The String Coffee Table

(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 $\mathbf{V}$ 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.

$\mathbf{V}$ 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 $B C$ the classifying space (geometric realization of the nerve of) $C$ and by $\pi_0 B C$ 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 $\mathcal{B} = \mathbf{V}$ we have $\pi_0 \mathbf{V} = \mathbb{N}$, 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 $A_\mathbf{V} = \mathbb{Z}$.

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 $(b_{ij})$ 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 $\mathcal{B} = \mathbf{V}$ 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. $(\mathrm{GL}_n(\mathcal{B}),\;\cdot\;, I_n)$ inherits a tensor structure from $M_n(\mathcal{B})$.

next:

2) Algebraic K-theory of Bipermutative Categories

3) $K(\mathbf{V})$ and Elliptic Cohomology