The String Coffee Table

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 $\sqcup_n B \mathrm{GL}_n(R)$ 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 $\mathbf{V}$ 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 $\left| \mathrm{GL}_n\left(\mathcal{B}\right) \right|$ is defined as follows.

Remark. $\left| \mathrm{GL}_n\left(\mathcal{B}\right) \right|$ 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 $j \neq i+1$ 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.