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

#### Posted by urs

Here is a transcript of part 1 of our first session.

**(1) Bipermutative Categories.**

*Definition.* A symmetric **bimonoidal category** is a category $\mathcal{B}$ with two bifunctors

as well as two special objects $1$ and $0$

- such that $1$ is the unit with respect to $\otimes $ and $0$ that with respct to $\oplus $

- such that $\oplus ,\otimes $ are associative, commutative and unital up to coherent natural isomorphism and

- such that $\otimes $ distributes over $\oplus $ up to coherent natural isomorphism.

*Definition.* A symmetric bimonoidal category is called a **bipermutative category** if all structure isomorphisms are identities, except for

and

These are required to make

commute.

*Example.* Let $V$ be the skeleton of the category of finite dimensional vector spaces over the complex numbers, with all morphisms being isomorphisms.

This means that

and we simply have

and $U(n)\oplus U(m)$ is the block sum of matrices and $U(n)\otimes U(m)$ the tensor product.

$V$ is clearly bipermutative.

*Remark.* As long as one can pick a “reasonable” skeleton, a similar construction will work for other abelian monoidal categories.

*Definition.* Let $C$ be an arbitrary small category. The set ${\pi}_{0}C$ of **path components** of $C$ is the set of equivalence classes of objects of $C$ under the equivalence relation

*Remark.*

1) Denote by $BC$ the **classifying space** (geometric realization of the nerve of) $C$ and by ${\pi}_{0}BC$ is 0th homotopy group, then

2) If $\mathcal{B}$ is a small bipermutative category, then ${\pi}_{0}\mathcal{B}$ is a commutative semiring. In this case we have

*Example.* For $\mathcal{B}=V$ we have ${\pi}_{0}V=\mathbb{N}$, with $\mathbb{N}$ regarded as a commutative semiring.

*Definition.* Let $\mathcal{B}$ be a small bipermutative category. Then ${M}_{n}(\mathcal{B})$ denotes the category of $n\times n$ matrices over $\mathcal{B}$. Here

We have a matrix multiplication functor

*Side remark.* We are secretly talking in a simplified way about the 2-category of Kapranov-Voevodsky’s 2-vector spaces, without wanting to make the 2-business explicit.

*Proposition.* Let ${I}_{n}$ be the obvious unit matrix object in ${M}_{n}(\mathcal{B})$. Then

is a tensor category. (“$\cdot $” is the above matrix multiplication functor.)

*Definition.* For $\mathcal{B}$ bipermutative, denote by ${A}_{\mathcal{B}}$ the **group completion** of the semiring ${\pi}_{0}\mathcal{B}$.

This is simply obtained by throwing in formal additive inverses to all elements in ${\pi}_{0}\mathcal{B}$. There is a natural injection

*Example.* Obviously we have ${A}_{V}=\mathbb{Z}$.

Now comes the **crucial definition** of part 1). We define a generalized notion of invertible objects in ${M}_{n}(\mathcal{B})$.

*Definition.*

1) Let $\mathcal{B}$ be a small bipermutative category. Denote by ${\mathrm{GL}}_{n}({A}_{\mathcal{B}})$ the ordinary general linear group over ${A}_{\mathcal{B}}$. Denote by ${\mathrm{GL}}_{n}({\pi}_{0}\mathcal{B})$ the *set* defined by this pullback:

2) Define ${\mathrm{GL}}_{n}(\mathcal{B})$ as the full subcategory of ${M}_{n}(\mathcal{B})$ whose objects $({b}_{\mathrm{ij}})$ are such that

In words, ${\mathrm{GL}}_{n}(\mathcal{B})$ is the subcategory of those matrices of vector spaces which would be invertible had we somehow allowed “virtual vector spaces”. i.e. formal inverses for $\oplus $.

*Example.* For $\mathcal{B}=V$ we have

This are all matrices $A\in {M}_{n}(\mathbb{N})$ which are invertible over $\mathbb{Z}$, i.e. those with $\mathrm{det}(A)=\pm 1$.

*Remark.*
$({\mathrm{GL}}_{n}(\mathcal{B}),\phantom{\rule{thickmathspace}{0ex}}\cdot \phantom{\rule{thickmathspace}{0ex}},{I}_{n})$ inherits a tensor structure from ${M}_{n}(\mathcal{B})$.

next:

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

**3) $K(V)$ and Elliptic Cohomology**

## 2-topologies

Hi Urs

Sorry to butt in on your wonderful exposition of interesting things, but, did that thesis on 2-topologies (that was supposed to be out at the end of ‘05) ever materialise?