Consider the definition of a $\mathcal{C}^{\infty}$-scheme given in Dominique Joyce's "Algebraic geometry over C-infty rings". As far as I uderstand (not being an expert either in C-infty rings nor in logic, nor in categories) the setting is the following. You have the Lawvere theory $\mathrm{Euc}$ given by objects $X^n:=\mathbb{R}^n$ where morphisms are dedfined by $\mathrm{Euc}(X^n,X^m):=\mathcal{C}^{\infty}(\mathbb{R}^n,\mathbb{R}^m)$. Define a $\mathcal{C}^{\infty}$-ring to be an algebra for (i.e. a model of) $\mathrm{Euc}$ in $\mathrm{Set}$. A C-infty ring $\mathfrak{C}$ (say $\mathfrak{C}:=F(\mathbb{R})$ for a product-preserving functor $F:\mathrm{Euc}\to\mathrm{Set}$) then acquires the structure of a commutative unital $\mathbb{R}$-algebra, essentially thanks to the fact that each $\mathcal{C}^{\infty}(\mathbb{R}^n,\mathbb{R})$ is, together with many other "smooth" operations coming from compositions of arbitrary smooth functions (i.e. possibly different from $(x,y)\mapsto x+y$ and $(x,y)\mapsto xy$ etc.). One can define local C-infty rings, as well as locally-C-infty ringed spaces and morphisms between such, like in the case of ordinary algebraic geometry. Then there is a notion of spectrum $\mathrm{Spec}(\mathfrak{C})$ of a C-infty ring, hence the notion of C-infty scheme as a locally C-infty ringed space which is locally "affine".

All of this -I think- doesn't really have to do with the fact that we're dealing with smooth functions, but just with the fact that we're given a Lawvere theory $\mathcal{T}$ such that $\mathcal{T}(X^n,X^1)$ is naturally a commutative unital ring.

Suppose we take as $\mathcal{T}$ the category that has objects $X^n:=\mathbb{A}^n$, the affine spaces over a field or ring, and as morphisms the scheme morphisms $\mathcal{T}(X^n,X^m):=\mathrm{Hom}_{\mathrm{Sch}}(\mathbb{A}^n,\mathbb{A}^m)$.

If we defined $\mathcal{T}$-schemes as "spaces with a sheaf of local $\mathcal{T}$-algebras that are locally affine" (in the analogous sense as with Joyce's definition), would we somehow get usual schemesschemes*?

• (at least in the only-closed-points-allowed definition, à la Serre if I'm not mistaken)
1

Can ordinary schemes be described as sheves of algebras/models for a certain Lawvere theory?

Consider the definition of a $\mathcal{C}^{\infty}$-scheme given in Dominique Joyce's "Algebraic geometry over C-infty rings". As far as I uderstand (not being an expert either in C-infty rings nor in logic, nor in categories) the setting is the following. You have the Lawvere theory $\mathrm{Euc}$ given by objects $X^n:=\mathbb{R}^n$ where morphisms are dedfined by $\mathrm{Euc}(X^n,X^m):=\mathcal{C}^{\infty}(\mathbb{R}^n,\mathbb{R}^m)$. Define a $\mathcal{C}^{\infty}$-ring to be an algebra for (i.e. a model of) $\mathrm{Euc}$ in $\mathrm{Set}$. A C-infty ring $\mathfrak{C}$ (say $\mathfrak{C}:=F(\mathbb{R})$ for a product-preserving functor $F:\mathrm{Euc}\to\mathrm{Set}$) then acquires the structure of a commutative unital $\mathbb{R}$-algebra, essentially thanks to the fact that each $\mathcal{C}^{\infty}(\mathbb{R}^n,\mathbb{R})$ is, together with many other "smooth" operations coming from compositions of arbitrary smooth functions (i.e. possibly different from $(x,y)\mapsto x+y$ and $(x,y)\mapsto xy$ etc.). One can define local C-infty rings, as well as locally-C-infty ringed spaces and morphisms between such, like in the case of ordinary algebraic geometry. Then there is a notion of spectrum $\mathrm{Spec}(\mathfrak{C})$ of a C-infty ring, hence the notion of C-infty scheme as a locally C-infty ringed space which is locally "affine".

All of this -I think- doesn't really have to do with the fact that we're dealing with smooth functions, but just with the fact that we're given a Lawvere theory $\mathcal{T}$ such that $\mathcal{T}(X^n,X^1)$ is naturally a commutative unital ring.

Suppose we take as $\mathcal{T}$ the category that has objects $X^n:=\mathbb{A}^n$, the affine spaces over a field or ring, and as morphisms the scheme morphisms $\mathcal{T}(X^n,X^m):=\mathrm{Hom}_{\mathrm{Sch}}(\mathbb{A}^n,\mathbb{A}^m)$.

If we defined $\mathcal{T}$-schemes as "spaces with a sheaf of local $\mathcal{T}$-algebras that are locally affine" (in the analogous sense as with Joyce's definition), would we somehow get usual schemes?