Motivic Tambara functors

Bachmann, Tom

doi:10.1007/s00209-020-02581-x

Cited by 2 publications

(1 citation statement)

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“…The decategorification of the above structure has been studied by Bachmann in [Bac21]. Working over a field, and restricting to a category of bispans between smooth schemes, , Bachmann proved that the structure of a normed algebra in the abelian category of homotopy modules (the heart of the so-called homotopy -structure on motivic spectra) is encoded by certain functors out of this bispan category to abelian groups (appropriately christened motivic Tambara functors ), at least after inverting the exponential characteristic of .…”

Section: Introductionmentioning

confidence: 99%

On distributivity in higher algebra I: the universal property of bispans

Elmanto,

Haugseng

2023

Compositio Math.

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Structures where we have both a contravariant (pullback) and a covariant (pushforward) functoriality that satisfy base change can be encoded by functors out of ( $\infty$ -)categories of spans (or correspondences). In this paper, we study the more complicated setup where we have two pushforwards (an ‘additive’ and a ‘multiplicative’ one), satisfying a distributivity relation. Such structures can be described in terms of bispans (or polynomial diagrams). We show that there exist $(\infty,2)$ -categories of bispans, characterized by a universal property: they corepresent functors out of $\infty$ -categories of spans where the pullbacks have left adjoints and certain canonical 2-morphisms (encoding base change and distributivity) are invertible. This gives a universal way to obtain functors from bispans, which amounts to upgrading ‘monoid-like’ structures to ‘ring-like’ ones. For example, symmetric monoidal $\infty$ -categories can be described as product-preserving functors from spans of finite sets, and if the tensor product is compatible with finite coproducts our universal property gives the canonical semiring structure using the coproduct and tensor product. More interestingly, we encode the additive and multiplicative transfers on equivariant spectra as a functor from bispans in finite $G$ -sets, extend the norms for finite étale maps in motivic spectra to a functor from certain bispans in schemes, and make $\mathrm {Perf}(X)$ for $X$ a spectral Deligne–Mumford stack a functor of bispans using a multiplicative pushforward for finite étale maps in addition to the usual pullback and pushforward maps. Combining this with the polynomial functoriality of $K$ -theory constructed by Barwick, Glasman, Mathew, and Nikolaus, we obtain norms on algebraic $K$ -theory spectra.

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Section: Introductionmentioning

confidence: 99%