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Stream: community: general

Topic: Garner on cartesian closed varieties

John Baez (Nov 10 2020 at 18:32):

From the secret (i.e., not on Zoom) meetings of the Australian Category Seminar:

Tomorrow we have one talk scheduled (locally there have been some suggestions on how to fill the other slot which will probably be taken up)

At 2:00pm well have Richard Garner (MQ) Speaking on: Cartesian closed varieties (part 3)

Abstract: In the previous talk in this series, I characterised the
category of (non-degenerate) finitary cartesian closed varieties as
being equivalent to the category of (non-degenerate) Boolean restriction
monoids.

In this final part, I explain how to drop the assumption of
finitariness, and show that (non-degenerate) cartesian closed varieties
correspond to (non-degenerate) strongly zero-dimensional join
restriction monoids. If I have time, I will also explain how models of a
cartesian closed theory relate to sheaves on the associated restriction
monoid.

John Baez (Nov 10 2020 at 18:33):

I assume "variety" is in the sense of universal algebra so he's talking about categories of models of Lawvere theories that happen to be cartesian closed.

John Baez (Nov 10 2020 at 18:34):

As you might notice if you reach for examples, it's pretty rare for such categories to be cartesian closed! Sets? Yes. Groups? No. Abelian groups? No. Rings? No.

John Baez (Nov 10 2020 at 18:35):

I tried to look up what a "Boolean restriction monoid" is. I still don't really know.

Nathanael Arkor (Nov 10 2020 at 18:36):

This sounds very interesting. It's a real pity these talks aren't open to a wider audience! (At first, I thought "restriction monoid" referred to restriction categories, but apparently there's a different kind of "restriction monoid", which is probably a more likely candidate.)

dusko (Nov 11 2020 at 08:50):

John Baez said:

I assume "variety" is in the sense of universal algebra so he's talking about categories of models of Lawvere theories that happen to be cartesian closed.

I think Anders Kock characterized such theories probably in the early 70s. All commutative theories (or monads) induce monoidal closed categories. That is where Linton started off with the notion of autonomous categories. Kock I think proved that a theory is commutative iff the variety is monoidal closed. So the question boils down to when the tensor coincides with the cartesian product. For that it is necessary that the free algebra over 1 generator is 1. (funny enough, that came up in an unrelated conversation here yesterday.) Which means that there must be no constants, and there any unary operation must be affine. I am not sure, but I think Kock proved that that is sufficient...

I hope I am not completely missing it. Maybe Garner is talking about those higher categories, I guess cartesian closed $(\infty,1)$-categories? They need those, in fact locally cartesian closed, for HOTT. But is there such a thing as $(\infty,1)$-algebraic theory?

Nathanael Arkor (Nov 11 2020 at 11:40):

In fact, this was the subject of a paper by Johnstone: Collapsed toposes and cartesian closed varieties, so I too am curious what extra understanding Garner's work provides.

Nathanael Arkor (Nov 11 2020 at 11:41):

There is such a thing as an $(\infty, 1)$-algebraic theory, though I would be surprised if this is to what the abstract referred.

John Baez (Nov 11 2020 at 15:43):

Thanks for the help, @dusko! I'm ignorant of these earlier results, but I'm (finally) at a point where they seem really interesting.

For that it is necessary that the free algebra over 1 generator is 1. (funny enough, that came up in an unrelated conversation here yesterday.) Which means that there must be no constants, and [that] any unary operation must be affine.

What does it mean for a unary operation to be affine? (We're already assuming the theory is commutative, right, so every unary operation "commutes" with, or preserves, all n-ary operations.)

John Baez (Nov 11 2020 at 15:45):

One thing about Garner's work is that he's apparently studying the whole categoryof finitary cartesian closed categories:

In the previous talk in this series, I characterised the category of (non-degenerate) finitary cartesian closed varieties as being equivalent to the category of (non-degenerate) Boolean restriction monoids.

John Baez (Nov 11 2020 at 15:47):

Nathanael Arkor said:

In fact, this was the subject of a paper by Johnstone: Collapsed toposes and cartesian closed varieties, so I too am curious what extra understanding Garner's work provides.

Thanks! One new thing, I guess, is that Garner is studying cartesian closed varieties not one at a time but en masse - he's studying the category of all of them:

In the previous talk in this series, I characterised the category of (non-degenerate) finitary cartesian closed varieties as being equivalent to the category of (non-degenerate) Boolean restriction monoids.

dusko (Nov 12 2020 at 01:05):

John Baez said:

What does it mean for a unary operation to be affine? (We're already assuming the theory is commutative, right, so every unary operation "commutes" with, or preserves, all n-ary operations.)

an n-ary f is affine when $f(x,x,\cdots,x) = x$. uh, i should have said all NON-unary operations must be affine. sorry.