Why is Voevodsky's motivic homotopy theory 'the right' approach?

Question

Morel and Voevoedsky developed what is now called motivic homotopy theory, which aims to apply techniques of algebraic topology to algebraic varieties and, more generally, to schemes. A simple way of stating the idea is that we wish to find a model structure on some algebraic category related to that of varieties or that of schemes, so as to apply homotopy theory in an abstract sense.

The uninitiated will find the name of the theory intriguing, and will perceive the simple idea presented above as a very fair approach, but upon reading the details of the theory, he might get somewhat puzzled, if not worried, about some of its aspects. We begin with the following aspects, which are added for completeness.

• The relevant category we start out with is not a category of schemes, but a much larger category of simplicial sheaves over the category of smooth schemes endowed with a suitable Grothendieck topology. A relevant MathOverflow topic is here.

• The choice of Grothendieck topology is the Nisnevich topology, the reasons being discussed right over here.

• The choice of smooth schemes rather than arbitrary schemes has been discussed here.

Let us now suppose the uninitiated has accepted the technical reasons that are mentioned inside the linked pages, and that he will henceforth ignore whatever aesthetic shortcomings he may still perceive. He continues reading through the introduction, but soon finds himself facing two more aspects which worries him even more.

• The resulting homotopy theory does not behave as our rough intuition would like. In particular, what we might reasonable want to hold, such as the fact that a space ought to be homotopy equivalent to the product of itself with the affine line, is false. We solve the issue by simply forcing them to be homotopy equivalent, and hope that whatever theory rolls out is more satisfactory.

• There are two intuitive analogues of spheres in algebraic geometry, and the theory does not manage to identify them. We solve the issue by just leaving both of them into the game, accepting that all homology and cohomology theories will be bigraded, and we hope that this doesn't cause issues.

The fact that the resulting theory is satisfactory, has proved itself over time. But hopefully the reader will not find it unreasonable that the uninitiated perceives the two issues mentioned above as a warning sign that the approach is on the wrong track, if not the 'wrong' one altogether; moreover, he will perceive the solutions presented as 'naive', as though it were but a symptomatic treatment of aforementioned warning signs.

Question. How would you convince the uninitiated that Voevodsky's approach of motivic homotopy theory is 'the right one'?

Answer 1

(Don't be afraid about the word "$\infty$-category" here: they're just a convenient framework to do homotopy theory in).

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I'm going to try with a very naive answer, although I'm not sure I understand your question exactly.

The (un)stable motivic ($\infty$-)category has a universal property. To be precise the following statements are true

Theorem: Every functor $E:\mathrm{Sm}_S\to C$ to a(n $\infty$-)category $C$ that

• is $\mathbb{A}^1$-invariant (i.e. for which $E(X\times \mathbb{A}^1)\to E(X)$ is an equivalence)
• satisfies "Mayer-Vietoris for the Nisnevich topology" (i.e. sends elementary Nisnevich squares to pushout squares)

factors uniquely through the unstable motivic ($\infty$-)category.

(see Dugger Universal Homotopy Theories, section 8)

Theorem: Every symmetric monoidal functor $E:\mathrm{Sm}_S\to C$ to a pointed presentable symmetric monoidal ($\infty$-)category that

• is $\mathbb{A}^1$-invariant
• satisfies "Mayer-Vietoris for the Nisnevich topology"
• sends the "Tate motive" (i.e. the summand of $E(\mathbb{P}^1)$ obtained by splitting off the summand corresponding to $E(\mathrm{Spec}S)\to E(\mathbb{P}^1)$) to an invertible object

factors uniquely through the stable motivic ($\infty$-)category.

(see Robalo K-Theory and the bridge to noncommutative motives, corollary 2.39)

These two theorems are saying that any two functors that "behave like a homology theory on smooth $S$-schemes" will factor uniquely through the (un)stable motivic ($\infty$-)category. Examples are $l$-adic étale cohomology, algebraic K-theory (if $S$ is regular Noetherian), motivic cohomology (as given by Bloch's higher Chow groups)... Conversely, the canonical functor from $\mathrm{Sm}_S$ to the (un)stable motivic ($\infty$-)category has these properties. So the (un)stable motivic homotopy theory is in this precise sense the home of the universal homology theory. In particular all the properties we can prove for $\mathcal{H}(S)$ or $SH(S)$ reflect on every homology theory satisfying the above properties (purity being the obvious example).

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Let me say a couple of words about the two aspects that "worry you"

• $\mathbb{A}^1$-invariance needs to be imposed. That's not surprising: we do need to do that also for topological spaces, when we quotient the maps by homotopy equivalence (or, more precisely, we need to replace the set of maps by a space of maps, where paths are given by homotopies: this more complicated procedure is also responsible for the usage of simplicial presheaves rather than just ordinary presheaves)

• Having more kinds of spheres is actually quite common in homotopy theory. A good test case for this is $C_2$-equivariant homotopy theory. See for example this answer of mine for a more detailed exploration of the analogy.

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Surprisingly, possibly the most problematic of the three defining properties of $SH(S)$ is the $\mathbb{A}^1$-invariance. In fact there are several "homology theories" we'd like to study that do not satisfy it (e.g. crystalline cohomology). I know some people are trying to find a replacement for $SH(S)$ where these theories might live. As far as I know there are some definitions of such replacements but I don't think they have been shown to have properties comparable to the very interesting structure you can find on $SH(S)$, so I don't know whether this will bear fruit or not in the future.

Answer 2

I just want to point out, with regards to your second question, that the fact that the $\mathbb P^1$ is not equivalent to a simplicial complex homotopic to the sphere built out of affine spaces (or whichever model for an un-Tate-twisted sphere) is exactly what you would expect from Grothendieck et. al.'s theory of motives.

In fact having a homology or cohomology theory without a notion of Tate twist would be very strong evidence that we are on the wrong track!

More carefully one might point out that we want a map to etale cohomology and a notion of Chern class, which requires the Tate twisting because Tate twists show up in the Chern class map in etale cohomology.