From: John Baez <john.c.baez@gmail.com>
Subject: Re: "compact", "rigid", or "autonomous"?
Date: Thu, 13 May 2010 10:37:58 -0700 [thread overview]
Message-ID: <E1OCuLU-0002QV-Qd@mailserv.mta.ca> (raw)
Mike wrote:
One reason I like "autonomous" to mean a symmetric monoidal category in
> which all objects have duals is that the only alternative names I have heard
> for such a thing convey misleading intuition to me. They are sometimes
> called "compact closed" or (I think) "rigid" monoidal categories...
Yes, I think "rigid" is traditional in algebraic geometry. Perhaps some
wiser head could explain how it originated!
Personally I often use "compact', since "compact closed" seems redudant.
And if I were king of the world, I'd use "with duals for objects".
Regarding the relation of this use of "compact" to the use in topology:
The only relationship I can think of is that a
>
compact subset of a Hausdorff space is closed, and a symmetric monoidal
> category with duals for objects is also automatically closed, but of course
> these two meanings of "closed" are totally different. Perhaps someone
> can enlighten me?
>
I don't know if this is what people were thinking when they first applied
"compact" to categories, or just my own rationalization, but:
A compact subset is closed, but it has a very nice property: its image under
any continuous map is again closed. Similarly a compact category is a
closed, but it has a very nice property: its essential image under any
symmetric monoidal functor is again compact.
I don't claim this justifies the terminology, but it helped me learn to live
with it.
Best,
jb
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next reply other threads:[~2010-05-13 17:37 UTC|newest]
Thread overview: 2+ messages / expand[flat|nested] mbox.gz Atom feed top
2010-05-13 17:37 John Baez [this message]
2010-05-14 15:34 ` Chris Heunen
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