Re: "compact", "rigid", or "autonomous"?

Re: "compact", "rigid", or "autonomous"?

[John Baez]

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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Re: "compact", "rigid", or "autonomous"?

[Chris Heunen]

> I don't know if this is what people were thinking when they first
> applied "compact" to categories

As far as I'm aware, the terminology "compact" for categories came about
via representation theory: the finite-dimensional unitary
representations of a group form a category with certain properties, and
the group can be reconstructed from that category when the group is
compact. It seems the name transferred from groups to such categories.

But I wouldn't claim historical correctness; perhaps someone has the
definitive word about the origin of this terminology?

Best,
Chris


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