivicHomotopyCategory}\hypertarget{MotivicHomotopyCategory}{} The \emph{motivic homotopy category} $\mathrm{H}(S)$ over $S$ is the [[homotopy localization]] at the [[affine line]] $\mathbb{A}^1$ \eqref{AffineLine} of the [[(∞,1)-topos of (∞,1)-sheaves]] on the [[Nisnevich site]] $Sm/S$. Objects of $\mathrm{H}(S)$ are called \emph{motivic spaces}. \end{defn} Thus, a motivic space over $S$ is an [[(∞,1)-presheaf]] $F$ on $Sm/S$ such that \begin{itemize}% \item $F$ is an [[(∞,1)-sheaf]] for the [[Nisnevich topology]] \item $F$ is \emph{$\mathbb{A}^1$-homotopy invariant}: for every $X\in Sm/S$, the projection $X\times\mathbb{A}^1\to X$ induces an [[equivalence]] $F(X)\simeq F(X\times\mathbb{A}^1$). \end{itemize} 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 \href{http://arxiv.org/pdf/1008.4915v1.pdf}{Spitzweck-\O{}stv\ae{}r, \emph{Motivic twisted K-theory}, pdf}) \hypertarget{MotivicSpheres}{}\subsubsection*{{Motivic spheres}}\label{MotivicSpheres} The \textbf{Tate sphere} is the [[pointed object]] of $\mathrm{H}(S)$ defined by \begin{displaymath} T {:=} \mathbb{A}^1/\mathbb{G}_m, \end{displaymath} 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 \emph{motivic sphere} given by its \emph{[[Thom space]]}: \begin{displaymath} S^V {:=} V/V^\times. \end{displaymath} Thus, $T=S^{\mathbb{A}^1}$. A crucial observation is that $T$ is the [[suspension]] of $\mathbb{G}_m$ (pointed at $1$): \begin{displaymath} T \simeq S^1\wedge\mathbb{G}_m. \end{displaymath} 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 $p\geq q\geq 0$, \begin{displaymath} S^{p,q} {:=} S^{p-q}\wedge \mathbb{G}_m^{\wedge q}. \end{displaymath} \begin{prop} \label{}\hypertarget{}{} There is a canonical equivalence of pointed motivic spaces $T\simeq (\mathbb{P}^1,1)$. \end{prop} \begin{proof} The cartesian square \begin{displaymath} \itexarray{ \mathbb{G}_m &\stackrel{}{\to}& \mathbb{A}^1 \\ \downarrow && \downarrow \\ \mathbb{A}^1 &\stackrel{}{\to}& \mathbb{P}^1 } \end{displaymath} 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 $(\mathbb{P}^1,1)\simeq S^1\wedge \mathbb{G}_m\simeq T.$ \end{proof} \hypertarget{StableCategory}{}\subsection*{{The stable motivic homotopy category}}\label{StableCategory} \begin{defn} \label{StableMotivicHomotopyCategory}\hypertarget{StableMotivicHomotopyCategory}{} The \emph{stable motivic homotopy category} $SH(S)$ over $S$ is the [[inverse limit]] of the tower of (∞,1)-categories \begin{displaymath} \dots \stackrel{\Omega_T}{\to} \mathrm{H}_*(S) \stackrel{\Omega_T}{\to} \mathrm{H}_*(S) \stackrel{\Omega_T}{\to} \mathrm{H}_*(S), \end{displaymath} 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 \emph{[[motivic spectrum]]} (or \emph{$T$-spectrum}). \end{defn} Thus, a motivic spectrum $E$ is a sequence of pointed motivic spaces $(E_0,E_1,E_2\dots)$ together with [[equivalences]] \begin{displaymath} \Omega_T E_{i+1}\simeq E_i. \end{displaymath} 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]]. \hypertarget{SymmetricMonoidalStructureAndUniversalProperty}{}\subsubsection*{{Symmetric monoidal structure and universal property}}\label{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]]: \begin{prop} \label{}\hypertarget{}{} Let $\mathcal{C}$ be a [[locally presentable (∞,1)-category|locally presentable]] [[symmetric monoidal (∞,1)-category|symmetric monoidal]] [[(∞,1)-category]]. The (∞,1)-functor \begin{displaymath} Fun^{\otimes, L}(SH(S),\mathcal{C}) \to Fun^{\otimes} (Sm/S, \mathcal{C}), \quad F\mapsto F\circ \Sigma_T^\infty(-)_+ \end{displaymath} (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: \begin{itemize}% \item [[Nisnevich site|Nisnevich excision]] \item $\mathbb{A}^1$-homotopy invariance \item \emph{$T$-stability}: the [[homotopy cofiber]] of $F(\mathbb{G}_m)\to F(\mathbb{A}^1)$ is $\otimes$-invertible. \end{itemize} \end{prop} This is (\hyperlink{Robalo}{Robalo, Corollary 5.11}). \begin{remark} \label{}\hypertarget{}{} Similar characterizations exist for [[noncommutative motives]], see at \emph{\href{noncommutative+motive#AsUniversalAditiveInvariant}{Noncommutative motive -- As the universal additive invariant}}. \end{remark} \hypertarget{StableMotivicSpheres}{}\subsubsection*{{Stable motivic spheres}}\label{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 \emph{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 r\^o{}le in the formalism of [[six operations]] in stable motivic homotopy theory (see \hyperlink{Ayoub}{Ayoub}). \hypertarget{CohomologyTheories}{}\subsubsection*{{Cohomology theories}}\label{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: \begin{displaymath} E^{p,q}(X) {:=} [\Sigma^\infty_T X_+, \Sigma^{p,q} E], \end{displaymath} as well as a bigraded homology theory: \begin{displaymath} E_{p,q}(X) {:=} [S^{p,q}, \Sigma^\infty_T X_+ \wedge E]. \end{displaymath} \hypertarget{main_features}{}\subsection*{{Main features}}\label{main_features} \hypertarget{SixOperations}{}\subsubsection*{{The six operations}}\label{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]] \begin{displaymath} f^* : SH(Y) \to SH(X) : f_* \end{displaymath} and, if $f$ is [[separated morphism|separated]] of finite type, a ([[direct image with compact support]] $\dashv$ [[exceptional inverse image]])-adjunction \begin{displaymath} f_! : SH(X) \to SH(Y): f^!, \end{displaymath} satisfying the properties listed at [[six operations]]. For more details see \hyperlink{Ayoub}{Ayoub}. \textbf{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 \emph{stable homotopy functor} is a contravariant (∞,1)-functor \begin{displaymath} D\colon Schemes^{op} \to PrStab (\infty,1) Cat, \end{displaymath} 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 $f_*$): \begin{enumerate}% \item $D(\emptyset)=0$. \item 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]]. \item (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 \begin{displaymath} \itexarray{ Y' &\stackrel{k}{\to}& X' \\ _h\downarrow && \downarrow \;_f \\ Y &\stackrel{g}{\to}& X } \end{displaymath} with $f$ smooth, there is a canonical equivalence $h_\sharp k^*\simeq g^* f_\sharp$. \item (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]]. \item (Homotopy invariance) If $p: X\times\mathbb{A}^1\to X$ is the projection, then $p^*$ is [[fully faithful (∞,1)-functor|fully faithful]]. \item ($T$-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]]. \end{enumerate} \begin{theorem} \label{}\hypertarget{}{} (Ayoub, Voevodsky) Every stable homotopy functor admits a formalism of four operations $f^\ast$, $f_\ast$, $f_!$, and $f^!$. \end{theorem} For a more precise statement, see \hyperlink{Ayoub}{Ayoub, Scholie 1.4.2}. \begin{theorem} \label{}\hypertarget{}{} $SH$ is a stable homotopy functor. \end{theorem} This is essentially proved in \hyperlink{MorelVoevodsky99}{Morel-Voevodsky 99}. In fact, something stronger is expected to be true: \begin{theorem} \label{}\hypertarget{}{} $SH$ is the [[initial object]] in the [[(∞,2)-category]] of stable homotopy functors. \end{theorem} This theorem has not been proved yet; however, \hyperlink{Ayoub}{Ayoub's thesis} shows that every stable homotopy functor factors through $SH$. \hypertarget{RealizationFunctors}{}\subsubsection*{{Realization functors}}\label{RealizationFunctors} \textbf{Complex realization.} The functor \begin{displaymath} Sm/\mathbb{C}\to SmoothMfd,\quad X\mapsto X(\mathbb{C}) \end{displaymath} 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 \emph{complex realization functor} \begin{displaymath} \mathrm{H}(\mathbb{C}) \to \infty Grpd. \end{displaymath} After $T$-stabilization, we obtain a functor from $SH(\mathbb{C})$ to the [[(∞,1)-category of spectra]]. \textbf{Real realization.} The functor \begin{displaymath} Sm/\mathbb{R}\to \mathbb{Z}/2-SmoothMfd,\quad X\mapsto X(\mathbb{C}) \end{displaymath} 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 \emph{Real realization functor} \begin{displaymath} \mathrm{H}(\mathbb{R}) \to PSh_\infty(\mathcal{O}_{\mathbb{Z}/2}), \end{displaymath} 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. \textbf{\'E{}tale realization.} Over a separably closed field $k$, we can consider the [[étale homotopy|étale homotopy type]] functor \begin{displaymath} Sh_{\infty}((Sm/k)_{Nis})\to Sh_{\infty}((Sm/k)_{et}) \stackrel{\Pi}{\to} Pro(\infty Grpd) \end{displaymath} (see [[shape of an (∞,1)-topos]]). However, it does not descend to $\mathrm{H}(k)$ because the \'e{}tale 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 \emph{\'e{}tale realization functor} \begin{displaymath} \mathrm{H}(k) \to Pro(\infty Grpd)^\wedge_{l}. \end{displaymath} \hypertarget{SliceFiltration}{}\subsubsection*{{The slice filtration}}\label{SliceFiltration} \emph{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 \emph{coniveau filtration} in [[algebraic K-theory]], the \emph{fundamental filtration} on [[Witt groups]], and the \emph{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 \emph{slices}) are modules over the \hyperlink{MotivicCohomology}{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 sequence]]s in that their first page consists of [[motivic cohomology]] groups. \hypertarget{A1PostnikovFiltration}{}\subsubsection*{{The $\mathbb{A}^1$-Postnikov filtration}}\label{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 \emph{$\mathbb{A}^1$-n-connected} if it is [[n-connected]] as a Nisnevich (∞,1)-sheaf, and the \emph{$\mathbb{A}^1$-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]]. \begin{remark} \label{}\hypertarget{}{} 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!). \end{remark} Intuitively, $\mathbb{A}^1$-connectedness corresponds to the topological connectedness of the \emph{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. \begin{theorem} \label{}\hypertarget{}{} 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. \end{theorem} This is Morel's \emph{connectivity theorem} (\hyperlink{Morel}{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 \emph{homotopy t-structure}. \hypertarget{relation_to_the_theory_of_motives}{}\subsubsection*{{Relation to the theory of motives}}\label{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. \hypertarget{relation_to_the_theory_of_symmetric_bilinear_forms}{}\subsubsection*{{Relation to the theory of symmetric bilinear forms}}\label{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 \emph{Grothendieck--Witt ring} $GW(k)$: this is the [[group completion]] of the [[semiring]] of isomorphism classes of nondegenerate symmetric [[bilinear form]]s over $k$ (\hyperlink{Morel}{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 $\mathbb{P}^1(\mathbb{R})\simeq S^1$. \hypertarget{EquivariantMotivicHomotopyTheory}{}\subsection*{{Equivariant motivic homotopy theory}}\label{EquivariantMotivicHomotopyTheory} A general theory of [[equivariant motivic homotopy theory|equivariant]] (unstable and stable) motivic homotopy theory was introduced in (\hyperlink{CJ2014}{Carlsson-Joshua 2014}) and further developed in (\hyperlink{Hoyois15}{Hoyois 15}). \hypertarget{applications_and_examples}{}\subsection*{{Applications and examples}}\label{applications_and_examples} \hypertarget{A1coverings}{}\subsubsection*{{$\mathbb{A}^1$-coverings and the $\mathbb{A}^1$-fundamental groupoid}}\label{A1coverings} Like the $\mathbb{A}^1$-\hyperlink{A1PostnikovFiltration}{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 \emph{$\mathbb{A}^1$-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 \begin{displaymath} \Pi_1^{\mathbb{A}^1}(X):=\tau_{\leq 1}X. \end{displaymath} \begin{theorem} \label{}\hypertarget{}{} 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: \begin{enumerate}% \item $\tilde X$ is the initial object in the category of pointed $\mathbb{A}^1$-coverings of $X$. \item $\tilde X$ is the unique pointed $\mathbb{A}^1$-covering of $X$ which is $\mathbb{A}^1$-simply connected. \item The Nisnevich sheaf of (unpointed) automorphisms $Aut_X(\tilde X)$ is canonically isomorphic to $\pi_1^{\mathbb{A}^1}(X)$. \end{enumerate} \end{theorem} This is \hyperlink{Morel}{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 $X$ (\hyperlink{Morel}{Morel, Lemma 6.5}). This can be used to compute some $\mathbb{A}^1$-fundamental groups, for example: \begin{prop} \label{}\hypertarget{}{} If $n\geq 2$, $\pi_1^{\mathbb{A}^1}(\mathbb{P}^n)=\mathbb{G}_m$. \end{prop} \begin{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, $\pi_1^{\mathbb{A}^1}(\mathbb{P}^n)=Aut_{\mathbb{P}^n}(\mathbb{A}^{n+1}-0)=\mathbb{G}_m$. \end{proof} \hypertarget{hcobordisms_and_the_classification_of_surfaces}{}\subsubsection*{{$\mathbb{A}^1$-h-cobordisms and the classification of surfaces}}\label{hcobordisms_and_the_classification_of_surfaces} An \emph{$\mathbb{A}^1$-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 \hyperlink{AsokMorel}{Asok-Morel}. \hypertarget{euler_classes_and_splittings_of_algebraic_vector_bundles}{}\subsubsection*{{Euler classes and splittings of algebraic vector bundles}}\label{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 \begin{displaymath} Vect_n(X) \cong [X,BGL_n], \end{displaymath} 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]] \begin{displaymath} \mathbb{A}^n-0 \to BGL_{n-1} \to BGL_n \end{displaymath} to obtain the following criterion: \begin{theorem} \label{}\hypertarget{}{} (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 \begin{displaymath} e(\xi)\in H^n_{Nis}(X,\pi_{n-1}^{\mathbb{A}^1}(\mathbb{A}^n-0)(\mathrm{det} \xi)) \end{displaymath} which vanishes if and only if $\xi$ splits off a trivial line bundle. \end{theorem} 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 \begin{displaymath} 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) \end{displaymath} 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 $\xi$. \hypertarget{MotivicCohomology}{}\subsubsection*{{Motivic cohomology}}\label{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 \emph{motivic Eilenberg--MacLane space}, such that, for every $X\in Sm/S$, \begin{displaymath} H^{p-r,q-s}(X,A) = [\Sigma^{r,s}X_+, K(A(q),p)]. \end{displaymath} Voevodsky's \emph{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 \emph{motivic Eilenberg--MacLane spectrum}, such that \begin{displaymath} H^{p,q}(X,A) = [\Sigma_T^\infty X_+, \Sigma^{p,q} H(A)]. \end{displaymath} 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. \hypertarget{algebraic_ktheory}{}\subsubsection*{{Algebraic K-theory}}\label{algebraic_ktheory} See [[algebraic K-theory spectrum]]. \hypertarget{AlgebraicCobordism}{}\subsubsection*{{Algebraic cobordism}}\label{AlgebraicCobordism} See [[algebraic cobordism]]. \hypertarget{references}{}\subsubsection*{{References}}\label{references} \begin{itemize}% \item [[Aravind Asok]], [[Fabien Morel]], \emph{Smooth varieties up to A1-homotopy and algebraic h-cobordisms}, Adv. Math. 227 (5) (2011), pp. 1990-2058 (\href{http://arxiv.org/abs/0810.0324}{arXiv}) \end{itemize} \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[equivariant homotopy theory]] \item [[motive]] \item [[motivic cohomology]] \item [[slice spectral sequence]] \item [[B1-homotopy theory]] \item 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 \emph{[[cohesion]]} for more details. \end{itemize} \hypertarget{references_2}{}\subsection*{{References}}\label{references_2} \hypertarget{general}{}\subsubsection*{{General}}\label{general} The original references are \begin{itemize}% \item [[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). \href{http://www.math.uiuc.edu/documenta/xvol-icm/00/Voevodsky.MAN.html}{web} \item [[Fabien Morel]], [[Vladimir Voevodsky]], \emph{$\mathbb{A}^1$-homotopy theory of schemes}, Publications Mathématiques de l'IHÉS, Volume 90 (1999), p. 45-143 (\href{http://www.numdam.org/item/?id=PMIHES_1999__90__45_0}{Numdam:PMIHES\_1999\_\_90\_\_45\_0} \href{http://www.math.uiuc.edu/K-theory/0305/}{K-Theory:0305} ) \item [[Fabien Morel]], \emph{$\mathbb{A}^1$-algebraic topology over a field}, LNM 2052, 2012, (\href{http://www.mathematik.uni-muenchen.de/~morel/A1TopologyLNM.pdf}{pdf}) \end{itemize} Readable introductions to the subject are: \begin{itemize}% \item [[Bjørn Ian Dundas]], [[Marc Levine]], [[Paul Arne Østvær]], [[Oliver Röndigs]], [[Vladimir Voevodsky]], \emph{Motivic Homotopy Theory: Lectures at a Summer School in Nordfjordeid, Norway, August 2002}, Springer, Universitext, (2006) (\href{https://www.springer.com/de/book/9783540458951}{doi:10.1007/978-3-540-45897-5}) \item [[Marc Levine]], \emph{Motivic Homotopy Theory}, Milan j. math (2008), (\href{http://www.math.unam.mx/javier/levine.pdf}{pdf}) \item [[Fabien Morel]], \emph{An introduction to $\mathbb{A}^1$ homotopy theory}, ICTP Trieste July 2002 (\href{http://publications.ictp.it/lns/vol15/vol15toc.html}{directory}, \href{http://www.ictp.it/%7Epub_off/lectures/lns015/Morel/Morel.pdf}{pdf}, \href{http://www.ictp.it/%7Epub_off/lectures/lns015/Morel/Morel.ps}{ps}) \item [[Fabien Morel]], \emph{On the motivic $\pi$ of the sphere spectrum} (\href{http://www.mathematik.uni-muenchen.de/~morel/Newton.ps}{ps}) \item [[Peter Arndt]], \emph{Abstract motivic homotopy theory}, thesis 2017 (\href{https://repositorium.ub.uni-osnabrueck.de/handle/urn:nbn:de:gbv:700-2017021015476?mode=full}{web}, \href{https://repositorium.ub.uni-osnabrueck.de/bitstream/urn:nbn:de:gbv:700-2017021015476/6/thesis_arndt.pdf}{pdf}, [[ArndtAbstractMotivic.pdf:file]]) exposition: lecture at \emph{\href{www.andrew.cmu.edu/user/fwellen/abstracts.html}{Geometry in Modal HoTT}}, 2019 (\href{https://www.youtube.com/watch?v=f0wpcNs8hQo}{recording I}, \href{https://www.youtube.com/watch?v=sTl8637a2Zo}{recording II}) \end{itemize} The [[model structure on simplicial presheaves]] on the [[Nisnevich site]] and its [[homotopy localization]] to [[A1-homotopy theory]] is in \begin{itemize}% \item [[Rick Jardine]], \emph{Motivic spaces and the motivic stable category} (\href{http://www.aimath.org/WWN/motivesdessins/motivic.pdf}{pdf}) \end{itemize} For more on the general procedure see [[homotopy localization]]. The universal property of $SH(S)$ is proved in \begin{itemize}% \item [[Marco Robalo]], \emph{Noncommutative Motives I: A Universal Characterization of the Motivic Stable Homotopy Theory of Schemes}, 2013 (\href{http://arxiv.org/pdf/1206.3645v3.pdf}{pdf}) \end{itemize} For the formalism of [[six operations]] see \begin{itemize}% \item [[Joseph Ayoub]], \emph{Les six op\'e{}rations de Grothendieck et le formalisme des cycles \'e{}vanescents dans le monde motivique}, Ast\'e{}risque 314-315 (2008) (\href{http://user.math.uzh.ch/ayoub/PDF-Files/THESE.PDF}{pdf}) \item [[Vladimir Voevodsky]], [[Pierre Deligne]], \emph{Voevodsky's lectures on cross functors} (\href{http://mat.uab.cat/~kock/tmp/delnotes01.pdf}{pdf}) \end{itemize} The slice filtration was defined in \begin{itemize}% \item [[Vladimir Voevodsky]], \emph{Open problems in the stable motivic homotopy theory} K-theory, 0392 (\href{http://www.math.uiuc.edu/K-theory/0392/}{web} \href{http://www.math.uiuc.edu/K-theory/0392/nowmovo.pdf}{pdf}) \end{itemize} Important representability results are in \begin{itemize}% \item [[Aravind Asok]], [[Marc Hoyois]], [[Matthias Wendt]], \emph{Affine representability results in $\mathbb{A} ^1$-homotopy theory I: vector bundles}, \href{http://arxiv.org/abs/1506.07093}{arXiv:1506.07093}. \end{itemize} and \begin{itemize}% \item [[Aravind Asok]], [[Marc Hoyois]], [[Matthias Wendt]], \emph{Affine representability results in $\mathbb{A} ^1$-homotopy theory II: principal bundles and homogeneous spaces}, \href{http://arxiv.org/abs/1507.08020}{arXiv:1507.08020}. \end{itemize} Discussion related to [[étale homotopy]] is in \begin{itemize}% \item [[Daniel Isaksen]], \emph{\'E{}tale realization of the $\mathbb{A} ^1$-homotopy theory of schemes}, Advances in Mathematics 184 (2004) \end{itemize} Discussion about [[thick ideals]] is in \begin{itemize}% \item [[Ruth Joachimi]], \emph{Thick ideals in equivariant and motivic stable homotopy categories}, \href{http://arxiv.org/abs/1503.08456}{arXiv:1503.08456}. \end{itemize} \hypertarget{motivic_homotopy_theory_in_other_contexts}{}\subsubsection*{{Motivic homotopy theory in other contexts}}\label{motivic_homotopy_theory_in_other_contexts} [[equivariant motivic homotopy theory]] is developed in \begin{itemize}% \item [[Gunnar Carlsson]], [[Roy Joshua]], \emph{Equivariant motivic homotopy theory}, \href{http://arxiv.org/abs/1404.1597v1}{arXiv}. \end{itemize} This was vastly generalized and studied more thoroughly in \begin{itemize}% \item [[Marc Hoyois]], \emph{The six operations in equivariant motivic homotopy theory}, \href{http://arxiv.org/abs/1509.02145}{arXiv:1509.02145}. \end{itemize} Motivic homotopy theory of [[derived noncommutative geometry|noncommutative spaces]] (associative dg-algebras) is studied in \begin{itemize}% \item [[Marco Robalo]], \emph{Noncommutative Motives I: A Universal Characterization of the Motivic Stable Homotopy Theory of Schemes}, 2013 (\href{http://arxiv.org/pdf/1206.3645v3.pdf}{pdf}) \end{itemize} Motivic homotopy theory of [[associative ring|associative]] [[nonunital rings]] is studied in \begin{itemize}% \item [[Grigory Garkusha]], \emph{Homotopy theory of associative rings}, \href{http://arxiv.org/abs/math/0608482v2}{arXiv:math/0608482}. \item [[Grigory Garkusha]], \emph{Algebraic Kasparov K-theory. II}, \href{http://arxiv.org/abs/1206.0178v2}{arXiv:1206.0178}. \end{itemize} See also \begin{itemize}% \item [[analytic motivic homotopy theory]] \item [[logarithmic motivic homotopy theory]] \end{itemize} [[!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]] \end{document}