There ought to be a symmetric monoidal functor from the symmetric monoidal $2$-groupoid whose objects are Morita-invertible real superalgebras (precisely the Clifford algebras), morphisms are invertible superbimodules, and $2$-morphisms are invertible superbimodule homomorphisms to the symmetric monoidal ($\infty$-)groupoid whose objects are invertible $KO$-module spectra, morphisms are invertible morphisms of $KO$-module spectra, etc. Everything I'm about to say should generalize to the complex case and $KU$ but let me only address the real case.

On $\pi_0$ this functor ought to induce a map from the graded Brauer group of $\mathbb{R}$ to the Picard group of $KO$, both of which turn out to be $\mathbb{Z}_8$; this directly relates Bott periodicity for Clifford algebras and for K-theory.

Recently I tried to write down this functor "explicitly" (to prepare for a talk and to confirm that I wasn't lying when I wrote this) and found that I couldn't quite get some details to work out the way I wanted them to. The idea should be, starting with a real (resp. complex) superalgebra $A$, to

  1. Construct the topological groupoid of finitely generated projective $A$-supermodules (automorphism groups have their usual topologies as Lie groups),
  2. Pass to the algebraic K-theory spectrum of this groupoid, regarded as a symmetric monoidal groupoid under direct sum,
  3. At some point during this process, kill off "trivial objects."

The correct version of this process should, when given $A = \mathbb{R}$ itself, reproduce $KO$, and more generally, when given the Clifford algebra $\text{Cliff}(p, q)$, reproduce a shift of $KO$ (as a $KO$-module spectrum) by $p - q$ either up or down depending on some sign conventions.

I ran into two problems getting this picture to work out:

  1. The construction I know of the algebraic K-theory spectrum of a symmetric monoidal groupoid produces a connective spectrum. For example, applied to $\text{Vect}_{\mathbb{R}}$ it reproduces $ko$ rather than $KO$.

  2. It seems like there are two inequivalent ways to kill off trivial objects, and they don't produce the same answer.

In detail, to fix the sign conventions, let $\text{Cliff}(p, q)$ be the Clifford algebra generated by $p$ anticommuting square roots of $1$ and $q$ anticommuting square roots of $-1$, so that $\text{Cliff}(1, 0) \cong \mathbb{R} \times \mathbb{R}$ and $\text{Cliff}(0, 1) \cong \mathbb{C}$ (as ungraded algebras). Let $K(\text{Cliff}(p, q))$ be the algebraic K-theory of the category of finite-dimensional $\text{Cliff}(p, q)$-supermodules (so some connective spectrum). There are two natural forgetful functors inducing maps

$$K(\text{Cliff}(p + 1, q)) \to K(\text{Cliff}(p, q)), K(\text{Cliff}(p, q + 1)) \to K(\text{Cliff}(p, q))$$

of connective spectra; as maps on categories, they pick out the supermodules admitting an additional odd automorphism squaring to $1$ or an odd automorphism squaring to $-1$ respectively. For example, when $p = q = 0$, these are two ways of identifying the supervector spaces you want to regard as trivial for the purposes of defining $KO$ in terms of supervector spaces.

I computed that the homotopy cofibers of these maps (in connective spectra) are $\Omega^{p-q} ko$ and $B^{p-q} ko$ respectively; that is, we get the infinite loop spaces making up $ko$, but in two different orders.

How can this construction be fixed so that it outputs $KO$-module spectra rather than $ko$-module spectra, and what's the "correct" way to kill trivial objects?

One desideratum for "correct" is that it should generalize cleanly to the case where we replace $\mathbb{R}$ with a discrete field $k$ and replace $KO$ with the algebraic K-theory spectrum $K(k)$ of $k$. The end result of the "correct" process is that it should naturally give a symmetric monoidal functor inducing a map from the graded Brauer group of $k$ to the Picard group of $K(k)$. But here there are not only two choices for what an odd automorphism could square to: instead the set of choices is parameterized by

$$k^{\times}/(k^{\times})^2 \cong H^1(\text{Spec } k, \mu_2).$$

So I don't know what to do. Maybe always pick squaring to $1$?

One thing that made me uncomfortable about the above is that it's conceptually important that the supermodule categories should be regarded as enriched and tensored over supervector spaces, but I ignored their enrichment over supervector spaces when I passed to algebraic K-theory spectra. The enrichment seems like it ought to be important for defining the "correct" way to kill of trivial objects.

  • $\begingroup$ Have you worked out the complex version (graded complex central simple algebras and invertible $KU$-module spectra) of this story? If so, perhaps you can go down to $KO$-module spectra by taking $\mathbb{Z}/2$-fixed points. $\endgroup$ – Benjamin Antieau Nov 6 '14 at 22:17
  • $\begingroup$ Update: awhile back I asked Lurie this question and his sense is that the functor I want isn't symmetric monoidal but at best lax / oplax. $\endgroup$ – Qiaochu Yuan Dec 25 '17 at 21:20

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