I am trying to understand Nikolai Durov's "New Approach to Arakelov Geometry" right now and it got me thinking about a particular thing.
Let $R$ be a commutative, associative ring with unit. We can associate two natural Lawvere theories to $R$:
- We can make $T^{lin}_R$, the Lawvere theory we get by defining all $n$-ary operations to be formal linear combinations $\sum_{i=1}^n a_i X_i$ in $n$ variables, with $a_i$ being elements of $R$. The only nullary operation is $0$.
- We can make $T^{poly}_R$, the Lawvere theory we get by defining all $n$-ary operations to be polynomials in $n$ variables with coefficients in $R$
A model for $T^{lin}_R$ is exactly a $R$-module, a model for $T^{poly}_R$ is exactly an $R$-algebra. In fact we can use that as the definition of modules over $R$ (or $R$-algebras, respectively), which I think of as a nice little fact. An R-algebra is, after all, exactly something in which polynomials with coefficients in $R$ "make sense", same with modules and linear combinations.
Note further, that $T^{lin}_R$ is a commutative theory, while $T^{poly}_R$ is, unless in trivial cases, not. We can recover the ring $R$ in both cases; for $T^{lin}_R$, $R$ is the same as $hom_{T^{lin}_R}(x,-)$; we can interpret $hom_{T^{lin}_R}(x,x)$ as the underlying set of the ring, the multiplication is given by $\circ$ and addition by the interpretation of the linear combination $X+Y$ under $hom_{T^{lin}_R}(x,-)$. $x$ here is the generic object of $T^{lin}_R$. For $T^{poly}_R$, the ring $R$ can be found by $hom_{T^{lin}_R}(1,-)$, addition by $X+Y$, multiplication by $XY$.
Durov tries to generalize algebraic geometry by the first approach, calling a general commutative theory a generalized ring, this includes for example the field with one element $\mathbb{F}_1$, which is the lawvere theory generated by a single nullary operation.
Now, the second approach has been closer to me, since it naturally leads to the functor of points approach, where we take a Lawvere theory $T$, define the category $\mathbb{Aff}_T$ of affine $T$ spaces as the dual to it's category of models, $\mathbb{Aff}_T := \textbf{Mod}(T,\text{Set})^{op}$, and then view presheaves/sheaves with regard to various Grothendieck topologies on $\mathbb{Aff}_T$ as it's generalized schemes. For $T^{poly}_R$ this gives the usual schemes, for the Lawvere theory CartSp with objects $\mathbb{R}^n$ and smooth mappings between them as morphisms we get $C^\infty$-schemes (of which smooth manifolds have a full embedding), taking $\mathbb{C}^n$ with holomorphic mappings we get a convenient category for complex differential geometry, etc.
My question is now the following: Is there a functor from commutative Lawvere theories to general Lawvere theories, which maps $T^{lin}_R$ to $T^{poly}_R$ for any commutative ring (or even better, rig) $R$ ? Can we view this functor as part of an adjunction?
