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.
Best Answer
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.