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

On this functor ought to induce a map from the graded Brauer group of to the Picard group of , both of which turn out to be ; 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 , to

1. Construct the topological groupoid of finitely generated projective -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 itself, reproduce , and more generally, when given the Clifford algebra , reproduce a shift of (as a -module spectrum) by 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 it reproduces rather than .

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 be the Clifford algebra generated by anticommuting square roots of and anticommuting square roots of , so that and (as ungraded algebras). Let be the algebraic K-theory of the category of finite-dimensional -supermodules (so some connective spectrum). There are two natural forgetful functors inducing maps

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

I computed that the homotopy cofibers of these maps (in connective spectra) are and respectively; that is, we get the infinite loop spaces making up , but in two different orders.

How can this construction be fixed so that it outputs -module spectra rather than -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 with a discrete field and replace with the algebraic K-theory spectrum of . 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 to the Picard group of . But here there are not only two choices for what an odd automorphism could square to: instead the set of choices is parameterized by

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

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.