The n-Category Café

Sorry – in the discussion of $n$-jet bundles, it ought to say that they had not previously been considered in the 1-categorical tangent category setting.

Thanks for these reactions, Tim. I guess the first thing I’d like to know is the relationship between BBC’s tangent $\infty$-categories and what nLab has as tangent (infinity,1)-category, as found in Lurie’s work.

The former concerns equipping an $(\infty, 1)$-category with one of possibly many tangent structures; the latter is a fixed construction (the fiberwise stabilization of the codomain fibration).

The latter construction applied to an $(\infty, 1)$-topos gives an $(\infty, 1)$-topos. Cohesiveness is retained. There are also higher jets.

One obvious question then is whether the latter tangent construction gives rise to one of BBC’s tangent structures.

In the second part of this paper, we construct a specific tangent ∞-category for which the tangent bundle functor is equivalent to that defined by Lurie, and which encodes the theory of Goodwillie calculus.

So what does setting this construction in the more general framework of a tangent structure offer? We don’t expect a manifold to have more than one tangent bundle.

The ABFJ approach is characterised here in terms of the tangent $(\infty, 1)$-topos construction.

I’m glad you asked, David, because it prompted met to take a closer look and I realized I missed one subtlety (which of course they’re careful to explain!).

First off, though, a tangent category is a category $C$ whose objects $X$ are thought of as “spaces”. $C$ is equipped with a “tangent endofunctor” $T: C \to C$, which sends a space $X$ to the total space $TX$ of its tangent bundle. There’s various extra data and coherence involved, such as a natural “projection” map $TX \to X$, a natural “zero section” map $X \to TX$, etc.

For example, the category of smooth manifolds is a tangent category, where $TX$ is the usual tangent space of the manifold $X$. This is the original motivating example for Rosicky. There are many other examples 1-categorically, e.g. any model of synthetic differential geometry is a tangent category, and the revival they’ve seen in recent years has to do with examples coming from computer science as I understand. In a more homotopical realm, BJORT use it as a framework to organize old tools and develop new tools in abelian functor calculus.

BBC extend this notion to $\infty$-categories. Their primary example is the $\infty$-category $Cat^{diff}$ of differentiable $\infty$-categories (i.e. $\infty$-categories with finite limits and sequential colimits which commute – these are pretty mild conditions for an $\infty$-category to satisfy!). $Cat^{diff}$ is a tangent $\infty$-category, where the tangent functor $T: Cat^{diff} \to Cat^{diff}$ sends $C$ to the tangent $\infty$-category $TC$ of $C$ in exactly the sense described at the nlab article (first envisioned by Goodwillie, developed by Lurie, etc.). This is the example studied by BBC.

So there’s room for confusion – a tangent category is a category whose objects have tangent bundles, but the example we’re interested in here has objects which are themselves categories, and the term “tangent category” is already used to refer to $TX$ itself within the tangent structure.

As you might guess, it’s natural then to think about what a tangent 2-category might be (in the sense of a 2-category $C$ equipped with a 2-functor $T: C \to C$, etc). In fact, in the paper BBC give a definition (Def 5.16 – there are no surprises in generalizing from 1-categories to 2-categories) and show that $Cat^{diff}$ is a tangent $(\infty,2)$-category.

For a tangent (2-)category $C$ whose objects $X$ are categories, it’s natural to suppose that the tangent structure on $C$ is related to some sort of cohesion on each object $X$, and some kind of compatibility between the cohesive structures on each $X \in C$. For instance, in the case $C = Cat^{diff}$, we know that the tangent structure on $Cat^{diff}$ is related to a cohesive structure on many $X \in Cat^{diff}$.

The 2-categorical angle also turns out to be related to jets. I initially misunderstood slightly and thought that BBC’s notion of $n$-jet (Def 11.22) was formulated for an arbitrary tangent $(\infty,1)$-category, but now I see that it requires a tangent $(\infty,2)$-structure, to formulate (and they are clear in pointing this out). At any rate, if $C$ is a tangent $(\infty,2)$-category and $X,Y \in C$ and $x \in X$ and certain representability properties are satisfied, they define an $\infty$-category $Jet^n_x(X,Y) \subseteq C(X,Y)$ of “$n$th-degree-at-$x$ equivalence classes of 1-morphisms” from $X \to Y$ which, in the case $C = Cat^{diff}$ recovers precisely Goodwillie’s category of $x$-based $n$-excisive functors from $X$ to $Y$.

What I think I wrongly said they do, but which they do not quite do, is to give a definition in the language of tangent $(\infty,2)$-categories which specializes to the notion of $n$-jet bundle as recorded on the nlab. However, from their work, it seems very likely that such a definition exists, because the definition appearing on the nlab is very closely related to the notion of $n$-excisive functor which they explicitly recover.

I should perhaps revisit the proposal by Tom Goodwillie, we began to discuss on the nForum here, concerning connections on tangent $\infty$-categories. Hopefully, I stored it on my office computer, though when I get access to that is an unknown.

For clarity, I will use the following terminology in this post:

• A Rosicky tangent category is a tangent category in the sense I discussed above – a category $C$ equipped with an endofunctor $T: C \to C$ and so forth. A Rosicky tangent $(\infty,1)$-category is the general notion introduced by BBC.

• “The” tangent category of a differentiable $\infty$-category $X \in Cat^{diff}$ is the category $TX$ of 1-excisive functors on $X$, as defined on the nlab, and as featured in the particular Rosicky tangent $\infty$-category structure that BBC construct on $Cat^{diff}$.

One point raised by Mike in that nForum discussion is the question (in the present language) of whether $Cat^{diff}$ forms a model of SDG. I think the answer clarifies the formal situation a bit.

One thing that Leung’s work clarified is that there is a close connection between Rosicky tangent categories and SDG. As I mentioned above, Leung constructed a certain monoidal category $Weil$ of Weil algebras and characterized Rosicky tangent categories as categories $C$ equipped with an action of $Weil$ preserving certain limits. This is related (and I think partly motivated?) by the fact that any model $C$ of SDG has just such a monoidal action of a larger monoidal category $\overline{Weil}$ of Weil algebras, and in fact this fact “almost” characterizes models of SDG. In particular, as mentioned above every model of SDG is a Rosicky tangent category. Conversely, a Rosicky tangent category can be thought of as a model of SDG with weaker “representability” properties.

As I mentioned earlier, BBC discuss in their introduction that the Rosicky tangent $\infty$-category $Cat^{diff}$ does fail certain representability properties – so I believe it does not model SDG. However, BBC point out that if one passes from $Cat^{diff}$ to the locally full subcategory $Topos$ of $\infty$-toposes, there is an induced Rosicky tangent category structure on $Topos$. They conjecture that the Rosicky tangent category $Topos$ has better representability properties, so that conjecturally it seems that $Topos$ might model SDG (or some version thereof) in this way.

A subtle point is this: one nice thing about the monoidal 1-category $Weil$ is that it is suitably cofibrant so that it does not need to be modified when it comes to defining Rosicky tangent $\infty$-categories as opposed to Rosicky tangent 1-categories. I suspect that larger categories of Weil algebras like $\overline{Weil}$ might not be cofibrant in the relevant sense, and so some modification might be required to talk about $\infty$-categorical models of SDG.

And Weil algebras in that sense (infinitesimally thickened point) point us to differential cohesion with all its rich structure (jet comonad and PDEs, etc.).

By the way, we have tangent bundle category in the Rosicky sense.

So I guess a useful thing to establish would be the relationship between equipping an $\infty$-topos, $\mathbf{H}$, with a Rosicky tangent structure, and locating an $\infty$-topos which is infinitesimally cohesive over $\mathbf{H}$.

So the tangent $\infty$-topos, $T \mathbf{H}$, is infinitesimally cohesive over $\mathbf{H}$ and this is an example of a tangent bundle structure on $\mathbf{H}$.

But then there’s also interest in finding such infinitesimal thickening for an $(\infty, 2)$-category, such as (∞,1)Topos, e.g., by taking all tangent $\infty$-categories together.

Related to the $n$Lab sense of a tangent category for $\infty$-toposes is the concept of a Joyal locus which is looking for situations where the $Grpd_{\infty}$-parameterized objects of a pointed presentable $(\infty, 1)$-category form an $(\infty, 1)$-topos (and similarly for any parameterizing $(\infty, 1)$-topos).

As we see in ‘Literature 2.10’ of

• David J. Myers, Hisham Sati, Urs Schreiber, QS: Quantum Programming via Linear Homotopy Types,

the $n$Lab sense of a tangent category for $\infty$-toposes is the special case of a more general construction for $E_{\infty}$-rings, yielding the concept of $R$-linear tangent $\infty$-topos.