### Topoi from Lawvere Theories

#### Posted by John Baez

Often we get a classifying topos by taking a finite limits theory $T$ and forming the category of presheaves $\hat{T} = Set^{T^{op}}$. 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.

## Re: Topoi from Lawvere Theories

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

Categorically, I suppose $\hat T$ 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 $\hat T$.

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.