nLab
jet (infinity,1)-category

Contents

Idea

Given a differentiable (āˆž,1)-category š’ž\mathcal{C}, then the (āˆž,1)-category of n-excisive functors from the pointed objects in āˆžGrpd to š’ž\mathcal{C} behaves like the bundles of order-nn Goodwillie derivatives over all objects of š’ž\mathcal{C}. Hence this is the analog of the nnth order jet bundle in Goodwillie calculus.

In particular for n=1n = 1 this is the tangent (āˆž,1)-category of š’ž\mathcal{C}.

Properties

Jet toposes

By the discussion at n-excisive functor ā€“ Properties ā€“ n-Excisive approximation, for H\mathbf{H} an (āˆž,1)-topos also its nnth jet (āˆž,1)(\infty,1)-category J nHJ^n \mathbf{H} is an (āˆž,1)(\infty,1)-topos, for all nāˆˆā„•n \in \mathbb{N}. For n=1n = 1 this is the tangent (āˆž,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}.

References

Section 7.1 of

Revised on December 20, 2013 06:50:41 by Urs Schreiber (89.204.139.131)