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December 5, 2019

Topoi from Lawvere Theories

Posted by John Baez

Often we get a classifying topos by taking a finite limits theory TT and forming the category of presheaves T^=Set T op\hat{T} = Set^{T^{op}}. For example, in their book, Mac Lane and Moerdijk get the classifying topos for commutative rings from the finite limits theory for rings this way. My student Christian Williams suggested an interesting alternative: what if we take the category of presheaves on a Lawvere theory?

For example, suppose we take the category of presheaves on the Lawvere theory for commutative rings. We get a topos. What is it the classifying topos for?

One possibility is that the act of taking presheaves supplies the Lawvere theory for commutative rings with the finite limits it doesn’t already have. This would presumably be due to some nice fact that I’ve never learned. Maybe taking presheaves on a Lawvere theory is the same as first fleshing it out to a finite limits theory and then taking presheaves on it. But this seems unlikely, both for mathematical reasons (I don’t see why it should work), and because Mac Lane and Moerdijk go to the trouble of first forming the finite limits theory of commutative rings before taking presheaves.

Another possibility is that the category of presheaves on the Lawvere theory of commutative rings can be seen as the classifying topos for some entity other than commutative rings.

Posted at December 5, 2019 4:24 PM UTC

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Re: Topoi from Lawvere Theories

Let TT be the Lawvere theory for commutative rings. If I recall correctly, T^\hat T is the classifying topos for the geometric theory of free commutative rings. This includes the additional geometric axioms such as (x+y=x)⇒(y=0)(x+y=x)\implies (y=0) that hold in a free commutative ring. Similarly for any other Lawvere theory.

Categorically, I suppose T^\hat T is the classifying topos for the geometric theory that comes from the finite limit theory that is formed by closing the representables under finite limits in T^\hat T.

I never found these things in print as above (and it’s been a while so I may have misremembered a detail). Having said this, Olivia Caramello and others have done a lot with classifying theories of presheaf toposes (e.g. here). For myself, I used presheaves over Lawvere theories in my work on parameterized algebraic theories, which is about strong monads on those presheaf categories.

Posted by: Sam Staton on December 5, 2019 8:02 PM | Permalink | Reply to this

Re: Topoi from Lawvere Theories

Thank you for your response. The paper on parameterized theories looks great; it comes as a surprise because I’ve had an idea which I believe is similar and connected. As I read, I may have questions for you, if that’s okay. Also, is there a fuller version somewhere? I can only seem to find an “extended abstract”. Thanks.

Posted by: Christian Williams on December 6, 2019 7:20 AM | Permalink | Reply to this

Re: Topoi from Lawvere Theories

Thanks Christian. I wrote a few more papers using the framework (quantum theory, beta/Bernoulli conjugacy, pi calculus and local store) but a longer article is still “in preparation”… Happy to discuss here or on email.

Posted by: Sam Staton on December 6, 2019 8:20 PM | Permalink | Reply to this

Re: Topoi from Lawvere Theories

The result follows from Thm 5.22

Peter T Johnstone and Gavin C Wraith. Algebraic theories in toposes. In Indexed categories and their applications, pages 141–242. Springer, 1978.

It is the classifying topos of flat rings. A flat ring is a filtered colimit of free rings. Flatness is a geometric notion.

I recently used this to compute what theory the topos of cubical sets classifies.

Posted by: Bas Spitters on December 5, 2019 9:09 PM | Permalink | Reply to this

Re: Topoi from Lawvere Theories

Nice! Thanks! And thanks to Sam, too.

Can someone show me the simplest example of a filtered colimit of free commutative rings that’s not free?

Posted by: John Baez on December 5, 2019 9:50 PM | Permalink | Reply to this

Re: Topoi from Lawvere Theories

Let A nA_n be the free commutative ring on a generator a na_n, for nn a positive integer. Define maps A n→A n+1A_n \rightarrow A_{n+1} by sending a na_n to 2*a n+12*a_{n+1}. The colimit of this filtered system of free commutative rings is not free, because it contains an element infinitely divisible by 2.

Posted by: Gavin Wraith on December 6, 2019 12:42 PM | Permalink | Reply to this

Re: Topoi from Lawvere Theories

Thank you Bas, John, Gavin, This is very nice. I wish I had read your paper properly the first time!

Still, I want to check whether my answer is also correct. Can we write down the axioms for the geometric theory of T^\hat T by just including the axioms that are true in all free algebras, as I thought?

Or is there a geometric axiom that is true in all free algebras but that fails in some flat algebra? For example I can’t see how to express finite divisibility by 2 in geometric logic.

(My informal argument is that the finite limit theory for T^\hat T is just the closure of representables under finite limits in T^\hat T, and hence determined by the properties of free algebras, but maybe I missed something.)

Posted by: Sam Staton on December 6, 2019 8:16 PM | Permalink | Reply to this

Re: Topoi from Lawvere Theories

An interesting special case that comes up as a model of HoTT is the cartesian cubical sets, cSet, defined as presheaves on the Lawvere theory of bipointed objects. In this case the free algebras are easy to describe: they are just the strictly bipointed objects, i.e. (A, a0, a1) with a0 /= a1, so it’s easy to say what cSet classifies, namely strictly bipointed objects. I don’t know of a simple description for any of the other other kinds of cubical sets, which classify things like “flat distributive lattices”, for which we seem to lack an axiomatization.

Posted by: Steve Awodey on December 7, 2019 5:47 AM | Permalink | Reply to this

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