### 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 $(\infty, 1)$-categorification of the former, comparison becomes more pressing:

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

and

- 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).

## 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 $(\infty,2)$-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 $(\infty,2)$-category theory.

I’m very interested to see how they end up getting the bundles of $n$-excisive functors as $n$-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.