It may be worthwhile to vary the question ever so slightly to read

**Question:** Is there a conceptual explanation for the tight relationship (or some aspect thereof) between K-theory and *[superalgebra](http://ncatlab.org/nlab/show/superalgebra)*? 

This seems to have a deep answer as follows:

1. The $\mathbb{Z}_2$-grading in superalgebra/supergeometry is usefully identified as the low degree pieces of _$\mathbb{S}$-grading_ for $\mathbb{S}$ the sphere spectrum;

1. every $E_\infty$ ring spectrum such as $KO$ and $KU$ is canonically $\mathbb{S}$-graded, or at least its group of units is.

This is discussed in a bit more detail on the nLab at _[superalgebra - Abstract idea](http://ncatlab.org/nlab/show/super+algebra#AbstractIdea)_. It provides, I think, a conceptual explanation of what might seem an unreasonable effectiveness of superalgebra in generalized cohomology theory.

A few weeks back I had written the following advertizement of this fact, which maybe I can reproduce here:

As we see amplified these days, homotopy theory,  instead of being the complicated edifice as which it was historically obtained, is actually simple in that it is foundational. That's the point of the new "[univalent foundations](http://ncatlab.org/nlab/show/Homotopy+Type+Theory+--+Univalent+Foundations+of+Mathematics)":

homotopy theory follows from formal logic the moment you stop insisting that you can decide if two things are actually equal and admit that you can only provide an explicit equivalence that exhibits the equal-ness. (What physicists call a gauge equivalence.. )

Algebra in homotopy theory is homotopical algebra, and in a way the most fundamental object here is the sphere spectrum, the free commutative infinity-ring on a single element. The sphere spectrum is in homotopy foundations what the integers is in traditional foundations.


Recently Kapranov (based on some pre-history of thoughts that are hard for me to track down precisely) amplifies a simple but striking [observation](http://ncatlab.org/nlab/show/super+2-algebra#References):

grading over the sphere spectrum is supersymmetry.

More in detail: a commutative oo-ring whose oo-group of units is graded over the sphere spectrum is a homotoy-analog/refinement of a superalgebra (see at _[superalgebra -- Abstract idea](http://ncatlab.org/nlab/show/super+algebra#AbstractIdea)_).

Hm, you think, so which commutative $\infty$-rings are graded in this way over the sphere spectrum?

Just about a year ago Sagave gives a striking reply to this: every single one is, and canonically so (see [here](http://ncatlab.org/nlab/show/infinity-group+of+units#AugmentedDefinition)).

In summary, there is a remarkable piece of magic that is happening here, magic in the sense that it opens our eyes to a truth that has been there all along without us noticing it: supersymmetry is right there in the new foundations. It's inevitable.

Hints of this have been seen all along of course. It is standard among mathematicians to praise the role that the idea of supersymmetry has played in pure mathematics, quite independently of its role in physics. Notably index theorems tend to have their most natural formulaton in superalgebra.

Hm, index theorems? What is an index theorem? An index theorem is the characterization of a push-forward in a cohomology theory. A cohomology theory, in turn, is just the theory of maps into a commutative infinity-ring. And there the circle closes.

For instance start with the abelian 2-group $B U(1)$ of ordinary line bundles. Its group ring (meaning: infinity-group infinity ring over the sphere spectrum) contains an element called the Bott element, quotienting that out yields the commutative infinity-ring known as KU, the one that gives the cohomology theory known as ordinary complex K-theory .

$KU = \mathbb{S}[BU(1)][Bott^-1]$

(This is [Snaith's theorem](http://ncatlab.org/nlab/show/Snaith+theorem)  ).

Now what is the $\infty$-group of units of KU? That's the 2-group of [super line 2-bundles](http://ncatlab.org/nlab/show/super+line+2-bundle).


In some disguise (see the references at the above link) this has been known for ages. This is why supergeometry is such a powerful tool in K-theory and index theory. But here we see that this nice "technical trick" as it may seem is but a shadow of something very deep, with "very deep" in the technical sense: just a handful of lines of code above the very univalent foundations of mathematics.

Kapranov combined with Sagave shows us that commutative algebra in homotopy theory is automatically and necessarily, in a "god given" way: superalgebra.