From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5790 Path: news.gmane.org!not-for-mail From: Michael Batanin Newsgroups: gmane.science.mathematics.categories Subject: Re: calculus, homotopy theory and more (corrected) Date: Thu, 13 May 2010 16:56:02 +1000 Message-ID: References: <4BE81F26.4020903@dm.uba.ar> Reply-To: Michael Batanin NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1273757523 17681 80.91.229.12 (13 May 2010 13:32:03 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Thu, 13 May 2010 13:32:03 +0000 (UTC) To: Andre Joyal Original-X-From: categories@mta.ca Thu May 13 15:32:01 2010 connect(): No such file or directory Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from mailserv.mta.ca ([138.73.1.1]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1OCYW8-0000Bu-AN for gsmc-categories@m.gmane.org; Thu, 13 May 2010 15:32:00 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1OCY36-0007DM-Ms for categories-list@mta.ca; Thu, 13 May 2010 10:02:00 -0300 In-Reply-To: Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5790 Archived-At: Dear Andre, thank you for your very nice posting. If I understood correctly your=20 proof of stabilization hypothesis it is based on classical Freudenthal theorem. I can not resist sketching another proof (this is joint work=20 with Clemens Berger and Denis-Charles Cisinski) from which Freudenthal=20 theorem is a consequence. It is based on the use of higher braided operads. Classically one can=20 consider nonsymmetric, braided and symmetric operads with the values in=20 a symmetric monoidal category V. If V is in addition a model category=20 one can speak about contractible operads. When V =3D Cat, for example,th= e=20 category of algebras of contractible nonsymmetric (braided, symmetric)=20 operads is (=3DQuillen equivalent) to the category of monoidal categories= =20 (braided, symmetric). This prompts a line of thinking that the category=20 of k-braided (weak) n-categories must be an algebra of a=20 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=20 compatible substitution). The contradiction here is that for a=20 contractible such operad to have a right homotopy type the quotient of=20 its m-th space by group actions must have the homotopy type of space=20 of (nonordered) configuration of m-points in R^k. And this is not a=20 K(\pi,1) space unless k=3D1,2,\infty. Nevertheless, one can overcome this difficulty if instead of group=20 action we consider an action of a category. Formally we consider a=20 category of quasibijections of k-ordinals (or maps of pruned k-trees,=20 which are bijections on the highest level) Q_k. The m-th connected=20 component of the nerve of this category has homotopy type of the space=20 of (nonordered) configuration of m-points in R^k. Then a k-collection is=20 a contravaliant functor Q_k --> V. One can construct a category of=20 operads O_n(V) which have k-collections as underlying collections: arXiv:0804.4165 Locally constant n-operads as higher braided operads. M.=20 A. Batanin. Finally, we define a k-braided operad to be an object of O_k(V) such=20 that the action of any quasibijection is a weak equivalence. Obviously a=20 contractible k-operad is a k-braided operad. We want, actually, a bit more and define a model category of k-braided=20 operads. This can be done by an appropriate Bousfield localization=20 O_k^loc(V) of the model category O_k(V). The fibrant objects in=20 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=20 a pair of Quillen functors between localised categories S^{loc}: O_{k+1}^{loc}(V) <=3D=3D> 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=20 operads). Proof. It follows from an explicit calculation of the derived left Kan=20 extension along the functor s: Q_k --> Q_{k+1} and the fact that s=20 induces an isomorphism of the m-th homotopy groups of the nerves when m < k-2 , k>2 (k=3D2 is special and corresponds to the canonical=20 homomorphism from braid groups to symmetric groups). Let us define the model category Br_k(V) of k-braided object in V as the=20 category of one object, one arrow, ..., one (k-1)-arrow algebras of a=20 contractible cofibrant k-operad. The (derived) Eckman-Hilton argument=20 shows that it is Quilen equivalent to the category E_k(V) of=20 E_k-algebras in V. Let V be an n-truncated symmetric monoidal model category and k is=20 greater or equal to n+2. Then the total left derived functor LF^{loc} map contractible=20 k-operads to contractible (k+1)-operads and Lsym map contractible=20 operads to E_{\infty}-operads. So we have a Quillen equivalence of the=20 categories E_k(V) --> Br_k(V) --> Br_{k+1}(Cat_n)--> E_{\infty}(V). Corollary 1[BBD stabilization hypothesis] Take V=3DCat_n (for example, Cat_n =3D Rezk n-categories). Corollary 2 [Freudenthal theorem] V =3D n-homotopy types. I have little understanding of Goodwillie calculus but I know that=20 operads play an important role in it. It would be very interested to see what corresponds in calculus to=20 operadic stabilization. with best regards, Michael. Joyal wrote: > Dear All, >=20 > The shift n-->n+1 which occurs in the terminologies >=20 > "n-braided monoidal category" =3D "(n+1)fold monoidal category" >=20 > "n-connected spaces" =3D "(n+1)fold loop spaces" >=20 > 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. >=20 > http://www.math.brown.edu/faculty/goodwillie.html > http://arxiv.org/abs/math/0310481 > http://ncatlab.org/nlab/show/Goodwillie+calculus >=20 > I would to describe the very elementary aspects of this theory. > I will also say a few things about the Breen-Baez-Dolan Stabilisation=20 > Hypothesis, > claiming that it is a theorem. >=20 > 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 (=3D 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). >=20 > K =3D=3D=3D the category of pointed sets >=20 > K[[x]]=3D=3D=3D the category of pointed homotopy types >=20 > x =3D=3D=3D the pointed circle. >=20 > The augmentation K[[x]]-->K > =3D=3D=3D the functor pi_0: pointed homotopy types ---> pointed sets >=20 > The augmentation ideal J > =3D=3D=3D the subcategory of pointed connected spaces. >=20 > The n+1 power of the augmentation ideal J^{n+1} > =3D=3D=3D the subcategory of pointed n-connected spaces. >=20 > The product of an element in J^{n+1} with an element of J^{m+1} > is an element of J^{n+m+2} > =3D=3D=3D the smash product of a n-connected space with > a m-connected space is (n+m+1)-connected. >=20 > Multiplication by x =3D=3D=3D the suspension functor. >=20 > Division by x =3D=3D=3D 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=3DxK[[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=20 > 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. >=20 >=20 >=20 > The quotient ring K[[x]]/J^{n+1} =3D=3D=3D the category of n-truncated=20 > homotopy types (=3Dn-types) >=20 > The sequence of approximations of a formal power series f(x)=3Da_0+a_1x= +... > a_0 > a_0+a_1x > a_0+a_1x+a_2x^2 > ... > ... >=20 > =3D=3D=3D 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. >=20 >=20 > The differences between f(x) and its successives approximations >=20 > R0 =3D f(x)-a_0 =3D a_1x+a_2x^2+a_3x^3+.... > R1 =3D f(x)-(a_0+a_1x) =3D a_2x^2+a_3x^3+a_4x^4+.... > R2 =3D f(x)-(a_0+a_1x+a_2x^2) =3D a_3x^3+a_4x^4+a_5x^5+.... >=20 > =3D=3D=3Dthe Whitehead tower of X, >=20 > C_0=3D[0;pi1X, pi2X, pi3X,....] > C_1=3D[0;0,pi2X,pi3X, pi4X,....] > C_2=3D[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=20 > the base point, > C_2 is the universal 2-cover of X constructed from paths starting the=20 > base point, etc. >=20 >=20 > Division by x is shifting down the coefficients of a power series > If f(x)=3Da_1x+a_2x^2+..., then f(x)/x=3D a_1+a_2x+... > Similarly, the loop space functor is shifting down the homotopy groups > of a pointed space: if X=3D[a_0;a_1,a_2,...] then Loop(X)=3D[a_1;a_2,..= ..]. >=20 > Unfortunately, the suspension functor does not shift up the homotopy=20 > groups of a space. > It is however shifting the first 2n homotopy groups of n-connected spac= e=20 > X (n geq 1) > by a theorem of Freudenthal: >=20 > http://en.wikipedia.org/wiki/Freudenthal_suspension_theorem > http://en.wikipedia.org/wiki/Hans_Freudenthal >=20 > For example, if X=3D[0;0,a_2, a_3,...] (n=3D1) then Susp(X)=3D[0;0,0,a_= 2,b_3...], > and if X=3D[0;0,0, a_3, a_4, a_5,...] (n=3D2) then Susp(X)=3D[0;0,0, 0,= a_3,=20 > a_4, b_5,...]. > In other words, the canonical map >=20 > X-->LoopSusp(X) >=20 > 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 >=20 > X[2n]-->LoopSusp(X)[2n]=3DLoop(Susp(X)[2n+1]). >=20 > It follows that if X is a n-connected 2n homotopy type then > we have a homotopy equivalence >=20 > X--->Loop(X') >=20 > where X'=3DSusp(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 >=20 > X=3DX_0, X_1, X_2,.... >=20 > such that X_n=3DLoop(X_{n+1}). In other words, >=20 > *a n-connected 2n homotopy type is an infinite loop space (canonically)= * >=20 > 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=20 > category of n-connected spaces and the category of group-like E(n+1)-sp= ace > (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 >=20 > *If a (n-1) homotopy type has the structure of a group-like E(n+1)-spac= e > then it has also the structure of an E(infty)-space (canonically)* >=20 > A nicer statement is obtained by shifting the index n by one. >=20 > * 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)* >=20 > The group-like condition can be dropped: >=20 > *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)* >=20 > This is a special case of the Stabilisation Hypothesis of Breen-Baez-Do= lan; >=20 > *If a n-category has the structure of an E(n+2)-category then it has th= e=20 > structure of > symmetric monoidal category (canonically)* >=20 > (Equivalently, *If a monoidal n-category is (n+1)-braided then it has=20 > the structure of > symmetric monoidal category (canonically)*) >=20 > It is not difficult to verify that these statements are formally equiva= lent. >=20 > The Breen-Baez-Dolan Stabilisation Hypothesis is a theorem. >=20 >=20 > Best, > Andr=E9 >=20 >=20 >=20 >=20 >=20 >=20 >=20 >=20 >=20 >=20 >=20 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]