1.1 Spans and commutative monoids

Before we explain what we mean by bispans, it is helpful to first recall the relation between commutative monoids and spans: if $\mathbb {F}$ denotes the category of finite sets, then we can define a (2,1)-category $\operatorname {Span}(\mathbb {F})$ where:

• • objects are finite sets;

• • morphisms from $I$ to $J$ are spans (or correspondences)

  

• • composition is given by pullback, i.e. the composite

  
  is the outer span in the diagram
  

• • 2-morphisms are isomorphisms of spans.

If $M$ is a commutative monoid in ${\text {Set}}$, we can use the monoid structure to define a functor

\[ \operatorname{Span}(\mathbb{F}) \to {\text {Set}}, \]

which takes $I \in \mathbb {F}$ to $M^{I} := \prod _{i \in I} M$ and a span $I \xleftarrow {f} S \xrightarrow {g} J$ to the composite $g_{\otimes }f^{*}$ where $f^{*}\colon M^{I} \to M^{S}$ is given by composition with $f$ (so $f^{*}\phi (s) = \phi (fs)$) and $g_{\otimes }$ is defined using the product on $M$ by

\[ g_{\otimes}(\phi)(j) = \prod_{s \in g^{-1}(j)} \phi(s). \]

This is compatible with composition of spans, since a pullback square gives a canonical isomorphism of fibres and we have $(gg')_{\otimes } = g_{\otimes }g'_{\otimes }$ as the multiplication is associative.

It can be shown that if $\mathbf {C}$ is any category with finite products, every functor $\Phi \colon \operatorname {Span}(\mathbb {F}) \to \mathbf {C}$ such that $\Phi (I) \cong \Phi (*)^{\times |I|}$ via the canonical maps arises in this way from a commutative monoid in $\mathbf {C}$. More precisely, we can identify commutative monoids in $\mathbf {C}$ with product-preserving functors $\operatorname {Span}(\mathbb {F}) \to \mathbf {C}$. (In other words, the homotopy category of $\operatorname {Span}(\mathbb {F})$ is precisely the Lawvere theory for commutative monoids.) This is also true homotopically as we now describe.

The earliest proof of this seems to be the in thesis of Cranch [Reference CranchCra10, Reference CranchCra11]; other proofs (as special cases of different generalizations) are due to Bachmann and Hoyois [Reference Bachmann and HoyoisBH21, Appendix C] and Glasman [Reference GlasmanGla17, Appendix A]. In addition, it appears in Harpaz [Reference HarpazHar20, § 5.2] as the bottom case of his theory of $m$-commutative monoids.

1.2 Bispans and commutative semirings

We can ask for a similar description for commutative semirings. In this case, we have two operations, addition and multiplication, so we want a $(2,1)$-category ${\text {Bispan}}(\mathbb {F})$ whose objects are again finite sets, with a morphism from $I$ to $J$ given by a bispan (or polynomial diagram)

(1)

If $R$ is a commutative semiring in ${\text {Set}}$, we want a functor

\[ {\text {Bispan}}(\mathbb{F}) \to {\text {Set}} \]

that takes a set $I$ to $R^{I}$ and the bispan (

) to $q_{\oplus }f_{\otimes }p^{*}$ where:

• • $p^{*} \colon R^{I}\to R^{X}$ is defined by composing with $p$,

  \[ p^{*}(\phi)(x) = \phi(px); \]

• • $f_{\otimes } \colon R^{X} \to R^{Y}$ is defined by multiplying in $R$ fibrewise,

  \[ f_{\otimes}(\phi)(y) = \prod_{x \in f^{-1}(y)} \phi(x); \]

• • $q_{\oplus }$ is defined by adding in $R$ fibrewise,

  \[ q_{\oplus}(\phi)(j) = \sum_{y \in q^{-1}(j)} \phi(y). \]

The question is then whether there is a way to define composition of bispans so that this gives a functor. Given a pullback square

(2)

in $\mathbb {F}$, we have identities $g_{\otimes }i^{*} = j^{*}f_{\otimes }$ and $g_{\oplus }i^{*} = j^{*}f_{\oplus }$ as before, but now we also need to deal with compositions of the form $v_{\otimes }u_{\oplus }$ for $u \colon I \to J$ and $v \colon J \to K$. Using the distributivity of addition over multiplication, for $\phi \colon I \to R$ and $k \in K$ we can write

(3)\begin{equation} v_{\otimes}u_{\oplus}(\phi)(k) = \prod_{j \in J_{k}} \sum_{i \in I_{j}} \phi(i) = \sum_{(i_{j}) \in \prod_{j \in J_{k}} I_{j}} \prod_{t \in J_{k}} \phi(i_{t}). \end{equation}

We can interpret this in terms of a distributivity diagram in $\mathbb {F}$: if we let $h \colon X \to K$ be the family of sets $X_{k} = \prod _{j \in J_{k}} I_{j}$ (so that $h = v_{*}u$ where $v_{*}$ is the right adjoint to pullback along $v$), then the pullback $v^{*}X$ has a canonical map to $I$ over $J$: on the fibre $(v^{*}X)_{j}$, which is the product $\prod _{j' \in J_{v(j)}} I_{j'}$, we take the projection to the factor $I_{j}$. This gives a commutative diagram

(4)

where the square is cartesian, and we can rewrite the distributivity relation (3) as

\[ v_{\otimes}u_{\oplus} = h_{\oplus}\tilde{v}_{\otimes}\epsilon^{*}. \]

This means we will get a functor ${\text {Bispan}}(\mathbb {F}) \to {\text {Set}}$ from the commutative semiring $R$ if we define the composition of two bispans

\[ I \xleftarrow{s} E \xrightarrow{p} B \xrightarrow{t} J, \]

\[ J \xleftarrow{u} F \xrightarrow{q} C \xrightarrow{v} K, \]

as the outer bispan in the following diagram:

(5)

Here we have used a distributivity diagram for $q$ and the pullback $\pi$.

An explicit construction of a $(2,1)$-category ${\text {Bispan}}(\mathbb {F})$ with this composition is given in the thesis of Cranch [Reference CranchCra10], where it is also proved that this has products (given by the disjoint union of sets), so that we can define commutative semirings in $\mathcal {S}$ as functors ${\text {Bispan}}(\mathbb {F}) \to \mathcal {S}$ that preserve finite products. Alternatively, one can relate bispans of finite sets to polynomial functors, which gives an easier definition of ${\text {Bispan}}(\mathbb {F})$ (as the complicated composition law (5) corresponds to the ordinary composition of such functors); this approach was carried out by Gambino and Kock [Reference Gambino and KockGK13], who also show that the homotopy category of ${\text {Bispan}}(\mathbb {F})$ is the Lawvere theory for commutative semirings, so that commutative semirings in an ordinary category $\mathbf {C}$ with finite products are equivalent to product-preserving functors

\[ {\text {Bispan}}(\mathbb{F}) \to \mathbf{C}. \]

We expect that the homotopical analogue of this statementFootnote ¹ is also true, but this has not yet been proved.

1.3 The universal property of spans

If $\mathcal {C}$ is a symmetric monoidal $\infty$-category such that $\mathcal {C}$ has finite coproducts and the tensor product preserves these in each variable, then we expect that $\mathcal {C}$ has a canonical semiring structure in ${\text {Cat}}_\infty$ with multiplication and addition given by the tensor product and coproduct, respectively. This followsFootnote ² from the universal property of an $(\infty,2)$-category of bispans in $\mathbb {F}$, which is a special case of our main result. Before we state this, it is convenient to first recall the simpler universal property of the $(\infty,2)$-category $\operatorname {SPAN}(\mathbb {F})$ of spans in $\mathbb {F}$, which can be used to prove that an $\infty$-category with finite coproducts has a canonical symmetric monoidal structure.

Here $\operatorname {SPAN}(\mathbb {F})$ has finite sets as objects, spans as morphisms, and morphisms of spans as 2-morphisms, i.e. 2-morphisms are commutative diagrams of the form

Suppose $\mathcal {X}$ is an $(\infty,2)$-category. A functor $\Phi \colon \mathbb {F}^{{\text {op}}} \to \mathcal {X}$ is called left adjointable if for every morphism $f \colon I \to J$ in $\mathbb {F}$, the morphism $f^{\circledast } := \Phi (f) \colon \Phi (J) \to \Phi (I)$ in $\mathcal {X}$ has a left adjoint $f_{\oplus }$, and for every pullback square (2) in $\mathbb {F}$, the canonical (Beck–Chevalley or mate) transformation

\[ g_{\oplus}i^{\circledast} \to j^{\circledast}f_{\oplus} \]

is an equivalence.

This is a special case of a recent result of Macpherson [Reference MacphersonMac22], which we review in more generality below in § 2.2. Another proof is sketched in the book of Gaitsgory and Rozenblyum [Reference Gaitsgory and RozenblyumGR17] where this universal property is used to encode a ‘six-functor formalism’ for various categories of quasicoherent sheaves on derived schemes. Lastly, we note that the analogous result for ordinary 2-categories seems to have been first proved by Hermida [Reference HermidaHer00, Theorem A.2].

1.4 The universal property of bispans

We now want to consider a 2-category ${\text {BISPAN}}(\mathbb {F})$ whose objects are finite sets, with morphisms given by bispans and 2-morphisms by commutative diagrams of the form

(6)

where the middle square is cartesian. If we look at the subcategory where the morphisms are bispans whose rightmost leg is invertible and with no non-trivial 2-morphisms, we get an inclusion $\operatorname {Span}(\mathbb {F}) \to {\text {BISPAN}}(\mathbb {F})$. A special case of our main result gives a universal property of ${\text {BISPAN}}(\mathbb {F})$ in terms of this subcategory as follows.

Here a functor $\Phi \colon \operatorname {Span}(\mathbb {F}) \to \mathcal {X}$ is distributive if:

• • for every morphism $f \colon I \to J$ in $\mathbb {F}$, the morphism $f^{\circledast } := \Phi (J \xleftarrow {f} I == {\,\,} I)$ in $\mathcal {X}$ has a left adjoint $f_{\oplus }$;

• • for every pullback square (2) in $\mathbb {F}$, the Beck–Chevalley transformation $g_{\oplus }i^{\circledast } \to j^{\circledast }f_{\oplus }$ is an equivalence;

• • for every distributivity diagram (4), the distributivity transformation

  \[ h_{\oplus}\tilde{v}_{\otimes}\epsilon^{\circledast} \to v_{\otimes}u_{\oplus}, \]
  which is defined as a certain composite of units and counits, is an equivalence in $\mathcal {X}$.

Note that the only property of $\mathbb {F}$ we have used in the definition of distributive functors is the existence of distributivity diagrams. These exist in any locally cartesian closed $\infty$-category, and more generally we can consider triples $(\mathcal {C}, \mathcal {C}_{F}, \mathcal {C}_{L})$ consisting of an $\infty$-category $\mathcal {C}$ with a pair of subcategories $\mathcal {C}_{F}$ and $\mathcal {C}_{L}$ such that:

• • pullbacks along morphisms in $\mathcal {C}_{F}$ and $\mathcal {C}_{L}$ exist in $\mathcal {C}$, and both subcategories are preserved under base change;

• • there exist suitable distributivity diagrams in $\mathcal {C}$ for any composable pair of morphisms $l \colon x \to y$ in $\mathcal {C}_{L}$, $f \colon y \to z$ in $\mathcal {C}_{F}$.

We can then generalize the notion of distributive functors above to that of $L$-distributive functors $\operatorname {Span}_{F}(\mathcal {C}) \to \mathcal {X}$, where $\operatorname {Span}_{F}(\mathcal {C})$ is the $\infty$-category of spans in $\mathcal {C}$ whose forward legs are required to lie in $\mathcal {C}_{F}$. Our main result in this paper is then the following generalization of Theorem 1.4.1.

Theorem 1.4.2 For $(\mathcal {C}, \mathcal {C}_{F}, \mathcal {C}_{L})$ as above, there exists an $(\infty,2)$-category

\[ {\text {BISPAN}}_{F,L}(\mathcal{C}) \]

such that:

• • objects are objects of $\mathcal {C}$;

• • morphisms are bispans

  \[ x \xleftarrow{p} e \xrightarrow{f} b \xrightarrow{l} y \]
  where $f$ is in $\mathcal {C}_{F}$ and $l$ is in $\mathcal {C}_{L}$;

• • 2-morphisms are diagrams of the form (6);

• • morphisms compose as in (5).

The $(\infty,2)$-category ${\text {BISPAN}}_{F,L}(\mathcal {C})$ has the universal property that restricting to the subcategory $\operatorname {Span}_{F}(\mathcal {C})$ gives for any $(\infty,2)$-category $\mathcal {X}$ an equivalence between functors ${\text {BISPAN}}_{F,L}(\mathcal {C}) \to \mathcal {X}$ and $L$-distributive functors $\operatorname {Span}_{F}(\mathcal {C}) \to \mathcal {X}$.

The analogue of this result for ordinary 2-categories (at least in the case where $\mathcal {C} = \mathcal {C}_{F} = \mathcal {C}_{L}$) is due to Walker [Reference WalkerWal19].

1.5 Equivariant and algebro-geometric bispans

We now look briefly at some examples of Theorem 1.4.2 beyond the case of finite sets, coming from equivariant and motivic homotopy theory and derived algebraic geometry. These examples are discussed in more detail in § 3.

Let us first consider the equivariant setting, over a finite group $G$. In all of our discussion above it is straightforward to replace the category $\mathbb {F}$ of finite sets with the category $\mathbb {F}_{G}$ of finite $G$-sets. The analogue of a commutative monoid is then a functor

\[ M \colon \operatorname{Span}(\mathbb{F}_{G}) \to {\text {Set}} \]

that preserves products. This is (essentiallyFootnote ³) the same thing as a Mackey functor [Reference DressDre71], an algebraic structure where for a subgroup $H \subseteq G$ we have restrictions $M^{G} \to M^{H}$ and transfers $M^{H} \to M^{G}$ satisfying a base change property that can be interpreted in terms of double cosets. Mackey functors play an important role in group theory, and every genuine $G$-spectrum $E$ has an underlying Mackey functor $\pi _{0}E$.

Similarly, the $G$-analogue of a commutative semiring is a product-preserving functor ${\text {Bispan}}(\mathbb {F}_{G}) \to {\text {Set}}$, which is essentiallyFootnote ⁴ a Tambara functor [Reference TambaraTam93, Reference StricklandStr12, Reference Blumberg and HillBH18]. This is a structure that has both an ‘additive’ and a ‘multiplicative’ transfer, satisfying a distributivity relation. If $E$ is a genuine $G$-$E_{\infty }$-ring spectrum, then $\pi _{0}E$ has the structure of a Tambara functor [Reference BrunBru07].

If we replace the category of sets with the $\infty$-category of spaces, a theorem of NardinFootnote ⁵ [Reference NardinNar16, Corollary A.4.1] shows that connective $G$-spectra are equivalently product-preserving functors $\operatorname {Span}(\mathbb {F}_{G}) \to \mathcal {S}$ that are grouplike, generalizing the classical description of connective spectra as grouplike commutative monoids in $\mathcal {S}$.Footnote ⁶

The analogue for ring spectra is also expected to hold: connective genuine $G$-$E_{\infty }$-ring spectra should be equivalent to product-preserving functors ${\text {Bispan}}(\mathbb {F}_{G}) \to \mathcal {S}$.

Now we turn to the ‘categorified’ versions of these structures: for $H$ a subgroup of $G$, the (additive) transfer from $H$-spectra to $G$-spectra is classical,Footnote ⁷ but the multiplicative transfer or norm was only introduced fairly recently by Hill, Hopkins, and Ravenel as part of the foundational setup for [Reference Hill, Hopkins and RavenelHHR16]. This inspired a plethora of work on equivariant symmetric monoidal structures [Reference Hopkins and HillHH16, Reference Guillou, May, Merling and OsornoGMMO20, Reference RubinRub17] and its relation to equivariant homotopy-coherent commutativity (in particular, [Reference Blumberg and HillBH15] and subsequent work on $N_{\infty }$-operads), culminating from our point of view in the approach of Barwick et al. [Reference Barwick, Dotto, Glasman, Nardin and ShahBDG⁺16], where a $G$-symmetric monoidal $\infty$-category can be viewed as a product-preserving functor

\[ \operatorname{Span}(\mathbb{F}_{G}) \to {\text {Cat}}_\infty, \]

i.e. a ‘categorified Mackey functor’.

If the contravariant (restriction) functors have left adjoints that satisfy base change and distributivity, Theorem 1.4.2 allows us to upgrade such $G$-symmetric monoidal structures to functors from ${\text {BISPAN}}(\mathbb {F}_{G})$, which encodes the distributive compatibility of multiplicative and additive transfers. We will see that this applies, in particular, to genuine $G$-spectra, giving a ‘categorified Tambara functor’ structure on $G$-spectra.

Next, we look at the motivic setting, where it is more instructive to first work in the categorified context. By this we mean Ayoub's construction of a functor from schemes to categories $X \mapsto \mathrm {SH}(X)$ which satisfies a full six functors formalism [Reference AyoubAyo07], vastly expanding Voevodsky's notes in [Reference VoevodskyVoe99]; we also refer the reader to the book of Cisinski and Déglise [Reference Cisinski and DégliseCD19] for another exposition, [Reference HoyoisHoy17] for an $\infty$-categorical enhancement of this construction in the more general motivic-equivariant setting, as well as the more recent [Reference Drew and GallauerDG22] for a universal property of this construction. Here $\mathrm {SH}(X)$ denotes the $\infty$-category of motivic spectra over a scheme $X$.

In this context, given a smooth morphism of schemes $f \colon X \rightarrow Y$ over a base $S$, the pullback functor $f^* \colon \mathrm {SH}(Y) \rightarrow \mathrm {SH}(X)$ admits a left adjoint, $f_{\sharp } \colon \mathrm {SH}(X) \rightarrow \mathrm {SH}(Y)$. This is a categorified version of the additive pushforward: if $f$ is the fold map $\nabla \colon Y^{\amalg I} \rightarrow Y$, then $\nabla _{\sharp }$ computes the $I$-indexed direct sum. The compatibility of $f_{\sharp }$ with pullbacks yields a functor

(7)\begin{equation} \mathrm{SH} \colon \operatorname{Span}_{\mathrm{sm}}(\mathrm{Sch}_S) \to {\text {Cat}}_\infty. \end{equation}

An important additional functoriality of $\mathrm {SH}$ was recently discovered by Bachmann and Hoyois in [Reference Bachmann and HoyoisBH21]: given a finite étale morphism $f \colon X \rightarrow Y$ we have the multiplicative pushforward or norm $f_{\otimes } \colon \mathrm {SH}(X) \rightarrow \mathrm {SH}(Y)$, which in the case when $f$ is the fold map computes the $I$-indexed tensor product. This also satisfies base change, and so can be encoded by a functor

(8)\begin{equation} \mathrm{SH} \colon \operatorname{Span}_{\text {f} {\unicode{x00E8}} \text{t}}(\mathrm{Sch}_{S}) \to {\text {Cat}}_\infty, \end{equation}

which leads to the correct notion of a coherent multiplicative structure in motivic homotopy theory, a normed motivic spectrum, as a section of the unstraightening of (8) that is cocartesian over the backwards maps in $\operatorname {Span}_{\text {f} {\unicode{x00E8}} \text{t}}(\mathrm {Sch}_{S})$.

The motivic bispan category should combine these two structures, giving an additive pushforward along smooth morphisms and a multiplicative pushforward along finite étale morphisms. For technical reasons (due to the non-existence of Weil restriction of schemes in general), for our motivic bispan categories we either have to restrict to morphisms between schemes that are smooth and quasiprojective or work with algebraic spaces. Thus, we consider 2-categories of the form ${\text {BISPAN}}_{\text {f} {\unicode{x00E8}} \text{t}},\mathrm {sm}(\mathrm {AlgSpc}_{S})$ where $\mathrm {AlgSpc}_{S}$ means the category of algebraic spaces over $S$, and we promote $\mathrm {SH}$ to a functor

(9)\begin{equation} \mathrm{SH} \colon{\text {BISPAN}}_{\text {f} {\unicode{x00E8}} \text{t}},\mathrm{sm}(\mathrm{AlgSpc}_{S}) \to {\text {Cat}}_\infty; \end{equation}

see Theorem 3.5.9.

The decategorification of the above structure has been studied by Bachmann in [Reference BachmannBac21]. Working over a field, and restricting to a category of bispans between smooth schemes, ${\text {Bispan}}_{\text {f} {\unicode{x00E8}} \text{t}},\mathrm {sm}(\mathrm {Sm}_{k})$, Bachmann proved that the structure of a normed algebra in the abelian category of homotopy modules (the heart of the so-called homotopy $t$-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 $k$.

We also note that there is a discrepancy with the classical and finite-equivariant story: finite étale transfers are a priori not sufficient to encode the structure of a motivic spectrum. Instead, the correct kind of transfers are framed transfers in the sense of [Reference Elmanto, Hoyois, Khan, Sosnilo and YakersonEHK⁺21]. In particular, the category of framed correspondences, where the backward maps encode framed transfers, is manifestly an $\infty$-category. In other words, the additive and multiplicative transfers are rather different in the motivic story; for example, we do not know if the space of units of a normed motivic spectrum has framed transfers (see [Reference Bachmann and HoyoisBH21, § 1.5] for a discussion). For us, this means that a more robust theory of bispans in the motivic setting which encodes framed transfers is open for future investigations.

Finally, we consider an example in the context of derived algebraic geometry: If $\mathrm {Perf}(X)$ denotes the $\infty$-category of perfect quasicoherent sheaves on a spectral Deligne–Mumford stack, Barwick [Reference BarwickBar17] has shown that the pullback and pushforward functors extend to a functor

\[ \mathrm{Perf} \colon \operatorname{Span}_{\mathcal{FP}}(\mathrm{SpDM}) \to {\text {Cat}}_\infty, \]

where $\mathrm {SpDM}$ is the $\infty$-category of spectral Deligne–Mumford stacks and $\mathcal {FP}$ is a certain class of morphisms (for which pushforwards preserve perfect objects and base change is satisfied). We promote this to a functor of $(\infty,2)$-categories

\[ {\text {BISPAN}}_{\text {f} {\unicode{x00E8}} \text{t}},{\mathcal{FP}'}(\mathrm{SpDM})^{2{\text {-op}}} \to {\text {CAT}}_{\infty}, \]

using a multiplicative pushforward functor for finite étale maps (which exists by results of Bachmann and Hoyois [Reference Bachmann and HoyoisBH21]), where $\mathcal {FP}'$ is a certain subclass of $\mathcal {FP}$ for which Weil restrictions exist.

1.6 Norms in algebraic $K$-theory

A combination of the present work and [Reference Barwick, Glasman, Mathew and NikolausBGMN21] produces concrete examples of Tambara functors valued in $\mathcal {S}$, the $\infty$-category of spaces, via (connective) algebraic $K$-theory, which we discuss in § 4. The motivation for our results traces back to classical representation theory: given a finite group $G$, the classical representation ring of $G$ is a certain Grothendieck ring:

(10)\begin{equation} \mathrm{Rep}(G,\mathbb{C}) \cong K_0({\text {Fun}}(BG, {\text {Vect}}^{\mathrm{fd}}_{\mathbb{C}})). \end{equation}

In particular, the tensor product of representations induces the multiplicative structure on $\mathrm {Rep}(G,\mathbb {C})$. It is natural to consider the formation of representation rings as a functor in $G$, where the functoriality encodes various operations in representation theory such as induction and restrictions. From this viewpoint, one can enhance the multiplicative structure on $\mathrm {Rep}(G,\mathbb {C})$ to one parametrized by cosets: if $K \subset G$ is a subgroup, then we have a map given by the operation of tensor induction:

\begin{align*} \mathrm{Rep}(K,\mathbb{C}) \rightarrow \mathrm{Rep}(G,\mathbb{C}), \quad V \mapsto \otimes_{G/K} V. \end{align*}

As reviewed in Example 3.4.18, all of this functoriality can be encoded as a functor out of a bispan category formed out of finite $G$-sets.

Representation theory with ‘fancy coefficients’ entails replacing the category $\mathrm {Vect}^{\mathrm {fd}}_{\mathbb {C}}$ with a more sophisticated symmetric monoidal ($\infty$-)category $\mathcal {C}$. This line of investigation arguably began with the subject of modular representation theory, which takes $\mathcal {C}$ to be $\mathrm {Vect}^{\mathrm {fd}}_{\mathbb {F}_p}$. This is especially subtle when $G$ is a $p$-group because of the failure of the category ${\text {Fun}}(BG, {\text {Vect}}^{\mathrm {fd}}_{\mathbb {F}_p})$ to be semisimple. More recently, Treumann has also considered replacing vector spaces with $KU$-module spectra, thus taking $\mathcal {C}$ to be $\mathrm {Perf}_{\mathrm {KU}}$ where $KU$ is the $\mathbb {E}_{\infty }$-ring spectrum representing complex topological $K$-theory [Reference TreumannTre15]; his work suggests that, up to $p$-completion, representation theory over $KU$ is, in a precise way, a smooth deformation of representation theory over the $p$-adic integers [Reference TreumannTre15, 1.7].

In light of the last example, which is homotopical in nature, it is natural to consider the $K$-theory space $\Omega ^{\infty }K({\text {Fun}}(BG, \mathcal {C}))$ for $\mathcal {C}$ a small stable $\infty$-category; its homotopy groups are the higher $K$-groups. Using the universal property of bispans, coupled with the main result of [Reference Barwick, Glasman, Mathew and NikolausBGMN21], we offer the following result concerning its functoriality in the variable $G$.

Theorem 1.6.1 Let $G$ be a finite group and $\mathcal {C}$ a symmetric monoidal $\infty$-category, then the presheaf on $G$-orbits:

\[ \Omega^{\infty}K_{G}(\mathcal{C})\colon\mathcal{O}_{G}^{{\text {op}}} \rightarrow \mathcal{S} \quad G/H \mapsto \Omega^{\infty}K({\text {Fun}}(BH, \mathcal{C})), \]

extends canonically as

The proof of Theorem 1.6.1 is quite simple given our main theorem: we use this to deduce that ${\text {Fun}}(BG,\mathcal {C})$ upgrades to a functor out of bispans into the $\infty$-category of $\infty$-categories. Using a general criterion which we detail in § 4.2, we prove that the multiplicative pushforward enjoys the property of being a polynomial functor in the sense of Goodwillie calculus. A recent breakthrough of Barwick et al. [Reference Barwick, Glasman, Mathew and NikolausBGMN21] proves that the formation of algebraic $K$-theory spaces is functorial in polynomial functors, which then gives Theorem 1.6.1.

We believe that Theorem 1.6.1 could have computational significance. Using an equivariant analogue of [Reference Bachmann and HoyoisBH21, Example 7.25], Theorem 1.6.1 produces genuine-equivariant power operations on these representation-theoretic gadgets. In particular, the coherence afforded by the language of bispans results in relations between these operations.

By exactly the same methods we also show that the algebraic $K$-theory of a genuine $G$-$E_{\infty }$-ring spectrum is also a Tambara functor valued in $\mathcal {S}$. The latter are expected to be precisely the connective $G$-$E_{\infty }$-ring spectra, and given this we prove the following result.

We believe this is a completely new structure on algebraic $K$-theory in equivariant homotopy theory that significantly extends several recent results in the literature; see Remark 4.3.8.

In the algebro-geometric context the same method also shows that, for instance, the algebraic $K$-theory of schemes (or more generally spectral Deligne–Mumford stacks) has multiplicative transfers along finite étale morphisms.

1.7 Notation

This paper is written in the language of $\infty$-categories. We use the following reasonably standard notation:

• • $\mathcal {S}$ is the $\infty$-category of spaces, i.e. $\infty$-groupoids;

• • ${\text {Cat}}_\infty$ is the $\infty$-category of $\infty$-categories;

• • ${\text {CAT}}_{\infty }$ is the $(\infty,2)$-category of $\infty$-categories;

• • ${\text {Cat}}_{(\infty,2)}$ is the $\infty$-category of $(\infty,2)$-categories;

• • if $\mathcal {C}$ and $\mathcal {D}$ are $\infty$-categories or $(\infty,2)$-categories, we write ${\text {Fun}}(\mathcal {C},\mathcal {D})$ for the $\infty$-category of functors from $\mathcal {C}$ to $\mathcal {D}$;

• • if $\mathcal {C}$ and $\mathcal {D}$ are $(\infty,2)$-categories, we write ${\text {FUN}}(\mathcal {C},\mathcal {D})$ for the $(\infty,2)$-category of functors from $\mathcal {C}$ to $\mathcal {D}$;

• • we write $(-)^{(1)} \colon {\text {Cat}}_{(\infty,2)} \to {\text {Cat}}_\infty$ for the functor taking an $(\infty,2)$-category to its underlying $\infty$-category (thus, $(-)^{(1)}$ is right adjoint to the inclusion of ${\text {Cat}}_\infty$ into ${\text {Cat}}_{(\infty,2)}$);

• • we write $(-)^{\simeq }$ for the functors ${\text {Cat}}_\infty \to \mathcal {S}$ and ${\text {Cat}}_{(\infty,2)} \to \mathcal {S}$ taking an $\infty$-category or $(\infty,2)$-category to its underlying $\infty$-groupoid;

• • if $\mathcal {C}$ is an $\infty$-category and $x$ and $y$ are objects of $\mathcal {C}$, we write ${\text {Map}}_{\mathcal {C}}(x,y)$ for the space of maps from $x$ to $y$ in $\mathcal {C}$;

• • if $\mathcal {C}$ is an $(\infty,2)$-category and $x$ and $y$ are objects of $\mathcal {C}$, we write ${\text {MAP}}_{\mathcal {C}}(x,y)$ for the $\infty$-category of maps from $x$ to $y$ in $\mathcal {C}$;

• • if $\mathcal {X}$ is an $(\infty,2)$-category we write $\mathcal {X}^{{\text {op}}}$ for the $(\infty,2)$-category obtained by reversing the morphisms in $\mathcal {X}$ and $\mathcal {X}^{2{\text {-op}}}$ for that obtained by reversing the 2-morphisms.

We also adopt the following standard notation for functors between slices of an $\infty$-category $\mathcal {C}$:

• • if $f \colon x \to y$ is a morphism in $\mathcal {C}$, we have a functor

  \[ f_!\colon \mathcal{C}_{/x} \rightarrow \mathcal{C}_{/y} \]
  such that $f_!(t \rightarrow x) = t \rightarrow x \rightarrow y$, i.e. given by composition with $f$;

• • if pullbacks along $f$ exist in $\mathcal {C}$, then $f_{!}$ has a right adjoint

  \[ f^*\colon \mathcal{C}_{/y} \rightarrow \mathcal{C}_{/x}; \]

• • if $f^{*}$ has a further right adjoint, this will be denoted by

  \[ f_* \colon \mathcal{C}_{/x} \to \mathcal{C}_{/y} \]
  (this right adjoint exists for all $f$ precisely when $\mathcal {C}$ is locally cartesian closed, for example if $\mathcal {C}$ is an $\infty$-topos).