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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(R):=ΩB( nBGL n(R)),

where Ω is the operation of forming the based loop space, while B is the operation of forming the classifying space. Note that nBGL n(R) is a topological monoid.

The homotopy groups

(2)K i(R):=π i(K(R))

of this space are the (ith) 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 is a bipermutative category, then the algebraic K-theory of the 2-category of finitely generated free -modules is

(3)K():=ΩB( nGL n()),

where GL n() is defined as follows.

Remark. GL n() should be nothing but the geometric realization of the nerve of GL n(), when regarded as a 2-category. During the talk, however, we were not fully sure about some details of this identification.

Definition. Let

(4)[p]={0 <1 <<p}

be isomorphism classes of finite ordered sets. Consider maps

(5)[p](M 0,1 ,M 1,2 ,,M p1 ,p)(GL n()) p,

i.e. ordered sets of p matrices in GL n().

(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

(7)f * : (GL n()) q (GL n()) p (M 0,1 ,,M q1 ,q) (M f(0 ),f(1 ),M f(1 ),f(2 ),,M f(p1 ),f(p)),

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

(8)M i,i=I n

(the n×n identity matrix in GL n()) and

(9)M i,i+k=M i,i+1 M i+1 ,i+2 M i+k1 ,i+k,

where, recall, “” is the matrix multiplication functor.

Next, denote by Δ p the standard p-simplex and by f *:Δ pΔ q the maps on simplices induced by order-preserving maps of ordered sets. Then, finally, we define

(10)GL n()= pΔ p×(GL n()) p/

where the equivalence relation that we divide out by is

(11)(f *S,A)(S,f *A)

with SΔ p, A(GL n()) q and f:[p][q] an order preserving map.

My personal remark. As indicated above, I think what is going on is that we regard GL n() as a 2-category with

- objects being natural numbers

- 1-morphisms nn 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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Read the post Seminar on 2-Vector Bundles and Elliptic Cohomology, I
Weblog: The String Coffee Table
Excerpt: Review of the 2-vector approach towards elliptic cohomology. Part I.
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