$ \newcommand\A{\mathrm{A}} \newcommand\C{\mathrm{C}} \newcommand\D{\mathrm{D}} \newcommand\E{\mathrm{E}} \newcommand\F{\mathrm{F}} \newcommand\G{\mathrm{G}} \newcommand\H{\mathrm{H}} \newcommand\h{\mathrm{h}} \newcommand\K{\mathrm{K}} \newcommand\L{\mathrm{L}} \newcommand\M{\mathrm{M}} \newcommand\t{\mathrm{t}} \newcommand{\bA}{\mathbf{A}} \newcommand{\bG}{\mathbf{G}} \newcommand{\bH}{\mathbf{H}} \newcommand{\bT}{\mathbf{T}} \newcommand{\bW}{\mathbf{W}} \newcommand{\Gm}{\bG_m} \newcommand\Ascr{\mathcal{A}} \newcommand\Cscr{\mathcal{C}} \newcommand\Dscr{\mathcal{D}} \newcommand\Escr{\mathcal{E}} \newcommand\Kscr{\mathcal{K}} \newcommand\Lscr{\mathcal{L}} \newcommand\Oscr{\mathcal{O}} \newcommand\Perfscr{\mathcal{P}\mathrm{erf}} \newcommand\Acscr{\mathcal{A}\mathrm{c}} \newcommand\heart{\heartsuit} \newcommand\cn{\mathrm{cn}} \newcommand\op{\mathrm{op}} \newcommand\gr{\mathrm{gr}} \newcommand\Gr{\mathrm{Gr}} \newcommand\fil{\mathrm{fil}} \newcommand\Ho{\mathrm{Ho}} \newcommand\dR{\mathrm{dR}} \newcommand\HH{\mathrm{HH}} \newcommand\HC{\mathrm{HC}} \newcommand\HP{\mathrm{HP}} \newcommand\TC{\mathrm{TC}} \newcommand{\bMap}{\mathbf{Map}} \newcommand{\End}{\mathrm{End}} \newcommand{\Mod}{\mathrm{Mod}} \newcommand{\coMod}{\mathrm{coMod}} \newcommand{\Fun}{\mathrm{Fun}} \newcommand{\bMap}{\mathbf{Map}} \newcommand\bE{\mathbf{E}} \newcommand\bZ{\mathbf{Z}} \newcommand\bAM{\mathbf{AM}} \newcommand\bLM{\mathbf{LM}} \newcommand\Spec{\mathrm{Spec}} \newcommand\CAlg{\mathrm{CAlg}} \newcommand\aCAlg{\mathfrak{a}\CAlg} \newcommand\dCAlg{\mathfrak{d}\CAlg} \newcommand{\Cat}{\mathrm{Cat}} \newcommand{\Sscr}{\mathcal{S}} \newcommand{\poly}{\mathrm{poly}} \newcommand{\perf}{\mathrm{perf}} $

The long-awaited paper [1] of Barwick, Glasman, Mathew, and Nikolaus has arrived. They prove that the algebraic $K$-theory space functor \(\K\colon\Cat_\infty^\perf\rightarrow\Sscr\) extends to a functor \(\Cat_\infty^\poly\rightarrow\Sscr,\) where $\Cat_\infty^\poly$ is the $\infty$-category of idempotent-complete stable $\infty$-categories and polynomial functors, in the sense of Goodwillie calculus. Moreover, the paper generalizes the main result of Blumberg–Gepner–Tabuada [2] to prove that $K$-theory admits a universal property among all polynomial functors to spaces.

Recall that [2] gives a universal property of algebraic $K$-theory as a functor on stable $\infty$-categories and exact functors. This result has been the bedrock of many of the major results in $K$-theory in the past 10 years. Of those, I give special mention to the work of Kerz–Strunk–Tamme proving Weibel’s conjecture, Tamme and Land–Tamme on excision, the collaboration of various among Clausen, Mathew, Naumann, and Noel on nilpotence and descent in algebraic $K$-theory, the work of Clausen, Mathew, and Morrow on descent and rigidity in algebraic $K$-theory, and finally the work of Land, Meier, Tamme (now with Mathew!) on telescopically localized $K$-theory.

One common feature of the works above is the interplay between the universal and the specific. While the universal property enjoyed by $K$-theory is used throughout, it is the study of what makes $K$-theory special among localizing or additive invariants that has attracted attention and made the area so interesting.

The present paper expands on this theme by endowing algebraic $K$-theory with additional functoriality, generalizing, for example, the existence the $\lambda$-ring structure on $\K_0(R)$ when $R$ is commutative. This functoriality is not present on other well-studied localizing invariants, such as topological Hochschild homology.

Let me say something about the precise statement of the theorem and the proof.

Definition. Let $\Cscr$ and $\Dscr$ be idempotent-complete stable $\infty$-categories. A functor $F\colon\Cscr\rightarrow\Dscr$ is polynomial of degree $0$ if it is constant. Inductively, $F$ is polynomial of degree $\leq n$ for some $n\geq 1$ if it preserves finite geometric realizations (i.e., geometric realizations of diagrams which are left Kan extended from $\Delta_{\leq d}^\op\subseteq\Delta^\op$ for some $d\geq 0$) and if for each $X\in\Cscr$ the functor

\[D_XF(-)=\mathrm{fib}(F(X\oplus(-))\rightarrow F(-))\]

is polynomial of degree $\leq n-1$.

Examples.

  • Exact functors are polynomial of degree $\leq 1$.
  • If $\Cscr$ is a symmetric monoidal idempotent-complete stable $\infty$-category, then the functor $X\mapsto X^{\otimes n}$ is polynomial of degree $\leq n$.
  • The derived functors $\L\mathrm{Sym}^n$, $\L\Lambda^n$, and $\L\Gamma^n$ on $\mathrm{Perf}(\bZ)$ of the symmetric, exterior, and divided powers are polynomial of degree $\leq n$.

Fix a regular uncountable cardinal $\kappa$ and let $\Cat_{\infty,\kappa}^\perf\subseteq\Cat_\infty^\perf$ be the full subcategory of $\kappa$-compact objects. Let $\Cat_{\infty,\kappa}^\poly\subseteq\Cat_{\infty}^\poly$ be the full subcategory on the objects which are $\kappa$-compact in $\Cat_\infty^\perf.$ One has a natural functor $\Cat_{\infty,\kappa}^\perf\rightarrow\Cat_{\infty,\kappa}^\poly$.

We are interested in product preserving functors $L$ on $\Cat_{\infty,\kappa}^\perf$ or $\Cat_{\infty,\kappa}^\poly$ with values in spaces (or, better, animae). Such functors have the property that $L(\Cscr)$ is an $\bE_\infty$-space (a highly structured version of an infinite loop space) and this is natural in exact functors, but not polynomial functors. We call such a functor additive if the abelian monoid $\pi_0(\Cscr)$ is an abelian group for every $\Cscr$ and if for any split exact sequence $\Cscr\rightarrow\Dscr\rightarrow\Escr$ in the sense of [2] the induced map $L(\Dscr)\rightarrow\L(\Cscr)\times\L(\Escr)$ is an equivalence. Note that this depends only on the restriction of the functor to $\Cat_{\infty,\kappa}^\perf$. If $L$ is a functor $\Cat_{\infty,\kappa}^\poly$, let $L^{\perf}$ denote its restriction to $\Cat_{\infty,\kappa}^\perf$.

There is a universal additivization $L^\mathrm{ad}$ of a product preserving functor on $\Cat_{\infty,\kappa}^\perf$ and a universal polynomial additivization $L^\mathrm{pad}$ of a product preserving functor on $\Cat_{\infty,\kappa}^\poly$. The main theorem relates the two.

Theorem. For any product preserving functor $L\colon\Cat_{\infty,\kappa}^\perf$, the natural map $L^{\perf,\mathrm{ad}}\rightarrow L^{\mathrm{pad},\perf}$ is an equivalence.

The theorem implies the polynomial functoriality of $K$-theory as follows. Let $\iota$ denote the functor which takes $\Cscr$ to its underlying space of objects. This is naturally a product preserving functor $\iota\colon\Cat_{\infty,\kappa}^\poly\rightarrow\Sscr$. The main result of [2] implies that $\iota^{\perf,\mathrm{ad}}\simeq\K$. Thus, the theorem above implies that $\K\simeq\iota^{\mathrm{pad},\perf}$. In particular, $\iota^{\mathrm{pad}}\colon\Cat_{\infty,\kappa}^\poly\rightarrow\Sscr$ is the algebraic $K$-theory space functor with its polynomial functoriality.

The strategy of the proof is as follows. First the authors produce polynomial functoriality on the additive Grothendieck group functor $\K_0^\oplus$. This goes back to work of Dold and Joukhovitski and relies on a simple observation about polynomial functions, namely, that if $M$ is an abelian monoid, $M^+$ its group completion, and $A$ an abelian group, the natural restriction map

\[\mathrm{Hom}_{\leq n}(M^+,A)\rightarrow\mathrm{Hom}_{\leq n}(M,A)\]

on polynomial functions of degree $\leq n$ is a bijection.

Second, one produces polynomial functoriality on the Grothendieck group functor $\K_0$, obtained from $\K_0^\oplus$ by further killing the classes $[B]-[A]-[C]$ whenever $A\rightarrow B\rightarrow C$ is an exact sequence. This leverages the fact that polynomial functors preserve finite geometric realizations and the Čech complex of $B\rightarrow C$ to reduce to the split case.

Third and finally, the authors show that a compatibility between the two additivization localizations. This relies on the notion of a universal $K$-equivalence $\Cscr\rightarrow\Dscr$ and the fact that a certain construction $\Gamma_n(-)$ preserves universal $K$-equivalences. Here, $\Gamma_n\Cscr$ is an idempotent-complete stable $\infty$-category with the property that exact functors $\Gamma_n\Cscr\rightarrow\Dscr$ correspond to polynomial functors $\Cscr\rightarrow\Dscr$ of degree $\leq n$. The point is that universal $K$-equivalences are checked in $\K_0$, so the previous polynomial functoriality on $\K_0$ completes the proof.

References

[1] Barwick, Glasman, Mathew, and Nikolaus, K-theory and polynomial functors, arXiv:2102.00936.

[2] Blumberg, Gepner, and Tabuada, A universal characterization of higher algebraic K-theory, Geom. Topol. 17 (2013), no. 2, 733–838.