Re: terminology: dagger and involution

[selinger@mathstat.dal.ca (Peter Selinger)]

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
>
>
>