The String Coffee Table

Sorry - I meant terminal object for the one point space.

Regarding terminology: it may not be standard, but the Aussie Cat theorists say

1-stack = sheaf

2-stack = stack a la Grothendieck

and so on, because one ought to standardise higher categorical indexing.

For a topological space $X$, the open sets form a lattice with $0 =$ the empty set and $1 = X$. A locale is just a generalisation of such a lattice, ie. one which is complete and distributive. Hence a point for a locale is a morphism

Frames are dual (opposite category) to locales. That’s why one needs initial objects now in order to define points.

From Mac Lane and Moerdijk - the definition of sober is that, for any proper open subset $U$ of the lattice $X$ satisfying

there is a unique point $x$ in the space such that $U = X - \overline{x}$ (where bar means closure).

If a space is sober then there is a homeomorphism between the space and the points of a locale.

Elsewhere (on PF) I mentioned Hilbert space lattices, which are no longer distributive because set theoretic union is replaced by the appropriate subspace construction.

Here I’m afraid I must disagree with John Baez on what is the way to approach NCG. Lattice theory tells us that even 1-Hilbert spaces are no longer 1-categorical. Hence my interest in 2-toposes (and higher).

Alvarez is at Riverside, no? See http://arxiv.org/PS_cache/math/pdf/0402/0402150.pdf He appears to be an expert on the Gelfand-Naimark side of things.

Maybe this helps:

http://mathworld.wolfram.com/NoncommutativeTopology.html

I’ll get around to saying more when I get time!

Heh! We are mostly talking about the same stuff after all!

…and morphisms INTO the initial object are ‘dual’ to morphisms OUT of a terminal object.

Objection to the Gelfand-Naimark idea:

Firstly, I was hoping we were (at least partly) talking about what NCG ‘should’ be, as opposed to what Connes et al call NCG.

The way I came to study lattice theory in the first place was in trying to understand the tetracategorical structure behind premonoidal higher categories. For simple physical reasons (see the quark confinement reference cited above) we require the pentagon condition to be broken. This occurs very naturally at the tetracategorical level of trimorphisms between tricategories - such as the well studied tricategories of representation theory. This is very nice, because of course we would like something special to happen in 4D.

Anyway. Things like the Swan theorem (but I’m far from an expert on NCG) can be discussed in a purely topos theoretic framework. For a ring $R$ one finds a space $X$ such that $R$ is recovered from the global sections of a sheaf of rings. That is, one studies a ring in the topos Sh($X$).

A finite dimensional vector space over the (two-sided Dedekind) reals in Sh($X$) is the same thing as a finite dimensional vector bundle on $X$. The Swan theorem says that for compact Hausdorff $X$ the category of vector bundles is equivalent to….whatever.

The point is, proofs are ‘intuitionistic’ - ie. they use topos logic. As much as one tries to avoid formulating everything in alternative logics - one seems to come crashing back to the inevitability of it!

Another example: in

Duality for Bounded Lattices G. Allwein C. Hartonas citeseer.ist.psu.edu/allwein93duality.html

the authors show that in order to describe the lattice duality including non-distributive lattices (such as subspaces of a Hilbert space) one MUST drop the axiom of choice. Oh no! No escape! (At some point I’ll drag some interesting PF posts across here, I guess)

I’m enjoying talking to you.

Let’s see. Perhaps you could explain a bit more about your spectral triple idea. I guess I should make more of an effort to understand it and not jump to the conclusion that it’s less fundamental than other stuff.

To recap: Hilbert spaces are not fundamental. The complex numbers are not fundamental. Even if we want $\mathbb{N}$, such as in the topos $\mathbf{Set}$, we need an axiom of infinity. Then we can define $\mathbb{Z}$ fairly easily, and then $\mathbb{Q}$ but by the time we get to the reals things start to get complicated.

The question I have for you is: if we could axiomatise (category theoretically) ALL the features of Hilbert spaces that are essential to physics, why do we need Hilbert spaces? They are but a model for the axioms. Moreover, if we want to do quantum gravity (or NCG) then the axioms of conventional QM will be modified, and it is no longer clear that Hilbert spaces have anything to do with it.

Let’s start with sets and subsets. There are two operations on subsets. $\wedge$ is intersection and $\vee$ is union. $\wedge$ is distributive over $\vee$. For any topos lattices are distributive.

Now consider linear subspaces. Set theoretic union $U \vee V$ must be replaced by the smallest subspace containing $U$ and $V$. Check for yourself that this breaks distributivity.

“Toposes, Triples and Theories” Michael Barr and Charles Wells, link

“Elementary Categories, Elementary Toposes” C. McLarty, Oxford Science (1992)

“Topoi” R. Goldblatt, North-Holland (1984)

Urs: In NCG one regards spaces in terms of properties of algebras. You seem to be saying that this is the same as regarding the space not in Set but in some other topos. Is that right?

No. Ordinary toposes are not good enough to do NCG. But NCG appears to be very ‘topos theoretic’. Some people study quantales, where one allows $\wedge$ to be non-commutative. I prefer the work of Street, Stubbe, Walters et al. (which will take me a long time to understand!) on alternative characterisations of $\mathbf{Sh}_{J}(\mathbf{C})$ (which is sheaves on a site with respect to a Grothendieck topology) because this is more category theoretic and will help us understand tetracats.

About breaking pentagons: yes, one can do this in 3D if one means the tricategorical (see Gordon, Power and Street) TD7 (a modification) which fills in the cube, five sides of which are the Mac Lane pentagon. But if, in addition, one wants freedom on the sixth face, then one is moving up to tetracategories.

By the way - there aren’t really any references for this. I’m just an old grad student with lots of crazy ideas that I’m being strongly discouraged from discussing!

EXCERPT FROM PF:

The objects of a tricategory T are labelled $p$, $q$ etc. For each pair of objects $p$ and $q$ there is a bicategory $T(p,q)$. The internalisation of weak associativity is a pseudonatural transformation $a$. The parity cube is labelled by bracketed words on four objects, which may be loosely thought of as the bicategories of ‘particles’ a la Heisenberg.

The Mac Lane pentagon lives on 5 sides of the cube. The new top face of the cube is the premonoidality deformation (of Joyce), which closes the horn. This square appears as a piece of data for a trimorphism, that is a map between tricategories.

The tetracategorical breaking of hexagons amounts to the breaking of topological invariance as described by Pachner moves for 4D spin foams. This view thus sort of explains why four dimensional gravity is not topological in the usual sense of the word.

Some useful REFERENCES on Stacks:

A. Grothendieck “Pursuing Stacks” available at http://www.math.jussieu.fr/~leila/mathtexts.php

Ross Street “Categorical and Combinatorial Aspects of Descent Theory” available at www.maths.mq.edu.au/~street/DescFlds.pdf

Oh, that’s interesting. I see that Baas is right into categories (I’ll meet him in July in Australia I guess) and Stacey is a young guy.

I wouldn’t be surprised if there were a LOT more people out there thinking along these lines.

Above I gave some hints on what I consider categorified NCG should be. I have thought more about it since then and there seems to be a nice illuminating example, being the categorification of the discrete NCG that I had discussed with Eric Forgy last year.

More thinking is required, but I would like to sketch some basic ideas:

Recall that the ordinary Gelfand-Naimark theorem says that every commutative $H^*$-algebra $H$ (usually this is formulated in terms of $\mathbb{C}^*$-algebras, but I follows HDA2 here) is isomorphic to a commutative $H^*$-algebra of functions from a set $\mathrm{Spec}(H)$ to $\mathbb{C}$.

This gets categorified by first sending

and then discussing how to reconstruct a groupoid from the 2-$H^*$-algebra of functors into $\mathbf{Hilb}$ from it.

I don’t feel like spelling all the details out right now, please see HDA2.

Actually, there only the special case where the underlying discrete 2-space is a groupoid is discussed. Without much ado I’ll assume in the following that one does not run into major difficulties by assuming that the entire theory can be generalized to base 2-spaces which are not necessarily groupoids. In these cases a functor into $\mathbf{Hilb}$ will assign finite Hilbert spaces $\mathbb{C}^N$ to objects and (not necessarily unitary) linear operators $\mathbb{C}^N \to \mathbb{C}^{N^\prime}$ to morphisms.

To fill the formalism a little more with life I’ll adopt the following point of view:

Given a base space $\mathcal{M}$ which is a groupoid, the category $\mathrm{Rep}(\mathcal{M})$ of functors

can be regarded as the category of discrete $U(N)$-connections on $\mathcal{M}$ for arbitrary $N$.

Each such functor assigns $\mathbf{C}^N$ to each vertex and an element of $U(N)$ to each morphism. That’s a connection in $U(N)$.

I will eventually think of the 2-space $\mathcal{M}$ as a configuration space of string. The string states which assign elements of $U(N)$ to each piece of string are precisely the boundary states for $N$ coinciding D9-branes with the given gauge field turned on (see the references given in hep-th/0408161).

Hence I’ll think of the $\mathbb{C}^N$ assigned to each vertex as inidciating the number of D-branes at that point and of the linear operators assigned to the edges as describing how strings stretch between the D-branes at two shuch vertices.

So, in this language, the functor

for the general non-unitary case assigns a number $N$ of D(-1)-branes to each vertex and a $n\times m$ matrix to each edge

describing how strings stretch between these stacks of D-instantons.

Anyway, this is just a picture I offer which I think makes good sense to be worked out in more detail. But if you don’t buy into this D-brane picture just consider it fancy language.

Now the the 2-NCG. The very point of $\mathrm{Rep}(\mathcal{M})$ is that it is a 2-$H^*$-category, which in particular means that its objects (functors $\mathcal{M} \to \mathbf{Hilb}$) form an algebra and can hence be multiplied and added and indeed form a categorified Hilbert space.

Clearly, this serves as the categorification of the bosonic function algbra that enters a commutative spectral triple.

Such an algebra is turned into a graded algebra with a graded nilpotent operator $\mathbf{d}$ acting on it by forming its universal abstract differential calculus obtained by generating the universal algebra generated by the elements $a_o (\mathbf{d}a_1)\dots(\mathbf{d}a_p)$ under a product which makes $\mathbf{d}$ nilpotent and graded Leibnitz and restricts on $A$ to the original algebra product. By throwing out certain elements consistently one gets any abstract differential calculus from this universal one.

So in particular consider the algebra of the point map of the functors $\mathcal{M} \to \mathbf{Hilb}$. They are essentially maps from vertices into the integers. Let $e_i$ be the function which assigns 1 to the $i$-th vertex and 0 to every other. Discrete differential 1-forms over this algebra are linear combinations of

This also encodes the presence of an edge $(i \to j)\in \mathcal{M}$.

We can do a similar thing to the algebra of functors which restricts to this operation under the source and target map. Let $E_{ij}$ be the functor which assigns $1: \mathbb{C}^1 \to \mathbb{C}^1$ to $(i \to j)\in \mathcal{M}$ and $\mathbb{C}^0$ to every other vertex and the trivial map to every other vertex.

Then $E_{ij}(\mathbf{d}E_{kl})$ denotes a discrete 1-form on discrete path space which encodes a surface element stretching between $i \to j$ and $k \to l$.

Let us now notationally identify this 1-form with the surface element that it is non-vanishing on. Similarly let us identify the the functor $E_{ij}$ notationally with $i \to j$ in the following. Then just like passing to the differential calculus on functions enlarged the set $S$ of points to the set of elementary discrete differential forms, the above process enlarges the category of the base 2-space to that of elementar discrete edge differential forms.

In order to make elementary discrete edge forms into functors we make the following natural definitions:

We declare the source of this edge 1-form to be $e_i (\mathbf{d}e_k)$ and the target to be $e_j (\mathbf{d}e_l)$.

Furthermore we define composition of such elementary discrete edge forms by functoriality of $\mathbf{d}$:

One can draw some pretty pictures illustrating this, but I won’t do that right now.

In fact, I am getting a little tired. More detailed discussion can be given, showing how this construction allows to construct a graded 2$H^*$-algebra $H$ of discrete path space differential forms, which would give rise to a 2-spectral triple $(A,H,\mathbf{d})$.

More later.

I really appreciate you explaining this, but as far as I can tell my objections all stand. A quote from the conclusions of HDAII: “While the general study of n-Hilbert spaces will require a deeper understanding of n-category theory…”

Exactly! On page 2 it is pointed out that Hilb is a ring category. These structures were studied in detail in the thesis and related papers of Joyce from 2000 onwards. Unfortunately I don’t even have journal references to give you, except for

“The Racah-Wigner category” W.P. Joyce, P.H. Butler, H.J. Ross; Can. J. Phys. 80 (2002) 613-632

Anyway, quoting HDAII: “so we expect it to play a role in 2Hilb theory similar to $\mathbb{C}$ in Hilbert space theory”

This is a simplified idea of categorification, a fact of which I am sure JB is well aware.

The sort of categories being discussed here are used by computer scientists, following the old work of Barr (LNM 752) on *-autonomous categories (symmetric monoidal closed with a duality structure). See for instance the recent seminal work on quantum computation:

“A categorical semantics of quantum protocols” Samson Abramsky, Bob Coecke; http://arxiv.org/abs/quant-ph/0402130

But, I’m digressing. On pages 52+ of HDAII there is a discussion of Categorified Fourier Transforms and Pontrjagin duality. I mentioned this above. Pontrjagin duality is secretly an example of Stone duality in a lattice theoretic context. This has to be more fundamental than the HDAII description because it recognises the importance of decribing the logic.

It seems to me that you just want to rewrite String theory in its natural language. I don’t see that there’s much physics in String theory. Now, I have a confession to make. I don’t pretend to understand it very well yet, but the reason I am studying tetracategories and lattices and so on is that I am quite convinced that one of the fundamental principles of Quantum Gravity (it makes no sense to talk about unification a la Strings because gauge theories aren’t at all fundamental) is something I call QGC. That is, “quantum general covariance”. General relativity must be respected. Strings don’t do this. QGC asks “what would spacetime-matter duality look like in a background independent quantum informational theory?” It seems there is an answer. There is one big thing on which Strings and Spin Foams agree: quantum gravity can interchange small and large scales. How does one represent this mathematically whilst respecting GR? I’ll let you think about this for a while, and then maybe I’ll continue trying to convince you that this pure category theoretic approach is absolutely essential.

In the end, everything is cohomological in the sense of Street et al descent theory, sometimes known as topos cohomology because, to begin with, one uses omega-cat objects in a topos.

By the way, a topos is really a generalised idea of a (commutative) space (this is the way category theorists view it, not just me). The categorical structure based on quantum logics should therefore describe NCG.

I probably wouldn’t bring it up here if it wasn’t for the fact that recent M-theory papers seem quite excited about the idea of a ‘purely cohomological’ M-theory (Sati? for instance).

So my question to you is: where is this spectral triple stuff heading PHYSICALLY?

Urs: But if you propose another notion of 2-Hilbert space which you find better suited I will be interested in being taught about it!

OK. Let’s give it a go. I’m guessing this conversation is going to go on for a while, and I apologise if sometimes I am busy doing other things such as things that I am supposed to be doing right now.

First, I guess I need to convince you that the important thing about Hilbert spaces is their lattice structure. The best reference for this, recently pointed out to me by Frank Valckenborgh, is the book

“Orthomodular Lattices” by G. Kalmbach (Academic 1983)

which begins with an historical introduction into some of von Neumann’s work.

A lattice is a poset with 0 and 1 and binary operations $\wedge$ and $\vee$. Traditionally, quantum logic is characterised by lattices that are also equipped with an orthocomplement which satisfies

(i) $U$ in $V$ implies $\neg V$ in $\neg U$

(ii) $\neg \neg U = U$

(iii) $U \vee \neg U = 1$

(iv) $U \wedge \neg U = 0$

These rules force Booleanness in topos logic. Orthomodular logic on the other hand replaces distributivity by the law

An orthomodular lattice is characterised by the rule “if $U$ in $V$ and $V \wedge \neg U = 0$ then $U = V$”.

Orthomodular lattices characterise Hilbert spaces. Inclusion of subspaces acts as an order relation, which we think of as a directed arrow. An orthogonal subspace $\neg U$ of a subspace $U$ of a Hilbert space $H$ satisfies the complement rules. Intersection represents $\wedge$, and $U \vee W$ is the smallest subspace of $H$ containing the union of $U$ and $W$. These operations are associative and commutative. The distinction between $\vee$ and set theoretic union breaks distributivity (consider three orthogonal subspaces).

It turns out that a lattice is orthomodular but not distributive iff it does contain sublattices that look like diamonds and pentagons, but does not contain a hexagonal sublattice made of distinct $\{ 0,1,U,V, \neg U , \neg V \}$.

Sorry, must go.

A two minute answer? OK. But I find your terminology difficult. When you say ‘categorify’ you’re talking about something that I don’t accept is essential. Anyway, thinking about the physics first:

Heisenberg said that in some sense every particle contains every other particle - hence particles are not fundamental. What is fundamental in quantum physics? Let us say it is a calculus of propositions.

A proposition $\phi$ is a statement that may be interpreted with respect to given variables to have truth values, which in the topos $\mathbf{Set}$ are the values true and false. Given a set $S$, for every proposition $\phi$ with restricted quantifiers (see McLarty’s book, or Mac Lane and Moerdijk) there exists a set $X$ with $x \in X$ if and only if $\phi (x)$ is true and $x \in S$. This is the comprehension scheme of restricted ZF set theory.

A long time ago Gray (see LNM 391 and references therein) considered the categorical analogue of this problem and was led to the development of 2-categories, in particular to the tensor product of 2-categories which has the novel feature of being dimension raising. Until one has a need for, say, tricategorical structures this dimensional instability may be completely ignored, and usually is in applications of higher categories to quantum gravity. But here the Gray tensor product is seen to have physical significance. The tensor product of more and more, let us call them, ‘fundamental systems’ requires higher and higher categorical dimension. This is why particle number should not be conserved in QFT.

You were wondering about $\mathbf{Cat}$. Which one? Normally people work in a series of Grothendieck universes. This is a way of getting around Russell type paradoxes and doing everything in set theoretic mathematics. Fix a set $U$. A ‘small’ category is one where the Hom sets are in the universe $U$. One can talk about the category of small categories. But this doesn’t include Top, for instance. So, one can talk about the category of large categories.

I’m an advocate of axiomatising things, because both physics and topos theory says to do this. That means we can do away with Grothendieck universes. Set is easy enough to axiomatise. Categories of categories are harder. McLarty has done some work on this, inspired by Lawvere and Benabou and others. No doubt there is a lot more out there than I am aware of.

This context suggests the following. Think of $\mathbf{Vect_{\mathbb{C}}}$ as a kind of ‘linear topos’ for now. The number field replaces the terminal object in $\mathbf{Set}$. Axiomatise this like a topos. Scalars, for instance, become arrows. Recall that monic arrows represent subobjects. The ‘lattices’ are the sort we want.

If you really want the 2-categorical analogue…..computer scientists have studied bilinear logic (ie. non-commutative linear logic). According to the above paragraph, this should correspond to something like braided monoidal categories with biproducts. This connection has been studied by Blute (see “Hopf Algebras and Linear Logic” and refs therein) under the heading Biautonomous Categories. It is shown that $\mathbf{Mod}(H)$ for a non-cocommutative Hopf algebra $H$ is biautonomous. More interesting is a topologised version of this (we need topologies to do cohomology) and a suitable restriction of the topologised reps gives Yetter’s (J. Sym. Logic 55,1 (1990) 41-64) bi*autonomous categories if the Hopf algebra has an involutive antipode.

In summary,

Definition: a 2-Hilbert space is an object in a cyclic bi*autonomous category.

This isn’t the only possible way of looking at it of course, but I hope it gives you an idea of what happens if one tries to respect the logic.

What do we see in fig 4.10 in Yetter’s paper? Looks a bit like a template. Something else I’d like to talk about…but not right now!

Hi Urs,

I might be busy this week, moving house.

Other target categories:

Within the framework that you are working in, I cannot see how to improve on Hilb and 2Hilb. Considering the question again anyway…

What ‘categorifies’ $\mathbb{C}$?

Let’s say that what you want, first of all, is something that ‘categorifies’ complex-valued functions on compact Hausdorff spaces. There is a book that I really should have referenced above, which is P.T. Johnstone’s “Stone spaces” (Cambridge 1982). Fact: the category of compact Hausdorff spaces is equivalent to a category of locales.

If I might revert to type… Did you know that the paper “A globalization of the Gelfand duality theorem”, B. Banaschewski and C.J. Mulvey 21pp shows that Gelfand duality (using locales) extends to any Grothendieck topos?

Following this paper…

Summary of Definition: a commutative $C^{\ast}$ algebra in a topos $E$ is a commutative Banach *-algebra $A$ such that for a norm $N$, which is a map from the positive rationals in $E$ into $\Omega^{A}$, the following condition holds: for each positive rational $q$ and $a \in A$, $a \in N(q)$ is the same as $a a^{\ast} \in N(q^{2})$. As an example, one has the algebra of complex numbers ‘in the topos $E$’, given by $N(q)$ = $z \in \mathbb{C}$ s.t. norm $z$ less than $q$.

The great thing about this is that it puts the complex numbers where they belong, as an object in a topos.

Spaces that are really spectra for finitely generated commutative algebras appear in a really cool commutative triangle, which I don’t believe I can draw here (can we use xypic here?), so in words: there is a functor ‘taking the spectrum’ from $\mathbf{Alg}^{op}$ into a topos called the Big Zariski topos (I don’t know who called it that - maybe Lawvere). The third vertex is the classifying topos $\mathbf{Set}^{\mathbf{Alg}}$. There is a functor from the Zariski topos into the classifying topos has an adjoint (ref. Kan’s theorem). The third functor is the Yoneda embedding into the classifying topos. In other words, the spaces you are using are really arrows in $\mathbf{CAT}$.

Buying into your idea of categorifying NCG:

We already agree that this is the way to go - but what you mean by categorified NCG and what I mean by it are two different things. To me ‘categorification’ means doing everything in purely category theoretic terms right from the start, including worrying about where the real numbers might come from in the reduction of the full theory to classical GR.