## May 29, 2006

### 2-Spectral Theory, Part I

#### Posted by Urs Schreiber I am doing some detective work in categorified linear algebra.

The goal is to understand the relation between Hecke-like operators ($\to$) and dualities in RCFT ($\to$). The hypothesized connection between the two is a certain condition known in RCFT, which characterizes disorder operators that induce CFT dualities (Kramers-Wannier, T-duality, etc). This condition is reminiscent of the condition on ordinary operators to be (semi)-normal. Therefore it might imply a categorified spectral theorem ($\to$). Therefore this condition might characterize disorder operators that have a basis of categorified eigenvectors ($\to$) - like the Hecke operator does.

Might.

In ordinary linear algebra and functional analysis, the condition on an operator $O$ to have something like a complete basis of eigenvectors, or, more precisely, to admit a spectral theorem ($\to$), is that it is normal ($\to$)

(1)$O{O}^{\star }-{O}^{\star }O=0\phantom{\rule{thinmathspace}{0ex}}.$

For instance $O$ might be unitary $O{O}^{*}=\mathrm{Id}$, or hermitean, $O={O}^{*}$.

More generally, there is spectral theory for operators which are semi-normal ($\to$), meaning that the above commutator is not necessarily vanishing, but trace class

(2)$\mathrm{Tr}\left(O{O}^{\star }-{O}^{\star }O\right)\le 0\phantom{\rule{thinmathspace}{0ex}}.$

There are various indications ($\to$, $\to$), that a suitable categorification of something like this helps to understand 2-dimensional (topological/conformal) field theory.

In that context, the field of complex numbers $ℂ$ gets replaced by some monoidal category $C$ and the category ${}_{ℂ}\mathrm{Mod}$ of complex vector spaces gets replaced by the 2-category ${}_{C}\mathrm{Mod}$ of $C$-module categories.

A $C$-module category plays hence the role of a categorified vector space, while 1-morphisms of $C$-module categories play the role of categorified linear maps.

In nice cases our $C$-module categories are equipped with an internal $\mathrm{Hom}$-functor ($\to$), which can be regarded as a categorified sesquilinear scalar product (items 3 and 4 in Ostrik’s Lemma 5 ($\to$)). In this case we can think of dealing with categorified Hilbert spaces (HDA II).

One easily sees that the category $\mathrm{BiMod}\left(C\right)$ of bimodules internal to $C$ sits inside ${\mathrm{Mod}}_{C}$, as does $C$ itself:

(3)$\Sigma \left(C\right)\to \mathrm{BiMod}\left(C\right)\to {\mathrm{Mod}}_{C}\phantom{\rule{thinmathspace}{0ex}}.$

In nice cases the second arrow is an equivalence ($\to$), which allows us to restrict attention to bimodules.

Moreover, at least in these nice cases we have a very explicit understanding of the internal $\mathrm{Hom}$ on all $C$-module categories, hence of the categorified scalar product.

As shown in FRS II (section 2.4, $\to$), we have, for $N$ and $N\prime$ being internal left (special symmetric Frobenius algebra-) $A$-modules, that the internal $\mathrm{Hom}$ of ${\mathrm{Mod}}_{A}\left(C\right)$ looks like

(4)$〈N,N\prime 〉:=\mathrm{IHom}\left(N,N\prime \right):={N}^{\vee }{\otimes }_{A}N\phantom{\rule{thinmathspace}{0ex}},$

where ${N}^{\vee }$ is the dual of $N$, as an object of $C$, with the canonical right $A$ action.

Notice how this realization of the internal $\mathrm{Hom}$ makes the categorified “sesquilinearity” of the scalar product manifest.

For $X\in \mathrm{Obj}\left(C\right)$ any object in $C$ (a categorified complex number, if you like), we have

(5)$〈\stackrel{⇀}{v}\otimes X\phantom{\rule{thickmathspace}{0ex}},\phantom{\rule{thickmathspace}{0ex}}\stackrel{⇀}{w}〉={X}^{*}\otimes {\stackrel{⇀}{v}}^{*}{\otimes }_{A}\stackrel{⇀}{w}={X}^{*}\otimes 〈\stackrel{⇀}{v}\phantom{\rule{thickmathspace}{0ex}},\phantom{\rule{thickmathspace}{0ex}}\stackrel{⇀}{w}〉$

and

(6)$〈\stackrel{⇀}{v}\phantom{\rule{thickmathspace}{0ex}},\phantom{\rule{thickmathspace}{0ex}}\stackrel{⇀}{w}\otimes X〉={\stackrel{⇀}{v}}^{*}{\otimes }_{A}\stackrel{⇀}{w}\otimes X=〈\stackrel{⇀}{v}\phantom{\rule{thickmathspace}{0ex}},\phantom{\rule{thickmathspace}{0ex}}\stackrel{⇀}{w}〉\otimes X\phantom{\rule{thinmathspace}{0ex}}.$

You should visualize all these formulas in the context of Kapranov-Voevodsky 2-vector spaces ($\to$). (which is, roughly, the right ambient category for topological 2D field theory).

In that context we have $C=\mathrm{Vect}$ over the ground field $ℂ$, say. Our internal algebras are ${ℂ}^{n}={\oplus }_{i=1}^{n}ℂ$ and a left ${ℂ}^{n}$-module is a vector whose $n$ entries are $ℂ$-vector spaces.

So for instance for $n=2$ we’d have a 2-vector of the form

(7)$\stackrel{⇀}{v}=\left[\begin{array}{c}{V}_{1}\\ {V}_{2}\end{array}\right]\phantom{\rule{thinmathspace}{0ex}}.$

Using the above definition of scalar product, we find its norm square to be

(8)$〈\left[\begin{array}{c}{V}_{1}\\ {V}_{2}\end{array}\right]\phantom{\rule{thickmathspace}{0ex}},\phantom{\rule{thickmathspace}{0ex}}\left[\begin{array}{c}{V}_{1}\\ {V}_{2}\end{array}\right]〉=〈\left[\begin{array}{cc}{V}_{1}^{*}& {V}_{2}^{*}\end{array}\right]{\otimes }_{{ℂ}^{2}}\left[\begin{array}{c}{V}_{1}\\ {V}_{2}\end{array}\right]〉={V}_{1}^{*}\otimes {V}_{1}\phantom{\rule{thickmathspace}{0ex}}\oplus \phantom{\rule{thickmathspace}{0ex}}{V}_{2}^{*}\otimes {V}_{2}\phantom{\rule{thinmathspace}{0ex}},$

as it should be.

It has been observed long before (HDA II) that for 2-vector spaces the notion of adjoint functor and adjoint 2-linear map coincide. That’s a direct consequence of interpreting the internal $\mathrm{Hom}$ as the categorified scalar product.

In the present context 2-linear maps are functors of $C$-module categories respecting the (right) $C$-action. If we work in terms of bimodule categories this are nothing but functors that act by (left) tensor-multiplication with a given bimodule.

Again, this is easiest sean in categorified matrix multiplication.

A ${ℂ}^{2}$-bimodule would be a $2×2$-matrix whose entries are vector spaces

(9)$B=\left[\begin{array}{cc}{B}_{11}& {B}_{12}\\ {B}_{21}& {B}_{22}\end{array}\right]\phantom{\rule{thinmathspace}{0ex}}.$

It acts on 2-vectors as

(10)$B\stackrel{⇀}{v}=\left[\begin{array}{cc}{B}_{11}& {B}_{12}\\ {B}_{21}& {B}_{22}\end{array}\right]{\otimes }_{{ℂ}^{2}}\left[\begin{array}{c}{V}_{1}\\ {V}_{2}\end{array}\right]=\left[\begin{array}{c}{B}_{11}\otimes {V}_{1}\oplus {B}_{12}\otimes {V}_{2}\\ {B}_{21}\otimes {V}_{1}\oplus {B}_{22}\otimes {V}_{2}\end{array}\right]\phantom{\rule{thinmathspace}{0ex}}.$

The adjoint of a 2-linear map ${}_{A}\mathrm{Mod}\left(C\right)\stackrel{B}{\to }{}_{A\prime }\mathrm{Mod}\left(C\right)$ (an $A\prime$-$A$ bimodule) is ${B}^{\vee }$, which is the dual of $B$ as an object of $C$, with the canonical $A$-$A\prime$ bimodule structure.

(11)$〈B{\otimes }_{A}{\stackrel{⇀}{v}}_{1}\phantom{\rule{thickmathspace}{0ex}},\phantom{\rule{thickmathspace}{0ex}}{\stackrel{⇀}{v}}_{2}〉\simeq \left({\stackrel{⇀}{v}}_{1}{\right)}^{*}{\otimes }_{A}{B}^{\vee }{\otimes }_{A}{\stackrel{⇀}{v}}_{2}\simeq 〈{\stackrel{⇀}{v}}_{1}\phantom{\rule{thickmathspace}{0ex}},\phantom{\rule{thickmathspace}{0ex}}{B}^{\vee }{\otimes }_{A}{\stackrel{⇀}{v}}_{2}〉\phantom{\rule{thinmathspace}{0ex}}.$

For instance

(12)${\left[\begin{array}{cc}{B}_{11}& {B}_{12}\\ {B}_{21}& {B}_{22}\end{array}\right]}^{\vee }=\left[\begin{array}{cc}{B}_{11}^{*}& {B}_{21}^{*}\\ {B}_{12}^{*}& {B}_{22}^{*}\end{array}\right]\phantom{\rule{thinmathspace}{0ex}},$

as it should be.

We could agree to write

(13)${B}^{†}:={B}^{\vee }$

to emphasize that the dual of a bimodule $B$ (with duality in the sense of duality of objects in $C$, which is supposed to be a monoidal category with all duals) plays the role of an adjoint linear operator.

Notice that vectors of vector spaces are nothing but vector bundles over finite sets. Hence, with little effort, we can generalize the above to a 2-linear algebra where 2-vectors are vector bundles (or locally free sheaves) over some space, and where 2-linear maps are correspondences between two base spaces.

We may imagine, in this categorified setup, to study all the questions familial from ordinary linear algebra or functional analysis. Are the 2-linear maps which have a complete basis of eigenvectors, for instance?

It seems that one nontrivial example for such a situation is that of Hecke operators ($\to$) arising in the context of geometric Langlands duality ($\to$). As Frenkel explains in section 4.4 of his lecture notes ($\to$), we may view Hecke operators as categorified differential operators whose eigen-2-vectors are like categorified exponential functions. The geometric Langlands duality (in the “classical limit” ($\to$) )is like a categorified Fourier transformation which exchanges exponential with delta-distributions.

If we try to abstract away from the particular example of geometric Langlands (which corresponds (only) to a very specific 2D TFT, as Kapustin and Witten explain), we may ask the general question under which conditions the category ${\mathrm{Mod}}_{C}$ associated with some 2D QFT admits disorder operators (2-linear maps) that have a spectral decomposition.

Apart from possibly being an interesting question by itself, we may also try to see if the condition to be found here has maybe already arisen in some guise in 2D field theory.

Indeed, maybe it has. In

J. Fröhlich, J. Fuchs, I. Runkel, C. Schweigert
Kramers-Wannier duality from conformal defects
cond-mat/0404051

results are announced which relate algebraic properties of bimodules associated to defect lines ($\to$) with various dualities of rational conformal field theory, among them order/disorder dualities like that found by Kramers-Wannier, but also for instance T-duality.

In brief, the condition for a bimodule $B$ to induce a duality is that

(14)$B{\otimes }_{A}{B}^{\vee }$

is a direct sum of simple invertible bimodules.

Now, given the above dictionary of 2-linear algebra, we notice that this is an equation relating a linear 2-map with its adjoint.

In fact, from a naïve categorification of the above mentioned condition on semi-normal operators, we would be tempted to ask what this duality condition tells us about possible isomorphisms

(15)$\mathrm{Tr}\left(B{\otimes }_{A}{B}^{\vee }\right)\simeq \mathrm{Tr}\left({B}^{\vee }{\otimes }_{A}B\right)\phantom{\rule{thinmathspace}{0ex}}.$

In order for that to make sense, we first need a notion of trace for 2-linear maps.

But luckily, one such notion which has several indications of being “right”, already exists. It has been proposed recently by Ganter and Kapranov ($\to$). that the right notion of 2-trace for 2-vector spaces is the functor which sends bimodules $B\otimes {B}^{\vee }$ to the 2-morphisms from the identity to $B\otimes {B}^{\vee }$:

(16)$\mathrm{Tr}\left(B{\otimes }_{A}{B}^{\vee }\right):={\mathrm{Hom}}_{2}\left({Id}_{A},B{\otimes }_{A}{B}^{\vee }\right)\phantom{\rule{thinmathspace}{0ex}}.$

They find that this trace is nothing but the Hochschild cohomology of $B{\otimes }_{A}{B}^{\vee }$ (see for instance this pdf for a quick definition of Hochschild cohomology for bimodules. Hm, I realize that I don’t understand precisely how the categorical trace leads to Hochschild cohomology here…).

Therefore, possibly

(17)$\mathrm{Tr}\left(B{\otimes }_{A}{B}^{\vee }\right)\simeq \mathrm{Tr}\left({B}^{\vee }{\otimes }_{A}B\right)$

is the right condition for $B$ to admit a categorified spectral decomposition, and possibly it should be interpreted as saying that the bimodule Hochschild cohomologies of $B{\otimes }_{A}{B}^{\vee }$ and ${B}^{\vee }{\otimes }_{A}B$ coincide.

Clearly, this is indeed the case for instance for those duality defects that satisfy the condition that $B{\otimes }_{A}{B}^{\vee }$ is a direct sum of simple invertible bimodules by simply being equal to the identity bimodule. This are the so-called group-like defects.

If it also holds more generally, I don’t know yet. If it really implies a categorified spectral decomposition as in Hecke operator theory, I don’t know yet.

Posted at May 29, 2006 1:00 PM UTC

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Read the post 2-Spectral Theory, Part II
Weblog: The String Coffee Table
Excerpt: Complete 2-bases of 2-eigenvectors (with an eye on Hecke eigensheaves and duality defects).
Tracked: May 30, 2006 9:59 PM

### Re: 2-Spectral Theory, Part I

I wonder how one should think of projective 2-transformations acting on projective 2-spaces. What would a 1-dimensional sub-2-vector space of a 2-vector space be?

Posted by: David Corfield on June 5, 2006 8:51 AM | Permalink | Reply to this

### Re: 2-Spectral Theory, Part I

What would a 1-dimensional sub-2-vector space of a 2-vector space be?

By assumption we have direct sums in our 2-vector space. So there is an obvious 2-notion of generating systems and of bases for 2-vector spaces.

Let’s look at the canonical example, Kapranov-Voevodsky.

Consider the KV 2-vector space ${\mathrm{Vect}}^{2}$. Its 2-vectors are ordered pairs of two ordinary vector spaces. Its morphisms are ordered pairs of linear maps between these, componentwise.

A basis for this 2-vector space is clearly given by the pair of objects

(1)$\left\{\left(\begin{array}{c}ℂ\\ 0\end{array}\right),\left(\begin{array}{c}0\\ ℂ\end{array}\right)\right\}\phantom{\rule{thinmathspace}{0ex}},$

in the sense that any 2-vector in this space is (non-canonically) isomorphic to a direct sum of tensor products of these two basis 2-vectors with vector spaces, i.e.

(2)$\left(\begin{array}{c}{V}_{1}\\ {V}_{2}\end{array}\right)\simeq \left(\begin{array}{c}ℂ\\ 0\end{array}\right)\otimes {V}_{1}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\oplus \phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\left(\begin{array}{c}0\\ ℂ\end{array}\right)\otimes {V}_{2}\phantom{\rule{thinmathspace}{0ex}}.$

So ${\mathrm{Vect}}^{2}$ is 2-dimensional, in this sense.

In a completely similar manner one would define 1-dimensional 2-vector subspaces.

I think a similar reasoning applies to more general 2-vector categories that are module categories for some suitable monoidal category $C$.

Though, certainly, for sufficiently involved examples the above simple identification of dimensions may possibly have to be replaced by something more refined. I don’t know.

Posted by: urs on June 5, 2006 6:30 PM | Permalink | Reply to this

### Re: 2-Spectral Theory, Part I

I guess forming the set, or category, of such one-dimensional sub-2-vector spaces, you see the effect of Vect being only a categorification of N, not ℂ.

Posted by: David Corfield on June 6, 2006 9:23 AM | Permalink | Reply to this

### Re: 2-Spectral Theory, Part I

you see the effect of Vect being only a categorification of $ℕ$, not $ℂ$.

Absolutely. This is something that changes for more sophisticated setups.

For instance, as I tried to indicate, when we pass from ${\mathrm{Vect}}^{n}$ to ${\mathrm{Vect}}^{X}$, for $X$ some topological space, and furthermore pass to the derived categories of these “infinite dimensional 2-vector spaces”, then there are apparently many more invertible linear 2-maps, since then there is a categorification in particular of things like Fourier transformations.

As I have tried to indicate here and there, passing to derived categories should be related to introducing additive inverses (passing from a categorification of $ℕ$ to something like that of $ℤ$).

The shift operator on derived categories (which shifts the degrees on all complexes by one) is then the analogue of multiplying with $-1$.

But it’s actually more that that, because these complexes are $ℤ$-graded instead of just ${ℤ}_{2}$-graded.

Posted by: urs on June 6, 2006 11:21 AM | Permalink | Reply to this

### Re: 2-Spectral Theory, Part I

Is it already the case that passage to the derived category of Vectn gives you many more invertible linear 2-maps, just as SL(n, Z) is infinite?

I wonder if there’s a simplish answer as to why SL(2, Z) appears in that paper of Shigeru Mukai you mention, p. 11.

Posted by: David Corfield on June 6, 2006 9:30 PM | Permalink | Reply to this

### Re: 2-Spectral Theory, Part I

Is it already the case that passage to the derived category of ${\mathrm{Vect}}^{n}$ gives you many more invertible linear 2-maps, just as $\mathrm{SL}\left(n,ℤ\right)$ is infinite?

Good question.

I wonder if there’s a simplish answer as to why $\mathrm{SL}\left(2,ℤ\right)$ appears in that paper of Shigeru Mukai you mention, p. 11.

That’s an interesting point.

Posted by: urs on June 7, 2006 10:00 AM | Permalink | Reply to this

### Re: 2-Spectral Theory, Part I

So, what’s the most straightforward flavour of 2-vector spaces such that each 2-vector space will, in general, have many different orthonormal bases? Is it this ${\mathrm{Vect}}^{X}$ example that Urs has mentioned?

Posted by: Jamie Vicary on August 13, 2007 1:43 PM | Permalink | Reply to this

### Re: 2-Spectral Theory, Part I

So, what’s the most straightforward flavour of 2-vector spaces such that each 2-vector space will, in general, have many different orthonormal bases? Is it this Vect X example that Urs has mentioned?

That’s a goiod question. I think the answer depends a little on how precisely you think of a basis, because on that it may depend what counts as a “2-basis” for a 2-vector space.

For many appolications, people are looking at semisimple abelian categories when they are thinking of 2-vector spaces.

Semisimple means that each object in the category is (isomorphic to) the direct sum of (finitely many) simple objects.

In that situation, we’d be entitled to address any collection of representatives of isomorphism classes of simple objects, one for each such isomorphism class, as a 2-basis.

But notice that this is a rather special case, as becomes apparent from the fact that with this notion of 2-basis, we generate the entire 2-vector space from the basis “over $ℕ$” (since we can form (only) all integer multiples of a given basis 2-vector).

There are other ways to look as bases, though. For instance it might be useful to regard an ordinary basis of an ordinary vector space $V$ as an isomorphism from $V$ to the vector space $\mathrm{Hom}\left(S,K\right)\phantom{\rule{thinmathspace}{0ex}},$ where $S$ is any (finite) set, $K$ is the ground field and the Hom is taken in the category of sets.

So, a 3-dimensional $K$-vector space is isomorphic to maps from the 3-element set to $K$. That’s all I mean here.

This has a nice generalization to 2-vector spaces:

replace the monoid $K$ by an abelian monoidal category $C$, and then consider a $C$-module category to be equipped with a basis if it comes with a chose equivalence to the $C$-module category $\mathrm{Hom}\left(\Sigma ,C\right)\phantom{\rule{thinmathspace}{0ex}},$ where now $\Sigma$ is some category.

For instance, if $C=\mathrm{Vect}$, if we take everything in sight to be $C$-enriched and if we take, for simplicity, $\Sigma$ to have just a single object, then a morphism $\Sigma \to C$ is nothing but a module for the algebra given by $\Sigma$. Hence we find here that we obtain the 2-vector space $\mathrm{Hom}\left(\Sigma ,C\right)\simeq {\mathrm{Mod}}_{A}\phantom{\rule{thinmathspace}{0ex}},$ where $A$ is the algebra given by the Hom-space of the one-object Vect-enriched category $\Sigma$.

This is good, because there are a couple of crucial applications where it is precisely categories of modules which we want to address as 2-vector spaces.

See for instance our recent discussion over at the $n$-Category Café, titled Quastion about Modules.

We hence have the nice result that all those $C$-module categories (2-vector spaces over $C$) which happen to be equivalent to categories of modules of algebras (or algebroids, more generally) internal to $C$ are precisely those 2-vector spaces over $C$ which admit a basis.

Posted by: Urs Schreiber on August 13, 2007 2:03 PM | Permalink | Reply to this

### Re: 2-Spectral Theory, Part I

Thanks Urs! I was worried nobody would spot my comment.

That’s a very elegant way to think about bases. So, objects in our $C$-module categories will have a unique basis (up to permutation) if $C$ has only 1 simple object, I can see that now. So this is how we add structure, by adding more simple objects, and defining an interesting tensor product algebra between them… is this a good way to think about it?

It’s a shame that $C=\mathrm{Vect}$ is so boring in this regard, I suppose, given Vect’s canonical status.

Posted by: Jamie Vicary on August 13, 2007 6:09 PM | Permalink | Reply to this

### Re: 2-Spectral Theory, Part I

So, objects in our $C$-module categories will have a unique basis (up to permutation) if $C$ has only 1 simple object,

Hold on, I am not sure I am following you with this statement.

Let’s make it concrete: for $C:=\mathrm{Vect}$ there is, up to isomorphism, a single simple object, namely the ground field itself.

An object in a $C$-module category though is a category with a $\mathrm{Vect}$-action on it (but here the terminology is slippery, with modules appearing in two different categorical dimensions. Probably here is where I am misunderstanding what you just said).

An example for a category with a $\mathrm{Vect}$-module structure is a category of ordinary modules for some algebra. (Because we may tensor any such module with a vector space and get again a module.)

But it is also true that it is interesting to replace $\mathrm{Vect}$ with some tensor category that has more than one simple object.

Oh, I have to run… more later.

Posted by: Urs Schreiber on August 13, 2007 6:55 PM | Permalink | Reply to this

### Re: 2-Spectral Theory, Part I

What I meant to ask is whether the only invertible linear endofunctors ${\mathrm{Vect}}^{\Sigma }\to {\mathrm{Vect}}^{\Sigma }$ arise from invertible endofunctors $\Sigma \to \Sigma$ — which means that the only basis choices are those obtained by `permuting’ the underlying basis category.

So I was getting muddled up before, sorry. I meant to say that our $C$-module categories themselves have a unique basis up to permutation if $C$ has a single iso class of simple object, not that objects in $C$-module categories do.

Posted by: Jamie Vicary on August 13, 2007 11:27 PM | Permalink | Reply to this

### Re: 2-Spectral Theory, Part I

What I meant to ask is whether the only invertible linear endofunctors ${\mathrm{Vect}}^{\Sigma }\to {\mathrm{Vect}}^{\Sigma }$ arise from invertible endofunctors $\Sigma \to \Sigma$

Okay, I see. Looks like a very good question. I don’t realkly know the answer.

In the case that $\Sigma$ is a group, you are asking about the automorphisms of $\mathrm{Rep}\left(G\right)$. There are those coming from automorphisms of $G$ itself.

There are also “uninteresting” ones, coming from tensoring with 1-dimensional vector spaces.

Then, the autoequivalences of the representation category come from auto-Moritaequivalences of the thing being represented.

Not sure if that is really relevant for mere groups. It certainly becomes relevant for algebras and groupoids, for instance.

That’s all I can say right now concerning this question. I wish I had a more complete answer.

Posted by: Urs Schreiber on August 15, 2007 2:56 PM | Permalink | Reply to this

### Re: 2-Spectral Theory, Part I

OK — I’m glad you agree with me! I’ll think about what you said about Morita equivalence.

What prompted me to post here was the fact that the 2D vector space $ℂ\oplus ℂ$ has many linear invertible endomorphisms (i.e., $\mathrm{GL}\left(2,ℂ\right)$ is very interesting.) Its linear automorphisms certainly don’t just arise from automorphisms from the underlying 2-element set. But the 2-categorical situation is different: the only linear automorphisms on the 2-vector space $\mathrm{Vect}\oplus \mathrm{Vect}$ are those arising from automorphisms of the 2-object discrete category.

This is what I mean by the fact that $ℂ\oplus ℂ$ has many inequivalent bases — that is, bases linked by nontrivial unitary transformations — but its ‘obvious’ categorification, $\mathrm{Vect}\oplus \mathrm{Vect}$, lacks this familiar and attractive property.

So, my question is: what is the simplest example of a 2-vector space which does have many inequivalent bases? Presumably we need this sort of behaviour to do categorified Fourier transforms, and so on — but is it possible to get something like this without all the machinery of derived categories?

Posted by: Jamie Vicary on August 15, 2007 3:46 PM | Permalink | Reply to this

### Re: 2-Spectral Theory, Part I

But maybe Morita equivalence is pointing in the right direction:

So, if we agree on conceiving a choice of equivalence $V\simeq {\mathrm{Mod}}_{A}$ of a 2-vector space $V$ with a category of modules for some algebra $A$ as a choice of “2-basis” of $V$, then we should in fact regard all algebras Morita equivalent to $A$ as different choices of basis of the same 2-vector space.

A simple but interesting and actually useful example is for instance the Morita equivalence $ℂ\simeq {M}_{n}\left(ℂ\right)$ for all $n\in ℕ$.

(The Morita equivalence is induced by the obvious $ℂ-{M}_{n}\left(ℂ\right)$-module structure on ${ℂ}^{n}$).

In fact, if done correctly we can let $n=\infty$, too, and arrive at the algebra of finite rank operators on some Hilbert space., being Morita equivalent to just the algebra of complex numbers itself.

All these algebras may be regarded as different bases of the one-dimensional 2-vector space ${\mathrm{Vect}}_{ℂ}$.

This is important in the study of rank-1 2-vector bundles:

their fibers are really categories equivalent to $\mathrm{Vect}$, but people like to think of them as $PU\left(H\right)$-bundles. How is that possible? The reason is that $PU\left(H\right)$ is the autmorphism group of finite rank operators on $H$, so looking at an abelian gerbe as a $PU\left(H\right)$ bundle essentially means looking at a rank-1 2-vector bundle by using a particular class of 2-bases.

Posted by: Urs Schreiber on August 15, 2007 4:19 PM | Permalink | Reply to this
Read the post Crane-Sheppeard on 2-Reps
Weblog: The String Coffee Table
Excerpt: On linear 2-reps as discussed by Crane and Sheppeard.
Tracked: June 9, 2006 2:27 PM
Read the post Kapustin on SYM, Mirror Symmetry and Langlands, III
Weblog: The String Coffee Table
Excerpt: The third part of the lecture.
Tracked: June 17, 2006 11:29 AM
Read the post Gukov on Surface Operators in Gauge Theory and Categorification
Weblog: The String Coffee Table
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Tracked: June 27, 2006 10:42 AM
Read the post Topology in Trondheim and Kro, Baas & Bökstedt on 2-Vector Bundles
Weblog: The n-Category Café
Excerpt: Baas, Dundas and Kro with new insights into 2-vector bundles.
Tracked: October 18, 2006 5:23 PM
Read the post Seminar on 2-Vector Bundles and Elliptic Cohomology, V
Weblog: The String Coffee Table
Excerpt: Part V of a seminar on elliptic cohomology and 2-vector bundles. Review of relations between elliptic cohomology and strings.
Tracked: May 9, 2007 10:44 PM

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