Abstract
Let k be a field and denote by mathcal {SH}(k) the motivic stable homotopy category. Recall its full subcategory mathcal {SH}(k)^{{text {eff}}heartsuit } (Bachmann in J Topol 10(4):1124–1144. arXiv:1610.01346, 2017). Write mathrm {NAlg}(mathcal {SH}(k)) for the category of {mathrm {S}mathrm {m}}-normed spectra (Bachmann-Hoyois in arXiv:1711.03061, 2017); recall that there is a forgetful functor U: mathrm {NAlg}(mathcal {SH}(k)) rightarrow mathcal {SH}(k). Let mathrm {NAlg}(mathcal {SH}(k)^{{text {eff}}heartsuit }) subset mathrm {NAlg}(mathcal {SH}(k)) denote the full subcategory on normed spectra E such that UE in mathcal {SH}(k)^{{text {eff}}heartsuit }. In this article we provide an explicit description of mathrm {NAlg}(mathcal {SH}(k)^{{text {eff}}heartsuit }) as the category of effective homotopy modules with étale norms, at least if char(k) = 0. A weaker statement is available if k is perfect of characteristic > 2.
Highlights
Norms and normed spectraIn [3], we defined for every finite étale morphism f : S → S of schemes a symmetric monoidal functor of symmetric monoidal ∞-categories f⊗ : SH(S ) → SH(S)
In this article we provide an explicit description of NAlg(SH(k)eff♥) as the category of effective homotopy modules with étale norms, at least if char (k) = 0
Remark 6 The cartesian square (1) can be used to elucidate the nature of motivic Tambara functors of the first kind: the category is equivalent to the category of triples (T, M, α) where T is a presheaf on a certain bispan category D, M is an effective homotopy module, and α is an isomorphism between the presheaves with finite étale transfers underlying T and M
Summary
In [3], we defined for every finite étale morphism f : S → S of schemes a symmetric monoidal functor of symmetric monoidal ∞-categories f⊗ : SH(S ) → SH(S). 7 we introduce yet another notion of motivic Tambara functors, called normed effective homotopy modules To do so we first note that there is a canonical functor ρ : NAlg(SH(k)eff♥) → T 2(k), where T 2(k) denotes the category of motivic Tambara functors of the second kind This just arises from the fact that, by construction, if M ∈ NAlg(SH(k)eff♥) M has certain norm maps, known to distribute over transfers. It is enough to prove that the induced morphism of monads is an isomorphism This reduces to showing that if X ∈ Smk and M denotes the free normed effective homotopy module on EX , M is the free motivic Tambara functor of the second kind on EX. In a 1-category, the ∞-categorical notions of colimits etc. reduce to their classical counterparts; so in many parts of this article the traditional-sounding language has the traditional meaning
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