### Seminar on 2-Vector Bundles and Elliptic Cohomology, III

#### Posted by urs

Transcript of part 2 of our first session.

**2) Algebraic K-theory of Bipermutative Categories**

*Reminder.* If $R$ is a commutative ring (to be thought of as a ring of functions over some space), then the **algebraic K-theory** of $R$ is the *space*

where $\Omega $ is the operation of forming the based loop space, while $B$ is the operation of forming the classifying space. Note that ${\bigsqcup}_{n}B{\mathrm{GL}}_{n}(R)$ is a topological monoid.

The homotopy groups

of this space are the ($i$th) **K-theory groups** of $R$.

*Strategy.* Try to mimic this with the ring $R$ replaced by the bipermutative category $V$ of finite dimensional vector spaces.

*Definition.* [BDR] If $\mathcal{B}$ is a bipermutative category, then the algebraic **K-theory of the 2-category of finitely generated free $\mathcal{B}$-modules** is

where $\mid {\mathrm{GL}}_{n}\left(\mathcal{B}\right)\mid $ is defined as follows.

*Remark.* $\mid {\mathrm{GL}}_{n}\left(\mathcal{B}\right)\mid $ should be nothing but the geometric realization of the nerve of ${\mathrm{GL}}_{n}\left(\mathcal{B}\right)$, when regarded as a 2-category. During the talk, however, we were not fully sure about some details of this identification.

*Definition.* Let

be isomorphism classes of finite ordered sets. Consider maps

i.e. ordered sets of $p$ matrices in ${\mathrm{GL}}_{n}(\mathcal{B})$.

(These matrices really are 1-morphisms in the 2-category of KV 2-vector spaces. In this sense these ordered tuples are nothing but ordered tuples of $p$ composable morphisms.)

Now, for each order-preserving map

we get a map

where on the right hand side ${M}_{i,j}$ for $j\ne i+1$ denotes the composition of the matrices with indices in the interval $[i,j]$. More precisely, we set

(the $n\times n$ identity matrix in ${\mathrm{GL}}_{n}(\mathcal{B})$) and

where, recall, “$\cdot $” is the matrix multiplication functor.

Next, denote by ${\Delta}^{p}$ the standard $p$-simplex and by ${f}_{*}:{\Delta}^{p}\to {\Delta}^{q}$ the maps on simplices induced by order-preserving maps of ordered sets. Then, finally, we define

where the equivalence relation that we divide out by is

with $S\in {\Delta}^{p}$, $A\in ({\mathrm{GL}}_{n}(\mathcal{B}){)}^{q}$ and $f:[p]\to [q]$ an order preserving map.

*My personal remark.* As indicated above, I think what is going on is that we regard ${\mathrm{GL}}_{n}(\mathcal{B})$ as a 2-category with

- objects being natural numbers

- 1-morphisms $n\to n$ being $n\times n$ matrices with entries in vector spaces (such that the determinant of their dimensions is $\pm 1$)

- 2-morphisms being matrices of invertible linear maps between the entries of the 1-morphsims.

Then we forget about the 2-morphisms and form the geometric realization of the nerve of the remaining 1-category as usual.