star versus dagger, compact versus closed
marcus said:
at first glance, it looks to me like what JB was calling a "star-category", Robert Coecke would prefer to call a "dagger-category"
I think so. A star-category is a category where any morphism
f: x -> y
can be "run in reverse" to give a morphism
f*: y -> x
and we have
f** = f
(fg)* = g*f*
Is this the same as Coecke's "dagger-category"?
I think someone with an ear for English will be apt to prefer "star-category" to "dagger-category" for several reasons. The phrase rings better---with a better assortment of vowels. It has fewer syllables.
That's true - it's also been in use longer! With no offense intended, I think people working on categories and quantum computation are reinventing certain concepts developed by people working on http://arxiv.org/abs/math.CT/9812040" . This is good: it means these concepts are really important. But, it causes some notational conflicts.
The concept is all about things like adjoint of an operator A, something often written A*, the complex conjugate transpose of a matrix.
Exactly, that's the main example - but physicists often write this with a dagger instead of a star.
So if Coecke insists on the nomenclature "compact" then a sensible compromise would be "compact star category"------instead of "dagger-compact category"----but we will just have to wait and see.
Compact categories have been around for a long, long time - they're categories where objects have nice duals. How did the term "compact" get used for this? Well, for one thing, compact categories are a special case of
closed categories, where given two objects x,y you have an object
HOM(x,y)
that acts like "the maps from x to y".
For example, consider the category of vector spaces. Given two vector spaces x and y, HOM(x,y) is the vector space of linear operators from x to y.
A closed category is called "closed" because normally we have a
set of morphisms from x to y, but now we have an
object of morphisms from x to y, so we don't have to leave our category to talk about "hom"! So, a closed category is like its own self-contained universe! Cool, huh?
Of course the classic example of a closed category is the category of sets, where there's a
set of functions from a set x to a set y. When they invented closed categories back in 1966,
http://citeseer.ist.psu.edu/context/20076/0" were trying to let other categories be "self-contained" like this. It's one step towards dethroning the category of sets - getting it to stop acting better than everyone else.
Every topos is a closed category... that was a later step towards dethroning the category of sets.
A
compact category is a special sort of closed category where
HOM(x,y) = x* tensor y
For example, this is true for the category of finite-dimensional vector spaces. It's not true for the category of sets, nor for most topoi. Topoi are cartesian closed, not compact, so they embody intuitiionistic logic, not quantum logic.
Now I can finally explain the term "compact" - this is taking long than I expected.
Since "compact" sets in a topological space are specially nice "closed" sets, when people discovered specially nice closed categories they decided to call them compact!
In other words, it's just an erudite joke, of the sort nobody finds funny except mathematicians.
By the way, I haven't given the actual precise definitions of closed and compact categories here, just the intuitions. I did give the precise definition of a star-category, though.
Personally, I often call a star-category a "category with duals for morphisms", and a compact category a "category with duals for objects".
In quantum mechanics we often want categories with both, which I call "categories with duals". I think Abramsky and Coecke call these "dagger-compact categories", or maybe "strongly compact categories". The terminology is more confusing than the actual ideas.
The method of descartes was for sure accepted by more than 0,2 percent of scientists.