Consider an arbitrary site (or an ∞-site) S. In fact, the constructions below only depend on the underlying topos (or ∞-topos) T of S, and not on S itself. Below “sheaf”, “∞-sheaf”, “stack”, and “∞-stack” are all synonyms for presheaves (of spaces) that satisfy homotopy descent.
The nth Weiss topology (n≥0 or n=∞) on T is defined by declaring a family {U_i→X} to be a covering family if its kth cartesian power {U_i^k→X^k} is a covering family of X^k in T for any 0≤k≤n. If m≤n, then the mth topology contains the nth topology. The category of n-polynomial functors is defined to be the category of sheaves in the nth Weiss topology. The 1st Weiss topology almost coincides with the original topology (for k=0 we see that the empty cover (of the intitial object) is excluded from the 1st Weiss topology), so a sheaf in the ordinary sense is a sheaf in the 1st Weiss topology that is reduced.
Given a presheaf F on T, i.e., a functor T^op→Spaces (one can also take Sets or any other nice target category), we define the nth Taylor approximation T_n(F) as the sheafification of F in the nth Weiss topology. We have a canonical tower F→T_∞(F)→⋯→T_n(F)→⋯→T_0(F).
If S=sSet^op, we recover the homotopy calculus.
If S is the site of manifolds (or the cartesian site), we recover the manifold calculus.
If S is the ∞-site of manifolds (i.e., enriched in spaces, and T=Sh(S,sSet)=Fun(S^op,sSet)_descent is its ∞-topos), then we recover the enriched manifold calculus, as defined by Boavida and Weiss.
If S=sSet^op, we recover the homotopy calculus, provided that we replace Čech covers with hypercovers as explained in https://nforum.ncatlab.org/discussion/6946/weiss-topology-and-goodwillie-calculus/.