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• CommentRowNumber1.
• CommentAuthorTim_van_Beek
• CommentTimeJun 23rd 2010

A $ℂ-$linear category is simply a category where every Hom(x, y) is a complex vector space and the composition of morphisms is bilinear. A *-category is a $ℂ-$linear category that has a *-operation on each Hom(x, y) (same axioms a for a *-algebra) and a ${C}^{*}-$category further has a norm on each Hom(x, y) that turns it into a Banach space with ${s}^{*}s=\mid s{\mid }^{2}$ and $\mid \mathrm{st}\mid \le \mid s\mid \mid t\mid$ for all arrows s, t (s and t composable).

Is there already a page on the nLab that describes this structure?

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeJun 23rd 2010

These are all examples of enriched categories of course. In the Examples-section there, Vect-enriched categories are referred to as algebroids.

Correspondingly, your other examples could be called $*$-algebroid and ${C}^{*}$-algebroid.

In some sense this terminology is more systematic, but the one you are asking about is of course more conventional. It is a bit curious that this latter terminology is a generalization of saying “linear monoid” for “algebra”, which few people would do.

But I think you should go ahead – if this is what you were going to – and create these entries. We can then still redirect to the other terminology and explain how it is related.

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeJun 23rd 2010

Oh, I just see that a stub for linear category does exist.

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeJun 23rd 2010

We also have ringoid for Ab-enriched category where other people might say pre-additive category (at least if there is a 0-object).

This oid-terminology may seem weird, compared to the standard terms, but I think it is quite useful and deserves to be more widely useed. For instance what is called Calabi-Yau category is really a Frobenius algebroid , which is much more descrptive of the concept (at least if one knows what Frobenius algebras are).

In this vein, notice – as I just remember – that some people proposed to call the formal dual of ${C}^{*}$-categories (i.e. ${C}^{*}$-algebroids) spaceoids!

• CommentRowNumber5.
• CommentAuthorTim_van_Beek
• CommentTimeJun 23rd 2010

Ohhhh…I think I’ll have to look at these before I can decide how to proceed. But I’m not scared by -oids (with the exception of killer androids, of course).

• CommentRowNumber6.
• CommentAuthorTobyBartels
• CommentTimeJun 23rd 2010

Is there any reason that we want to have two articles, linear category and algebroid ?, similarly Ab-enriched category and ringoid ?. After all, the definitions are precisely the same, so that one doesn’t even have to translate between them. I’d be inclined, in each case, to rewrite the larger article to be more neutral between the terms, then redirect the smaller article to the larger.

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeJun 23rd 2010

@Tim: I don’t think there is much to be looked at. You should go ahead and write your entries.

@Toby: yes, I think it would be good to merge the material, with discussion of the terminology included. I forget what exactly happened when we created these. That was at the very beginning of the nLab.

• CommentRowNumber8.
• CommentAuthorTobyBartels
• CommentTimeJun 23rd 2010

Actually, it appears that linear category and algebroid do not mean the same thing; the former must be an additive category, while the latter need not be. (In particular, therefore, an algebra is always an algebroid but —except in the degenerate case of the zero algebra— never of a linear category.)

• CommentRowNumber9.
• CommentAuthorJohn Baez
• CommentTimeJun 23rd 2010
What you're calling a *-category, Tim, is a special case of a dagger-category. The term *-category was popular in algebraic quantum field theory circles long before people working on quantum information theory popularized the term dagger-category. But nowadays I think it's crucial to emphasize their relation, since they both show up when studying quantum theory with the help of categories.

Also, C-linear categories are a special case of enriched categories.

So, I hope you note these facts:

What you're calling a *-category is precisely a dagger-category that is also C-linear, and for which the dagger operation (= star operation) is conjugate linear.

Also: what you're calling a C-linear category is the same as a category enriched over the category of complex vector spaces. I've added this example to page on enriched categories.
• CommentRowNumber10.
• CommentAuthorUrs
• CommentTimeJun 23rd 2010

Well, but then maybe it should be merged with pre-additive category. I think there we have a remark on algebroids, even, but can’t check right now.

• CommentRowNumber11.
• CommentAuthorUrs
• CommentTimeJun 23rd 2010

I’ve added this example to page on enriched categories.

Strange, i could swear that I had added it. Do we have it duplicated now?

• CommentRowNumber12.
• CommentAuthorUrs
• CommentTimeJun 23rd 2010

But I have to admit that I missed that Tim was of course talking about dagger-categories.

• CommentRowNumber13.
• CommentAuthorTobyBartels
• CommentTimeJun 23rd 2010

I have combined ringoid with Ab-enriched category. (I also started to combine linear category with algebroid but reversed this when I noticed that the definitions were not in fact the same. But I did end up editing the introductory material of each.)

• CommentRowNumber14.
• CommentAuthorTobyBartels
• CommentTimeJun 23rd 2010
• (edited Jun 23rd 2010)

Strange, i could swear that I had added it. Do we have it duplicated now?

Yes, it’s there twice now. Edit: John has fixed this.

• CommentRowNumber15.
• CommentAuthorTobyBartels
• CommentTimeJun 23rd 2010

Questions remaining for Tim:

• For your purposes, should $*$-categories and other $ℂ$-linear categories be additive? That is, should they have direct sums? (It seems to me that ${C}^{*}$-categories do not usually have that requirement.)
• Do you want to write star-category and C-star-category? We do not have pages on those yet.

Questions for everybody:

• Is it reasonable that the term ‘linear category’ should parallel the term ‘additive category’ in that these are required to have biproducts? A search of the arXiv suggests that this is often assumed of linear categories in practice but that it doesn’t seem to be part of the definition of the term.
• If not, is there any term for an additive algebroid? Since linear categories are often assumed to be additive in practice, such a term might be handy.
• CommentRowNumber16.
• CommentAuthorJohn Baez
• CommentTimeJun 23rd 2010
• (edited Jun 23rd 2010)
Who says that a k-linear category needs to have biproducts? I always thought it was just a category enriched over Vectk!

See for example this reference - not a very good one, but the first that came to hand. Davydov works at Macquarie, and is a smart guy, so I don't think this is just a fluke.

(People often talk about k-linear abelian categories, which of course have biproducts... but that's because of the abeliannness.)

I just edited the examples over at enriched category. This page said that an algebroid is the same as a k-linear category. Whatever you decide, please make sure this page winds up being consistent with everything else!
• CommentRowNumber17.
• CommentAuthorJohn Baez
• CommentTimeJun 23rd 2010
I don't think we want to assume that *-categories to have direct sums. First, and most importantly, I don't think people who use this term usually assume that. Second, I would like a *-algebra to be a *-category with one object.
• CommentRowNumber18.
• CommentAuthorTobyBartels
• CommentTimeJun 23rd 2010

I would like a *-algebra to be a *-category with one object.

Why do you want that? I would want a $*$-algebra to be a $*$-algebroid with one object.

I don’t think people who use this term usually assume that.

But this is decisive.

• CommentRowNumber19.
• CommentAuthorMike Shulman
• CommentTimeJun 24th 2010

I feel that “ringoid” and “Ab-enriched category” have different connotations. In particular, I would feel a little uncomfortable calling a large Ab-enriched category a “ringoid.” But it’s still probably a good idea to have just one page.

I think Tim’s original question isn’t phrased quite right to be talking about a $†$-category, since the $†$ operation on a $†$-category is not an “operation on each Hom(x, y)” but rather an operation taking Hom(x,y) to Hom(y,x). However, it seems likely that a $†$-category is what he actually had in mind, because the equation ${s}^{*}s={\mid s\mid }^{2}$ doesn’t make much sense if ${s}^{*}$ lives in the same homset as $s$. Is that right Tim?

• CommentRowNumber20.
• CommentAuthorTim_van_Beek
• CommentTimeJun 24th 2010

Thanks for all the feedback and sorry if I’m a bit slow to catch up (have some urgent business to take care of).

Do you want to write star-category and C-star-category? We do not have pages on those yet.

Yes, as soon as I can afford the time.

For your purposes, should *-categories and other ℂ-linear categories be additive? That is, should they have direct sums? (It seems to me that C *-categories do not usually have that requirement.)

I’d like to have a name for a *-category that is additive, but that can be a different one, if appropriate.

A little background: My question was triggered by the paper Algebraic Quantum Field Theory, chapter “the category of localized transportable endomorphisms”. This is about the Doplicher-Roberts reconstruction theorem, that says that some crucial information can be reconstructed from the Haag-Kastler net in AQFT under certain circumstances. The gadgets used in the proof naturally form a (symmetric tensor *- ….) category, and the use of category theory leads to a considerable simplification of the proof (if you already know enough category theory, of course, if not, the streamlined version of the original proof by Baumgärtel is probably easier). An exposition of this proof is what I am after.

More later.

• CommentRowNumber21.
• CommentAuthorTim_van_Beek
• CommentTimeJun 24th 2010

However, it seems likely that a $\dagger$-category is what he actually had in mind, because the equation $s^* s = {|s|}^2$ doesn’t make much sense if $s^*$ lives in the same homset as $s$. Is that right Tim?

That’s right, I should have said “the *-operation is from Hom(x,y) to Hom(y,x)”.

• CommentRowNumber22.
• CommentAuthorTobyBartels
• CommentTimeJun 24th 2010

OK, linear category now redirects to algebroid, and nothing left on the Lab should suggest that a ‘linear category’ must be additive.

• CommentRowNumber23.
• CommentAuthorJohn Baez
• CommentTimeJun 24th 2010
Toby: of course I'd want a *-algebra to be a *-algebroid with one object, but apparently some shortsighted people working on quantum theory decreed long ago that *-categories are inevitably linear. So, to them, a *-algebra is a *-category with one object.

Then, later, Peter Selinger introduced the term †-category without any presumption of linearity.

By the way: for decades Jim Dolan and I used the word *-category without any presumption of linearity - and I don't think it was just us.

I wish we had won! Then a *-monoid would be a *-category with one object. A *-algebroid would be a linear *-category (linear over some *-field or commutative *-ring) for which the * operation is *-linear. A *-algebra would be a *-algebroid with one object. And the world would be perfect.

But the world inherits defects through the wrinkly process of history.
• CommentRowNumber24.
• CommentAuthorTobyBartels
• CommentTimeJun 24th 2010

I still don’t understand why you wrote:

I would like a *-algebra to be a *-category with one object.

It seems to me that you meant that, due to the wrinkly process of history, it is an established fact that a $*$-algebra is a $*$-category with one object. It seems clear from what you wrote above that you do not like this!

• CommentRowNumber25.
• CommentAuthorJohn Baez
• CommentTimeJun 25th 2010

I certainly don’t love it. I just meant that if people adopted a definition of *-category where additivity was built in, a *-category with one object would be a very degenerate thing, so I’d prefer leaving out additivity and get a *-category with one object to be something useful like a *-algebra. But you’re right, maybe that doesn’t make sense: maybe I should want *-category to mean something weird, so I can introduce the term *-algebroid to mean what it obviously should mean.

• CommentRowNumber26.
• CommentAuthorTobyBartels
• CommentTimeJun 25th 2010

Actually, it sounded to me like you wanted ‘$*$-category’ to mean a many-object $*$-monoid, not an additive linear $*$-category at all. That would make a lot of sense!

• CommentRowNumber27.
• CommentAuthorTim_van_Beek
• CommentTimeJun 29th 2010

Created the shortest possible versions of star-category and C-star-category and linked to them from DHR category.

I hope I understood John’s remark #23 correctly in the sense that he admits defeat, and that it’s Ok to say that a *-category is linear (i.e. an algebroid).

The next steps would be to show that the DHR category has direct sums and is a braided tensor category. Well, It’s Halvorson who uses “tensor category”, but as the nLab page suggests I should use braided monoidal category instead.

• CommentRowNumber28.
• CommentAuthorTodd_Trimble
• CommentTimeJun 29th 2010

I hope I understood John’s remark #23 correctly in the sense that he admits defeat

John can speak for himself of course, but that’s not how I interpreted it. He says, “I wish we had won!… But the world inherits defects through the wrinkly process of history.” To me that could easily mean, “I wish we had won by now” but that he’ll continue fighting the fight.

• CommentRowNumber29.
• CommentAuthorTim_van_Beek
• CommentTimeJun 29th 2010

No problem :-)

I won’t fight any changes, I simply tried to merge the terminology of the Halvorson paper with the discussion of this thread.

• CommentRowNumber30.
• CommentAuthorMike Shulman
• CommentTimeJun 29th 2010

I think it’s a shame that we can’t just call these “C-linear †-categories.” It’s bad enough already that †-categories are named after their notation, and $*$-categories as defined here have nothing to do with, say, *-automonous categories. It’s also confusing because in a compact dagger-category, I think that $*$ is sometimes used to denote the dualization, not the †-involution, i.e. for $f:A\to B$ we have ${f}^{†}:B\to A$ and ${f}^{*}:{B}^{*}\to {A}^{*}$ and ${f}^{*†}:{A}^{*}\to {B}^{*}$. But if there are lots of people using “$*$-category” this way then there’s probably not much we can do about it.

• CommentRowNumber31.
• CommentAuthorTim_van_Beek
• CommentTimeJun 29th 2010

I hope no one is being considerate of me, I’m completly indifferent about the definition of $*-$category. I’m already translating both the Halvorson paper and the Baumgärtel book to the nomenclature of the nLab, one more vocable to take care of does not pose a problem.

But if there are lots of people using “*-category” this way then there’s probably not much we can do about it.

As long as the nLab is consistent, why not side with the minority if there are good reasons to do so?

• CommentRowNumber32.
• CommentAuthorJohn Baez
• CommentTimeJun 29th 2010
• (edited Jun 29th 2010)

Mike wrote:

I think it’s a shame that we can’t just call these “C-linear †-categories.”

If we’re going to talk to people like Peter Selinger, Bob Coecke, Samson Abramsky, and the large group at Oxford, that’s what we’re going to have to say. There ain’t no way these folks are gonna call these things “*-categories”.

I don’t know how insistent the algebraic field theorists are, or how many there are. Maybe we can take ’em on.

Todd wrote:

To me that could easily mean, “I wish we had won by now” but that he’ll continue fighting the fight.

If there’s anyone who wants to keep fighting this fight, I’ll join in. But I’m not sure what the goal of the $n$Lab is, or whether anybody has decided: to describe how people talk, or to reform how they talk.

It’s possible, just possible, that we have a unique opportunity here to establish consistent conventions for talking about categories and $n$-categories… as long as we don’t get too idealistic and crazy and diverge too far from normal practices. If so, it would be a shame to blow it.

• CommentRowNumber33.
• CommentAuthorTim_van_Beek
• CommentTimeJun 29th 2010

I don’t know how insistent the algebraic field theorists are, or how many there are. Maybe we can take ’em on.

There are 125.

:-)

It’s possible, just possible, that we have a unique opportunity here to establish consistent conventions for talking about categories and n-categories.

IMHO it would be Ok to either use consistently the dagger variant or to provide the link of the two concepts like it is done on the star-category page right now. But you all have to tell me what to do, you are the experts…

• CommentRowNumber34.
• CommentAuthorUrs
• CommentTimeJun 29th 2010
• (edited Jun 29th 2010)

We had these kinds of issues elsewhere already. Usually we decide like this:

• we use the notation and terminology that we think is good.

• we explain how it relates to the way other people use notation and terminology.

Every page should try to tell the reader which terminology it is using and which other choices there are. In a wiki on which many people work and many more people will work in the unforeseeable future, it is hopeless to try to establish a global convention that has to be adhered to in all entries. We can never guarantee that the next random person who decides to create some new page is aware of any one convention that some of us may have agreed on somewhere at some point.

There are 125.

At most. That list contains a bunch of people who are certainly not AQFT researchers, and some that maybe were at some point but no longer are.