The n-Category Café

Morphisms and Tangent Vectors

There are many different ways to relate ($n$-)categories with spaces, one way or another.

The way to think of an ($n$-)category as a space for the purpose of our arrow-theoretic differential theory is the immediate generalization of the way orbifolds may be regarded as groupoids.

So we should be thinking of the space in question as being actually the space of objects of our category. The morphisms of the category, on the other hand, encode something like neighbourhood relations among the points in that space. Tangency relations, if you wish.

In the following precise sense (details for $n=1$ and $n=2$ and everything strict are available, and for $n \geq 3$ are expected to follow straightforwardly, though possibly tediously):

Every $n$-groupoid $$C$$ comes with an $n$-groupoid $$T C \,,$$ called, here, its tangent $n$-bundle, which is essentially defined to be the space of maps of the fat point $\mathbf{pt}$ into $C$.

This $n$-groupoid $T C$ has the following remarkable properties:

a) $T C$ is indeed an $n$-bundle over the space underlying $C$ in that there is a canonical projection functor $$p : T C \to \mathrm{Obj}(C)$$

Even more is true: $T C$ is a “deformation retract” of the space underlying $C$, in that we have an equivalence of $n$-groupoids $$T C \simeq \mathrm{Obj}(C) \,.$$ (I am grateful to David Roberts for emphasizing the importance of this fact, regarded this way, and to David Roberts and Jim Stasheff for discussion on this point.)

b) The $n$-bundle $T C$ fits into the short exaxt sequence $$\mathrm{Mor}(C) \to T C \to C$$ of $n$-groupoids.

c) There is a canonical monomorphism $$\Gamma(T C) \hookrightarrow T_{\mathrm{Id}_C}(\mathrm{End}(C))$$ of the space (an $(n-1)$-category, really) of sections of $T C$ into the categorical tangent space at the identity in the space of endomorphisms of $C$.

The latter is the space of categorical flows on $C$. This is naturally an $(n+1)$-group. And by this embedding the space of sections $\Gamma(T C)$, too, receives the structure of an $(n+1)$-group.

Endofunctors and Vector Fields

To appreciate the concept formation above, it is helpful to see how ordinary vector fields – and then their generalizations – arise from this point of view.

Definition. For $C$ any $n$-groupoid and $G_{(n)}$ any $n$-group, we say that an $n$-group homomorphism $$v : G_{(n)} \to \Gamma(T C)$$ is a $G_{(n)}$-flow on $C$.

We shall hence think of $$\mathrm{Hom}(G_{(n)}, \Gamma (T C))$$ as the space of $G_{(n)}$-vector fields on $C$.

Example. For $X$ a smooth manifold and $P_1(X)$ its path groupoid, ordinary vector fields on $X$ are smooth $\mathbb{R}$-flows on $P_1(X)$.

Hence the space of $\mathbb{R}$-vector fields on $P_1(X)$ is indeed the space of ordinary vector fields on $X$: $$\mathrm{Hom}(\mathbb{R}, \Gamma T(P_1(X))) \simeq \Gamma(T X) \,.$$ (As before, I should admit that I have a strict proof so far only for the inclusion of the right hand side into the left hand side. The converse is a little more technical.)

But noticing that for any Lie group $G$ we have $$\mathrm{Hom}(\mathbb{R}, G) \simeq \mathrm{Lie}(G)$$ we should be able to restate this as $$\mathrm{Lie}( \Gamma T(P_1(X))) \simeq \Gamma(T X) \,.$$ This is actually essentially just saying that vector fields on $X$ form the Lie algebra of diffeomorphisms on $X$ connected to the identity.

Maps of categorical tangent bundles

With this in hand, there is now a straightforward definition of categorical maps of vector fields.

Definition.

For $C$ and $D$ $n$-groupoids and with some $n$-group $G_{(n)}$ fixed, we say that a morphism of their tangent $n$-bundles $$f : T C \to T D$$ is a map of $G_{(n)}$-vector fields if $f$ respects $G_{(n)}$ flows on $C$ and $D$ in the following, essentially obvious sense:

Recall what the “arrow theory” behind these concepts actuallly looks like:

Under our embedding $\Gamma(T C) \hookrightarrow T_{\mathrm{Id}_C}(\mathrm{End}(C))$ a section of the categorical tangent bundle is a “categorical tangent” in $T_{\mathrm{Id}_C}(\mathrm{End}(C))$ to the identity map from $C$ to itself, hence a transformation $$\array{ & {}^{\;\;}\nearrow \searrow^{\mathrm{Id}} \\ C &\;\;\Downarrow^{\mathrm{exp}(v)}& C \\ & {}_{\;\;\;\;\;\;}\searrow \nearrow_{\mathrm{Ad}_{\mathrm{exp}(v)}} } \,.$$

As we precompose this with any map $$x : \mathrm{point} \to C$$ from the point $$\mathrm{pt} = \{\bullet\}$$ into $C$, we obtain an element in $$T_x C \,,$$ namely $$\array{ & {}^{\;\;}\nearrow \searrow^{x} \\ C &\;\;\Downarrow^{}& C \\ & {}_{\;\;\;\;\;\,}\searrow \nearrow_{\mathrm{exp}(v)(x)} } \;\;\; := \;\;\; \array{ & & & {}^{\;\;}\nearrow \searrow^{\mathrm{Id}} \\ \mathrm{pt} & \stackrel{x}{\to}& C &\;\;\Downarrow^{\mathrm{exp}(v)}& C \\ & & & {}_{\;\;\;\;\;\;}\searrow \nearrow_{\mathrm{Ad}_{\mathrm{exp}(v)}} } \,.$$

This just means picking out the value of the given section of $T C$ at the point $x \in \mathrm{Obj}(C)$.

This element of $T_x C$ may then be sent with with our map $f$ to $T_{f(x)} D$. And clearly, for $f$ to be a map of $G_{(n)}$-vector fields, we should require that this procedure takes $G_{(n)}$-flows to $G_{(n)}$-flows, i.e. $$f \;\;\;\; : \;\;\;\; \array{ & & & {}^{\;\;}\nearrow \searrow^{\mathrm{Id}} \\ \mathrm{pt} & \stackrel{x}{\to}& C &\;\;\Downarrow^{\mathrm{exp}(v(t))}& C \\ & & & {}_{\;\;\;\;\;\;\;\;}\searrow \nearrow_{\mathrm{Ad}_{\mathrm{exp}(v(t))}} } \;\;\;\; \mapsto \;\;\;\; \array{ & & & {}^{\;\;}\nearrow \searrow^{\mathrm{Id}} \\ \mathrm{pt} & \stackrel{f(x)}{\to}& D &\;\;\Downarrow^{\mathrm{exp}(w(t))}& D \\ & & & {}_{\;\;\;\;\;\;\;\;}\searrow \nearrow_{\mathrm{Ad}_{\mathrm{exp}(w(t))}} }$$ for a suitable $w$.

This last sentence is intentionally left a little vague. I am still thinking about how to formulate this best. There is an issue here to be dealt with whenever $C$ and $D$ have more than one object.

But, actually, my main point here, to which I will now finally come, is what happens in the case that $D$ has just a single object, i.e. in which $$D = \Sigma H_{n}$$ is the suspension of an $n$-group. In that case $$T D = \mathrm{INN}_0(H_{(n)})$$ is the inner automorphism $(n+1)$-group of $D$. Then our tangent vector map $f$ will take values in the inner automorphism $(n+1)$-group of $H_{(n)}$. Since that’s the tangent space to the single point of $D$. (Compare this to the notion of tangent stacks, courtesy of David Ben-Zvi.)

So looking at it from this point of view, we find that a connection $n$-form for given structure $n$-group $H_{(n)}$ is actually a map $$\mathbf{A} : T P_n(X) \to T \Sigma H_{(n)}$$ hence a map $$\mathbf{A} : T P_n(X) \to \mathrm{INN}_0(H_{(n)}))$$ hence should come, differentially, from a Lie $(n+1)$-algebra morphism $$A^* : \mathrm{inn}(h_{(n)})^* \to \Omega^\bullet(X) \,.$$ Since $\mathrm{inn}(h_{(n)})$ is trivializable, any such map will be homotopic to the trivial such map – but as discussed at great length at various places now, it the choice of trivializing homotopy which encodes the expected information. And second order homotopies of that encode gauge transformations, and so on.

My suggestion, therefore: maybe we are being told here we are not supposed to be looking at homotopies of $L_\infty$ morphisms unless the target is $\mathrm{inn}$ of something.

While, clearly, this requires further thinking, I thought I’d mention this observation here.