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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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.
Well, I agree with Andre's argument but it does not convince me to use Andre's terminology nor John's terminology (see my objections below). The shift of numbers in Andre's terminmology is annoying when you try to prove stabilisation hypothesis using higher braided operads. I hope to talk about this proof in Genoa in a couple of months but it follows readily from another atabilization theorem for n-braided operads. It is where I was more or less forced to call braided operads 2-braided operads despite violation of ("foo" = "1-foo"). Another argument in favor of this terminology is that it provides a uniform terminology in higher dimensions which agrees with E_n-algebra point of view developed by Lurie and also his proof of stabilization hypothesis (see Urs's message). I agree that it creates some clash in low dimensions but I think it is not a big deal since classical terminology does not have numbers (nobody calls a monoidal category 0-braided or symmeteic monoidal category 2-braided monoidal). The low dimensional cases are important but they are not always good models for higher dimension. As an example, -2 and -1 categories as Baez and Dolan pointed out can be understood as one pointed set and two pointed set correspondingly. Should we shift the numbers and call category a 3-category?
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.
The trouble is that n-monoidal categories already exist. They were introduced my Balteanu, Fioderowicz, Shwantzl and Vogt. This is why I also see n-tuply monoidal as confusing. I do not say that they sound identical but certainly very close to each other.
But I will not lose sleep if Andre uses "k-braided" as a synonym for "(k+1)-tuply monoidal".
I am glad to join John. I am also grateful to everybody participating in this discussion. Terminology is a very important issue but I do not think it is a crime to use a different one if the clarity of exposition dictates it and if one acknowledges the existence of an alternative. I think I will continue to use my own terminology but I am going to give more explanation in the introduction for those who like a different one. with best regards, Michael. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Michael Batanin wrote in part:
I agree that it creates some clash in low dimensions but I think it is not a big deal since classical terminology does not have numbers (nobody calls a monoidal category 0-braided or symmeteic monoidal category 2-braided monoidal). The low dimensional cases are important but they are not always good models for higher dimension. As an example, -2 and -1 categories as Baez and Dolan pointed out can be understood as one pointed set and two pointed set correspondingly. Should we shift the numbers and call category a 3-category?
No, but it seems to me that you are doing something very much like this. The concept of n-category makes sense for n as low as -2, so it would be nice to renumber this so that we start at n = 0. However, if we do so, then we need a word other than "-category"; if "category" = "3-category", then this violates "foo" = "1-foo". Similarly, the concept of k-braided MC makes sense for k = -1, so it would be nice to renumber this so that we start at k = 0. However, if we do so, then we need a word other than "-braided MC"; if "braided MC" = "2-braided MC", then this violates "foo" = "1-foo". So either we stick with Andre's numbering, inelegant as may be, or we change Andre's "-braided MC" to John's "-tuply MC". But you say, no, we do not need "foo" = "1-foo", simply renumber so that "braided MC" = "2-braided MC". That is like saying, renumber so that "category" = "3-category". While it is a more elegant numbering, it is likely to be confusing. I will say no more about it. I will be happy to read your papers, as long as you explain your terminology up front, as we all should. I may grumble to myself at your violation of "foo" = "1-foo", but I will nevertheless understand since you have explained. (But if you later post to the categories list about it, then I may be confused if you don't recall the numbering.) --Toby [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Should we shift the numbers and call category a 3-category?
No, but it seems to me that you are doing something very much like this.
Not at all. It may be was not a good example. A better example would be categories. If we follow the principle "foo = 1 foo" and want to agree with historical low dimensional terminology we should call categories 2-sets. Set = 1 Set. So, categories = 2 Set. Nobody will do it I guess. There are many other examples like stack, gerbes and so on. I agree with Mike Shulman that this is a byproduct of categorification. But we can survive with it. Concerning n-braided categories versus (n+1)-fold categories. Yes, I would be happy to use (n+1)-fold terminology but it also clashes with iterated monoidal categories of BFSW as I said. Michael. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Micheal Batanin wrote
If we follow the principle "foo = 1 foo" and want to agree with historical low dimensional terminology we should call categories 2-sets. Set = 1 Set. So, categories = 2 Set. Nobody will do it I guess.
A few thoughts about terminology. Categories are tradidionally named according to the nature of their objects, not the nature of their morphisms. We say "the category of sets" not "the category of functions". This convention is not respected in the case where the category has only one object: we call it a monoid, not because it is a mono-object category (maybe we should) but because it has only one binary operation in contrast with a ring. Like monoids, operads are collections of abstract operations closed under composition. Classical operads have only one object, one color. But multi-colored operads are often called muti-categories, especially when they are big. A set is a discrete homotopy type, a 0-type. This why I like to give the category of sets rank 0. I like to denote the quasi-category of n-types by U[n]. Best, André -------- Message d'origine-------- De: categories@mta.ca de la part de Michael Batanin Date: jeu. 13/05/2010 19:09 À: Toby Bartels Objet : categories: Re: bilax_monoidal_functors?=
Should we shift the numbers and call category a 3-category?
No, but it seems to me that you are doing something very much like this.
Not at all. It may be was not a good example. A better example would be categories. If we follow the principle "foo = 1 foo" and want to agree with historical low dimensional terminology we should call categories 2-sets. Set = 1 Set. So, categories = 2 Set. Nobody will do it I guess. There are many other examples like stack, gerbes and so on. I agree with Mike Shulman that this is a byproduct of categorification. But we can survive with it. ... [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Thanks for this list, Peter! I have put much of its content on the nLab at http://ncatlab.org/nlab/show/category+with+duals (and Mike has already put more on pages linked from there), so feel free to speak up again (here or by editing those pages) if something is wrong. --Toby [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
I had written:
My last comment is that, unlike what Jeff Egger claimed, "autonomous category" is not a special case of "*-autonomous category", because no symmetry is assumed in autonomous categories. Unless of course one first drops symmetry from the definition of *-autonomous categories, as Jeff has also suggested. As it stands, neither of "autonomous" and "*-autonomous" implies the other, which is perfectly fine in my opinion, since they are two different words.
I would like to clarify that Jeff himself did not say anything false, because in the context in which he said it, he had in fact assumed the non-symmetric definition of *-autonomous category (of [Barr 1995]). Sorry if it sounded like I was accusing him. My intention was only to point out that the statement "autonomous categories are a special case of *-autonomous categories" cannot be quoted out of context, because it is false under the original definition of *-autonomous category that includes symmetry (of [Barr 1979]). Since it had already been quoted out of context when I wrote the above, I just wanted to point out how the potential confusion. I think this is a very apt illustration of what happens if a term with an existing meaning is redefined to mean something else. Henceforth it is impossible for anybody to use the term (with either meaning) without first giving a definition. That's no problem in a math paper, where definitions are usually given or cited anyway, and therefore terminology is in principle arbitrary. But it does tend to hobble everyday discussion. -- Peter P.S.: since I have a demonstrated ability to put my foot in my mouth, I'd like to clarify that I am not accusing Mike Barr of anything either. His 1995 paper is clearly entitled "Non-symmetric *-autonomous categories", and the inside of the paper clearly explains the distinction. It is only in subsequent use that any confusion arises. The usual solution, of putting either (non-symmetric) or (symmetric) in parentheses the first time the term is used, and omitting it for subsequent uses, is perfectly adequate. I am very happy with the statement "an autonomous category is a special case of a (non-symmetric) *-autonomous category". M. Barr (1979). "*-Autonomous Categories", Lectures Notes in Mathematics 752. Springer. M. Barr (1995). "Non-symmetric *-autonomous categories". Theoretical Computer Science 139:115–130. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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.
The trouble is that n-monoidal categories already exist. They were introduced my Balteanu, Fioderowicz, Shwantzl and Vogt. This is why I also see n-tuply monoidal as confusing. I do not say that they sound identical but certainly very close to each other.
This is a strong point. Obviously n-tuply monoidal category should mean category with n "compatible" monoidal structures; but there many possible meanings of "compatible". One choice leads to a single monoidal structure with an (n-1)-braiding; but a different choice leads to the notion of BFSV. In fact, I think that even the BFSV notion is too strict---it forces all the units to be the same, where I think one should allow them to be different (in general). That is, I think it would be reasonable to use "doubly monoidal category" to mean (pseudo)monoid internal to LAX (rather than STRONG, or even NORMAL). Cheers, Jeff. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (9)
-
Andre Joyal -
Fred E.J. Linton -
Jeff Egger -
John Baez -
Joyal, André
-
Michael Batanin -
Michael Shulman -
selinger＠mathstat.dal.ca -
Toby Bartels