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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?


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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 : (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

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