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September 3, 2007

Arrow-Theoretic Differential Theory IV: Cotangents

Posted by Urs Schreiber

With everybody growing more comfortable with the idea of tangent categories, in good part due to the various cross-relations which we found in More on Tangent Categories, the obvious open question became more urgent:

What about cotangent categories?

(asked by David Corfield here and by Jim Stasheff here).

Defining a cotangent means first of all agreeing on a codomain in which it takes values. That’s the reason why one gets a headache when trying to define cotangent categories in the same manner as tangent categories.

I think a good idea is to consider the arrow-theoretic analogue of vector valued differential forms. This should include scalar-valued differential forms then as an obvious special case. So we should be after morphisms of tangent categories, or better: morphisms of flows on categories.

Tangent categories, flows on categories and vector fields

I’ll adopt Todd Trimble’s nice description of tangent categories:

Fix some nn and some notion of nn-categories.

Let 2:={} 2 := \{\bullet \to \circ\} be the nn-category with one nontrivial 1-morphism. For C C any nn-category, write C 2:=Hom nCat(2,C). C^2 := \mathrm{Hom}_{n\mathrm{Cat}}(2,C) \,. The tangent nn-category TCT C of CC is that subcategory of this which arises as the pullback TC C 2 dom C 0 C. \array{ T C &\to& C^2 \\ \downarrow && \downarrow^{\mathrm{dom}} \\ C_0 &\to& C } \,. (Here C 0:=Obj(C)C_0 := \mathrm{Obj}(C).)

I used to put this, less elegantly, as :

The tangent nn-category is that maximal sub nn-category of all maps of the fat point into CC which collapses to a 0-category as we pull back along the inclusion {}{}. \{\bullet\} \to \{\bullet \to \circ\} \,.

I still find this a very useful heuristics to think about what is going on here: suppose we are in the world of nn-groupoids. Then the “fat point” {}\{\bullet \to \circ\} is equivalent to the “point” {}\{\bullet\}. Then:

The tangent nn-groupoid is the space of those images of the fat point in CC which become “invisble” as soon as we stop distinguishing the fat point from the ordinary point.

I read this as:

A tangent vector is a path which becomes invisible as soon as we stop distinguishing the infinitesimal from the vanishing.

At this point, though, something interesting happens. A couple of people have complained to me that TCT C itself is not at all “infinitesimal” in any sense.

That’s true. But it’s supposed to be a feature, not bug. The point is (or that’s at least how I am thinking about it) that we are not only generalizing the notion of tangent spaces here, but even that of infinitesimality itself : we may get ordinary tangent vectors using the tangent category construction, but we may also get more exotic things, like “integrated odd vector fields”, for instance.

To see this, first consider the following important properties of TCT C (checked up to n=2n=2 using the Gray category of 2-categories, but supposedly true more generally):


a) TCT C is an nn-bundle over C 0C_0: p:TCC 0 p : T C \to C_0 which is a “deformation retract” of C 0C_0 in that TCC 0. T C \simeq C_0 \,.

b) there is a canonical inclusion Γ(TC)T Id C(End nCat(C)) \Gamma (T C ) \to T_{\mathrm{Id}_C} ( \mathrm{End}_{n\mathrm{Cat}}( C) ) of sections of the tangent category into the categorical tangents to the identity endomorphism on CC. This equips Γ(TC)\Gamma (T C) with a monoidal structure.

Flows and vector fields

It is this monoidal structure on Γ(TC)\Gamma(T C) which establishes the contact to the ordinary notion of infinitesimal: we may find images of different groups inside Γ(TC)\Gamma(T C). Smooth images of the additive group of real numbers correspond to ordinary vector fields. Images of other groups may correspond to exotic “vector fields” like they appear in supergeometry.

Definition (GG-flow and GG-tangent vector field) For CC any nn-category and GG any group, we say that a group homomorphism GΓ(TC) G \to \Gamma(T C) is a GG-flow on CC. Equivalently: a GG-vector field on C 0C_0 relative to CC.

To make contact to ordinary vector fields, set G=G = \mathbb{R} and consider everything in the world of smooth spaces. Then

a) For XX a smooth manifold and C=Π 1(X)C = \Pi_1(X) be its fundamental groupoid or C=X×XC = X \times X the pair groupoid, \mathbb{R}-flows on CC are ordinary vector fields on XX: Γ(TX)Hom(,Γ(TC)). \Gamma(T X ) \simeq \mathrm{Hom}(\mathbb{R}, \Gamma (T C)) \, . This is the archetypical example. Keeping the inclusion Γ(TC)T Id CEnd(C)\Gamma (T C ) \to T_{\mathrm{Id}_C \mathrm{End}(C)} in mind, notice that an \mathbb{R}-flow sends real numbers to transformations that connect an automorphism of CC to the identity:

t Id C exp(v)(t) C Ad exp(v)(t). t \;\;\, \mapsto \;\;\; \array{ & {}^{\;\;\;\;}\nearrow\searrow^{\mathrm{Id}} \\ C &\Downarrow \exp(v)(t)& C \\ &{}_{\;\;\;\;} \searrow\nearrow_{\mathrm{Ad}_{\exp(v)(t)}} } \,. For vv any vector field on XX, the component of this transformation at any xXx \in X is the (class of the) path which follows the flow line of parameter length tt along vv starting at xx.

( I think the same statement is true if we replace Π 1(X)\Pi_1(X) by the path groupoid P 1(X)P_1(X). But in this case it is a little more suble to show that Γ(TX)Hom(,Γ(TC))\Gamma(T X ) \simeq \mathrm{Hom}(\mathbb{R}, \Gamma (T C)) is not just an inclusion, but actually surjective.)

b) More generally, for CC any Lie groupoid, \mathbb{R}-flows on CC are, I think, nothing but sections of the Lie algebroid corresponding to CC Γ(Lie(C))Hom(,Γ(TC)). \Gamma(\mathrm{Lie}(C)) \simeq \mathrm{Hom}(\mathbb{R}, \Gamma (T C)) \,. I mentioned this for the special case that CC is the Atiyah groupoid of a principal bundle here.

Notice that any groupoid CC is a groupoid over the codiscrete groupoid over its space of objects C(C 0×C 0). C \to (C_0 \times C_0) \,. This means that we canonically have a projection Γ(TC)ΓT(C 0×C 0). \Gamma (T C) \to \Gamma T (C_0 \times C_0) \,. Under passing to \mathbb{R}-flows, this is the anchor map ΓT(Lie(C))Γ(TC 0) \Gamma T(\mathrm{Lie}(C)) \to \Gamma (T C_0) of the Lie algebroid.

Moreover, we canonically have a bracket on \mathbb{R}-flows, obtained by taking the usual group commutator in INN(C):=T Id CEnd(C)\mathrm{INN}(C) := T_{\mathrm{Id}_C \mathrm{End}(C)}. This is the bracket on the sections of the Lie algebroid corresponding to CC.

c) In Supercategories I argued that, analogously, a category of super objects should be a category on which we have not an ordinary vector field as above, being an \mathbb{R}-flow, but a 2\mathbb{Z}_2-vector field . (Or analogously with 2\mathbb{Z}_2 replaced by \mathbb{Z} or × 2\mathbb{Z} \times \mathbb{Z}_2 or the like.)

It turns out that my definitions 3 and 4, which define categories with 2\mathbb{Z}_2-flow as above and the notion of a compatible grading, are in fact equivalent to the conditions on categories of graded D-branes, as described by Lazaroiu.

I take this as evidence that we may usefully think of the structure on these graded D-brane categories as being equipped with the “flow of an odd vector field”.

d) Apart from generalizing the parameter group GG, the most important generalization is certainly that to higher categorical dimension. For CC any Lie nn-groupoid, its \mathbb{R}-flows should be nothing but the sections of the corresponding Lie nn-algebroid. I discuss some examples of that here.


With a concept of tangents in hand, one needs two things in order to define cotangents:

a) a notion of morphisms of tangents

b) a choice of “valuation object”, i.e. of codomain.

Here I shall take the point of view that a cotangent vector field is nothing but a morphism of tangent vector bundles, with the codomain possibly being a particularly degenerate tangent vector bundle.

First of all, the tangent nn-category construction is functorial:

For F:CD F : C \to D and nn-functor between nn-categories, we get an nn-functor TF:TCTD T F : T C \to T D between the corresponding tangent categories in the obvious way. (Again, this is easily made explicit for low nn and a sufficiently strict notion of nn-categories, but it should be true for any choice of nn and any notion of nn-category.)

But, given the above discussion, this TFT F is not yet quite what we would address (not by itself, at least) as a vector valued covector field.

Instead, we should want to map GG-flows on CC to GG-flows on DD, clearly.


Definition (cotangent vector fields) For CC and DD nn-categories and GG a group, a TDT D-valued covector field on CC is an (n+1)(n+1)-functor ΣT Id C(End(C)) = ΣINN(C) ω ΣT Id D(End(C)) = ΣINN(D) \array{ \Sigma T_{\mathrm{Id}_C}(\mathrm{End}(C)) &=& \Sigma \mathrm{INN}(C) \\ \downarrow^\omega \\ \Sigma T_{\mathrm{Id}_D}(\mathrm{End}(C)) &=& \Sigma \mathrm{INN}(D) } which sends (smooth, if required) GG-flows on CC to (smooth, if required) GG-flows on DD.

(Notice that none of my definitions here are meant to be carved in stone. There are a couple of obvious and a couple of not so obvious modifications and refinements that come to mind. But the following examples are supposed to indicate that these definitions do go in the right direction.)

We already know that smooth nn-functors on path nn-groupoids encode differential nn-forms. For instance for XX any smooth space and P n(X)P_n(X) the smooth path nn-groupoid of that space, we have that smooth nn-functors P n(X)Σ n(U(1)) P_n(X) \to \Sigma^n(U(1)) are in bijection with smooth nn-forms on XX.

So if the above definition is any good, it should subsume mere nn-functors as “cotangent nn-vectors” of sorts. And it does as follows.

Let At:=P n(X)×Σ n(U(1)) \mathrm{At} := P_n(X) \times \Sigma^n(U(1)) be the Atiyah Lie nn-groupoid of the trivial Σ (n1)U(1)\Sigma^{(n-1)} U(1)-nn-bundle over XX. It comes with the canonical projection p:AtP n(X) p : \mathrm{At} \to P_n(X) discussed at some length in On Hess and Lack on Bundles of Categories.

This projection in fact extends to a cotangent vector field which sends vector fields on At\mathrm{At} to vector fields on P n(X)P_n(X):

The element Id At V At Ad(V) \array{ {}^{\;\;\;} &\nearrow \searrow^{\mathrm{Id}} \\ \mathrm{At} &\Downarrow^V& \mathrm{At} \\ {}^{\;\;\;} & \searrow \nearrow^{\mathrm{Ad}(V)} } of INN(At)\mathrm{INN}(\mathrm{At}) is sent to the element Id P n(X) v P n(X) Ad(v) \array{ & {}^{\;\;\;}\nearrow \searrow^{\mathrm{Id}} \\ P_n(X) &\Downarrow^v& P_n(X) \\ & {}^{\;\;\;}\searrow \nearrow^{\mathrm{Ad}(v)} } of INN(P n(X))\mathrm{INN}(P_n(X)) determined by the condition that Id At V At Ad(V) p P n(X)=At p Id P n(X) v P n(X) Ad(v). \array{ &{}^{\;\;\;}\nearrow \searrow^{\mathrm{Id}} \\ \mathrm{At} &\Downarrow^V& \mathrm{At} \\ &{}^{\;\;\;} \searrow \nearrow^{\mathrm{Ad}(V)} & \downarrow^{p} \\ && P_n(X) } \;\;\;\;\;\; = \;\;\;\;\;\; \array{ \mathrm{At} \\ \downarrow^{p} & {}^{\;\;\;}\nearrow \searrow^{\mathrm{Id}} \\ P_n(X) &\Downarrow^v& P_n(X) \\ & {}^{\;\;\;}\searrow \nearrow^{\mathrm{Ad}(v)} } \,.

(This uses the obvious notion of morphisms of categories with fixed chosen GG-flow indiced by functors, as considered for instance in Supercategories.)

This provides a morphism ΣINN(At)ΣINN(P n(X)). \Sigma \mathrm{INN}(\mathrm{At}) \to \Sigma \mathrm{INN}(P_n(X)) \,.

We may try to “split” this by finding a one-sided inverse ΣINN(At)traΣINN(P n(X)) \Sigma \mathrm{INN}(\mathrm{At}) \stackrel{\mathrm{tra}}{\leftarrow} \Sigma \mathrm{INN}(P_n(X)) which sends vector fields in the base to vector fields on the Atiyah nn-groupoid.

I think (but haven’t proven) that for n=1n=1 all such morphisms are in fact induced from functors tra:P 1(X)At \mathrm{tra} : P_1(X) \to \mathrm{At} by Id At V At tra Ad(V) P 1(X)= At Id tra P 1(X) v P 1(X) Ad(v). \array{ &{}^{\;\;\;}\nearrow \searrow^{\mathrm{Id}} & \\ \mathrm{At} &\Downarrow^V& \mathrm{At} \\ \uparrow^{\mathrm{tra}} &{}^{\;\;\;} \searrow \nearrow^{\mathrm{Ad}(V)} & \\ P_1(X) } \;\;\;\;\;\; = \;\;\;\;\;\; \array{ &&\mathrm{At} \\ & {}^{\;\;\;}\nearrow \searrow^{\mathrm{Id}} & \uparrow^{\mathrm{tra}} \\ P_1(X) &\Downarrow^v& P_1(X) \\ & {}^{\;\;\;}\searrow \nearrow^{\mathrm{Ad}(v)} } \,.

Let At\mathrm{At} be the Atiyah groupoid of the trivial \mathbb{R}-bundle (or U(1)U(1)-bundle) over XX, and this is nothing but a 1-form ωΩ 1(X)\omega \in \Omega^1(X). Similarly for higher nn.

For n>1n\gt 1 things become more interesting. The point of the entry The GG and the BB was to note that for At\mathrm{At} the Atiyah 2-groupoid of the trivial ΣU(1)\Sigma U(1)-2-bundle, there are more TAtT \mathrm{At}-valued 1-forms on the base than come from smooth 2-functors P 2(X)AtP_2(X) \to \mathrm{At}, which in turn come from smooth 2-forms on XX. It turned out that now also a Riemannian metric on XX induces an TAtT \mathrm{At}-valued 1-form on XX.

One should go on here. But I feel I am running out of energy. I’ll quit by passing directly to some…

Open questions

There are lots of them.

For instance:

a) while I have the feeling that the gap is narrowing, I still don’t know how to nicely express the wedge product in this context.

b) The diagrams above are beginning to look like the tin can diagrams for pseudonatural transformations of 2-functors. Is that what we are supposed to think of? Maybe not. I find it striking that by rotating these by 90 degrees, we get the structure of the diagrams discussed in QFT of Charged n-Particle: Algebra of Observables: there we have a bigon sitting on the above vertical functors, while the bigons on the horizontal morphisms are suppressed.

Somehow the picture of “arrow-theoretic differential theory” developed here should merge with the discussion of the quantum nn-particle, where the categorical tangents act as “quantum observables” on the quantum states, which are morphisms into our vertical functors.

If you don’t like to consider the “quantum” aspect in this context: this is equivalently asking how we can understand the “arrow-theory” of tangent vectors acting on sections of vector bundles.

Hm, actually, merging the above pictures with those in Algebra of Observables the respective diagrams begin to look like those of a modification of pseudonatural transformations.


Posted at September 3, 2007 1:57 PM UTC

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Re: Arrow-Theoretic Differential Theory IV: Cotangents

Just a brief comment: tangents and vector fields are geometry, cotangents and 1-forms are algebra, though we can make vector fields algebra, i.e. as derivations of algebras (of functions).

Cf. supermanifolds and formal manifolds which pretend to be geometry but are really just algebra :-)

Posted by: jim stasheff on September 4, 2007 1:39 AM | Permalink | Reply to this

Re: Arrow-Theoretic Differential Theory IV: Cotangents

Urs wrote:

Defining a cotangent means first of all agreeing on a codomain in which it takes values. That’s the reason why one gets a headache when trying to define cotangent categories in the same manner as tangent categories.

Right — this is why I reacted a bit grumpily when David first raised the problem.

Perhaps a more articulate form of grumpiness would have been more useful.

The cotangent bundle relies on the notion of the dual of a vector bundle, which in turn relies on the notion of the dual of a vector space.

(By “vector space”, let’s mean “finite-dimensional vector space”.)

Duals of vector spaces are important largely because vector spaces form a compact closed category. This is a precise way of saying that the space of linear maps from a vector space VV to a vector space WW is isomorphic to V *WV^* \otimes W:

Hom(V,W)V *W Hom(V,W) \cong V^* \otimes W

In other words, to study morphisms from VV to any vector space WW, we only need to study morphisms from VV to the ground field kk: tensoring will do the rest:

Hom(V,W)Hom(V,k)W Hom(V,W) \cong Hom(V,k) \otimes W

So, the dual V *=Hom(V,k)V^* = Hom(V,k) plays a special role when studying morphisms out of VV.

Similarly, the category of finite-dimensional vector bundles over a manifold forms a complact closed category, so ‘WW-valued 1-forms’ on a manifold MM are just sections of the cotangent bundle tensored with WW:

Ω 1(M,W)Γ(T *MW)\Omega^1(M,W) \cong \Gamma(T^* M \otimes W) But, there’s nothing quite like this when we’re working in the category of sets, or the 2-category of categories, or various bundley generalizations of these.

In the simplest case:

There is no ‘dual’ S *S^* of a set SS such that

Hom(S,T)S *×THom(S,T) \cong S^* \times T

for all sets TT. Similarly for categories.

So, instead of trying to find ‘the cotangent category’ T *CT^* C of a category CC, it seems we should study morphisms from the tangent category TCT C to all possible bundles of categories ECE \to C.

Posted by: John Baez on September 6, 2007 1:52 AM | Permalink | Reply to this

Re: Arrow-Theoretic Differential Theory IV: Cotangents

The cotangent bundle relies on the notion of the dual of a vector bundle, which in turn relies on the notion of the dual of a vector space.

Not necessarily. Just define the cotangent bundle as the quotient of functions that vanish at the point by functions that vanish to first order.

Posted by: Aaron Bergman on September 6, 2007 1:59 AM | Permalink | Reply to this

Re: Arrow-Theoretic Differential Theory IV: Cotangents

Amen! and that way it is algebra right away.
Avoid duals where possible; they sometimes are unnatural.

Posted by: jim stasheff on September 6, 2007 12:51 PM | Permalink | Reply to this

Re: Arrow-Theoretic Differential Theory IV: Cotangents

I do think that you are secretly making a choice of valuation object here, too: while no dual of a vector space appears manifestly, you are making a choice of what counts as a dual of a point.

Posted by: Urs Schreiber on September 6, 2007 1:02 PM | Permalink | Reply to this

Re: Arrow-Theoretic Differential Theory IV: Cotangents

Compare `pointless’ geomoetry! ;-D

Posted by: jim stasheff on September 6, 2007 2:28 PM | Permalink | Reply to this

Re: Arrow-Theoretic Differential Theory IV: Cotangents

Compare ‘pointless’ geomoetry! ;-D

Right. I get your point.

Seriously, though:

I believe the issue here is that we are dealing with a process that involves three steps:

1) we start with a notion of space

2) then we figure out what is dual to that (algebras of functions, etc.)

3) then we notice that these have a life of their own and we may generalize this algebra by forgetting about the original notion of space.

My entry above is in a way about understanding all this with “space” replaced by “n-category”. Or maybe “n-groupoid”.

And I am trying to understand step 2) in that context.

Once we have understood 2), it would be fun to try 3). Personally, I wouldn’t dare to try to guess the right answer to 3) before having understood 2). So that’s what I am doing here.

You seem to indicate that you would try to jump to 3) directly. I’d be very interested in seeing approaches to that extent. But I think for the moment I’ll restrict myself to trying to understanding 2) in the world of nn-categories.

Posted by: Urs Schreiber on September 6, 2007 3:53 PM | Permalink | Reply to this

Re: Arrow-Theoretic Differential Theory IV: Cotangents

I am convincing myself that a problem Urs is having formulating the wedge product “arrow theoretically” could be that the wedge product is “sick” as usually presented *gasp*

I talked about this a little here

Wedge Product vs DMHSF Product

The product taking forms to cochains is not an algebra map, as Jim Stasheff pointed out in a comment to the link above.

However, you can define a NEW continuum product that is an algebra map

αβ=W[R(α)R(β)],\alpha\wedge'\beta = W\circ [R(\alpha) \smile R(\beta)],


R:Ω(M)C(M)R:\Omega(M)\to C(M)

is the de Rham map and

W:C(M)Ω(M)W: C(M)\to\Omega(M)

is the Whitney map satisfying

RW=Id.R\circ W = \mathrm{Id}.

I can’t promise, but I suspect that this product DOES have a nice arrow theoretic formulation and preserves most, if not all, the desirable properties of the wedge product.

The wedge product puts too much emphasis on smooth continuum spaces, whereas the new product works equally well for continuum and discrete spaces. The fact that discrete spaces admit products that are associative, but non-commutative or non-associative, but (skew) commutative is a hint that I do not think we’ve taken seriously enough.

Posted by: Eric on December 15, 2008 11:04 PM | Permalink | Reply to this

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