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1. • CommentRowNumber1.
   • CommentAuthorDavid_Corfield
   • CommentTimeJul 23rd 2010
   • (edited Jul 23rd 2010)
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexWe talk of a 'homogeneous linear functor' at [Goodwillie calculus](http://ncatlab.org/nlab/show/Goodwillie+calculus), a functor which maps homotopy pushout squares to homotopy pullback squares. There are also higher degree homogeneous functors which map $(n+1)$-dimensional cubical homotopy pushout diagrams to $(n+1)$-dimensional cubical homotopy pullback diagrams. These allow polynomial approximation in the functor calculus. We also have [linear functor](http://ncatlab.org/nlab/show/linear+functor) and [polynomial functor](http://ncatlab.org/nlab/show/polynomial+functor). I take it that these latter two are unrelated to each other, and to the functor calculus terms. I think we need some disambiguation. Does anyone know why in the Goodwillie calculus those functors are called linear? Perhaps [this](http://www.birs.ca/events/2011/5-day-workshops/11w5058) helps: >At the heart of Algebraic Topology is the study of geometric objects via algebraic invariants. One would like such invariants to be subtle enough to capture interesting geometric information, while still being computable in the sense of satisfying some sort of local-to-global properties. >A simple and familiar example of this is the Euler characteristic $e(X)$, where the local-to-global property for good decompositions takes the form $e(U \union V) = e(U) + e(V) - e(U \cap V)$. A more sophisticated invariant is homology, where the local-to-global equation above is replaced by the Meyer--Vietoris sequence. Finally one can consider the functor $S P^{\infty}: Top \to Top$, assigning to a based topological space, its infinite symmetric product. This functor has the property that it takes homotopy pushout squares (i.e. good decompositions) to homotopy pullback squares. As the Dold-Thom theorem tells us that the homotopy groups $\pi_*(SP^{\infty}(X)) = H_*(X)$, the Meyer-Vietoris sequence for homology is thus a consequence of applying $\pi_*(-)$ to the homotopy pullback square. >It was the insight of Tom Goodwillie in the 1980's that such "linear" functors $F: Top \to Top$ form just the beginning of a hierarchy of polynomial functors, where a polynomial functor of degree $n$ takes appropriate sorts of $(n+1)$-dimensional cubical homotopy pushout diagrams to $(n+1)$-dimensional cubical homotopy pullback diagrams. Furthermore, many important functors admit good approximations by a Taylor tower of polynomial approximations.

   We talk of a ’homogeneous linear functor’ at Goodwillie calculus, a functor which maps homotopy pushout squares to homotopy pullback squares. There are also higher degree homogeneous functors which map $(n+1)$-dimensional cubical homotopy pushout diagrams to $(n+1)$-dimensional cubical homotopy pullback diagrams. These allow polynomial approximation in the functor calculus.

   We also have linear functor and polynomial functor. I take it that these latter two are unrelated to each other, and to the functor calculus terms. I think we need some disambiguation.

   Does anyone know why in the Goodwillie calculus those functors are called linear? Perhaps this helps:

   At the heart of Algebraic Topology is the study of geometric objects via algebraic invariants. One would like such invariants to be subtle enough to capture interesting geometric information, while still being computable in the sense of satisfying some sort of local-to-global properties.

   A simple and familiar example of this is the Euler characteristic $e(X)$, where the local-to-global property for good decompositions takes the form $e(U \union V) = e(U) + e(V) - e(U \cap V)$. A more sophisticated invariant is homology, where the local-to-global equation above is replaced by the Meyer–Vietoris sequence. Finally one can consider the functor $S P^{\infty}: Top \to Top$, assigning to a based topological space, its infinite symmetric product. This functor has the property that it takes homotopy pushout squares (i.e. good decompositions) to homotopy pullback squares. As the Dold-Thom theorem tells us that the homotopy groups $\pi_*(SP^{\infty}(X)) = H_*(X)$, the Meyer-Vietoris sequence for homology is thus a consequence of applying $\pi_*(-)$ to the homotopy pullback square.

   It was the insight of Tom Goodwillie in the 1980’s that such “linear” functors $F: Top \to Top$ form just the beginning of a hierarchy of polynomial functors, where a polynomial functor of degree $n$ takes appropriate sorts of $(n+1)$-dimensional cubical homotopy pushout diagrams to $(n+1)$-dimensional cubical homotopy pullback diagrams. Furthermore, many important functors admit good approximations by a Taylor tower of polynomial approximations.

2. • CommentRowNumber2.
   • CommentAuthorChris SchommerPries
   • CommentTimeJul 23rd 2010
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   Author: Chris SchommerPries
   Format: MarkdownItexThere are several reasons to think of Goodwillie's linear functors as linear, but here is my favorite. Any such linear functor from spaces to spaces is a generalized cohomology theory. More precisely, there is a model category on the functors from spaces to spaces called the model category of W-spaces. Really I should be using pointed spaces here. This model category is one of the standard models for the category spectra and so the fibrant objects can be thought of as the (co)homology theories. The fibrant objects are precisely those functors which are linear in Goodwillie's sense. The example you mention $SP^\infty$ corresponds to ordinary cohomology (well there is a $\pi_0$ issue, but let's ignore that). In general evaluating the linear functor E on a space X gives you a space which should be thought of as the smash product of E and X. So now your question is why should spectra/cohomology theories be thought of as linear functors? Well if you think of spectra as analogous to abelian groups, then applying a spectrum to a space (i.e. smashing with it) is a linearization of that space. Following this analogy, if we now have any old functor from space to spaces we can take its fibrant replacement in W-spaces. This is a linear functor which is the best approximation to the original functor. So it is like taking a derivative of a function. Goodwillie's insight was to extend this analogy to encompass the rudiments of calculus. There is in fact a whole series of model categories on functors from spaces to spaces where the fibrant objects are Goodwillie's polynomial functors of degree n.

   There are several reasons to think of Goodwillie’s linear functors as linear, but here is my favorite. Any such linear functor from spaces to spaces is a generalized cohomology theory. More precisely, there is a model category on the functors from spaces to spaces called the model category of W-spaces. Really I should be using pointed spaces here. This model category is one of the standard models for the category spectra and so the fibrant objects can be thought of as the (co)homology theories. The fibrant objects are precisely those functors which are linear in Goodwillie’s sense. The example you mention $SP^\infty$ corresponds to ordinary cohomology (well there is a $\pi_0$ issue, but let’s ignore that). In general evaluating the linear functor E on a space X gives you a space which should be thought of as the smash product of E and X.

   So now your question is why should spectra/cohomology theories be thought of as linear functors? Well if you think of spectra as analogous to abelian groups, then applying a spectrum to a space (i.e. smashing with it) is a linearization of that space.

   Following this analogy, if we now have any old functor from space to spaces we can take its fibrant replacement in W-spaces. This is a linear functor which is the best approximation to the original functor. So it is like taking a derivative of a function. Goodwillie’s insight was to extend this analogy to encompass the rudiments of calculus. There is in fact a whole series of model categories on functors from spaces to spaces where the fibrant objects are Goodwillie’s polynomial functors of degree n.

3. • CommentRowNumber3.
   • CommentAuthorDavid_Corfield
   • CommentTimeJul 23rd 2010
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexThat's very good thanks, and it should be included in an nLab page. But before setting up a new page on linear functors in this sense, is it just possible that there is a connection to the other linear functors, as may be the [case](http://www.math.ntnu.no/~stacey/Vanilla/nForum/comments.php?DiscussionID=1635&Focus=14391#Comment_14391)? >...if you think of spectra as analogous to abelian groups,... I suppose abelian groups form another tangent category at $\mathbb{Z}$ in $CRing$.

   That’s very good thanks, and it should be included in an nLab page. But before setting up a new page on linear functors in this sense, is it just possible that there is a connection to the other linear functors, as may be the case?

   …if you think of spectra as analogous to abelian groups,…

   I suppose abelian groups form another tangent category at $\mathbb{Z}$ in $CRing$.

4. • CommentRowNumber4.
   • CommentAuthorTobyBartels
   • CommentTimeJul 23rd 2010
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   Author: TobyBartels
   Format: MarkdownItexI added a dismabiguation hatnote to [[linear functor]].

   I added a dismabiguation hatnote to linear functor.