nLab jet (infinity,1)-category

Idea

Given a differentiable (∞,1)-category $\mathcal{C}$, then the (∞,1)-category of n-excisive functors from the finite pointed objects in ∞Grpd to $\mathcal{C}$ behaves like the bundles of order-$n$ Goodwillie derivatives over all objects of $\mathcal{C}$. Hence this is like an analog in Goodwillie calculus of the $n$th order jet bundle in differential geometry.

In particular for $n = 1$ the “1-jet $\infty$-category” of $\mathcal{C}$ is the tangent (∞,1)-category of $\mathcal{C}$.

Properties

Jet toposes

By the discussion at n-excisive functor – Properties – n-Excisive approximation, for $\mathbf{H}$ an (∞,1)-topos also its $n$th jet $(\infty,1)$-category

$$J^n \mathbf{H} \coloneqq Exc^n(\infty Grpd_{fin}^{\ast/}, \mathbf{H})$$

is an $(\infty,1)$-topos, for all $n \in \mathbb{N}$. For $n = 1$ this is the tangent (∞,1)-topos $J^1 \mathbf{H} = T \mathbf{H}$ (see also at tangent cohesion). If $\mathbf{H}$ is cohesive, so too is $J^n \mathbf{H}$.

Related entries

• twisted cohomology

References

• Jacob Lurie, section 6.1 of Higher Algebra