Homotopy coherence datum for composition of Becker-Gottlieb transfers

Question

I have a question about certain detail in following answer by Denis Nardin adressing the concept of presheaves with transfer (mostly known in constructions in motivic homotopy theory) from viewpoint of classical stable homotopy theory.

The rough summary is that the notion of a "transfer" appears already (... or phrased differently has it's counterpart) in classical SHT in guise of so called Becker-Gottlieb transfer which to a map $f: X \to Y$ of top spaces (#Correction: (thanks to John Rognes' remark) BG is not defined for arbitrary $f$; but let assume for those maps and their compositions we are considering in this question in is assumed to be defined)
associates in category $\text{Sp}$ of spectra certain map $f^*: \Sigma^\infty_+Y\to \Sigma^\infty_+X$ for which for composition $X\xrightarrow{f}Y\xrightarrow{g} Z$ the equation $g^*f^*=(gf)^*$ is literally an equality only after passing to homotopy category (...and so "rectifiyng" the functoriality). In other words, one may think of this equality (...in spirit of $\infty$-categorical philosophy) as a rather compliciated datum of a homotopy between the two maps (then with coherent homotopies of homotopies on the "next" layer, and so on ...); at all a rather complicated datum.

But instead of working directly with spectra (which may be regarded as space-valued homology functors $\mathrm{Top}\to \mathrm{Top}_*$ & some consistence conditions), one could also study functors $\mathrm{Cor}\to \mathrm{Top}_* $ (from the category $ \mathrm{Cor}$ of correspondences & some consistence conditions; see the linked discussion for details), which turns of to be given as $D(\mathbb{Z})$, the derived cat of $\mathbb{Z}$ endowed with natural forgetful functor $D(\mathbb{Z})\to \mathrm{Sp}$ to the cat of spectra identifying $D(\mathbb{Z})$ with spectra carying additionally $H\mathbb{Z}$-module structure.

The pun seems to be that if we restrict to the coherence conditions for compositions of Becker-Gottlieb transfers given above to those spectra inside $D(\mathbb{Z})$, then according to the statement in the linked answer these "hold on the nose".

Question: And exactly this point I not completely understand. What does here mean that these coherences conditions on the transfers "hold on the nose" if we restrict ourself to $D(\mathbb{Z})$.

In the answer is stated that the mapping spaces to $D(\mathbb{Z})$ is given dy $X \mapsto C_*(X)$ (ie associating ato $X$ it's complex and pass to derived cat of complexes).

The latter formulation suggests that working inside $D(\mathbb{Z})$ makes the coherences conditions on the composition of Becker-Gottlieb transfers in certain way more "amenable"/ "nicer"/ better behaved/controllable in certain sense (?) justifying the formulation that in $D(\mathbb{Z})$ these hold more or less "on the nose" in contrast to the "messy" datum if we work in $\text{Sp}$ instead.

What are the advantages of this viewpoint and in which sense in $D(\mathbb{Z})$ the coherence relations hold more or less "on the nose"? Does this mean that in $D(\mathbb{Z})$ these can be written down explicitly , while in $\text{Sp}$ they exist only abstractly?

Answer 1

$\newcommand{\S}{\mathcal S} \newcommand{\Top}{\mathrm{Top}} \newcommand{\Z}{\mathbb Z} \newcommand{\Span}{\mathrm{Span}} \newcommand{\Fin}{\mathrm{Fin}} \newcommand{\Fun}{\mathrm{Fun}}$ This is a bit too long for a comment, but maybe not a full answer.

First, let me make a small comment about Becker-Gottlieb (=BG) transfers : they are not defined for arbitrary maps, and even in the generality in which they are defined, they are not even known to compose up to homotopy (as John says in the comments, progress has been made in this direction, but the general question is still open). Denis' answer was about the case of finite covering maps ("more generally" maps with homotopy fibers given by finite sets), where they are known to indeed compose, and in fact, up to higher coherences and so on.

Now, about coherences. Before answering about BG transfers, let me deal with a simpler case: maps between finite sets. Let $\Span(\Fin)$ denote the category of spans of finite sets - its objects are finite sets and morphisms spans. In that case, $E_\infty$-groups (and indeed, monoids) can be encoded as functors $\Span(\Fin)\to \S$ preserving finite products - here I mean $\infty$-functor, so with homotopies, coherences etc.

But such a functor $\Span(\Fin)\to \Top$, the $1$-category of topological spaces, corresponds to a classical commutative monoid in topological spaces, i.e. topological monoid, and say, group if I assume an extra "grouplikeness" condition. Now, any such thing, when postcomposed along $\Top\to\S$ corresponds to a $\Z$-module in $E_\infty$-groups, and in particular is an Eilenberg-MacLane space, while there are (very much) non EM $E_\infty$-monoids.

So here, strictifying a functor amounts to making it have a $\Z$-linear structure. This is the general slogan underlying Denis' answer : if you ask for too much strictification on your higher alebraic structures, you'll end up asking for $\Z$-linearity.

You can make this a bit more precise: the $\infty$-category $\Fun^\times(\Span(\Fin),\Top)[W^{-1}]$ where $W$ is the class of pointwise weak-equivalences (and $\Fun^\times$ means preserving products up to homeomorphism) is equivalent to $D_{\geq 0}(\Z)$. Well, not quite - I haven't imposed the grouplikeness condition, but if I do, then it becomes true. Without the grouplikeness condition, this is $\mathrm{Mod}_\mathbb N(\mathrm{CMon}(\S))$. On the other hand, $\Fun^\times(\Span(\Fin),\S)$ is equivalent to $\mathrm{CMon}(\S)$ or, with the grouplike condition, to $\mathrm{CGrp}(\S)\simeq\mathrm{Sp}_{\geq 0}$.

In this sense, while (connective) spectra are like "commutative groups in spaces where all the axioms hold up to coherent homotopy", (connective) $\Z$-modules are like "commutative groups in spaces where all the axioms hold 'strictly'". Of course this is a bit hard to express intuitively without details because there are still (/there can still be) homotopies and higher coherences, but the idea is that they can be strictified. A key example to always keep in mind is the example of $\eta\in\pi_1(\mathbb S)$. Let me make that more precise: one algebraic and ahistorical interpretation of $\eta$ is as a default in "strict commutativity": if $x\in \pi_0(X)$, where $X$ is an $E_\infty$-group, then $\eta x\in \pi_1(X)$ can be interpreted as follows : there is a commutativity homotopy $h_{y,z} : yz \simeq zy$ coming from the $E_\infty$-structure on $X$. Plugging in $(y,z)=(x,x)$ gives you a specific homotopy $h_{x,x} : x^2\simeq x^2$ which is typically not the trivial loop, and hence gives you an element of $\pi_1(X)$ (using grouplikeness to make it a loop around $0$ rather than $x^2$). If $X$ could be strictified, because $\eta$ maps to $\eta$ under functors, it would follow that $\eta x = 0$ for all $x$. This "corresponds" to the fact that $\eta\mapsto 0$ under $\mathbb S\to \Z$ and that strictifiable commutative monoids admit a $\Z$-module structure.

Hopefully, for $\Span(\Fin)$, this makes the claim in Denis' answer somewhat clearer.

For the category of correspondences with covering maps as backwards maps replacing $\Span(\Fin)$, the story becomes a bit more complicated because you have to impose certain conditions to ensure some kind of "descent" - I guess Denis' answer puts this under the rug with "satisfying some conditions", but the moral story should be the same.

EDIT to answer the comments below:

• In the above answer, I should have specified that $\S$ denotes the $\infty$-category of spaces, not of spectra. $E_\infty$-monoids or groups are then defined as functors $\Fin_*\to \S$ satisfying certain so-called Segal conditions; or equivalently product-preserving functors $\Span(\Fin)\to \S$. The equivalence is not trivial, but true nonetheless. Here, $\Span(\Fin)$ is not the nerve of a $1$-category : $\Fin$ is, but in the span category one is supposed to allow for composition (given by pullbacks) to be defined only up to equivalence. In particular, $\Span(\Fin)$ is a $(2,1)$-category - for general $C$ with pullbacks, $\Span(C)$ is defined and studied in Barwick's Spectral Mackey functors and algebraic K-theory (I)

• A precise formulation of strictification is what I mentioned in the 4th paragraph: a lift of the product-preserving functor $\Span(\Fin)\to\S$ to a (product preserving) functor $\Span(\Fin)\to \Top$ - note that since $\Top$ is a $1$-category, this means among other things that this will factor through $\mathrm{ho}(\Span(\Fin))$ and correspond to a classical functor $\mathrm{ho}(\Span(\Fin))\to \Top$, that is, one where everything commutes strictly. In particular, this is not saying that the functor $\Span(\Fin)\to \S$ itself involves diagrams that "commute strictly", but it can be lifted to an ordinary functor between $1$-categories where everything does commute strictly.