nLab manifold calculus

Idea

Let $M$ be a smooth manifold without boundary and denote by $\mathcal{O}(M)$ the poset of open subsets of $M$, as defined in Wei99, ordered by inclusion. Manifold calculus is a way to study (say, the homotopy type of) contravariant functors $F$ from $\mathcal{O}(M)$ to spaces which take isotopy equivalences to (weak) homotopy equivalences. In essence, it associates to such a functor a tower - called the Taylor tower - of polynomial approximations which in good cases converges to the original functor, very much like the approximation of a function by its Taylor series. (BriWei)

In BriWei the authors develop an enriched version.

Related concepts

• Goodwillie calculus
• orthogonal calculus
• Weiss topology

References

• Michael Weiss, Embeddings from the point of view of immersion theory I, Geom. Topol. 3 (1999), 67–101

• Thomas Goodwillie, Michael Weiss, Embeddings from the point of view of immersion theory, Part II, Geometry and Topology 3 (1999), 103-118.

• Pedro Boavida de Brito, Michael Weiss, Manifold calculus and homotopy sheaves, Homology, Homotopy and Applications, vol. 15(2), 2013, pp.361–383 (arXiv:1202.1305)

• Hiro Lee Tanaka, Manifold calculus is dual to factorization homology, pdf

• Brian Munson, Introduction to the manifold calculus of Goodwillie-Weiss (arXiv:1005.1698)

• Thomas Willwacher, Configuration spaces of points and real Goodwillie-Weiss calculus, talk at Isaac Newton Institute, 2018.

• Kensuke Arakawa, A Context for Manifold Calculus, (arXiv:2403.03321)