Re: calculus, homotopy theory and more (corrected)

[Michael Batanin <mbatanin@ics.mq.edu.au>]

Dear Andre,

thank you for your very nice posting. If I understood correctly your 
proof of stabilization hypothesis it is based on classical Freudenthal
theorem. I can not resist sketching another proof (this is joint work 
with Clemens Berger and Denis-Charles Cisinski) from which Freudenthal 
theorem is a consequence.

It is based on the use of higher braided operads. Classically one can 
consider nonsymmetric, braided and symmetric operads with the values in 
a symmetric monoidal category V. If V is in addition a model category 
one can speak about contractible operads.  When V = Cat, for example,the 
category of  algebras of contractible nonsymmetric (braided, symmetric) 
operads is (=Quillen equivalent) to the category of monoidal categories 
(braided, symmetric). This prompts a line of thinking that the category 
   of  k-braided (weak) n-categories must  be an algebra of a 
contractible k-braided operad with values in Cat_n.

It is well known that we can not define k-braided operads in classical
terms (i.e. a sequence of objects with action of some groups  and a 
compatible substitution). The contradiction here is that for a 
contractible such operad to have a right homotopy type the  quotient of 
   its m-th space  by group actions must have the homotopy type of space 
of (nonordered) configuration of m-points in R^k. And this is not a 
K(\pi,1) space unless k=1,2,\infty.

Nevertheless, one can overcome this difficulty if instead of group 
action we consider an action of a category. Formally we consider a 
category of quasibijections of k-ordinals (or maps of pruned k-trees, 
which are bijections on the highest level) Q_k. The m-th connected 
component of the nerve of this category has homotopy type of the space 
of (nonordered) configuration of m-points in R^k. Then a k-collection is 
a contravaliant functor Q_k --> V. One can construct a category of 
operads  O_n(V) which have k-collections as underlying collections:

arXiv:0804.4165 Locally constant n-operads as higher braided operads. M. 
A. Batanin.

Finally, we define a k-braided operad to be an object of O_k(V) such 
that the action of any quasibijection is a weak equivalence. Obviously a 
contractible k-operad is a k-braided operad.

We want, actually, a bit more and define a model category of k-braided 
operads. This can be done by an appropriate Bousfield localization 
O_k^loc(V) of the model category O_k(V). The fibrant objects in 
O_k^loc(V) are precisely fibrant k-braided operads.

There is also a simple functor

    S: O_{k+1}(V) --> O_k(V)

induced by inclusion

s: Q_k --> Q_{k+1}

(add a tail to any pruned tree of height k).

This functor has a left adjoint, moreover, this pair of adjoint induces 
a pair of Quillen functors between localised categories
   S^{loc}: O_{k+1}^{loc}(V) <==> O_k^{loc}(V):F^{loc}

Theorem [Operadic stabilisation]:
Let V be n-truncated as a model category and k is greater or equal to n+2.
Then the Quillen adjunction
         S^{loc} |- F^{loc}
   is a Quillen equivalence. Moreover,
in this case the symmetrisation functor

sym:O_k^{loc}(V) --> SO(V)

is the left Quillen equivalence (SO(V) is the category of symmetric 
operads).

Proof. It follows from an explicit calculation of the derived left Kan 
extension along the functor s: Q_k --> Q_{k+1} and the fact that s 
induces an isomorphism of the m-th homotopy groups of the nerves when
m < k-2 , k>2 (k=2 is special and corresponds to the canonical 
homomorphism from  braid groups to symmetric groups).


Let us define the model category Br_k(V) of k-braided object in V as the 
category of one object, one arrow, ..., one (k-1)-arrow algebras of a 
contractible cofibrant k-operad. The (derived) Eckman-Hilton argument 
shows that it is Quilen equivalent to the category E_k(V) of 
E_k-algebras in V.


Let V be an n-truncated symmetric monoidal model category and k is 
greater or equal to n+2.
Then the total left derived functor  LF^{loc}   map contractible 
k-operads to contractible (k+1)-operads and Lsym map contractible 
operads to E_{\infty}-operads. So we have a Quillen equivalence of the 
categories

    E_k(V) --> Br_k(V) --> Br_{k+1}(Cat_n)--> E_{\infty}(V).

Corollary 1[BBD stabilization hypothesis]
Take V=Cat_n (for example, Cat_n = Rezk n-categories).

Corollary 2 [Freudenthal theorem]
   V = n-homotopy types.

I have little understanding of Goodwillie calculus but I know that 
operads play an important role in it.
It would be very interested to see what corresponds in calculus to 
operadic stabilization.

with best regards,
Michael.





Joyal wrote:
> Dear All,
> 
> The shift n-->n+1 which occurs in the terminologies
> 
> "n-braided monoidal category" = "(n+1)fold monoidal category"
> 
> "n-connected spaces" = "(n+1)fold loop spaces"
> 
> is very natural. A similar shift occurs in calculus.
> The analogy between calculus and homotopy theory is far reaching.
> It is the basis of the theory of analytic functors of Goodwilie.
> 
> http://www.math.brown.edu/faculty/goodwillie.html
> http://arxiv.org/abs/math/0310481
> http://ncatlab.org/nlab/show/Goodwillie+calculus
> 
> I would to describe the very elementary aspects of this theory.
> I will also say a few things about the Breen-Baez-Dolan Stabilisation 
> Hypothesis,
> claiming that it is a theorem.
> 
> Let me denote by K[[x]] the ring of formal power series in one
> variable over a field K. The ring K[[x]] bears some ressemblance
> with the category of pointed homotopy types (= pointed spaces up
> to weak homotopy equivalences). The category of pointed
> homotopy types is a ring (the product is the smash product
> and the sum is the wedge).
> 
> K === the category of pointed sets
> 
> K[[x]]=== the category of pointed homotopy types
> 
> x === the pointed circle.
> 
> The augmentation K[[x]]-->K
> === the functor  pi_0: pointed homotopy types ---> pointed sets
> 
> The augmentation ideal J
> === the subcategory of pointed connected spaces.
> 
> The n+1 power of the augmentation ideal J^{n+1}
> === the subcategory of pointed n-connected spaces.
> 
> The product of an element in J^{n+1} with an element of J^{m+1}
> is an element of J^{n+m+2}
> === the smash product of a n-connected space with
> a m-connected space is (n+m+1)-connected.
> 
> Multiplication by x === the suspension functor.
> 
> Division by x === the loop space functor.
> Notice here the difference: the loop functor is right adjoint to
> the suspension functor, not its inverse. Moreover,
> the loop space of a space has a special structure (it is a group).
> The ideal J=xK[[x]] is isomorphic to K[[x]] via division by x.
> Similarly, the category of pointed connected spaces is equivalent to
> the category of topological groups via the loop space functor
> (it is actually an equivalence of model categories).
> More generally, the ideal J^{n+1} is isomorphic to K[[x]] via division 
> by x^{n+1}.
> Similarly, the category of n-connected space is equivalent to
> the category of (n+1)-fold topological group (it is actually an
> equivalence of model categories) via the (n+1)-fold loop space functor.
> 
> 
> 
> The quotient ring K[[x]]/J^{n+1} === the category of n-truncated 
> homotopy types (=n-types)
> 
> The sequence of approximations of a formal power series f(x)=a_0+a_1x+...
> a_0
> a_0+a_1x
> a_0+a_1x+a_2x^2
> ...
> ...
> 
> === the Postnikov tower of a pointed homotopy type X:
> [pi0X]
> [pi0X;pi1X]
> [pi0X;pi1X,pi2X]
> ...
> ...
> Here, pi0X is the set of components of X,
> [pi0X;pi1X] is the fundamental groupoid of X,
> [pi0X;pi1X,pi2X] is the fundamental 2-groupoid of X, etc.
> 
> 
> The differences between f(x) and its successives approximations
> 
> R0 = f(x)-a_0               = a_1x+a_2x^2+a_3x^3+....
> R1 = f(x)-(a_0+a_1x)        =      a_2x^2+a_3x^3+a_4x^4+....
> R2 = f(x)-(a_0+a_1x+a_2x^2) =             a_3x^3+a_4x^4+a_5x^5+....
> 
> ===the Whitehead tower of X,
> 
> C_0=[0;pi1X, pi2X, pi3X,....]
> C_1=[0;0,pi2X,pi3X, pi4X,....]
> C_2=[0;0,0,pi3X,pi4X,pi4X,....]
> ....
> ....
> Here, C_0 is the connected component of X at the base point,
> C_1 is the universal cover of X constructed by from paths starting at 
> the base point,
> C_2 is the universal 2-cover of X constructed from paths starting the 
> base point, etc.
> 
> 
> Division by x is shifting down the coefficients of a power series
> If f(x)=a_1x+a_2x^2+..., then f(x)/x= a_1+a_2x+...
> Similarly, the loop space functor is shifting down the homotopy groups
> of a pointed space: if X=[a_0;a_1,a_2,...] then Loop(X)=[a_1;a_2,....].
> 
> Unfortunately, the suspension functor does not shift up the homotopy 
> groups of a space.
> It is however shifting the first 2n homotopy groups of n-connected space 
> X (n geq 1)
> by a theorem of Freudenthal:
> 
> http://en.wikipedia.org/wiki/Freudenthal_suspension_theorem
> http://en.wikipedia.org/wiki/Hans_Freudenthal
> 
> For example, if X=[0;0,a_2, a_3,...] (n=1) then Susp(X)=[0;0,0,a_2,b_3...],
> and if X=[0;0,0, a_3, a_4, a_5,...] (n=2) then Susp(X)=[0;0,0, 0,  a_3, 
> a_4, b_5,...].
> In other words, the canonical map
> 
> X-->LoopSusp(X)
> 
> is a 2n-equivalence if X is n-connected (n geq 1).
> If X[2n] denotes the 2n-type of X (the 2n-truncation of X),
> then we have a homotopy equivalence
> 
> X[2n]-->LoopSusp(X)[2n]=Loop(Susp(X)[2n+1]).
> 
> It follows that if X is a n-connected 2n homotopy type then
> we have a homotopy equivalence
> 
> X--->Loop(X')
> 
> where X'=Susp(X)[2n+1]. The space X' is said to
> be a *delooping* of X. By iterating this construction
> we can construct an infinite sequence of spaces
> 
> X=X_0, X_1, X_2,....
> 
> such that X_n=Loop(X_{n+1}). In other words,
> 
> *a n-connected 2n homotopy type is an infinite loop space (canonically)*
> 
> The (n+1)-fold loop space of a n-connected space
> is an E(n+1)-space (a E(n)-space is a model of the little n-cubes
> operad of Boardman and Vogt, a E(1)-space is a monoid,
> a E(2)-space is a braided monoid,...).
> The (n+1)-fold loop space functor induces an equivalence between the 
> category of n-connected spaces and the category of group-like E(n+1)-space
> (a monoid M is said to be group-like if pi0(M) is a group).
> Observe that the (n+1)-fold loop space of a 2n-type is a (n-1)-type.
> Freudenthal theorem implies that
> 
> *If a (n-1) homotopy type has the structure of a group-like E(n+1)-space
>   then it has also the structure of an E(infty)-space (canonically)*
> 
> A nicer statement is obtained by shifting the index n by one.
> 
> * If a n-type has the structure of a group-like E(n+2)-space then it
> has also the structure of an E(infty)-space (canonically)*
> 
> The group-like condition can be dropped:
> 
> *If a n-type has the structure of an E(n+2)-space then it has
> the the structure of an E(infty)-space (canonically)*
> 
> This is a special case of the Stabilisation Hypothesis of Breen-Baez-Dolan;
> 
> *If a n-category has the structure of an E(n+2)-category then it has the 
> structure of
> symmetric monoidal category (canonically)*
> 
> (Equivalently, *If a monoidal n-category is (n+1)-braided then it has 
> the structure of
> symmetric monoidal category (canonically)*)
> 
> It is not difficult to verify that these statements are formally equivalent.
> 
> The Breen-Baez-Dolan Stabilisation Hypothesis is a theorem.
> 
> 
> Best,
> André
> 
> 
> 
> 
> 
> 
> 
> 
> 
> 
> 


[For admin and other information see: http://www.mta.ca/~cat-dist/ ]