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February 2, 2021

Tangent ∞-Categories and Cohesion

Posted by David Corfield

I’ve been wondering for a while about the relationship between Robin Cockett, Geoff Cruttwell, and colleagues’ categorical approach to differential calculus and differential geometry, and similar constructions possible in the setting provided by cohesive (∞,1)-toposes.

Now with the appearance of a (,1)(\infty, 1)-categorification of the former, comparison becomes more pressing:

  • Kristine Bauer, Matthew Burke, Michael Ching, Tangent \infty-categories and Goodwillie calculus (arXiv:2101.07819)


  • Michael Ching, Dual tangent structures for infinity-toposes, (arXiv:2101.08805).

In the first of these the authors write

we might speculate on how the Goodwillie tangent structure fits into the much bigger programme of ‘higher differential geometry’ developed by Schreiber [Sch13, 4.1], or into the framework of homotopy type theory [Pro13], though we don’t have anything concrete to say about these possible connections. (p. 13)

Presumably we’d need cohesive HoTT/linear HoTT.

Anyone interested might take a look also at nLab: infinitesimal cohesive (∞,1)-topos, nLab: tangent cohesive (∞,1)-topos, nLab: twisted cohomology, nLab: jet (∞,1)-category.

There’s modal HoTT work in this area, here.

No doubt useful too is

  • Mathieu Anel, Georg Biedermann, Eric Finster, André Joyal, Goodwillie’s Calculus of Functors and Higher Topos Theory (arXiv:1703.09632).
Posted at February 2, 2021 8:12 AM UTC

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Re: Tangent ∞-Categories and Cohesion

This is very exciting. I remember hearing Michael Ching speak about this maybe a year or two ago and thinking that it sounded like such an ambitious project, I wouldn’t expect to see it completed for quite some time. Kudos to them for managing this feat!

I always found it amusing that although Lurie wrote the book on doing Goodwillie calculus in \infty-categories, most of it does not appear in his infinity,2-Categories and the Goodwillie Calculus which is primarily a trailblazing foundational text on (,2)(\infty,2)-categories. (The Goodwillie calculus appears in Higher Algebra, Ch. 6). So it’s very interesting to see at the end of the abstract that Bauer, Burke, and Ching do end up leveraging what sounds like some serious (,2)(\infty,2)-category theory.

I’m very interested to see how they end up getting the bundles of nn-excisive functors as nn-jet bundles in the sense of tangent categories, deriving it just from the notion of 1-excisive functor which goes into the tangent structure. Maybe they use some earlier work of Ching which does do something like that at least in a stable setting.

Posted by: Tim Campion on February 8, 2021 9:41 PM | Permalink | Reply to this

Re: Tangent ∞-Categories and Cohesion

The introduction to the paper is very clear, and at the end they include some nice discussion of conjectures and possibilities for future work. I can’t resist sharing some thoughts after reading it. I’ll refer to the authors as “BBC” by their last initials. They mention the work of BJORT as an inspiration for the project, which seems apt because before that paper, the gulf between what 1-category theorists were doing with tangent categories and the sorts of things \infty-category theorists were doing seemed vast.

  • As an outsider to the theory, I’ve always been a bit intimidated by the theory of tangent categories partly because the number of pieces of data involved in the definition is fairly large, and accordingly so is the number of coherence diagrams. This sort of situation is really a bit of a red flag for generalization to higher categories, where generally every piece of data and coherence data will entail adding in all sorts of higher coherence, magnifying such complexity very quickly. But there is a beautiful, elegant solution already in the literature which BBC are able to leverage: Poon Leung’s repackaging of all this data and coherence into a simple definition: A tangent structure on XX is simply an action of a certain monoidal category WeilWeil of Weil algebras on XX. WeilWeil is a certain category of finitely-generated commutative semirings, and if I understand the introduction correctly, BBC are able to use this category “out of the box” to define their \infty-categorical notion of a tangent category. They didn’t, for instance, have to replace the semirings involved with fancy E E_\infty-algebras. I would like to understand why this works better. I guess I’ll have to read the paper!

  • The precise tangent \infty-category related to Goodwillie calculus has underlying \infty-category the \infty-category of differentiable infinity-categories in Lurie’s sense – defined to have the bare minimum of structure to get Goodwillie calculus off the ground (although one could imagine getting away with less at the cost of requiring bigger filtered colimits to exist). The tangent \infty-category of such an \infty-category XX is the \infty-category of parameterized spectrum objects in XX, i.e. the \infty-category of (unpointed) excisive functors S * finXS_\ast^{fin} \to X. This is right in line with the idea, going back to Goodwillie, that the cateogry of parameterized spectrum objects should be thought of as (the total space of) the tangent bundle to a category.

  • They develop a fraction of the apparatus which has been developed 1-categorically in tangent categories by many different authors, now in the \infty-categorical setting, and also develop nn-jet bundles, which had not previously been considered in the \infty-categorical setting. The idea, naturally motivated by the geometric analogy and recovering if of course when specialized to the tangent category of smooth manifolds, is to define two functors F,G:XYF,G: X \rightrightarrows Y to have the same nn-jet at an object xXx \in X if they induce the same map on the nn-fold tangent space T x nXT^n_x X at xx. And this recovers the notion of n-jet bundle of an \infty-category which David alluded to, which I assume also goes back to Goodwillie, and which is closely related to Goodwillie calculus (and the proof of this features the established machinery of Goodwillie calculus). This is a much neater story than I’d have thought we had any right to expect!

  • Their ideas for future work are inspiring. Among quite a few other theings, they discuss a bit the restriction of the above tangent structure to \infty-toposes, relating to the work of ABFJ that David linked to. Specifically, they have an interesting conjecture: although the tangent structure on differentiable \infty-categories is not “representable” by an “infinitesimal object” in Rosicky’s sense, they conjecture that after restricting to \infty-toposes and passing to an appropriate “dual”, the tangent structure is representable, by the \infty-topos of parameterized spectra.

For awhile now, I’ve been confused about the relationship between \infty-toposes and Goodwillie calculus. Because Goodwillie calculus exploits a geometric picture where you think of a category as a “directed fundamental groupoid” of a space – objects are points, and morphisms are paths. Whereas topos theory exploits a totally different analogy, where objects of a topos are etale covers of the corresponding space, and morphisms are maps between etale covers. When they come together, I end up with two seemingly unrelated analogies running in my head, and I’m never quite sure which one to focus on, or how to get them to talk to each other.

  • Lots more interesting ideas there. It feels a shame to gloss over them!

One thing I’d be interested in seeing is whether there is a tangent structure related to Dotto and Moi’s Equivariant Excision Dotto’s Higher Equivariant Excision in the same way that the Goodwillie differential structure is related to ordinary excision. One might even wonder about an analog in motivic homotopy theory. These are places where one has an interesting notion of “stabilization” which is slightly more sophisticated than ordinary stabilization.

Posted by: Tim Campion on February 8, 2021 11:05 PM | Permalink | Reply to this

Re: Tangent ∞-Categories and Cohesion

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

Posted by: Tim Campion on February 8, 2021 11:09 PM | Permalink | Reply to this

Re: Tangent ∞-Categories and Cohesion

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 (,1)(\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 (,1)(\infty, 1)-topos gives an (,1)(\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.

Posted by: David Corfield on February 10, 2021 5:39 PM | Permalink | Reply to this

Re: Tangent ∞-Categories and Cohesion

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

Posted by: David Corfield on February 10, 2021 6:06 PM | Permalink | Reply to this

Re: Tangent ∞-Categories and Cohesion

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

For example, the category of smooth manifolds is a tangent category, where TXTX is the usual tangent space of the manifold XX. 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 diffCat^{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 diffCat^{diff} is a tangent \infty-category, where the tangent functor T:Cat diffCat diffT: Cat^{diff} \to Cat^{diff} sends CC to the tangent \infty-category TCTC of CC 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 TXTX 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 CC equipped with a 2-functor T:CCT: 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 diffCat^{diff} is a tangent (,2)(\infty,2)-category.

For a tangent (2-)category CC whose objects XX are categories, it’s natural to suppose that the tangent structure on CC is related to some sort of cohesion on each object XX, and some kind of compatibility between the cohesive structures on each XCX \in C. For instance, in the case C=Cat diffC = Cat^{diff}, we know that the tangent structure on Cat diffCat^{diff} is related to a cohesive structure on many XCat diffX \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 nn-jet (Def 11.22) was formulated for an arbitrary tangent (,1)(\infty,1)-category, but now I see that it requires a tangent (,2)(\infty,2)-structure, to formulate (and they are clear in pointing this out). At any rate, if CC is a tangent (,2)(\infty,2)-category and X,YCX,Y \in C and xXx \in X and certain representability properties are satisfied, they define an \infty-category Jet x n(X,Y)C(X,Y)Jet^n_x(X,Y) \subseteq C(X,Y) of “nnth-degree-at-xx equivalence classes of 1-morphisms” from XYX \to Y which, in the case C=Cat diffC = Cat^{diff} recovers precisely Goodwillie’s category of xx-based nn-excisive functors from XX to YY.

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 (,2)(\infty,2)-categories which specializes to the notion of nn-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 nn-excisive functor which they explicitly recover.

Posted by: Tim Campion on February 10, 2021 10:13 PM | Permalink | Reply to this

Re: Tangent ∞-Categories and Cohesion

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.

Posted by: David Corfield on February 11, 2021 7:46 AM | Permalink | Reply to this

Re: Tangent ∞-Categories and Cohesion

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 CC equipped with an endofunctor T:CCT: C \to C and so forth. A Rosicky tangent (,1)(\infty,1)-category is the general notion introduced by BBC.

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

One point raised by Mike in that nForum discussion is the question (in the present language) of whether Cat diffCat^{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 WeilWeil of Weil algebras and characterized Rosicky tangent categories as categories CC equipped with an action of WeilWeil preserving certain limits. This is related (and I think partly motivated?) by the fact that any model CC of SDG has just such a monoidal action of a larger monoidal category Weil¯\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 diffCat^{diff} does fail certain representability properties – so I believe it does not model SDG. However, BBC point out that if one passes from Cat diffCat^{diff} to the locally full subcategory ToposTopos of \infty-toposes, there is an induced Rosicky tangent category structure on ToposTopos. They conjecture that the Rosicky tangent category ToposTopos has better representability properties, so that conjecturally it seems that ToposTopos might model SDG (or some version thereof) in this way.

A subtle point is this: one nice thing about the monoidal 1-category WeilWeil 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 Weil¯\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.

Posted by: Tim Campion on February 11, 2021 2:44 PM | Permalink | Reply to this

Re: Tangent ∞-Categories and Cohesion

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.

Posted by: David Corfield on February 11, 2021 6:00 PM | Permalink | Reply to this

Re: Tangent ∞-Categories and Cohesion

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

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

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

Posted by: David Corfield on February 13, 2021 8:35 AM | Permalink | Reply to this

Re: Tangent ∞-Categories and Cohesion

Actually – motivic homotopy theory might be a great application for this. There one has a notion of “stable” which is fancier than ordinary stabilization, but (so far as I know) nobody has worked out a theory of higher excision or “genuinely motivic Goodwillie calculus” (even in the equivariant case, though the basics of the higher story have been worked out by Dotto, I’m not aware of it having been fully exploited to this point). So an application of the theory here could be to define a “motivic tangent \infty-category” and derive from it a notion of nn-jet bundles which would serve as the motivic notion of higher excision.

Posted by: Tim Campion on February 8, 2021 11:14 PM | Permalink | Reply to this

Re: Tangent ∞-Categories and Cohesion

Have people thought about the tangent \infty-category of Sh (ProFinSet)Sh_\infty(ProFinSet), which as extension by its stabilization, is presumably composed of some kind of parameterized condensed spectra?

I see pyknotic spectra are discussed in sec 3.1 of Pyknotic objects, I. Basic notions.

Posted by: David Corfield on February 12, 2021 7:31 AM | Permalink | Reply to this

Re: Tangent ∞-Categories and Cohesion

To add to the rich cocktail we now have a modal type theory which allows reasoning in infinitesimal cohesive settings such as tangent \infty-toposes:

  • Mitchell Riley, Eric Finster, Daniel R. Licata, Synthetic Spectra via a Monadic and Comonadic Modality, (arXiv:2102.04099)
Posted by: David Corfield on February 12, 2021 12:55 PM | Permalink | Reply to this

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