jet (infinity,1)-category



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

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


Jet toposes

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

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

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


Revised on February 10, 2016 06:35:58 by Urs Schreiber (