## February 5, 2006

### 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

(1)$K\left(R\right):=\Omega B\left({\bigsqcup }_{n}B{\mathrm{GL}}_{n}\left(R\right)\right)\phantom{\rule{thinmathspace}{0ex}},$

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

The homotopy groups

(2)${K}_{i}\left(R\right):={\pi }_{i}\left(K\left(R\right)\right)$

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 $ℬ$ is a bipermutative category, then the algebraic K-theory of the 2-category of finitely generated free $ℬ$-modules is

(3)$K\left(ℬ\right):=\Omega B\left({\bigsqcup }_{n}\mid {\mathrm{GL}}_{n}\left(ℬ\right)\mid \right)\phantom{\rule{thinmathspace}{0ex}},$

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

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

Definition. Let

(4)$\left[p\right]=\left\{0<1<\cdots

be isomorphism classes of finite ordered sets. Consider maps

(5)$\left[p\right]↦\left({M}_{0,1},{M}_{1,2},\cdots ,{M}_{p-1,p}\right)\in \left({\mathrm{GL}}_{n}\left(ℬ\right){\right)}^{p}\phantom{\rule{thinmathspace}{0ex}},$

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

(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

(6)$f:\left[p\right]\to \left[q\right]$

we get a map

(7)$\begin{array}{ccccc}{f}^{*}& :& \left({\mathrm{GL}}_{n}\left(ℬ\right){\right)}^{q}& \to & \left({\mathrm{GL}}_{n}\left(ℬ\right){\right)}^{p}\\ & & \left({M}_{0,1},\dots ,{M}_{q-1,q}\right)& ↦& \left({M}_{f\left(0\right),f\left(1\right)},{M}_{f\left(1\right),f\left(2\right)},\dots ,{M}_{f\left(p-1\right),f\left(p\right)}\right)\end{array}\phantom{\rule{thinmathspace}{0ex}},$

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 $\left[i,j\right]$. More precisely, we set

(8)${M}_{i,i}={I}_{n}$

(the $n×n$ identity matrix in ${\mathrm{GL}}_{n}\left(ℬ\right)$) and

(9)${M}_{i,i+k}={M}_{i,i+1}\cdot {M}_{i+1,i+2}\cdots \cdot {M}_{i+k-1,i+k}\phantom{\rule{thinmathspace}{0ex}},$

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

(10)$\mid {\mathrm{GL}}_{n}\left(ℬ\right)\mid ={\bigsqcup }_{p}{\Delta }^{p}×\left({\mathrm{GL}}_{n}\left(ℬ\right){\right)}^{p}/\sim$

where the equivalence relation that we divide out by is

(11)$\left({f}_{*}S,A\right)\sim \left(S,{f}^{*}A\right)$

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

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

- objects being natural numbers

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

- 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.

Posted at February 5, 2006 11:55 AM UTC

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