October 19, 2006

Algebras as 2-Categories and its Effect on Algebraic Geometry

Posted by Urs Schreiber Here is another question from Bruce Bartlett which really deserves to be a post of its own (see also his last question of this type)… it’s about categories of algebras, algebras as categories and the possible implications for non-commutative algebraic geometry.

As adequate for a deep question, this one extends over three seperate emails.

Hi Urs,

Today at Sheffield we had a colloquium by Alexey Bondal (Steklov Mathematics Institute) on “Noncommutative deformations of algebraic varieties and Poisson brackets”.

In the beginning of the program he outlined the grand scheme (excuse the pun) of “noncommutative algebraic geometry”. Namely, (1) first set up ordinary commutative algebraic geometry in a nice categorical framework, and then (2) apply this to noncommutative algebras.

To do (1), it seems the best way is to set up ordinary algebraic geometry from a “functor of points” perspective. (At this point, I am going to become very vague and factually incorrect, since I am still learning the basics of algebraic geometry. But you could probably fix up mentally what I’m saying.)

I believe that in this perspective, you simply define the category of affine schemes to be the opposite of the category of commutative algebras. Then you put a Grothendieck topology on it - which allows you to glue things together, amongst other things - and then you play the game of sheaves and stacks over this “site”. In particlar, at some point one considers “prestacks” on this category which are simply weak 2-functors : $F : \mathrm{CommAlg}^\mathrm{op} \to \mathrm{Groupoids}$ Here we are considering $\mathrm{CommAlg}$ as a 2-category with trivial 2-morphisms. (By the way, I get this “stacky” stuff from notes from a “chromatic homotopy” seminar going on at Sheffield - go to “An introduction to stacks”).

Now it seems easy to perform step (2) : simply cross out the word “commutative” in the above paragraph. So the category of “noncommutative schemes” is the opposite of the category $\mathrm{Alg}$.

The trouble is, as he explained, is that its quite difficult to work with this - there are all sorts of technical problems that arise, and one is forced to use a kind of work-around by using derived categories of coherent sheaves, etc.

Anyway, here is what made me think:

Simon [Willerton, -urs] taught me a month ago that the category of algebras is really a 2-category. An $n$-category cafe regular such as yourself will agree that an algebra is best defined as a kind of one object category. Thus morphisms are functors, and 2-morphisms… are natural transformations. What are these? Well work it out : if $A$ and $B$ are algebras, and $f$ and $g$ are homomorphisms $A \to B$, then a 2-morphism $\phi : f \Rightarrow g$ is an element $b$ of $B$ such that $b f(a) = g(a) b$ for all $a \in A$.

Okay… so if Alg is really a 2-category, then how does this affect algebraic geometry? Well, one thing all kindergarten algebraic geometers (like me) know is that the category of integral domains is equivalent to the opposite of the category of affine varieties… give or take some technicalities.

How does the notion “Alg is really a 2-category” affect this? Well, suppose $A$ and $B$ are integral domains, with $f,g : A \to B$. What is a 2-morphism? Well we look at the formula above, and use the fact that the algebra is commutative to get : $b[ f(a) - g(a) ] = 0.$ Now we use the fact that it is an integral domain, to conclude that $f(a) - g(a) = 0$… that is, $f$ and $g$ must be the same map, and there is no condition on $b$ at all! In other words, the 2-morphisms are trivial precisely for integral domains - there is no information in them at all!

Thus we elegantly arrive at the fact that the theory of “ordinary” affine varieties does not need to care about the 2-morphisms… they are trivial… precisely when the algebras are integral domains! It is reassuring to see how the notion “integral domain” kind of “emerges” from a 2-category point of view.

Here is my point : when we change our attention to non-integral domains, or more seriously, noncommutative algebras, then the 2-morphisms become important! We must take $\mathrm{Alg}$ seriously as a 2-category. This will change the way we set up sheaves, stacks, etc. This might fix up a lot of the fundamental problems that we are experiencing.

For example, a prestack on $\mathrm{Alg}^\mathrm{op}$ will now be a weak 2-functor: $F : \mathrm{Alg}^\mathrm{op} \to \mathrm{Groupoids}$ but where we are taking seriously the 2-morphisms inside $\mathrm{Alg}^\mathrm{op}$ - we haven’t just set them to be trivial. Thus this definition of a stack is a bit stricter and also richer.

What do you think?

Regards Bruce

P.S. To me, a lot of the problems in going from commutative to noncommutative geometry seems to be linked to the following. To an “old-fashioned” algebraist, there is no difference in principle between a noncommutative algebra and a commutative one - the latter just obeys an extra equation.

To an $n$-category café patron, there is a world of difference : a (non)commutative algebra is a 1-object category, while a commutative algebra is a 1-object, 1-morphism 2-category… and this affects our viewpoint quite a lot.

$\;$

Hi Urs,

In my last mail I wrote a P.S…. which made me rethink a lot of what I said.

Namely, a commutative algebra is really a one-object, one-morphism 2-category… which means that $\mathrm{CommAlg}$ is really a 3-category : objects are commutative algebras, morphisms are weak 2-functors, 2-morphisms are transformations, and 3-morphisms are modifications.

Its a bit late… but I seem to have worked out that, give or take some stuff:

A 2-functor in this sense is just an ordinary algebra homomorphism. (I should check this… being a weak 2-functor might introduce extra data).

A transformation is just an ordinary natural transformation in the previous email sense; i.e. if $f, g : A \to B$ then a natural transformation from $f$ to $g$ is an element $b$ of $B$ such that $b [ f(a) - g(a)] = 0 \; \forall\; a \in A.$ If $b, b' : f \Rightarrow g$ is a transformation then a 3-morphism from $b$ to $b'$ is an element $c$ of $B$ such that $c [ b - b'] = 0.$ Anyhow… you get the picture. It is an interesting thing to think about - in the Baez/Dolan stabilization hypothesis, it is normally stated that , e.g. “1-object 1-morphism 2-categories are commutative algebras”. But we should check this for the arrows as well : do we introduce any interesting new data into the game?

For instance, it is supposed to stabilize, so that a “one object, one 1-morphism, one 2-morphism 3-category” is also just a commutative algebra. But then we can consider 4-morphisms… what will they turn out to be?

In other words -> do the morphisms also stabilize? I hope they do, but my calculation above seems to say otherwise, as it suggests that modifications bring new data into the game.

Perhaps its true that somehow, if we think of commutative algebras as “one object, one 1-morphism, one 2-morphism, one 3-morphism, …., one 9-morphism 10 categories” then this will introduce all sorts of extra morphisms inside $\mathrm{CommAlg}$ $\to$ it will become a 10-category! And the game can be continued…

What is going on here?

Regards, Bruce

$\;$

Hi Urs,

I’m not so sure anymore about integral domains forcing the 2-morphisms to be totally uninteresting… after all, having around $B$’s worth of 2-morphisms from $f \Rightarrow f$ for every $f: A\to B$ might be significant - what do they correspond to one the affine varieties side? I kind of worked that out though, but how to motivate it geometrically?

What seems to be somehow true though is that, if considering commutative algebras as a “one object, one 1-morphism, …, n-category” really does throw in new morphisms into the game (which I’m not certain of at all), then they seem to be all of the form $c [ a - b]$ so they’re kind-of-trivial precisely when you have an integral domain. Said more succinctly : $\mathrm{Alg}$ collapses to a 1-category precisely when you have commutative integral domains. Otherwise it seems to be some kind of infinity-category. (Gulp! Can that be?? Think I’m making a blunder here :-))

Regards, Bruce

Posted at October 19, 2006 12:15 PM UTC

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Re: Algebras as 2-Categories and its Effect on Algebraic Geometry

Hi Bruce,

I’ll comment on one aspect of your remarks.

One of your points is, paraphrased, that whenever we use an algebra or some other monoid in some context, we might want to ask ourselves:

“How would our perception of that context change, if we systematically used the fact that a monoid is really a certain kind of category? For instance: would we think differently about the natural notions of morphisms that we are dealing with?”

Specifically, the context of interest here is, vaguely speaking, the op-description of spaces in terms of their representations, and the notions of generalized spaces derived from that.

Here I say “representations” instead of “functions”, in order to conjure up the picture which we expect # to be the one that categorifies nicely.

Using this point of view as a guiding principle, it might be fruitful to consider the following aspect of your remark:

A complex function on a space $X$

(1)$f : X \to \mathbb{C}$

maps points of $X$ to elements of $\mathbb{C}$. Obviously.

But $\mathbb{C}$ is not just any old set. It is in fact a monoid. Following your remark, we want to be very sophisticated and instead think of $\Sigma(\mathbb{C})$: the category with a single object and $\mathbb{C}$-worth of morphisms, with composition rule of morphisms given by the product in $\mathbb{C}$.

From this point of view, $f$ is a actually a map to morphisms of a category:

(2)$f : X \to \mathrm{Mor}(\Sigma(C)) \,.$

A triviality.

On top of this triviality, I’ll now demand that we do with the domain what we did with the codomain and think of $f$ as a map

(3)$f : \mathrm{Obj}(\mathrm{Disc}(X)) \to \mathrm{Mor}(\Sigma(\mathbb{C})) \,,$

where $\mathrm{Disc}(X)$ denotes the discrete category on the set $X$.

This reformulation is utterly trivial and might seem void.

But your remark suggests that we take it serious and try to figure out where this leads us.

Okay, so what can a map from objects of one category to morphisms of another possibly be, if we demand everything to have nice category-theoretic interpretation?

Answer: it must be a natural transformation of functors.

That’s what it means to take a complex-valued function and be serious about the fact that complex numbers are really to be thought of as morphisms.

Now, I claim, the above trivialities that I went through lead to something a little more interesting:

there is not much of a choice concerning the functors that $f$ could be a natural transformation for.

For instance, let

(4)$\mathrm{tra}^0 : P_1(X) \to 1D\mathrm{Vect}_\mathbb{C}$

be a trivial line bundle with trivial connecton

(5)$\mathrm{tra}^0 : (x \stackrel{\gamma}{\to} y) \mapsto (V \stackrel{\mathrm{Id}}{\to} V ) \,.$

Then $f$ would be a morphism

(6)$f : \mathrm{tra}^0 \to \mathrm{tra}^0 \,.$

While still not really non-trivial, this is somewhat pleasing. Because this now has lead us automagically from

- replacing the monoid $\mathbb{C}$ by the category $\Sigma(\mathbb{C})$

to

- replacing the monoid $[X,\mathbb{C}]$ by the category $\simeq \Sigma([X,\mathbb{C}])$, whose single object we regard as $\mathrm{tra}^0$ and whose morphisms are functions $f$, regarded as natural transformations

(7)$f : \mathrm{tra}^0 \to \mathrm{tra}^0 \,.$

The product in the algebra of functions on $X$ is now nothing but composition of natural transformations

(8)$f g : \mathrm{tra}^0 \stackrel{f}{\to} \mathrm{tra}^0 \stackrel{g}{\to} \mathrm{tra}^0 \,.$

Notice that this picture nicely categorifies: from the categorified Gelfand-Naimark theorem # we know that the right categorification of the algebra of complex functions on spaces $X$ is the 2-algebra of Hermitean vector bundles with connection on $X$.

But the 2-algebra of vector bundles with connections is nothing but the monoid of endomorphisms of the trivial (bundle-)gerbe on $X$!

(There is a nice diagrammatic way to say this, using pseudonatural transformations of trivial 2-functors, but I’ll refrain from reproducing this right now.)

Now I would claim that this way of looking at things helps us decide which answers to the problems you mentioned are more natural than others.

But I have to end here for the moment.

Posted by: urs on October 20, 2006 2:14 PM | Permalink | Reply to this

Re: Algebras as 2-Categories and its Effect on Algebraic Geometry

I wrote:

But I have to end here for the moment.

Here is a little more:

One general remark concerning your discussion is that quite frequently, morphisms of algebras are best thought of not in terms of functors between 1-object categories, but in terms of bimodules for these algebras.

Every algebra homomorphism yields a bimodule, but there are more bimodules than algebra homomorphisms, so that’s a slightly more general notion of morphism.

There are various ways to motivate this. Here are two:

An algebra is a $\mathrm{Vect}$-enriched category on a single object.

One can think of enrichment this way: A $C$-enriched category with set of objects $S$ is the same as a lax functor from the pair groupoid of $S$ to $\Sigma(C)$.

This lax functor will send the identity morphism to a possibly non-identity morphism in $\Sigma(C)$, together with a compositor on it. That’s an algebra internal to $C$.

So one way to define an algebra is to say that it is a lax functor

(1)$\mathrm{Disc}(\bullet) \to \Sigma(\mathrm{Vect}) \,.$

But if there are more objects around, we get further structure. The presence of the unitor and the compatibility of the compositor with the unitor says for instance that a lax functor from the pair groupoid on two objects $\{ \bullet \leftrightarrow \circ\}$ to $\Sigma(C)$ is an algebra $A_\bullet$ internal to $C$, an algebra $A_\circ$ and a $A_\bullet$-$A_\circ$ bimodule $N_{\bullet \to \circ}$.

Thinking along these lines, we might want to take the bicategory $\mathrm{Alg}(C)$ that you were talking about as such that

- objects are algebras internal to $C$

- morphisms are bimodules internal to $C$, with composition being the tensor product over the respective algebra

- 2-morphisms being bimodule homomorphisms.

It’s easily seen that the 2-catgory of algebras, algebra morphisms and natural transformations of these sits inside this $\mathrm{Alg}(C)$.

I claim that including general bimodules as morphisms is a “good thing”, because it yields the right notion of equivalence.

In a decent setup, equivalence of the objects we are dealing with should be ordinary equivalence in the category they live in.

But when thinking in terms of algebras as op-incarnations of spaces, we want equivalence to be Morita equivalence. This makes us want to take morphisms of algebras to be bimodules.

But I want to offer yet another view on why modules and bimodules of algebras are important in our context:

Once we start talking about the algebra

(2)$A = [X,\mathbb{C}]$

of (suitably well-behaved) complex functions on $X$, which I said before can naturally be though of as the monoid

(3)$\mathrm{End}(\mathrm{tra}^0) \,,$

we want to talk about (finitely generated projective) modules of this algebra - since these will encode the vector bundles on the space corresponding to $A$ - more precisely, the modules are the spaces of sections of these vector bundles.

This desire is very naturally satisfied from the point of view of $\mathrm{End}(\mathrm{tra}_0)$.

You might recall that I am claiming that a useful notion of the space of sections of an $1$-bundle (with connection) represented by a transport functor $\mathrm{tra}$ is as the space of morphisms

(4)$\array{ \mathrm{Conf}_1 &\stackrel{\mathrm{tra}^0_*}{\to}& [p_1,\mathrm{Vect}] \\ & \;\Downarrow e \\ &\searrow \nearrow_{\mathrm{tra}_*} } \,,$

where $p_1 = \{\bullet\}$ is the 1-object no-nontrivial morphism category and $\mathrm{Conf}_1 = \mathrm{Disc}(\mathrm{Obj}([p_1,P_1(X)]))$.

You can check that this says nothing but that $e$ is indeed a section of the vector bundle.

And it makes nicely manifest how the space of these sections is a module for our monoid $A$. An element

(5)$f: \mathrm{tra}^0 \to \mathrm{tra}^0$

of our algebra of complex functions acts on sections simply by pre-composition:

(6)$\array{ & \nearrow \searrow^{\mathrm{tra}^0_*} \\ & \;\Downarrow f_* \\ \mathrm{Conf}_1 &\stackrel{\mathrm{tra}^0_*}{\to}& [p_1,\mathrm{Vect}] \\ & \;\Downarrow e \\ &\searrow \nearrow_{\mathrm{tra}_*} } \,.$

Of course this notation here is certainly overkill. But that’s the point. The claim is that this is the right sort of overkill, following your suggestion to seriously consider the overkill of regarding

(7)$\mathbb{C}$

as

(8)$\Sigma({\mathbb{C}}) \,.$

In particular, the above formulation seamlessly categorifies.

Notice that I am essentially saying that

- the algebra of functions on $X$ is best thought of as $\mathrm{End}(\mathrm{tra}^0_*)$

- a module for that is $\mathrm{Hom} (\mathrm{tra}^0_*,\mathrm{tra}_*)$

for a nontrivial $\mathrm{tra}_*$ .

While all that still doesn’t explicitly address the issue of commutative versus noncommutative op-spaces that you raised, I think it is clear that this way of looking at the situation clearly suggests certain answers.

Posted by: urs on October 20, 2006 4:12 PM | Permalink | Reply to this

sectionology

Once we start talking about the algebra $A=[X,\mathbb{C}]$ of (suitably well-behaved) complex functions on $X$, which I said before can naturally be though of as the monoid $\mathrm{End}(\mathrm{tra}^0)$,

I have now started including the general notion of algebra of observables associated to a given $n$-vector transport with connection in my notes on transport of sections (originally posted here).

(currently def. 4 and 5).

Thanks to your remarks, Bruce, this is now getting ever closer to a decent $n$-categorical generalization of spectral triples.

Posted by: urs on October 20, 2006 5:07 PM | Permalink | Reply to this

Re: Algebras as 2-Categories and its Effect on Algebraic Geometry

this way of looking at the situation clearly suggests certain answers.

I will have to catch my train in a minute, but maybe I should make this slightly more explicit:

the way my argument runs here, it leads to a perspective like this:

talking about spaces $X$ in terms of algebras is talking about spaces in terms of algebras of observables of a quantum system with configuration space $X$.

Accordingly, this algebra is commutative in as far as it is the algebra of observables of a particle charged under something abelian (or not charged at all).

It becomes noncommutative in particular if the particle couples to something non-abelian.

In both cases, the algebra of observables comes to us manifestly in the incarnation of a category with a single object, namely the category of endomorphisms of the totally trivial bundle that the particle might be coupled to.

And these statements hold analogously for $n$-particles with $n \gt 1$.

So this leaves us with something like a Connes point of view, albeit with slight changes of conceptual perspective and a straightforward option for categorification.

So this is all still about the case where the algebra is globally defined (the “affine case”), the only case that ever appears in spectral (non-commutative) geometry a la Connes.

The next step would hence to be to think about what it means to sheafify the perspective and consider gluing the above setup piecewise.

I based everything on the observation that the algebra of functions with values in a monoid is to be thought of as a natural endo-transformation of a trivial functor.

It is defined globally in as far as that trivial functor is defined globally.

So: if this functor were itself just a 2-transition function of a 2-functor, then everything I said would turn into a description in terms of algebras on patches glued on intersections.

But I have to run now. And this monologue of mine is too long anyway.

Posted by: urs on October 20, 2006 5:47 PM | Permalink | Reply to this

Re: Algebras as 2-Categories and its Effect on Algebraic Geometry

I am trying to compare the following idea of yours:

I based everything on the observation that the algebra of functions with values in a monoid is to be thought of as a natural endo-transformation of a trivial functor.

… with the viewpoint given in secions 2.1, 2.2 and 2.3 of Simon’s paper on The twisted Drinfeld double of a finite group via gerbes and finite groupoids.

In this paper, one thinks of a 0-form on a groupoid $\mathcal{G}$ as a locally-constant function on $\mathcal{G}$. One thinks of a 1-form as a functor $\mathcal{G} \rightarrow \Sigma(U(1))$. Alternatively, one can think of a 1-form on $\mathcal{G}$ as a section of the trivial line bundle $\mathcal{G} \times \Sigma (U(1)) \rightarrow \mathcal{G}$ over $\mathcal{G}$.

And one thinks of a 2-form as a “way to twist the trivial 2-vector bundle”. Namely, given a 2-form $\theta$ on $\mathcal{G}$, one can twist the trivial 2-vector bundle $G \times \Vect \rightarrow G$ by defining a new composition:

$(\phi_2, g_2) \circ (\phi_1, g_1) = (\theta(g_2, g_1)\phi_2 \circ \phi_1, g_2 \circ g_1).$

The beauty of this approach is that a “$\theta$-twisted representation of $\mathcal{G}$” is then nothing but a section of the 2-vector bundle above.

These two approaches (yours and Simon’s) seem related, but I can’t quite put my finger on it.

Also, about what you said about thinking in terms of Bim instead of Alg… I think you’re spot on there. Simon has also pointed this out to me before.

Amusingly I see even “normal” (i.e. not manifestly n-category enthusiasts) mathematicians are beginning to appreciate this point : in this paper on Hopfish Algebras’ by Tang, Weinstein and Zhu, they basically ask the question : “What is a Hopf algebra inside Bim (instead of inside Alg)?” Actually they don’t use the 2-category structure on Bim (their morphisms are isomorphism classes of bimodules), mentioning that the 2-category viewpoint is left “for the future”. They seem to also be inspired by noncommutative geometry.

An excellent context in which to discuss these issues - for instance, about stuff related to the stability of the periodic table of n-categories - will be during Eugenia Cheng’s lectures at the Fields Institute in Canada next year on n-categories with duals and TQFT. I’m really looking forward to that.

Posted by: Bruce Bartlett on October 23, 2006 5:07 PM | Permalink | Reply to this

p-forms as p-functors

I can’t quite put my finger on it.

Consider this:

We know that ordinary $p$-forms on a smooth space $X$ are the same as strict smooth $p$-functors from $P_n(X)$ to $\Sigma^p(U(1))$, where $P_n(X)$ is the strict $n$-groupoid of thin-homotopy classes of $n$-paths and $n \geq p$:

(1)$\Omega^p(X) \simeq [P_p(X),\Sigma^p(U(1))] \,.$

(You can replace $U(1)$ with $(\mathbb{R},+)$ here, if you like).

We can think of any such $p$-form as a connection on a trivial $p$-bundle.

The exterior derivative is defined on these functors as follows:

We can regard any $p$-functor $\omega$ with values in $\Sigma^p(U(1))$ as a $p$-functor with values in $\mathrm{Mor}_{p+1}(\Sigma^{p+1}(U(1)))$.

This allows to interpret it as a pseudonatural transformation from the trivial $(p+1)$-functor to a nontrivial one. This nontrivial one is the exterior derivative $d \omega$ of $\omega$.

For instance a 0-functor $f$ with values in $U(1)$ can be considered as a map

(2)$\mathrm{Obj}(P_n(X)) \to \mathrm{Mor}(\Sigma(U(1)))$

which can be interpreted as a natural transformation of the trivial 1-functor coming from the naturality square

(3)$\array{ \bullet &\stackrel{f(x)}{\to}& \bullet \\ \mathrm{Id}\downarrow\;\; && \;\;\; \downarrow d f(\gamma) \\ \bullet &\stackrel{f(y)}{\to}& \bullet } \,,$

where $x\stackrel{\gamma}{\to}y \in \mathrm{Mor}_1(P_n(X))$ and $d f (\gamma)$ is really shorthand for the functor

(4)$\gamma \mapsto \exp(\int_\gamma d f) \,.$

I think one can set up quite a bit of differential geometry this way by replacing $p$-forms by suitable smooth functors everywhere and suitably reformulating all other notions (like contraction of $p$-forms with vectors, etc).

Now, there is an obvious generalization: For an arbitrary (strict) $n$-category $C$, we may define $p$-forms on it as $p$-functors

(5)$C \to \Sigma^p(U(1)) \,.$

(We might want to put additional conditions on these functors, similarly to how we had demanded smoothness above.)

For instance, take $C$ to be the $n$-category freely generated from a finite directed graph.

Objects are the vertices of the graph, 1-morphisms those generated from the edges, 2-morphisms those generated from triangles of edges, etc.

Then, defining a $p$-form on $C$ as a $p$-functor to $\Sigma^p(U(1))$ reproduces the “discrete differential calculus” that was mentioned in another thread.

Or take $C$ to be a groupoid, but throw in a unique 2-morphism filling every triangle, subject to the relation that all tetrahedra of these triangles 2-commute.

Then a 2-form on $C$ in my sense should be the same as a 2-form on $C$ in the sense used in Simon Willerton’s paper, the cocycle relation being the compatibility with the tetrahedron law.

I think.

Posted by: urs on October 23, 2006 9:29 PM | Permalink | Reply to this

Re: p-forms as p-functors

I wrote:

I think one can set up quite a bit of differential geometry this way by replacing $p$-forms by suitable smooth functors everywhere and suitably reformulating all other notions (like contraction of $p$-forms with vectors, etc).

There are some fun games one can play here:

if a 1-form on $X$ is a functor

(1)$P_1(X) \to \Sigma(U(1)) \,,$

what is a vector field on $X$?

I think the answers is this:

consider the category $\mathrm{Flow}(X)$ whose single object is $P_1(X)$, and whose morphisms are natural transformations of the form

(2)$\array { & \nearrow \searrow^\mathrm{Id} \\ P_1(X) & \Downarrow& P_1(X) \\ & \searrow \nearrow_{\exp(v)} } \,,$

where $\exp(v)$ is just a name for any morphism $P_1(X) \to P_1(X)$. Composition is horizontal composition of natural transformations.

Then I claim that a vector field on $X$ is the same as a (smooth) functor

(3)$v : \Sigma(\mathbb{R},+) \to \mathrm{Flow}(X) \,.$

Either I am dreaming (I should be in bed already), or you can easily check this by writing out what this means.

Now, there is a nice and obvious way to evaluate a 1-form on a vector field to get a 0-form in terms of this arrow theory.

Recall that my 1-form is a functor

(4)$\omega : P_1(X) \to \Sigma(U(1))$

and that my 0-form $f$ is a natural transformation

(5)$\array{ & \nearrow \searrow^{\mathrm{tra}_0} \\ \mathrm{Disc}(\mathrm{Obj}(X)) & \;\Downarrow f& \Sigma(U(1)) \\ & \searrow \nearrow_{\mathrm{tra}_0} } \,.$

(As before, $\mathrm{Disc}(S)$ denotes the discrete category on the set $S$.) There is an obvious way to pair $\omega$ with $v$ to get such a 0-form, namely as the composition:

(6)$\array { &&& \nearrow \searrow^\mathrm{Id} \\ \mathrm{Disc}(\mathrm{Obj}(X)) &\to& P_1(X) & \Downarrow& P_1(X) &\stackrel{\omega}{\to}& \Sigma(U(1)) \\ &&& \searrow \nearrow_{\exp(v)(t)} } \,.$

For finite $t$ this is in fact (at each $x$ in $X$) the exponentiated integral of $\omega$ along the flow line of $V$ starting at $x$ and of parameter length $t$. So in the smooth setup we need to differentiate $\frac{\partial}{\partial t}|_{t=0}$ to get the ordinary $x \mapsto \omega(v)(X)$.

On the other hand, if we replace $(\mathbb{R},+)$ with $(\mathbb{Z},+)$, this yields flows lines and form/vector pairing for instance on finite graphs.

Posted by: urs on October 23, 2006 11:10 PM | Permalink | Reply to this

Re: p-forms as p-functors

It might be simpler to simplify your approach above slightly as follows.

Define an $n$-form $\omega$ on a smooth space $X$ to be a strict smooth n-functor $\omega : P_n(X) \rightarrow \Sigma^n (U(1))$, where $P_n(X)$ is the strict $n$-groupoid of thin homotopy classes of $n$-paths.

The exterior derivative $d\omega : P_{n+1} \rightarrow \Sigma^{n+1} (X)$ is defined by Stokes theorem.

For instance, suppose $\omega$ is a one-form. Then $d \omega : P_2(X) \rightarrow \Sigma^2 (X)$ is the 2-functor which assigns to 2-morphisms in $P_2(X)$ (that is, smooth patches of surfaces with given boundary $f \circ g^{op}$) the value $\omega(f \circ g^{op})$.

This approach is nothing but an “integrated” form of the ordinary De Rham theory of differential forms. But thinking about $n$-forms as $n$-functors makes it clear how their values are supposed to behave under gluing of paths (surfaces, etc.).

In fact, you can think of it as a baby form of TQFT - except the ordinary De Rham calculus corresponds to a bog-standard TQFT, not an *extended* one.

Instead of functors $nCob^1 \rightarrow \nHilb$ from the ordinary cobordsim category (*not* n-category) $nCob^1$ to $Hilb$, we are thinking about functors $P_n (X) \rightarrow \Sigma^n (X)$.

The interesting bit is trying to understand how we need to think of differential forms, in order to think of them as a baby version of *extended* TQFT. For instance, one needs to find the analogue of “nHilb” for differential forms.

But quite possibly it doesn’t make sense to think of differentiable forms in an extended’ way.

Posted by: Bruce Bartlett on October 28, 2006 12:29 PM | Permalink | Reply to this

Re: p-forms as p-functors

The exterior derivative [on $p$-functors to $\Sigma^p(U(1))$] is defined by Stokes theorem.

Yes! This is indeed what the prescription I described yields for target $\Sigma^p(U(1))$.

The way I formulated it, in terms of isomorphisms from the trivial $p$-form functor, was supposed to be more intrinsic to category theory. But here it produces precisely what you spelled out.

The advantage of the description in terms of morphisms of functors is that it generalizes. To nonabelian $p$-forms, for instance, where it yields what is sometimes called the nonabelian Stokes theorem.

Or to the discrete setup, where it correctly yields a discreteized version of Stokes theorem.

Or, in fact, to the “extended” case, where the target contains nontrivial objects and morphisms beneath top level.

The interesting bit is trying to understand how we need to think of differential forms, in order to think of them as a baby version of *extended* TQFT. For instance, one needs to find the analogue of “nHilb” for differential forms.

We can think of a vector bundle with connection as an (extended) 1dTFT defined on 1-cobordisms equipped with maps into base space. Points are sent to vector spaces, 1-cobordisms to linear maps between these.

We can think of a (abelian or nonabelian) gerbe with connection as an extended 2dTFT defined on 2-cobordisms equipped with maps into target space. Points are sent to algebras, 1-dimensional cobordisms between points to bimodules and 2-dimensional cobordisms between 1-dimensional things to bimodule homomorphism.

(Seeing that this really is taking values in 2-vector spaces (or 2-Hilbert spaces) amounts to realizing how the bicategory of bimodules sits inside the bicategory of $\mathrm{Vect}$ module categories.)

This reduces to ordinary 1- and 2-forms after we trivialize the gerbe (locally).

A transition 1-bundle with connection (represented by a 1-functor) is then related to the gerbe with connection (represented by a 2-functor) precisely by Stokes theorem (locally).

Posted by: urs on October 29, 2006 2:27 PM | Permalink | Reply to this
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