Fred E.J. Linton wrote in part:

Steve Lack wrote:

...

Such a T is called a symmetric monoidal functor.

Thanks for helping dispel my illusion that all monoidal functors might necessarily be thus symmetric :-) :

Something like this is true, however. First, every monoidal natural transformation is symmetric monoidal (assuming that it goes between symmetric monoidal functors at all). Also, there is the concept of braided monoidal categories that lies between monoidal categories and symmetric monoidal categories. And every braided monoidal functor is symmetric monoidal (assuming that it goes between symmetric monoidal categories at all). Each of these facts is trivial by itself; for example, the definition of symmetric monoidal functor that you wrote down makes sense for a functor between braided monoidal categories; it is simply the definition of braided monoidal functor, and there is nothing more to add when the braiding is symmetric. But the entire pattern is interesting: PC -- PF -- PNT -- ENT MC -- MF -- MNT -- ENT BMC -- BMF -- MNT -- ENT SMC -- BMF -- MNT -- ENT SMC -- BMF -- MNT -- ENT SMC -- BMF -- MNT -- ENT (etc) (To fit this all on the screen, I have used initialisms: "Categories", "Functors", "Natural transformations", "Equality of", "Pointed", "Monoidal", "Braided", "Symmetric".) The thing to notice is that each column stabilises one row earlier than the column before it. The columns stabilise because there is nothing more to write down. * John Baez, Some definitions everyone should know. http://math.ucr.edu/home/baez/qg-winter2001/definitions.pdf (This discusses strong monoidal functors between weak monoidal categories, but it is easy enough to generalise to lax monoidal functors or to specialise to strict monoidal categories.) It's possible that the columns stabilise only through our ignorance (as once we were ignorant that BMC were there between MC and SMC). However, there is a general theory of k-tuply monoidal n-categories which confirms the pattern, although some of that is still conjecture. * nLab, k-tuply monoidal n-categories http://ncatlab.org/nlab/show/k-tuply+monoidal+n-category --Toby [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Andre Joyal wrote in part:

We should use a similar terminology for spaces and maps. E-n space <--> E-n map Also for (higher) categories and functors. monoidal category <---> monoidal functor braided monoidal category <----> braided monoidal functor 2-braided monoidal category <--> 2-braided monoidal functor 3-braided monoidal category <--> 3-braided monoidal functor ...... symmetric monoidal category <--> symmetric monoidal functor

I agree, one should say "symmetric monoidal functor"; if nothing else, that indicates that the source and target are symmetric (not merely braided) monoidal categories. I only put "BMF" in my table to show a particular pattern. Depending on how you write down the definitions, that a braided monoidal functor between symmetric monoidal categories is the same thing as a symmetric monoidal functor between them is either an utter triviality or a deep and interesting theorem; but in either case, we need the words to state it. (I do agree with John about preferring "k-tuply monoidal", but I'll let him make that argument.)

A (n+1)-braided monoidal n-category is symmetric by the stabilisation hypothesis. I believe that a (n+1)-braided monoidal functor between (n+1)-braided monoidal n-categories is symmetric.

I think that you mean to say (which is even stronger) that an n-braided monoidal functor between SM n-categories is symmetric. More generally, a k-braided monoidal l-transfor between SM n-categories is symmetric as long as k + l is greater than or equal to n. (A 0-transfor is a functor, a 1-transfor is a natural transformation, etc. This numbering is due to Sjoerd Crans; feel free to argue that it's off.) More generally yet, a k-braided monoidal l-transfor between m-braided monoidal n-categories is m-braided, as long as k + l >= n, regardless of the value of m (although we need m >= k for the antecedent to make sense).

Is this part of the official stabilisation hypothesis?

I don't know what's official, but I'll claim the conjecture above as mine if nobody else has written it down yet. (^_^) --Toby [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Dear All, In the chapter 3 of their book "Monoidal functor, species and Hopf algebras" http://www.math.tamu.edu/~maguiar/ Aguiar and Mahajan introduces 4 kinds of monoidal functors: 1) strong monoidal 2) lax monoidal 3) colax monoidal 4) bilax monoidal A monoid in a monoidal category C is a lax monoidal functor 1-->C, a comonoid is a colax monoidal functor 1-->C and a bimonoid is a bilax monoidal functor 1-->C. I wonder who first introduced the notion of bilax monoidal functor and when? An example of bilax monoidal functor is the singuler chain complex functor from spaces to chain complexes. The bilax structure is provided by the Eilenberg-MacLane map together with the Alexander-Whitney map. Best, AJ -------- Message d'origine-------- De: categories@mta.ca de la part de Steve Lack Date: jeu. 06/05/2010 19:02 À: Fred E.J. Linton; categories Objet : Re: categories: Q. about monoidal functors Dear Fred, Such a T is called a symmetric monoidal functor. Example: let _A_ be Set with the cartesian monoidal structure. Let M be a monoid and let T be the functor Set->Set sending X to MxX (which I'll write as MX). This functor T is monoidal via the map MXMY->MXY sending (m,x,n,y) to (mn,x,y). It is symmetric monoidal iff M is commutative. Steve Lack. On 6/05/10 4:01 PM, "Fred E.J. Linton" <fejlinton@usa.net> wrote:

Suppose _A_ is a symmetric monoidal category in the sense of the Eilenberg-Kelley La Jolla paper, and T: _A_ --> _A_ a monoidal functor.

What, if anything, is known, where τ: X ⊗ Y --> Y ⊗ X is the symmetry structure on the (symmetric) tensor product ⊗, as to whether

[T_X,Y: TX ⊗ TY --> T(X ⊗ Y)] and [T(τ_X,Y): T(X ⊗ Y) --> T(Y ⊗ X)]

have the same composition as have

[τ_TX,TY: TX ⊗ TY --> TY ⊗ TX] and [T_Y,X: TY ⊗ TX --> T(Y ⊗ X)] ?

TIA for any relevant information and/or references thereto.

Cheers, -- Fred

[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

John Baez could not recall whether bilax and Frobenius monoidal functors = are the same. The answer is no, in the usage I'd been familiar with, bilax meant = simply equipped with both lax and oplax structures, while a Frobenius = monoidal functor satisfies additional coherence relation which = generalize the relations between the multiplication and comultiplication = in a Frobenius algebra. A bilax monoidal functor from the one-object monoidal category to VECT = would be a vector-space with both an algebra and a coalgebra structure = on it (no coherence relations relating them), while a Frobenius monoidal = functor would be a Frobenius algebra. =20 Aguiar (with good reason), on the other hand, reserves bilax for = functors equipped with coherence relations generalizing the relations = between the operations and cooperations in a bialgebra, so that a bilax = functor from the one-object monoidal category to VECT would be a = bialgebra. This notion, however, only makes sense in the presence of = braidings on the source and target. I think Aguiar's usage should prevail, though we also need a name for = functors between general monoidal categories which are simultaneously = lax and oplax. Best Thoughts, David Yetter= [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

2010/5/9 Dusko Pavlovic <Dusko.Pavlovic@comlab.ox.ac.uk>: Asks

is there any reason why words should be taken seriously?

That just depends on whether or not you want to be understood by people who do not already know everything you are going to say. best, Colin [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Hi Dusko,

i am reluctant call them dagger star autonomous categories, because it is a mouthful.

Perhaps it's a symptom of growing up in a country where "Kangiqsualujjuaq" is considered a perfectly acceptable name for a village, but I don't think that "dagger star- autonomous" is a mouthful. It's only one syllable longer than "sesquipedalian", and one less than "linearly distributive", neither of which I would hesitate to use in day-to-day conversation, should the occasion arise. It even scans nicely. Moreover, it communicates something (at least to me); for better or worse, both "dagger" and "star-autonomous" are both established terms, and I can see how they might be combined. Agglutination, though often mocked, is often effective.

so now, what should we call those "dagger star autonomous categories" if we don't want to type 30 characters each time we mention them?

One of the many curious features of the English language is that adjectives are never inflected; assuming you use TeX, why not take advantage of this fact in your source code? \def\dsa{dagger star-autonomous}

peter suggests DSA-categories.

If you're publishing in a print journal, or a conference proceedings with a hard page-limit, then that seems sensible (though I'd drop the hyphen). Otherwise, do us all a favour and stick to the long form: pixels are cheap, as editors of TAC are wont to say.

(maybe someone will abbreviate them to D-categories...)

What's the point of that? D-category could stand for (just plain old) dagger category, or differential category, or any number of other things. But maybe someone some day will \def\dsa{Pavlovic}. Cheers, Jeff. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Dear John and Michael, It all depends on where you start counting. For americans, the first floor of a buiding is the ground floor but for most europeans, it is the floor right above: http://en.wikipedia.org/wiki/Storey#Numbering We sometime need to recall in which part of the world we are when we take an elevator! But a ten stories building is the same for everyone. More seriously, John wrote:

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.

Michael wrote:

I am using a mixture of your terminologies: monoidal = 1-braided braided = 2-braided sylleptic = 3-braided

I understand your ideas both. Along the same line we could also use: E1-category = Monoidal E2-category = Braided monoidal E3-category = ..... ..... John wrote:

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!

My goal is to have a public discussion on terminology. It can be very difficult to agree upon because adopting one is like commiting to a rule of law, to a moral code, possibly to a social code. There is an emotional and social aspect to this commitment. There is also a psychological aspect because a terminology looks natural if you use it long enough (it is a matter of a few days). I hope that a public discussion can help peoples choosing their terminology. I do think that my terminology for higher braided monoidal categories is quite good. Let me say a few things in its defense: First, it extends naturally a terminology which is used by the mathematical community since many decades. Only a specialist can truly appreciate E(k)-categories or k-tuply monoidal categories. Second, a braiding is a commutation structure. 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. Third, a n-braided (topological or simplicial) group is exactly what you need to describe the homotopy type of an n-connected space (n\geq 1). I wonder who introduced the notion of E(n)-space and the terminology? Best regards, André [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Dear David Thanks for clarifying the notion of Frobenius functor. In the chapter 4 of the latest version of their book http://www.math.tamu.edu/~maguiar/a.pdf Aguiar and Mahajan introduce a notion of P-monoidal functor for P is an operad. If P is the Ass operad (whose models are monoids). then a P-monoidal functor is a lax monoidal functor, and if P is the Com operad (whose models are commutative monoids), then a P-monoidal functor is a symmetric lax monoidal functor. Their examples include a notion of Lie-monoidal functor (in the enriched case). Dually, they introduce a notion of P-comonoidal functor with the examples of colax (=oplax) monoidal functors and of symmetric oplax monoidal functors. But a bilax monoidal functor is not a P-monoidal functor in the sense of Aguiar and Mahajan because the notion of bialgebra is defined by a PROP, not by an operad. Similarly a Frobenius monoidal functor is not a P-monoidal functor because the notion of Frobenius algebra is defined by a PROP, not by an operad. The notion of P-monoidal functor for P a PROP is not defined in their book. Any idea? Best regards, André -------- Message d'origine-------- De: categories@mta.ca de la part de David Yetter Date: ven. 07/05/2010 21:05 À: Categories Objet : categories: bilax monoidal functors John Baez could not recall whether bilax and Frobenius monoidal functors = are the same. The answer is no, in the usage I'd been familiar with, bilax meant = simply equipped with both lax and oplax structures, while a Frobenius = monoidal functor satisfies additional coherence relation which = generalize the relations between the multiplication and comultiplication = in a Frobenius algebra. A bilax monoidal functor from the one-object monoidal category to VECT = would be a vector-space with both an algebra and a coalgebra structure = on it (no coherence relations relating them), while a Frobenius monoidal = functor would be a Frobenius algebra. =20 Aguiar (with good reason), on the other hand, reserves bilax for = functors equipped with coherence relations generalizing the relations = between the operations and cooperations in a bialgebra, so that a bilax = functor from the one-object monoidal category to VECT would be a = bialgebra. This notion, however, only makes sense in the presence of = braidings on the source and target. I think Aguiar's usage should prevail, though we also need a name for = functors between general monoidal categories which are simultaneously = lax and oplax. Best Thoughts, David Yetter= [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Dear Andre,

My goal is to have a public discussion on terminology.

It is good that you provoke us into having such discussions!

It can be very difficult to agree upon because adopting one is like commiting to a rule of law, to a moral code, possibly to a social code. There is an emotional and social aspect to this commitment.

I don't understand this at all. A co-author of mine recently commented (complained?) that I seem to "change [my] notation as often as [my] underwear"; and I am not that much better with terminology. Indeed, I am overtly anarchist in this respect, and instinctively resist all attempts at codifying language. Most people would agree that the most important concepts deserve the shortest names; but people frequently (honestly) disagree over which concept is the most important. More significantly, attitudes often change with time! It is frustrating, then, that people will cling to archaic terminology for the sake of an emotional and social commitment. [A wonderful counter-example to this phenomenon is when Mike Barr gave his opinion that the meaning of star- autonomous category, which initially included symmetry, should not do so. I should also say that I think young mathematicians are generally worse at this than older ones. Indeed, the most extreme version of (what I perceive to be) the same phenomenon is that of the undergrad who cannot differentiate z=t^2 "because there is no x".] My objection to the phrase "autonomous category" (which Dusko brought up) has less to do with defending Fred Linton's original usage of that phrase than the fact that "autonomous category" is a special case (and, from one point of view, a rather uninteresting special case) of "star-autonomous category", whereas it sounds like "star-autonomous category" should mean an "autonomous category" with some extra structure. (And, of course, this once was the case, w.r.t. the older terminology.) This is confusing; hence one term or the other should be changed. I am, in fact, open to all suggestions, though I cannot help but prefer that "star-autonomous" be kept and "autonomous" changed. Cheers, Jeff. P.S. A propos of your first email in this thread, why bother with all those "lax"s? If you used

1) strong monoidal 2) monoidal 3) comonoidal 4) bimonoidal

instead, then you would have

A monoid is a monoidal functor 1-->C, a comonoid is a comonoidal functor 1-->C and a bimonoid is a bimonoidal functor 1-->C.

and you could even substitute

5) ambimonoidal for "Frobenius", since "ambialgebra" has been used for "Frobenius algebra".

Dare I point out that a strong monoidal functor 1-->C is a _trivial_ monoid? ;) [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Dear Urs and John, I see no real conflict between your terminology and mine. I do use the notion of n-fold monoid in my work, for example in my "Notes on Quasi-categories". An *algebraic theory* is defined to be a (quasi-)category with finite products. The n-fold tensor power of the theory of monoids M is the theory of n-fold monoids = E(n)-monoids for every n. I am sketching a proof of the Stabilisation Hypothesis at section 43.5 of my notes. The hypothesis is formulated in terms of an equivalence of theories: <The theory of (n+2)-fold monoidal n-categories is equivalent to the theory of symmetric monoidal n-categories>. It follows that the quasi-category of (n+2)-fold monoidal n-categories is equivalent to the quasi-category of symmetric monoidal n-categories. Best, André -------- Message d'origine-------- De: categories@mta.ca de la part de Urs Schreiber Date: lun. 10/05/2010 06:28 À: John Baez Objet : categories: Re: bilax monoidal functors

An n-tuply monoidal k-category is (conjecturally) a special sort of (n+k)-category

By the way, some progress on this is available from John Francis' and Jacob Lurie's discussion of k-tuply monoidal (n,1)-categories as those equipped with a little k-cubes action. In particular there is a proof of the stabilization hypothesis for (n,1)-categories this way, and an analog of the May recognition theorem for parameterized oo-groupoids, i.e. (oo,1)-sheaves. Some of this is summarized with pointers to references here: http://ncatlab.org/nlab/show/k-tuply+monoidal+n-category Best, Urs [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

On Thu, May 13, 2010 at 6:09 PM, Michael Batanin <mbatanin@ics.mq.edu.au> wrote:

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.

No one has suggested "(n+1)-fold monoidal" categories for that very reason. The terminology being suggested is "(n+1)-tuply monoidal." Mike [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

On Fri, May 14, 2010 at 8:05 PM, Joyal, Andre <joyal.andre@uqam.ca> wrote:

I guess that in the category of R-modules over a commutative ring R, a module M has a (good) dual iff it is finitely generated projective iff the endo-functor functor Hom(M,-) preserves all colimits (M is *compact* in a strong sense).

Indeed, but in this case it is the objects of the category which are "compact," not the category itself. So if this is the argument, then a more natural term would be "locally compact" (clashing with "locally small," of course, but agreeing with "locally presentable" categories in which all objects are presentable). (I am *not* proposing to *actually* use "locally compact" -- I don't want to introduce yet another name for something that already has at least four names, even if none of the existing four are optimal.) Mike [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Argh, Michael, you have managed to make a mess of the existing terminology. The terminology is confusing, but it is actually settled. While many concepts have more than one name, thankfully no name refers to more than one concept so far (and I am working hard to keep it that way - for example by discouraging redefinitions of "autonomous"). Here are, for reference, the four most common notions of (1-)categories with duals: (1) An "autonomous category" is a monoidal category where every object has a left dual and a right dual. Note that it is not assumed to be symmetric. There is also the notion of a "left autonomous category", where only left duals are assumed, and analogously "right autonomous category". Note that duals, where they exist, are unique up to isomorphism, so being autonomous is a property of monoidal categories, not an additional structure. "Rigid category" is a synonym of "autonomous category", preferred by certain communities of authors. (2) A "pivotal category" is an autonomous category equipped with a monoidal natural isomorphism A -> A**. (A right autonomous category with such an isomorphism is automatically left autonomous too, so the right/left distinction does not apply to pivotal categories). "Sovereign category" is a synonym of "pivotal category" used by Freyd and Yetter in one paper, but it does not seem to have caught on. It was a word play suggesting something that is even more than autonomous. (3) A "tortile category" is a braided pivotal (equivalently balanced autonomous) category satisfying theta* = theta (where theta is the twist). "Ribbon category" is a synonym of "tortile category", preferred by certain communities of authors. (4) A "compact closed category" is a tortile category that is symmetric (as a balanced monoidal category), or equivalently, an autonomous symmetric monoidal category. Of course (4) => (3) => (2) => (1). There are a number of in-between concepts, which are generally less natural and of interest primarily for technical reasons. Please see my recent survey "A survey of graphical languages for monoidal categories" for a far more detailed discussion (http://arxiv.org/abs/0908.3347). Particularly the table on p.60 shows the whole taxonomy on one page. I will briefly mention two of the "less natural" notions: * A "braided autonomous" category is a monoidal category that is both braided and autonomous (with no axioms relating the two structures). This notion is entirely uninteresting, except to note that a braided left autonomous category is automatically right autonomous, due to the existence of isomorphisms A -> A**, and to note that it is NOT automatically pivotal, because said isomorphism is not monoidal. * A "braided pivotal category" is a monoidal category that is both braided and autonomous (again with no axioms relating the two structures). This notion is also completely uninteresting, except to note that a braided pivotal category is exactly the same thing as a balanced autonomous category (because on a braided autonomous category, giving a pivotal structure is precisely equivalent to giving a balanced structure). Such categories were studied by Freyd and Yetter, but arguably they were superseded by the better notion of tortile categories. These categories have a graphical language up to "regular isotopy", which means that one of the three Reidemeister moves fails. I have come to the opinion that it is a very good thing that notions (1)-(3) above have distinct names, and are not just distinguished by adjectives. It would be tempting to call a pivotal category a "[something] autonomous category", and to call a tortile category for example "[something else] braided pivotal" or "[something else] balanced autonomous". But the most natural adjective for [something] would be "pivotal", and the most natural adjective for [something else] would be "tortile", which would only make the names longer without adding any information. I do believe that the term (4) "compact closed" is something of an oddity, since "symmetric autonomous" would be similarly succinct, more systematic, and much more descriptive - in fact, it requires no additional definition if "symmetric" and "autonomous" have already been defined. Also, as Michael has pointed out, the name "compact" here has little to do with its usual meaning in mathematics. If this concept were invented today, one should certainly call it "symmetric autonomous". But in light of the fact that "compact closed" was historially the first of notions (1)-(4) defined, and that the term "compact closed" is already extremely well-known and wide-spread, this is one case where I believe it is better to stick with the existing terminology rather than trying to force it into a taxonomy. That doesn't mean that slow incremental change is not possible. For example, it seems reasonable to write "note that a compact closed category is the same as a symmetric autonomous category" whenever giving the definition. Perhaps after a few years, people will write "note that a symmetric autonomous category is also known as a compact closed category", and maybe after many more years, the term "symmetric autonomous" will even become standard. But such changes should come about through repeated and incremental use by a community, and not by unilateral choices. As a general rule, I think it is good manners when changing terminology (or inventing new unsystematic terminology) to give the old (or systematic) terminology in parentheses at least once per paper. In Oxford, compact closed categories are nowadays called "compact categories". I try not to follow this convention because it replaces one bad term with a shorter, but equally bad one. It would also clash with the standard meaning of "compact" in cases where the category was actually a topological space. But it seems like a benign enough change and is catching on rapidly. 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. -- Peter Michael Shulman wrote:

On Mon, May 10, 2010 at 2:28 PM, Jeff Egger <jeffegger@yahoo.ca> wrote:

...

the fact that "autonomous category" is a special case (and, from one point of view, a rather uninteresting special case) of "star-autonomous category", whereas it sounds like "star-autonomous category" should mean an "autonomous category" with some extra structure.

I agree, it does sound like that, but there is at least a long tradition of such names in mathematics (not that that makes them a good thing). (http://ncatlab.org/nlab/show/red+herring+principle)

One reason I like "autonomous" to mean a symmetric monoidal category in which all objects have duals is that the only alternative names I have heard for such a thing convey misleading intuition to me. They are sometimes called "compact closed" or (I think) "rigid" monoidal categories, but "compact" and "rigid" are words with definite and inapplicable intuitive meanings for me. Compact means small, finite, bounded, inaccessible by directed joins, etc. and "rigid" means "having few automorphisms," and I don't see that there is anything very compact or rigid about such categories. The only relationship I can think of is that a compact subset of a Hausdorff space is closed, and a symmetric monoidal category with duals for objects is also automatically closed, but of course these two meanings of "closed" are totally different. Perhaps someone can enlighten me?

Mike

[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Argh, Michael, you have managed to make a mess of the existing terminology. The terminology is confusing, but it is actually settled. While many concepts have more than one name, thankfully no name refers to more than one concept so far (and I am working hard to keep it that way - for example by discouraging redefinitions of "autonomous"). Here are, for reference, the four most common notions of (1-)categories with duals: (1) An "autonomous category" is a monoidal category where every object has a left dual and a right dual. Note that it is not assumed to be symmetric. There is also the notion of a "left autonomous category", where only left duals are assumed, and analogously "right autonomous category". Note that duals, where they exist, are unique up to isomorphism, so being autonomous is a property of monoidal categories, not an additional structure. "Rigid category" is a synonym of "autonomous category", preferred by certain communities of authors. (2) A "pivotal category" is an autonomous category equipped with a monoidal natural isomorphism A -> A**. (A right autonomous category with such an isomorphism is automatically left autonomous too, so the right/left distinction does not apply to pivotal categories). "Sovereign category" is a synonym of "pivotal category" used by Freyd and Yetter in one paper, but it does not seem to have caught on. It was a word play suggesting something that is even more than autonomous. (3) A "tortile category" is a braided pivotal (equivalently balanced autonomous) category satisfying theta* = theta (where theta is the twist). "Ribbon category" is a synonym of "tortile category", preferred by certain communities of authors. (4) A "compact closed category" is a tortile category that is symmetric (as a balanced monoidal category), or equivalently, an autonomous symmetric monoidal category. Of course (4) => (3) => (2) => (1). There are a number of in-between concepts, which are generally less natural and of interest primarily for technical reasons. Please see my recent survey "A survey of graphical languages for monoidal categories" for a far more detailed discussion (http://arxiv.org/abs/0908.3347). Particularly the table on p.60 shows the whole taxonomy on one page. I will briefly mention two of the "less natural" notions: * A "braided autonomous" category is a monoidal category that is both braided and autonomous (with no axioms relating the two structures). This notion is entirely uninteresting, except to note that a braided left autonomous category is automatically right autonomous, due to the existence of isomorphisms A -> A**, and to note that it is NOT automatically pivotal, because said isomorphism is not monoidal. * A "braided pivotal category" is a monoidal category that is both braided and autonomous (again with no axioms relating the two structures). This notion is also completely uninteresting, except to note that a braided pivotal category is exactly the same thing as a balanced autonomous category (because on a braided autonomous category, giving a pivotal structure is precisely equivalent to giving a balanced structure). Such categories were studied by Freyd and Yetter, but arguably they were superseded by the better notion of tortile categories. These categories have a graphical language up to "regular isotopy", which means that one of the three Reidemeister moves fails. I have come to the opinion that it is a very good thing that notions (1)-(3) above have distinct names, and are not just distinguished by adjectives. It would be tempting to call a pivotal category a "[something] autonomous category", and to call a tortile category for example "[something else] braided pivotal" or "[something else] balanced autonomous". But the most natural adjective for [something] would be "pivotal", and the most natural adjective for [something else] would be "tortile", which would only make the names longer without adding any information. I do believe that the term (4) "compact closed" is something of an oddity, since "symmetric autonomous" would be similarly succinct, more systematic, and much more descriptive - in fact, it requires no additional definition if "symmetric" and "autonomous" have already been defined. Also, as Michael has pointed out, the name "compact" here has little to do with its usual meaning in mathematics. If this concept were invented today, one should certainly call it "symmetric autonomous". But in light of the fact that "compact closed" was historially the first of notions (1)-(4) defined, and that the term "compact closed" is already extremely well-known and wide-spread, this is one case where I believe it is better to stick with the existing terminology rather than trying to force it into a taxonomy. That doesn't mean that slow incremental change is not possible. For example, it seems reasonable to write "note that a compact closed category is the same as a symmetric autonomous category" whenever giving the definition. Perhaps after a few years, people will write "note that a symmetric autonomous category is also known as a compact closed category", and maybe after many more years, the term "symmetric autonomous" will even become standard. But such changes should come about through repeated and incremental use by a community, and not by unilateral choices. As a general rule, I think it is good manners when changing terminology (or inventing new unsystematic terminology) to give the old (or systematic) terminology in parentheses at least once per paper. In Oxford, compact closed categories are nowadays called "compact categories". I try not to follow this convention because it replaces one bad term with a shorter, but equally bad one. It would also clash with the standard meaning of "compact" in cases where the category was actually a topological space. But it seems like a benign enough change and is catching on rapidly. 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. -- Peter Michael Shulman wrote:

On Mon, May 10, 2010 at 2:28 PM, Jeff Egger <jeffegger@yahoo.ca> wrote:

...

the fact that "autonomous category" is a special case (and, from one point of view, a rather uninteresting special case) of "star-autonomous category", whereas it sounds like "star-autonomous category" should mean an "autonomous category" with some extra structure.

I agree, it does sound like that, but there is at least a long tradition of such names in mathematics (not that that makes them a good thing). (http://ncatlab.org/nlab/show/red+herring+principle)

One reason I like "autonomous" to mean a symmetric monoidal category in which all objects have duals is that the only alternative names I have heard for such a thing convey misleading intuition to me. They are sometimes called "compact closed" or (I think) "rigid" monoidal categories, but "compact" and "rigid" are words with definite and inapplicable intuitive meanings for me. Compact means small, finite, bounded, inaccessible by directed joins, etc. and "rigid" means "having few automorphisms," and I don't see that there is anything very compact or rigid about such categories. The only relationship I can think of is that a compact subset of a Hausdorff space is closed, and a symmetric monoidal category with duals for objects is also automatically closed, but of course these two meanings of "closed" are totally different. Perhaps someone can enlighten me?

Mike

[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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An n-tuply monoidal k-category is (conjecturally) a special sort of (n+k)-category

By the way, some progress on this is available from John Francis' and Jacob Lurie's discussion of k-tuply monoidal (n,1)-categories as those equipped with a little k-cubes action. In particular there is a proof of the stabilization hypothesis for (n,1)-categories this way, and an analog of the May recognition theorem for parameterized oo-groupoids, i.e. (oo,1)-sheaves. Some of this is summarized with pointers to references here: http://ncatlab.org/nlab/show/k-tuply+monoidal+n-category Best, Urs [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Dear John and Andre, I am using a mixture of your terminologies: monoidal = 1-braided braided = 2-braided sylleptic = 3-braided ...... In this way the numbers in stabilization hypothesis are standarts. Another advantage, at least for me, is the connection to n-operads. n-braided higher category is an algebra of an n-braided operad , which is a special sort of an n-operad. It is convenient in the proof of stabilisation hypothesis. What do you think about this version? Michael. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]