# Seeing stacks in the Calculus of Functors

Recently I was told (by an algebraic geometer) that when algebraic geometers look at the Calculus of Functors, they think of stacks.

When I look at the Calculus of Functors, I see a categorification of polynomial approximation. While I am at best a beginner at algebraic geometry, I would like to understand why he is saying this.

My motivation is twofold. First, I want to know why he is saying this, and second, because I am beginning to learn about stacks, and I want to come at it with some intuition. I have pursued the obvious routes of reading about them in general (such as Tolland's Blog Post).

Specifically, my question is

How does one see Calculus of Functors as stacks?

A secondary question,

Is there some highly degenerate way to look at stacks to see polynomial approximation?

Thanks!

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I have no clue about question one, although of course I am curious. In question two, what do you mean by "degenerate"? – Tom Goodwillie Jul 27 '10 at 0:44
You're probably more likely to get answer if you give some idea of what the calculus of functors is. – JBorger Jul 27 '10 at 1:34
Calculus of functors is an organizing principle in homotopy theory. It is named for an analogy with (differential) calculus. Calculus is concerned with approximating functions by linear functions; functor calculus is concerned with approximating functors of a certain kind by a special kind of functor that may be called linear. Linearity of functions is a lot like a sheaf condition, in a way, so I can see that it might suggest descent. Like calculus, functor calculus has nth degree Taylor polynomials, not just for n=1. That's the sense of "polynomial approximation" here. – Tom Goodwillie Jul 27 '10 at 2:50
There's a pretty good wikipedia page on the topic here : en.wikipedia.org/wiki/Calculus_of_functors (n.b. : the above exchange between Bischof and Goodwillie may become clearer once the reader learns that the calculus of functors is often known as the Goodwillie Calculus). – Andy Putman Jul 27 '10 at 4:22
The "homotopy calculus" of functors from Top to Top (or to Spectra) doesn't look a whole lot like stacks to me, but the "manifold calculus" of space-valued functors on some poset of subspaces of a manifold M does look very much like stacks to me. When I last thought about this (which was during Tom's talks at the Georgia topology conference), it looked kind of as though there was a hierarchy of different Grothendieck topologies on that poset of subspaces, and an nth degree polynomial functor was a stack relative to the nth topology. – Mike Shulman Jul 27 '10 at 4:27

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.

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Sounds great. Should this answer come with a reference or with the announcement of a reference? – Urs Schreiber Feb 9 at 11:00
@UrsSchreiber: I learned this material from Hiro Tanaka's notes Manifold calculus is dual to factorization homology. It is more-or-less explained in this way in the paper of Boavida and Weiss. – Dmitri Pavlov Feb 9 at 11:58
Incredible, Dmitri, this question has been open for so long, and this seems like a great reply. – B. Bischof Feb 9 at 20:18
Could we check the claim for Goodwillie calculus? Boavida-Weiss do not quite speak about that explicitly, do they. What exactly is the claim meant to be for this case? – Urs Schreiber Feb 13 at 13:56
@UrsSchreiber: I answered on nForum. – Dmitri Pavlov Feb 13 at 15:10