terminology: dagger and involution
Dear Peter, what you are calling a "dagger-category", i.e. a category equipped with a contravariant involutive endofunctor, which is the identity on objects, has been called "a category with involution", at least from Burgin 1969 to Lambek 2001. "Involutive category" has also been used, if less. (The main object of these papers, or most of them, is: categories of relations.) I think it would be better to come back to the old term, which is meaningful, translatable, and old. With best regards Marco
[To Peter Selinger. Cc: categories] Dear Peter, what you are calling a "dagger-category", i.e. a category equipped with a contravariant involutive endofunctor, which is the identity on objects, has been called "a category with involution", at least from Burgin 1969 to Lambek 2001. "Involutive category" has also been used, if less. (The main object of these papers, or most of them, is: categories of relations.) I think it would be better to come back to the old term, which is meaningful, translatable, and old. With best regards Marco
Marco wrote:
what you are calling a "dagger-category", i.e.
a category equipped with a contravariant involutive endofunctor, which is the identity on objects,
has been called "a category with involution", at least from Burgin 1969 to Lambek 2001. "Involutive category" has also been used, if less.
I think it would be better to come back to the old term, which is meaningful, translatable, and old.
There's also a body of work, mainly from mathematical physics, that calls these categories "star-categories". But, by now there's enough literature using the term "dagger-categories" that the genie is out of the bottle. Best, jb
Dear John, Sooner or later somebody will call them "sharp" categories, or "tilde" categories... What you are saying is a good argument in favour of a sensible, well established name. Also, on a more general ground, should we have a different terminology in, say: - category theory, - category theory applied to computer science, - category theory applied to physics? Funny names, like quark, can be good and typographical names can be useful, when there is no better substitute. Eg, I do not know of any good substitute for "comma category". But I see no reason to replace a sensible name with a meaningless one; or, even worse, many meaningless ones. --------- Dear Jeff, The problem you are mentioning is essentially based on terminology for different dualities in higher categories. I do not think there is a way of finding a coherent terminology for them, which would not clash with some well established, quite sensible use, already existing in some particular case. Therefore, I would not be surprised if the contravariancy of an involution should assume different meanings in different contexts. --------- All the best Marco
John Baez wrote:
Marco wrote:
what you are calling a "dagger-category", i.e.
a category equipped with a contravariant involutive endofunctor, which is the identity on objects,
has been called "a category with involution", at least from Burgin 1969 to Lambek 2001. "Involutive category" has also been used, if less.
I think it would be better to come back to the old term, which is meaningful, translatable, and old.
There's also a body of work, mainly from mathematical physics, that calls these categories "star-categories".
But, by now there's enough literature using the term "dagger-categories" that the genie is out of the bottle.
Best, jb
Dear John, just my view: this is not a good argument. I do not know about these dagger categories though i read about the compact closed ones. So may be I miss the point but, if this is the case, why introducing a new terminology if the concepts are not? That just creates confusion. Best, Vincent
On Thu, Mar 01, 2007 at 09:21:55AM +0000, V. Schmitt wrote:
John Baez wrote:
by now there's enough literature using the term "dagger-categories" that the genie is out of the bottle.
Dear John, just my view: this is not a good argument.
It's not an argument - I'm just reporting on what I see. I don't really like the term "dagger-categories", and I gently tried to get people to stop using it, but it didn't work. They're already comfortable with it.
I do not know about these dagger categories though i read about the compact closed ones. So may be I miss the point but, if this is the case, why introducing a new terminology if the concepts are not? That just creates confusion.
I hope this is clear: "dagger-categories" are completely different from "compact closed categories". We need *some* term for them; we're just arguing about whether to call them "star-categories", "dagger-categories", or "categories with involution". I like "star-categories", because in analysis and quantum topology the special case of "C*-categories" is very important. But, I doubt we'll reach any sort of agreement! Best, jb
On Thu, 1 Mar 2007, John Baez wrote:
I hope this is clear: "dagger-categories" are completely different from "compact closed categories". We need *some* term for them; we're just arguing about whether to call them "star-categories", "dagger-categories", or "categories with involution". I like "star-categories", because in analysis and quantum topology the special case of "C*-categories" is very important. But, I doubt we'll reach any sort of agreement!
You are completely right, of course - but one thing was clear from the start: naming a structure from the notation used is rarely a smart move; instead one should try to capture the essence of the structure in the name. (For that reason, "star-categories" isn't a whole lot better than "dagger-categories", though admittedly, it's hard to think of a worse name! However, "star-categories" is likely to make folks think "dagger = star", and that would be unhelpful. That is probably partially why getting a good name was tricky - after all, "dagger- categories" sounds like the act of a desparate person failing to come up with a good name.) But by now, too many folks are probably unwilling to change (and there isn't really an obvious better name anyway), and their collegues and students will probably follow suit, making a name revision even less likely. Pity though ... -= rags =- -- <rags@math.mcgill.ca> <www.math.mcgill.ca/rags>
Hi Marco and John, thanks for your comments. Although I am not sure how many people this will interest, I should probably try to defend my choice of terminology. I originally invented the term "dagger category" because I was looking for a flexible term that could be used both as an adjective and an adverb. I wanted a term that could be applied not just to categories, but also to many other categorical notions ("dagger categories", "dagger functor", "dagger biproducts", "dagger subobject", "dagger idempotent", "to dagger-split" etc). Abramsky and Coecke had used the term "strongly compact closed category", but "strongly" couldn't be applied in most of these contexts. If I had known about Burgin's erstwhile term "involutive category", I would have probably used it. As it is, I have now been publicly using the term "dagger categories" for over two years, including on this list (first 8 Jun 2005), and the terminology has not drawn any criticism until now (except from John Baez, see below). By now, the term has found its way into published papers, and other have picked it up. So, as John has already pointed out, the proverbial genie has left the bottle. Despite due respect for historical terminology, I have to say that I don't much like the term "involutive category". Most importantly, this leaves no good terminology for categories with an involution that is not identity-on-objects, or not contravariant. I don't much like terminologies that use the name "A" to mean "has properties A, B, and C", just because the first example someone studied happened to have those additional properties. Also, a functor between involutive categories cannot be called an "involutive functor" for obvious reasons. Similarly, one cannot say "involutive idempotent", "involutive biproduct", etc. I think the "dagger" terminology is elegant. As John Baez has pointed out, the term "star category" has ample precedent, and indeed, this shares all the useful grammatical properties of "dagger category". Aside from the fact that star categories are often assumed to satisfy additional properties, the two terminologies are equivalent to each other. The difference comes about because mathematicians write "f^*" for the adjoint of a linear map, whereas physicists write "f^\dagger". So why am I siding with the physicists? The choice was forced by the fact that category theorists have long ago decided to write f^* : B^* -> A^* for the transpose of a linear map f : A -> B (in compact closed categories). This is good notation, because functors should be written the same way on objects as on morphisms. However, this makes it impossible to also write f^* for the adjoint B -> A. So one has no choice but to use f^\dagger : B -> A. The difference between the transpose f^* : B^* -> A^* and the adjoint f^\dagger : B -> A is probably the single most common source of confusion about Hilbert spaces for category theorists and others. Both functors are contravariant, and they have little else in common. Sticking to the term "*-category" would have compounded these problems. Fortunately, the symbol $\dagger$ doesn't already have other meanings in related contexts. So its adoption, at least, should not contradict existing terminology. It is better to have two names for one concept than to have one name for two different concepts. Moreover, since $\dagger$ is only a symbol, and not a dictionary word, there is nothing that prevents it from being pronounced differently by different people. I propose that $\dagger$ can be pronounced (and even translated) as "involutive" by those who prefer to do so. This way, time-honored terminology can be used without a change of notation. -- Peter John Baez wrote:
On Thu, Mar 01, 2007 at 09:21:55AM +0000, V. Schmitt wrote:
John Baez wrote:
by now there's enough literature using the term "dagger-categories" that the genie is out of the bottle.
Dear John, just my view: this is not a good argument.
It's not an argument - I'm just reporting on what I see.
I don't really like the term "dagger-categories", and I gently tried to get people to stop using it, but it didn't work. They're already comfortable with it.
I do not know about these dagger categories though i read about the compact closed ones. So may be I miss the point but, if this is the case, why introducing a new terminology if the concepts are not? That just creates confusion.
I hope this is clear: "dagger-categories" are completely different from "compact closed categories". We need *some* term for them; we're just arguing about whether to call them "star-categories", "dagger-categories", or "categories with involution". I like "star-categories", because in analysis and quantum topology the special case of "C*-categories" is very important. But, I doubt we'll reach any sort of agreement!
Best, jb
Dear John, Sooner or later somebody will call them "sharp" categories, or "tilde" categories... What you are saying is a good argument in favour of a sensible, well established name. Also, on a more general ground, should we have a different terminology in, say: - category theory, - category theory applied to computer science, - category theory applied to physics? Funny names, like quark, can be good and typographical names can be useful, when there is no better substitute. Eg, I do not know of any good substitute for "comma category". But I see no reason to replace a sensible name with a meaningless one; or, even worse, many meaningless ones. --------- Dear Jeff, The problem you are mentioning is essentially based on terminology for different dualities in higher categories. I do not think there is a way of finding a coherent terminology for them, which would not clash with some well established, quite sensible use, already existing in some particular case. Therefore, I would not be surprised if the contravariancy of an involution should assume different meanings in different contexts. --------- All the best Marco
participants (5)
-
John Baez -
Marco Grandis -
Robert Seely -
selinger＠mathstat.dal.ca -
V. Schmitt