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.