homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
models: topological, simplicial, localic, …
see also algebraic topology
Introductions
Definitions
Paths and cylinders
Homotopy groups
Basic facts
Theorems
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 $C$. 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
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
Let $S$ be a fixed Noetherian base scheme, and let $Sm/S$ be the category of smooth schemes of finite type over $S$.
The motivic homotopy category $\mathrm{H}(S)$ over $S$ is the homotopy localization of the (∞,1)-topos of (∞,1)-sheaves on the Nisnevich site $Sm/S$ equipped with the interval object $\mathbb{A}^1$. 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
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. This allows us to regard any presheaf on (or object of) $Sm/S$ as an object of $\mathrm{H}(S)$.
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)
The Tate sphere is the pointed object of $\mathrm{H}(S)$ defined by
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:
Thus, $T=S^{\mathbb{A}^1}$.
A crucial observation is that $T$ is the suspension of $\mathbb{G}_m$ (pointed at $1$):
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 $p\geq q\geq 0$,
There is a canonical equivalence of pointed motivic spaces $T\simeq (\mathbb{P}^1,1)$.
The cartesian square
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 $SH(S)$ over $S$ is the inverse limit of the tower of (∞,1)-categories
where $H(S)$ is the ordinary motivic homotopy category from def. 1, 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
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.
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:
Let $\mathcal{C}$ be a locally presentable symmetric monoidal (∞,1)-category. The (∞,1)-functor
(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:
This is (Robalo, Corollary 5.11).
Similar characterizations exist for noncommutative motives, see at Noncommutative motive – As the universal additive invariant.
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).
Any motivic spectrum $E\in SH(S)$ gives rise to a bigraded cohomology theory for smooth $S$-schemes and more generally for motivic spaces:
as well as a bigraded homology theory:
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
and, if $f$ is separated of finite type, a (direct image with compact support $\dashv$ exceptional inverse image)-adjunction
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
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 $f_*$):
$D(\emptyset)=0$.
If $i: X\to Y$ is an immersion of schemes, then $i_*\colon D(X)\to D(Y)$ is fully faithful.
(Smooth base change/Beck-Chevalley condition) If $f$ is smooth, then $f^*$ admits a left adjoint $f_\sharp$. Moreover, given a cartesian square
with $f$ smooth, there is a canonical equivalence $h_\sharp k^*\simeq g^* f_\sharp$.
(Locality) If $i: Z\hookrightarrow X$ is a closed immersion with open complement $j: U\hookrightarrow X$, then the pair $(i^*,j^*)$ is conservative.
(Homotopy invariance) If $p: X\times\mathbb{A}^1\to X$ is the projection, then $p^*$ is fully faithful.
($T$-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.
(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.
$SH$ is a stable homotopy functor.
This is essentially proved in Morel-Voevodsky. In fact, something stronger is expected to be true:
$SH$ 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 $SH$.
Complex realization. The functor
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
After $T$-stabilization, we obtain a functor from $SH(\mathbb{C})$ to the (∞,1)-category of spectra.
Real realization. The functor
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
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.
Étale realization. Over a separably closed field $k$, we can consider the étale homotopy type functor
(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
For a non-separably closed field $k$, there is a $Gal(k^{sep}/k)$-equivariant realization analogous to the Real realization.
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.
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.
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.
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.
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.
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 $\mathbb{P}^1(\mathbb{R})\simeq S^1$.
A general theory of equivariant (unstable and stable) motivic homotopy theory was introduced in (Carlsson-Joshua 2014) and further developed in (Hoyois 15).
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
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:
$\tilde X$ is the initial object in the category of pointed $\mathbb{A}^1$-coverings of $X$.
$\tilde X$ is the unique pointed $\mathbb{A}^1$-covering of $X$ which is $\mathbb{A}^1$-simply connected.
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:
If $n\geq 2$, $\pi_1^{\mathbb{A}^1}(\mathbb{P}^n)=\mathbb{G}_m$.
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, $\pi_1^{\mathbb{A}^1}(\mathbb{P}^n)=Aut_{\mathbb{P}^n}(\mathbb{A}^{n+1}-0)=\mathbb{G}_m$.
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.
Let $k$ be a perfect field. If $X$ is a smooth affine $k$-scheme, Morel proved that
where $[-,-]$ denote homotopy classes of maps in the motivic homotopy category $\mathrm{H}(k)$, def. 1. 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
to obtain the following criterion:
(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
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
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 $\xi$.
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 $X\in Sm/S$,
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
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.
See algebraic K-theory spectrum.
See algebraic cobordism.
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.
The original “textbook” references are
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
Fabien Morel, Vladimir Voevodsky, $\mathbb{A}^1$-homotopy theory of schemes K-theory, 0305 (web pdf)
Readable introductions to the subject are:
Bjørn Ian Dundas, Marc Levine, P.A. Østvær?, Oliver Röndigs, Vladimir Voevodsky, Motivic Homotopy Theory: Lectures at a Summer School in Nordfjordeid, Norway, August 2002, Springer, Universitext, (2006).
Marc Levine, Motivic Homotopy Theory, Milan j. math (2008), (pdf)
Fabien Morel, An introduction to $\mathbb{A}^1$ homotopy theory, ICTP Trieste July 2002 (directory, pdf, ps)
Fabien Morel, On the motivic π₀ of the sphere spectrum (ps)
The model structure on simplicial presheaves on the Nisnevich site and its homotopy localization to A1-homotopy theory is in
For more on the general procedure see homotopy localization.
The universal property of $SH(S)$ is proved in
For the formalism of six operations see
Joseph Ayoub, Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique, Astérisque 314-315 (2008) (pdf)
Vladimir Voevodsky, Pierre Deligne, Voevodsky’s lectures on cross functors (pdf)
The slice filtration was defined in
Important representability results are in
and
Discussion related to étale homotopy is in
Discussion about thick ideals is in
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
Motivic homotopy theory of associative nonunital rings is studied in
Grigory Garkusha, Homotopy theory of associative rings, arXiv:math/0608482.
Grigory Garkusha, Algebraic Kasparov K-theory. II, arXiv:1206.0178.
See also