Re: bilax_monoidal_functors

[Jeff Egger <jeffegger@yahoo.ca>]

Dear Andre,

> My goal is to have a public discussion on terminology.

It is good that you provoke us into having such discussions!

> It can be very difficult to agree upon because
> adopting one is like commiting to a rule of law,
> to a moral code, possibly to a social code.
> There is an emotional and social aspect to this
> commitment.

I don't understand this at all.  A co-author of mine
recently commented (complained?) that I seem to "change
[my] notation as often as [my] underwear"; and I am not
that much better with terminology.  Indeed, I am overtly
anarchist in this respect, and instinctively resist all
attempts at codifying language.  Most people would agree
that the most important concepts deserve the shortest
names; but people frequently (honestly) disagree over
which concept is the most important.  More significantly,
attitudes often change with time!  It is frustrating,
then, that people will cling to archaic terminology for
the sake of an emotional and social commitment.

[A wonderful counter-example to this phenomenon is when
Mike Barr gave his opinion that the meaning of star-
autonomous category, which initially included symmetry,
should not do so.  I should also say that I think young
mathematicians are generally worse at this than older
ones.  Indeed, the most extreme version of (what I
perceive to be) the same phenomenon is that of the
undergrad who cannot differentiate z=t^2 "because there
is no x".]

My objection to the phrase "autonomous category" (which
Dusko brought up) has less to do with defending Fred
Linton's original usage of that phrase than the fact
that "autonomous category" is a special case (and, from
one point of view, a rather uninteresting special case)
of "star-autonomous category", whereas it sounds like
"star-autonomous category" should mean an "autonomous
category" with some extra structure.  (And, of course,
this once was the case, w.r.t. the older terminology.)
This is confusing; hence one term or the other should
be changed.  I am, in fact, open to all suggestions,
though I cannot help but prefer that "star-autonomous"
be kept and "autonomous" changed.

Cheers,
Jeff.

P.S. A propos of your first email in this thread, why
bother with all those "lax"s?  If you used

> 1) strong monoidal
> 2) monoidal
> 3) comonoidal
> 4) bimonoidal

instead, then you would have

> A monoid is a monoidal functor 1-->C,
> a comonoid is a comonoidal functor 1-->C
> and a bimonoid is a bimonoidal functor 1-->C.

and you could even substitute
> 5) ambimonoidal
for "Frobenius", since "ambialgebra" has been used for
"Frobenius algebra".

Dare I point out that a strong monoidal functor 1-->C
is a _trivial_ monoid?  ;)





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