Re: calculus, homotopy theory and more

Re: calculus, homotopy theory and more

[Fred E.J. Linton]

Two more boringly garden-variety [n - n+1] shifts, analogous to

> The shift n-->n+1 which occurs [in Joyal's missive] in the terminologies
> 
> "n-braided monoidal category" = "(n+1)fold monoidal category" ...

French "deuxieme etage" == American "two flights up" == "third floor";
ordinal number 2 == second ordinal number past 0 == third ordinal  number.

HTH. Cheers, -- Fred





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RE : bilax monoidal functors

[John Baez]

André Joyal wrote:

I am using the following terminology for
> higher braided monoidal (higher) categories:
>
> Monoidal< braided < 2-braided <.......<symmetric
>
> A (n+1)-braided n-category is symmetric
> according to your stabilisation hypothesis.
>
> Is this a good terminology?
>

I use "k-tuply monoidal" to mean what you'd call "(k-1)-braided".  This
seems preferable to me, not because it sounds nicer - it doesn't - but
because it starts counting at a somewhat more natural place.  I believe that
counting monoidal structures is more natural than counting braidings.

For example, a doubly monoidal n-category, one with two compatible monoidal
structures, is a braided monoidal n-category.    I believe this is a theorem
proved by you and Ross when n = 1.  This way of thinking clarifies the
relation between braided monoidal categories and double loop spaces.

Various numbers become more complicated when one counts braidings rather
than monoidal structures:

An n-tuply monoidal k-category is (conjecturally) a special sort of
(n+k)-category... while an n-braided category is a special sort of
(n+k+1)-category.

Similarly: n-dimensional surfaces in (n+k)-dimensional space are n-morphisms
in a k-tuply monoidal n-category... but they are n-morphisms in an
(k-1)-braided n-category.

And so on.

On the other hand, if it's braidings that you really want to count, rather
than monoidal structures, your terminology is perfect.

By the way: I don't remember anyone on this mailing list ever asking if
their own terminology is good.  I only remember them complaining about other
people's terminology.  I applaud your departure from this unpleasant
tradition!

Best,
jb


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Re: bilax_monoidal_functors?=

[Michael Shulman]

I think it is the least confusing for everyone if when "foo"s start
being decorated with numbers, a "1-foo" is the same thing as what an
unadorned "foo" used to be.  So I definitely have to agree that an
ordinary braided monoidal category should be called "1-braided" if the
naming scheme is going to go by decorating "braided" with numbers.

On the other hand, occasionally it seems to happen that after "foo"s
have been studied for a while, someone introduces a categorified "foo"
and calls it a "bar," and then later someone else comes along and
categorifies again but now starts introducing numbers with "2-bar,"
"3-bar," and so on.  So what really should have been called a "2-foo"
is called a "bar," what really should have been called a "3-foo" is
called a "2-bar," and so on with the numbers all off by one.  As John
points out, the use of "braided = 1-braided" and then "2-braided,"
etc. could be viewed this way, with "monoidal" as the basic "foo" that
we should have started numbering at.

(One other example of this that comes to mind is the original use of
"stack" to mean essentially "2-sheaf," leading to "2-stack" for
something that is really a 3-categorical object, and so on.
Fortunately this particular trend seems to be reversing somewhat.)

However, in the case at hand, it seems to me that there is also an
advantage to the term "braided" over "doubly monoidal."  To give a
category a braided monoidal structure may be *equivalent* to giving it
two interchanging monoidal structures, but that's only true because in
the latter case, the interchange law forces the two monoidal
structures to be essentially the same.  In practice, I find that I
very rarely think about a braided monoidal category as if it were
equipped with two monoidal structures; rather I think of it as having
one monoidal structure together with an extra structure called a
"braiding."  So there are arguments on both sides of this issue, and
as John says probably neither usage will create any confusion.

Mike

On Mon, May 10, 2010 at 1:16 PM, John Baez <john.c.baez@gmail.com> wrote:
> Eduardo wrote:
>
>
>> Andre points out:
>>
>> "To call a monoidal category 1-braided is kind of confusing because there
>> is no commutation structure on a general monoidal category. A monoidal
>> category is 0-braided."
>>
>> Being an outsider, with no previous neither usage or opinion on this
>> terminology beyond just monoidal and/or tensor category, this seems to me
>> definitive, and more than enough to settle the question.
>
>
> I'm glad that's enough to convince you that Michael Batanin's terminology
> "monoidal = 1-braided" is inferior to Andre's "monoidal = 0-braided".
>
> But I think "braided = doubly monoidal" is even better.  After all, a
> monoidal category has one tensor product; a braided monoidal category has
> two compatible tensor products, and a symmetric monoidal category has three.
>
>
> But I will not lose sleep if Andre uses "k-braided" as a synonym for
> "(k+1)-tuply monoidal".  I don't see it causing any confusion. I just think
> it will create more +1's in various formulas.  E.g.: the classifying space
> of a k-braided n-category is a (k+1)-fold loop space.
>
> Best,
> jb
>


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calculus, homotopy theory and more

[Andre Joyal]

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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