Just Categories now - sci.physics.research

Message from discussion Just Categories now

john baez

More options Dec 10 1998, 3:00 am

Newsgroups: sci.physics.research

From: b...@galaxy.ucr.edu (john baez)

Date: 1998/12/10

Subject: Re: Just Categories now

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Once upon a time, I wrote:

>Hmm, again I'm unhappy for the same sort of nitpicky reason. Again,
>it's bad to care if U is injective on objects, because this property
>is not preserved by natural isomorphisms. I believe the politically
>correct substitute for this property is called "reflecting isomorphisms":
>we say a functor U: C -> D "reflects isomorphisms" if U(f) being an
>isomorphism in D implies that f is an isomorphism in C. In particular,
>nonisomorphic objects in C can't get sent to isomorphic objects in D
>by a functor that reflects isomorphisms.

Jim Dolan kindly pointed out that the last sentence is in error.
For example, if D is a category with lots of isomorphisms, and C is
the category with the same objects and only identity morphisms,
there's an obvious functor U: C -> D. This reflects isomorphisms
but maps nonisomorphic objects in C to isomorphic ones in D.

However, if U: C -> D reflects isomorphisms and is also full, it can't
map nonisomorphic objects to isomorphic ones. In the context of my
remark, this fact is all we really need. Recall that we defined objects
of C to be objects of D "with extra properties" if U: C -> D was full
and faithful. This implies that U reflects isomorphisms. So it also
implies that U can't send nonisomorphic objects to isomorphic ones.

And that's reassuring, because we expect that forgetting extra properties
can't make nonisomorphic objects isomorphic --- though forgetting extra
*structure* can.

But enough of this --- back to physics! Has anyone read Wilczek's paper
"Beyond the Standard Model: This Time for Real"? What do you think?
It argues that the recent neutrino oscillation results support a
supersymmetric SU(5) or SO(10) grand unified theory. Do people really
believe this?

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