dagger vs involutive

dagger vs involutive

[tholen]

Here is an outsider's view on the debate which is all about a
formalistic (not to say meaningless) vs a meaningful name. There seem
to be only very few occasions in mathematics when the formalistic name
won, C*-algebras being a prominent example. In category theory, one is
reminded of the hot debate of triples vs monads of the 60s and 70s. I
guess that at the time of the "Zurich triple book" (SLNM 80) most
people would have predicted that triples had already won the race. Mac
Lane's book CWM appeared only 2 or 3 years later, after a vast amount
of literature on triples. But he consistently used the meaningful name
monad, even though (as far as I know) he had never directly published
on the subject. You be the judge who won!

Walter Tholen.

Re: dagger vs involutive

[Paul B Levy]

Oh dear, I think I might have recently increased the set of synonyms.

A couple of years ago, in conversation with Weng Kin Ho, I suggested the
following terminology.

Involutive category:

a category C, with a functor c : C^op --> C and an isomorphism alpha : c^2
--> id_C

Strictly involutive category:

a category C with a functor c : C^op --> C such that c^2 = id_C

Locally involutive category:

a category C with an identity-on-objects functor c : C^op --> C such that
c^2 = id_C.

Weng Kin used this terminology in his PhD thesis (pp 17-18)

http://www.cs.bham.ac.uk/~wkh/papers/thesis.pdf

I wasn't aware of the other terminologies.

Paul

Re: dagger vs involutive

[Zinovy Diskin]

On 3/5/07, tholen@mathstat.yorku.ca <tholen@mathstat.yorku.ca > wrote:
>
> Here is an outsider's view on the debate which is all about a
> formalistic (not to say meaningless) vs a meaningful name. There seem
> to be only very few occasions in mathematics when the formalistic name
> won, C*-algebras being a prominent example. In category theory, one is


why only few? Recall the Poisson bracket, or Dirac's delta-function, or
quaternions (though as a shorthand for 4D complex number it's probably more
meaningful than triples) or, say, derivative, which is a basic notion in
calculus yet is, in fact, quite a formalistic name.  If to talk about
general tendencies, then it seems the  winner would be a formalistic term
(unfortunately). Consider a competition between a meaningful  yet too long,
or hard to pronounce, or not smooth in some sense term and a meaningless yet
short and energetic term, who would win? Many attempts to make terminology
and notation in a particular domain entirely consistent failed as soon as
they went beyond some reasonable level of consistency.
Zinovy Diskin

And aren't left-right adjoints, vertical-horizontal morphisms in fibrations
of purely typographical ("blackboardial") origin?




reminded of the hot debate of triples vs monads of the 60s and 70s. I
> guess that at the time of the "Zurich triple book" (SLNM 80) most
> people would have predicted that triples had already won the race. Mac
> Lane's book CWM appeared only 2 or 3 years later, after a vast amount
> of literature on triples. But he consistently used the meaningful name
> monad, even though (as far as I know) he had never directly published
> on the subject. You be the judge who won!
>
> Walter Tholen.
>
>
>

Re: dagger vs involutive

[Prof. Peter Johnstone]

Walter is of course quite right about triples vs. monads. But it is
interesting to compare that with the truly awful example of the
term "comma category" (and of course the 2-categorical notion of
"comma object" which it has spawned). The awfulness derives from the
fact that the term is derived not just from a particular notation,
but from an obsolete notation (Mac Lane, for example, despite his
sterling efforts to kill off "triple", uses the term "comma category"
in his book, even though his notation for it doesn't involve a comma).
How is it that we have never managed to find a more descriptive name
for this concept?

While I'm on the subject, does anyone out there know who invented the
terms "pullback" and "pushout"? They have always seemed to me to be
splendid examples of descriptive terminology, but I've never seen
them attributed to a particular person. (And yes, I know that Peter
Freyd invented "Doolittle diagram"; but that joke wouldn't have been
possible if "pullback" and "pushout" hadn't already been established
terminology.)

Peter Johnstone

On Mon, 5 Mar 2007 tholen@mathstat.yorku.ca wrote:

> Here is an outsider's view on the debate which is all about a
> formalistic (not to say meaningless) vs a meaningful name. There seem
> to be only very few occasions in mathematics when the formalistic name
> won, C*-algebras being a prominent example. In category theory, one is
> reminded of the hot debate of triples vs monads of the 60s and 70s. I
> guess that at the time of the "Zurich triple book" (SLNM 80) most
> people would have predicted that triples had already won the race. Mac
> Lane's book CWM appeared only 2 or 3 years later, after a vast amount
> of literature on triples. But he consistently used the meaningful name
> monad, even though (as far as I know) he had never directly published
> on the subject. You be the judge who won!
>
> Walter Tholen.
>
>
>

Re: dagger vs involutive

[Eduardo Dubuc]

>
> In category theory, one is
> reminded of the hot debate of triples vs monads of the 60s and 70s. I
> guess that at the time of the "Zurich triple book" (SLNM 80) most
> people would have predicted that triples had already won the race. Mac
> Lane's book CWM appeared only 2 or 3 years later, after a vast amount
> of literature on triples. But he consistently used the meaningful name
> monad, even though (as far as I know) he had never directly published
> on the subject. You be the judge who won!
>
> Walter Tholen.
>

"after a vast amount of literature on triples"

you should recall that also
                    after a vast amount of literature on monads


e.d.