The n-Category Café

Tangent categories, flows on categories and vector fields

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

Fix some $n$ and some notion of $n$-categories.

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

I used to put this, less elegantly, as :

The tangent $n$-category is that maximal sub $n$-category of all maps of the fat point into $C$ 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 $n$-groupoids. Then the “fat point” $\{\bullet \to \circ\}$ is equivalent to the “point” $\{\bullet\}$. Then:

The tangent $n$-groupoid is the space of those images of the fat point in $C$ 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 $T C$ 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 $T C$ (checked up to $n=2$ using the Gray category of 2-categories, but supposedly true more generally):

Properties:

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

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

Flows and vector fields

It is this monoidal structure on $\Gamma(T C)$ which establishes the contact to the ordinary notion of infinitesimal: we may find images of different groups inside $\Gamma(T C)$. 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 ($G$-flow and $G$-tangent vector field) For $C$ any $n$-category and $G$ any group, we say that a group homomorphism $$G \to \Gamma(T C)$$ is a $G$-flow on $C$. Equivalently: a $G$-vector field on $C_0$ relative to $C$.

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

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

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

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

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

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

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

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 $\mathbb{Z}_2$-vector field . (Or analogously with $\mathbb{Z}_2$ replaced by $\mathbb{Z}$ or $\mathbb{Z} \times \mathbb{Z}_2$ or the like.)

It turns out that my definitions 3 and 4, which define categories with $\mathbb{Z}_2$-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 $G$, the most important generalization is certainly that to higher categorical dimension. For $C$ any Lie $n$-groupoid, its $\mathbb{R}$-flows should be nothing but the sections of the corresponding Lie $n$-algebroid. I discuss some examples of that here.

Cotangents

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 $n$-category construction is functorial:

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

But, given the above discussion, this $T F$ 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 $G$-flows on $C$ to $G$-flows on $D$, clearly.

So

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

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

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

Let $$\mathrm{At} := P_n(X) \times \Sigma^n(U(1))$$ be the Atiyah Lie $n$-groupoid of the trivial $\Sigma^{(n-1)} U(1)$-$n$-bundle over $X$. It comes with the canonical projection $$p : \mathrm{At} \to P_n(X)$$ 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 $\mathrm{At}$ to vector fields on $P_n(X)$:

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

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

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

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

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

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

For $n\gt 1$ things become more interesting. The point of the entry The $G$ and the $B$ was to note that for $\mathrm{At}$ the Atiyah 2-groupoid of the trivial $\Sigma U(1)$-2-bundle, there are more $T \mathrm{At}$-valued 1-forms on the base than come from smooth 2-functors $P_2(X) \to \mathrm{At}$, which in turn come from smooth 2-forms on $X$. It turned out that now also a Riemannian metric on $X$ induces an $T \mathrm{At}$-valued 1-form on $X$.

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 $n$-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.

Hm…