calculus, homotopy theory and more

[Andre Joyal <joyal.andre@uqam.ca>]

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 ressemblances
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_1x^2+...
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,...] then Susp(X)=[0;0,0,a_2,b_3...],
and if X=[0;0,0, a_3, a_4, a_5,...] 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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