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 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/.