Presheaves with transfer are a main technical ingredient in Voevodsky's construction of his category of mixed motives. From the perspective of motivic homotopy theory, the only difference between SH and DM is that objects in the latter have transfers.

I'd like to know some non-artificial examples of presheaves with transfer. I know that all qfh sheaves have transfers, but I'm looking for concrete motivational examples.

Is algebraic K-theory a presheaf with transfer? I've seen people talking about transfer maps in K-theory, but K-theory does not live in DM(k) (while it lives in SH(k)). What's going on there?

Mainly, I'd like to understand how Suslin and Voevodsky realized that motivic cohomology was supposed to have transfers.

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    $\begingroup$ I actually think that "presheaves with transfers" is a bit of a misnomer. Even objects in $SH$ have transfers (google Motivic Becker-Gottlieb transfer for details). The actual difference is the coherence conditions that these transfers are required to satisfy: in $DM$ the transfer need to satisfy a strong version of the Mackey decomposition formula where all higher coherences are collapsed. It is roughly the same difference that there is in homotopy theory between $D(\mathbb{Z})$ and the category of spectra $\endgroup$ Jan 18, 2017 at 8:38
  • $\begingroup$ Another part of the answer is that motives "have transfers" (i.e., action of correspondences) from the very moment when they were introduced by Grothendieck. However, it certainly was not easy to pass from action of Chow groups (in Grothendieck's definition) to that of Voevodsky's SmCor. $\endgroup$ Jan 18, 2017 at 21:24
  • $\begingroup$ As for K-theory: it does not live in DM(k); yet it makes some sense to put it into certain K-motives (that can probably be defined using quite different constructions). So, it has "better transfers" than general SH-representable theory; yet transfers of this sort are "not compatible with DM". $\endgroup$ Jan 18, 2017 at 21:27

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I hope someone can provide a better answer with more of an eye towards the motivic world, but for now let me outline that the exact same phenomenon exists in classical stable homotopy theory. Here the analogue of $SH$ is the category $Sp$ of spectra. There are plenty of ways to define it, so let me assume it exists for now. One way to think about them is as space-valued homology theories, that is functors $\mathrm{Top}\to \mathrm{Top}_*$ satisfying some conditions.

This is a symmetric monoidal category, with a unit $\mathbb{S}$ called the sphere spectrum. For any space $X$ we can associate a spectrum, called its pointed suspension spectrum $\Sigma^\infty_+X$. Then we get a homology theory called stable homotopy defined by $$\pi_n^sX := [\Sigma^n\mathbb{S}, \Sigma^\infty_+X]$$ This is an interesting cohomology theory, and in fact it does have transfer maps: for any map $X\to Y$ with finite and discrete homotopy fibers we get a map called Becker-Gottlieb transfer [1] $\Sigma^\infty_+Y\to \Sigma^\infty_+X$ which, roughly speaking, sends a point $y\in Y$ to the sum of the points of the fiber. However this transfer has a very complicated functoriality. Intuitively the problem is that the for $X\xrightarrow{f}Y\xrightarrow{g} Z$ the equation $g^*f^*=(gf)^*$ takes place in the space $\mathrm{Map}(\Sigma^\infty_+Z,\Sigma^\infty_+X)$, and so instead of a simple equality you should think of it as the datum of a homotopy between the two maps, and then you need to add coherences to these homotopies when you try to write down associativity and so on and so forth...

But sometimes we would like these equalities to hold in a more strict sense. So let us try to make them do so. Instead of spaces let us work with the category $\mathrm{Cor}$ of correspondences of spaces. Its objects are spaces, but now the maps are closed subspaces $Z\subseteq X\times Y$ where the projection to $X$ is a covering space map.

Then we can try to repeat the construction of spectra using the category of correspondence of spaces, that is studying functors $\mathrm{Cor}\to \mathrm{Top}_*$ satisfying some conditions. It turns out that the category we get doing this is $D(\mathbb{Z})$, the derived category of $\mathbb{Z}$ and that the obvious forgetful functor $D(\mathbb{Z})\to \mathrm{Sp}$ is the functor providing the identification of $D(\mathbb{Z})$ with $H\mathbb{Z}$-modules in $\mathrm{Sp}$. The functor from spaces to $D(\mathbb{Z})$ realizing the "strict stable homotopy type" is just the functor sending $X$ to $C_*(X)$, its complex of chains, and the resulting homology theory is just ordinary homology. Now the coherences conditions on the transfers hold on the nose.

In the motivic world the story is more complicated, but the idea is essentially the same. The reason we are introducing transfers in the definition is not to get transfers (we would have them anyway!) but to force their behaviour with respect to functoriality. The reason why this is a reasonable idea to get motivic homology is that when you do it in the classical setting you get ordinary homology.

[1] In fact the Becker-Gottlied transfer exists every time the fibers have finite stable homotopy type, but this introduces a lot more issues in the functoriality.

  • $\begingroup$ Thanks! This is very helpful. Do you have a reference for topological correspondences, where I can find a proof of your claim about D(Z)? $\endgroup$ Jan 18, 2017 at 10:11
  • $\begingroup$ @FabianCarlström I don't know of any reference, this is more of a folklore result (in fact I played a bit fast and loose with the definitions, you'll notice that I never described what I mean with "do the same thing you do to define spectra"). I'll try to think if I can write down a reasonably self-contained proof of the statement without invoking too many sledgehammers. $\endgroup$ Jan 18, 2017 at 10:17
  • $\begingroup$ that would be great if you do that. This must be well-known to topologists, are there applications of existence of transfers in singular cohomology? $\endgroup$ Jan 19, 2017 at 9:24
  • $\begingroup$ Another thing I still wonder about. Looking at some references on Becker-Gottlied transfers I see that it uses Atiyah duality. For motives we have that for smooth and proper things, right? Does this mean that in SH we have transfers for smooth proper maps--not for finite surjections like in DM? $\endgroup$ Jan 19, 2017 at 9:27
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    $\begingroup$ One related thing is that various kinds of transfers characterize various motivic categories: SH can be built using "framed transfers" (some algebraic version of framed cobordism), if you drop the framings you get MGL modules, if you allow all finite flat transfers you get kgl modules, and if you allow all finite transfers in the sense of voevodsky you get HZ modules (= Voevodsky motives). [sorry I didn't realize this was an ancient question] $\endgroup$ Oct 24, 2020 at 6:44

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