The n-Category Café

Would it be too much trouble to give an example

I did! This is what I wrote above:

Here is how we reproduce the ordinary tangent bundle of an ordinary manifold $X$ this way:

we need to model $X$ as a category. Take it to be the 2-groupoid $$P_2(X)$$ whose objects are the points of $X$, whose morphisms are piecewise smooth parameterized paths in $X$ and whose 2-morphisms are thin homotopy classes of homotopies between all those paths whose tangent vectors at source and target coincide.

Then $$(T P_2(X))_\sim = T X \,,$$ where $(\cdot)_\sim$ here denotes dividing out 2-isomorphisms.

That’s also the motivating idea behind the concept of tangent space here: we take all morphisms emanating at a given object and then let the 2-morphisms between them identify all those which “have the same tangent” at that object.

There is also the following alternative formulation of the space of vector fields $\Gamma(T X)$ in terms of this language of tangent categories:

Let $X$ be any manifold and $P_1(X)$ its path groupoid (object space is $X$, morphisms are thin homotopy classes of paths in $X$).

Then a categorical tangent to the identity map on $X$ $$\mathbf{v} \in T_{\mathrm{Id}_{P_1(X)}}(\mathrm{Mor}(\mathrm{Cat}))$$ is a “path field” on $X$: an smooth assignment of a path starting at $x$ for each $x \in X$.

(This is essentially what Chris Isham calls an “arrow field”.)

So one tangent vector in $T_{\mathrm{Id}_{P_1(X)}}(\mathrm{Mor}(\mathrm{Cat}))$ is a tangent vector in $T_x P_1(X)$ for each $x$ in $X$.

But the crucial difference is that on $T_{\mathrm{Id}_{P_1(X)}}(\mathrm{Mor}(\mathrm{Cat}))$ there is a monoidal structure: we may compose two path fields to yield another path field.

This implies that an ordinary vector field on $X$ (as opposed to a path field) is a smooth functor $$v : \Sigma \mathbb{R} \to \Sigma T_{\mathrm{Id}_{P_1(X)}}(\mathrm{Mor}(\mathrm{Cat})) \,.$$ as also described here.

(Well, I should admit that it is easy to see that every vector field on $X$ gives injectively rise to such a smooth functor and that this is I think it is also bijective, but haven’t written out the proof of the converse statement yet, which is a little more technical.)

Urs,

There are a couple of ways to answer this. First, one can ask about the tangent complex to pt/G. More generally for a quotient stack X/G (with X smooth) we can calculate its tangent complex as $\g_X \to T_X$, where $\g_X$ is the action Lie algebroid (ie the trivial bundle with fiber the Lie algebra) and the map is the action (aka anchor map). So in this case it’s just $\g \to 0$. This needs to be interpreted as a complex of vector bundles on the quotient, ie we consider $\g$ (and $0$) not as vector spaces but as representations of $G$.

As for the tangent stack, it’s not very interesting here I think, it’s just pt/G itself - there are no new tangent directions in the quotient. Another way to say this: we can naively define the tangent stack as Spec of $H^0$ of the symmetric algebra of the cotangent complex (dual to the above tangent complex), which in this case is Spec of $H^0$ of the symmetric algebra on $g^*$ in degree one, which is just $pt/G$.

However, one does get the interesting answer one feels like one deserves if you look at the derived cotangent stack (ie pass to the world of stacks over commutative dgas, rather than over commutative rings). In other words while there are no new degree zero cotangent directions in pt/G, there is a g* worth of degree 1 cotangent directions…. said another way, Spec of the symmetric algebra on the tangent complex defines a (differential) graded vector bundle over pt/G, which is just g* in degree 1 (again considered as the coadjoint representation of G so as to define a vector bundle on $pt/G$). Maybe that fits better with what you would like?

(Note I switched from tangent to cotangent to be a little careful with gradings: the world of derived stacks is not symmetric with respect to positive and negative gradings – of course one can pass to the super-world, and remember only $\Z/2$ gradings, if one so desires, and this distinction goes away - or perhaps another stable setting..)

More generally if we pass to higher (smooth) stacks we’ll get tangent complexes concentrated in an arbitrary range of negative degrees I believe, and derived cotangent stacks which are nonnegatively graded “vector bundles” over them.

The magic ink did work. Before I call it quits today, I’ll share this, a neat (I think) refinement of the concept of $n$-curvature using the concept of tangent categories.

Curvature from categorical differentials of transport

Abstract. Using the concept of tangent categories, a notion of differential of functors can be defined. We indicate how for the case that the functor is the parallel transport functor of a bundle with connection, its differential is the corresponding curvature 2-functor.

The point is that the tangent category construction in fact gives, for each category $C$, a functor $$T C : C^{\mathrm{op}} \to \mathrm{Cat}$$ which sends any morphism $$f : a \leftarrow b$$ to the functor $$T_a C \stackrel{T_f C}{\to} T_b C$$ between the tangent categories over the source and target, which simply precomposes with $f$.

(I wouldn’t be surprised if everything I am doing here already runs under some standard name and is well known. Rather, I am surprised that nobody has told me so yet. So if you are an expert and shaking your head at my naïvety, please drop me a note and release me from my ignorance!)

Using this, one finds that for every functor between categories $$F : C \to D$$ the composition $$T_* F : C \stackrel{F}{\to} D \stackrel{T D}{\to} \mathrm{Cat}$$ has a unique extension to a 2-functor $$d\mathrm{tra} : C_2 \to \mathrm{Cat}$$ when $D$ is a groupoid. Here $$C_2 = \mathrm{Codisc}(C)$$ is the 2-category whose Hom-categories are the codiscrite groupoids over the Hom-sets of $C$.

Applying all this to the special case that $C$ is some category of paths in some space and that $D$ is some codomain where parallel transport takes values in, this reproduces all the stuff about curvature 2-transport which I was going on about.

In particular, notice that if we have parallel transport $$\mathrm{tra} : P_1(X) \to \Sigma G$$ in a trivial $G$-bundle (possibly the local trivialization of some nontrivial one) then the curvature 2-transport $$d\mathrm{tra} := \mathrm{curv} : P_2(X) \to \mathrm{Cat}$$ sends each point to a fiber of the form $$T \Sigma G := \mathrm{INN}(G)$$ over it.

A nice aspect of this concise formulation of 2-curvature is that it unifies the two concepts of morphisms into the curvature 2-functor which I have been emphasizing so much.

The point is that $\mathrm{tra}$ is trivializable in various ways (closely related to the fact that $\mathrm{INN}(G)$ is equivalent to the trivial category.) But the choice of trivialization encodes interesting information:

a) Stokes’ theorem is essentially the fact that the transport 1-functor is the component map of an isomorphism $$d I \stackrel{\mathrm{tra}}{\to} \mathrm{curv} \,,$$ where $I$, in the above simple example, is the trivial $G$-bundle with trivial connection.

b) A section of the original $G$-bundle, together with its covariant derivative under the given connection, is an equivalence $$e : d J \stackrel{\sim}{\to} \mathrm{curv} \,,$$ where $J$ is the trivial point-bundle.

If you think about it, that’s pretty neat, I’d say. It’s the concept of holography, really.

the 2 morphisms seem a little forced

Let’s see.

First to clarify, in case I wasn’t being clear on this:

notice that there can be nontrivial 2-morphisms in $T C$ only if $C$ is a 2-category to start with. If $C$ is an $n$-category, then so is $T C$.

Then concerning your point:

what made me finally think that $T C$ is a good idea is precisely that its (higher) morphisms are not “forced” at all.

In fact, in the early stages I defined these higher morphisms by hand, such that they’d do for me what it seemed I needed to get done.

Then, after a little thinking, it occurred to me that what I did by hand was a actually just spelling out a very natural basic condition:

the higher morphisms in $T C$ are just higher morphisms of $n$-functors – of $n$-functor that start at the “fat point”.

Such higher morphisms of $n$-functors in general have components given by lots of “cylinders” in the codomain.

$T C$ is defined to be that maximal sub $n$-category of all these higher morphisms of $n$-functors where these cylinders have the identity morphism over one of the two objects of the fat point.

I liked to phrase this in the following suggestive (I belive) way:

$T C$ is that maximal sub $n$-category of $\mathrm{Hom}_{n\mathrm{Cat}}(\mathbf{pt},C)$ which collapses to a 0-category when pulled back along the inclusion of the point into the fat point.

I am thinking of this condition as the “arrow-theoretic analogue” of “infinitesimality”:

a tangent vector is something which becomes invisible when I stop distinguishing between a point and a “fat point”.

Of course this last sentence by itself is just heuristsics, which you may or may not find a helpful guide for thinking along the lines that I am proposing here. The real proof of principle is in checking whether $T C$ the way I defined it usefully captures what it is supposed to in applications.