## February 3, 2006

### 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 $ℬ$ with two bifunctors

(1)$\oplus ,\otimes :ℬ×ℬ\to ℬ$

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

(2)$\begin{array}{ccc}A\otimes B& \stackrel{{c}^{\otimes }}{\to }& B\otimes A\\ A\oplus B& \stackrel{{c}^{\oplus }}{\to }& B\oplus A\end{array}$

and

(3)$A\otimes \left(B\oplus C\right)\to \left(A\otimes B\right)\phantom{\rule{thickmathspace}{0ex}}\oplus \phantom{\rule{thickmathspace}{0ex}}\left(A\otimes C\right)\phantom{\rule{thinmathspace}{0ex}}.$

These are required to make

(4)$\begin{array}{ccc}A\otimes \left(B\oplus C\right)& \to & \left(A\otimes B\right)\oplus \left(A\otimes C\right)\\ {c}^{\otimes }↓\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}& & \phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}↓{c}^{\otimes }\oplus {c}^{\otimes }\\ \left(B\oplus C\right)\otimes A& \to & \left(B\otimes A\right)\oplus \left(C\otimes A\right)\end{array}$

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

(5)$\mathrm{Obj}\left(V\right)=ℕ$
(6)${\mathrm{Mor}}_{V}\left(n,m\right)=\left\{\begin{array}{cc}U\left(n\right)& \mathrm{for}\phantom{\rule{thickmathspace}{0ex}}n=m\\ \varnothing & \mathrm{for}\phantom{\rule{thickmathspace}{0ex}}n\ne m\end{array}$

and we simply have

(7)$\begin{array}{ccc}n\oplus m& =& n+m\\ n\otimes m& =& n\cdot m\end{array}$

and $U\left(n\right)\oplus U\left(m\right)$ is the block sum of matrices and $U\left(n\right)\otimes U\left(m\right)$ 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

(8)$c\stackrel{{\pi }_{0}}{\sim }d\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}⇔\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}{\mathrm{Mor}}_{C}\left(c,d\right)\ne \varnothing \phantom{\rule{thickmathspace}{0ex}}\mathrm{or}\phantom{\rule{thickmathspace}{0ex}}{\mathrm{Mor}}_{C}\left(d,c\right)\ne \varnothing \phantom{\rule{thinmathspace}{0ex}}.$

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

(9)${\pi }_{0}C={\pi }_{0}BC\phantom{\rule{thinmathspace}{0ex}}.$

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

(10)$\begin{array}{ccc}\left[b{\right]}_{{\pi }_{0}}\cdot \left[b\prime {\right]}_{{\pi }_{0}}& =& \left[b\otimes b\prime {\right]}_{{\pi }_{0}}\\ \left[b{\right]}_{{\pi }_{0}}+\left[b\prime {\right]}_{{\pi }_{0}}& =& \left[b\oplus b\prime {\right]}_{{\pi }_{0}}\end{array}\phantom{\rule{thinmathspace}{0ex}}.$

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

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

(11)$\begin{array}{ccccc}\mathrm{Obj}\left({M}_{n}\left(ℬ\right)\right)& =& \left\{\left({b}_{\mathrm{ij}}{\right)}_{i,j=1}^{n}& \mid & {b}_{\mathrm{ij}}\in \mathrm{Obj}\left(ℬ\right)\right\}\\ \mathrm{Mor}\left({M}_{n}\left(ℬ\right)\right)& =& \left\{\left({b}_{\mathrm{ij}}\stackrel{{\varphi }_{\mathrm{ij}}}{\to }b{\prime }_{\mathrm{ij}}{\right)}_{i,j=1}^{n}& \mid & {\varphi }_{\mathrm{ij}}\in \mathrm{Mor}\left(ℬ\right)\right\}\end{array}\phantom{\rule{thinmathspace}{0ex}}.$

We have a matrix multiplication functor

(12)$\begin{array}{ccc}{M}_{n}\left(ℬ\right)×{M}_{n}\left(ℬ\right)& \stackrel{\cdot }{\to }& {M}_{n}\left(ℬ\right)\\ \left({b}_{\mathrm{ij}}\right)×\left({c}_{\mathrm{ij}}\right)& ↦& \left({\oplus }_{k=1}^{n}{b}_{\mathrm{ik}}\otimes {c}_{\mathrm{kj}}\right)\end{array}\phantom{\rule{thinmathspace}{0ex}}.$

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 ${M}_{n}\left(ℬ\right)$. Then

(13)$\left({M}_{n}\left(ℬ\right)\right),\phantom{\rule{thickmathspace}{0ex}}\cdot \phantom{\rule{thickmathspace}{0ex}},{I}_{n}\right)$

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

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

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

(14)${\pi }_{0}ℬ\to {A}_{ℬ}\phantom{\rule{thinmathspace}{0ex}}.$

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

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

Definition.

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

(15)$\begin{array}{ccc}{\mathrm{GL}}_{n}\left({\pi }_{0}ℬ\right)& \to & {\mathrm{GL}}_{n}\left({A}_{ℬ}\right)\\ ↓& & ↓\\ {M}_{n}\left({\pi }_{0}ℬ\right)& \to & {M}_{n}\left({A}_{ℬ}\right)\end{array}\phantom{\rule{thinmathspace}{0ex}}.$

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

(16)$\left(\left[{b}_{\mathrm{ij}}{\right]}_{{\pi }_{0}}\right)\in {\mathrm{GL}}_{n}\left({\pi }_{0}ℬ\right)\phantom{\rule{thinmathspace}{0ex}}.$

In words, ${\mathrm{GL}}_{n}\left(ℬ\right)$ 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 $ℬ=V$ we have

(17)${\mathrm{GL}}_{n}\left({\pi }_{0}V\right)={\mathrm{GL}}_{n}\left(ℕ\right)\phantom{\rule{thinmathspace}{0ex}}.$

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

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

next:

2) Algebraic K-theory of Bipermutative Categories

3) $K\left(V\right)$ and Elliptic Cohomology

Posted at February 3, 2006 4:15 PM UTC

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### 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?

Posted by: Kea on February 4, 2006 1:06 AM | Permalink | Reply to this

### Re: 2-topologies

did that thesis on 2-topologies (that was supposed to be out at the end of ‘05) ever materialise?

You mean the thesis by Igor Bakovic? Not yet. It is still in the making. I plan to report on this when possible. Igor plans to visit Hamburg in the near future.

Posted by: urs on February 4, 2006 5:37 PM | Permalink | Reply to this
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