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
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