### Jet Categories at the nForum

#### Posted by David Corfield

Some people I talk to who have noticed a slackening off at the Café in recent months, and who know that some of this is due to John’s energy passing to his Azimuth project, don’t seem aware that another chunk of the energy didn’t just vanish, but got transmitted to the nForum. This venue has the advantage of democratically allowing anyone to initiate a discussion, but the disadvantage that people don’t seem to want to wade through every announcement of any alteration to an nLab entry for the occasional interesting nugget. For whatever reason, we don’t get visited much nowadays by some of the prestigious visitors of yesteryear. Still, if you want to see the day to day movements of Urs sweeping up great clumps of mathematical physics into a glorious synthetic package, the nForum is the place to be. Or, if you prefer to read the finished product, see his site.

Something we occasionally suggest to each other is a periodic digest of what’s happening at the $n$Lab, but I believe we’ve only managed two to date (I, II). Let me try something less ambitious.

One idea that’s interested me of late over at the nForum is to generalise the observation that an $(\infty, 1)$-category, $C$, has a tangent $(\infty, 1)$-category, $T C$. If $C$ is an $(\infty, 1)$-topos, then so too is $T C$. If further it is cohesive, so too is $T C$ (tangent cohesion). Since $(\infty, 1)$-toposes are the natural homes for cohomologies, and cohesive ones for differential cohomology, we would expect tangent $(\infty, 1)$-toposes to provide a version of such cohomologies, and they do, namely, twisted cohomology.

Now, given that tangent spaces are the natural environments for linear approximations of functions, one might expect there to be a parallel to the higher tangency provided by jet spaces. So we’ve been talking about the idea of a $k$th jet $(\infinity, 1)$-category of $C$, which will be a topos and cohesive again when $C$ is. There ought then to be cohomologies corresponding to these intermediate approximations.

There’s still so much I don’t understand about this general area, Goodwillie calculus, even with Charles Rezk helping us out, and so much still needs to be added to the $n$Lab (orthogonal calculus, manifold calculus,…). Recruits are always welcome.

## Typo

Your link to Azimuth is not followable, it needs an http://.