Re: bilax_monoidal_functors

[Jeff Egger <jeffegger@yahoo.ca>]

>> I guess that in the category of R-modules over a
> > commutative ring  R,
> > a module M has a (good) dual iff it is finitely
> > generated projective
> > iff the endo-functor functor Hom(M,-) preserves all
> > colimits
> > (M is *compact* in a strong sense).

Obviously this is correct.  But, on the other hand, Rel is a 
compact closed category (also: V-Prof, for suitable choice 
of V).  So it is not necessarily the case that every object 
of a compact closed category is small/finite/compact.  

> Indeed, but in this case it is the objects of the category
> which are
> "compact," not the category itself.  So if this is the
> argument, then
> a more natural term would be "locally compact" (clashing
>  with "locally
> small," of course, but agreeing with "locally presentable"
> categories
> in which all objects are presentable).

Hmmm, even that last point is pretty tenuous...  A locally 
presentable category may have the property that every object 
is presentable, but the converse is false.  For example, Sup 
(the category of complete lattices and supremum-preserving 
maps) is not locally presentable; but it is monadic over Set
and therefore has the property in question. 

> (I am *not* proposing to *actually* use "locally compact"
> -- I don't
> want to introduce yet another name for something that
> already has at
> least four names, even if none of the existing four are
> optimal.)

I disagree with this line of argument: if good terminology
can be found, it will kill off its rivals PDQ.  In fact, I 
have not been able to stop myself from thinking about this
issue, and would like to propose "simply closed category" as 
a replacement for "autonomous category" (in the sense of: 
monoidal category in which every object has a left and a 
right dual).  The point is that such a monoidal category is 
(both left and right) closed; moreover, it is one in which 
the "closed structure" (i.e. the pair of internal homs) 
admits an unusually simple description.  

One possible objection, aside from that which Mike has 
already made, is that the word  "simple" already has an 
established mathematical meaning.  My rebuttal to this is 
that there are precedents for using an adverb independently 
of the corresponding adjective.  For example, I see no 
connection between the "completely" in "completely positive 
map" and any of the standard meanings of "complete".  

Cheers,
Jeff.





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