The n-Category Café

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.