Topological characterization of various types of \(\mathcal{C}^{\infty}\)-rings (Q2515007)

A small category \(\mathbb T\), having all finite direct products, such that every object in \(\mathbb T\) is a finite Cartesian power of one chosen \(T\in\mathbb T\) is called an algebraic theory. A \(\mathbb T\)-algebra in a category \({\mathcal M}\) is a product-preserving functor \(\mathbb T\to{\mathcal M}\), and the category of \(\mathbb T\)-algebras in \({\mathcal M}\) is \(\mathbb T({\mathcal M})\). A morphism between algebraic theories is a functor \(\mathbb T\to\mathbb T'\) that preserves finite products and maps \(T\) to \(T'\). Any such morphism induces a functor \(\mathbb T'({\mathcal M})\to\mathbb T({\mathcal M})\). A category \(\mathfrak T\), having all small direct products, such that every object in \(\mathfrak T\) is a finite Cartesian power of one chosen \(T\in\mathfrak T\) is called an equational theory. A \(\mathfrak T\)-algebra in a category \({\mathcal M}\) is a product-preserving functor \(\mathfrak T\to{\mathcal M}\), and the category of \(\mathfrak T\)-algebras in \({\mathcal M}\) is \(\mathfrak T({\mathcal M})\). A morphism between equational theories is a functor \(\mathfrak T\to\mathfrak T'\) that preserves finite products and maps \(T\) to \(T'\). Any such morphism induces a functor \(\mathbb T'({\mathcal M})\to\mathbb T({\mathcal M})\). There is a functor: \(\text{Equational theories}\to\text{Algebraic theories}\). The theory of smooth functions \({\mathcal C}^\infty\) has \(\{\mathbb R^n\}\) as objects, and smooth maps between them as morphisms. \({\mathcal C}^\infty\)-algebras in Set are called \({\mathcal C}^\infty\)-rings, and the category of such rings is denoted by \({\mathcal L}\). The example of a \({\mathcal C}^\infty\)-ring is the ring \(C^\infty(\mathbb R^n)\) of smooth functions on \(\mathbb R^n\). Rings having not only polynomial operations, but also all possible smooth functions as operations, are applied when using algebraic-geometric approach to deal with smooth manifolds and singular \(C^\infty\)-spaces. It is very important to choose correctly the right subcategory of the category \({\mathcal L}\). This choice is usually made by using topological or analytic properties of the rings \(C^\infty(\mathbb R^n)\). In this paper, the author shows how to do it. As a result he obtains the categories of germ determined, near-point determined, and point determined \({\mathcal C}^\infty\)-rings, without requiring the rings to be finitely generated. It is proven that any \(\mathbb R\)-algebra morphism, without requiring continuity, into a near-point determined \({\mathcal C}^{\infty}\)-ring is a \({\mathcal C}^{\infty}\)-morphism, and hence continuous. To prove this, the author introduces the notion of a semi-topological theory, which is a theory together with a topology, such that composition of operations is separately continuous in each variable. Three topologies on the theory of smooth functions \({\mathcal C}^\infty\) are defined. Basic open sets for each one of the topologies are obtained by fixing germs or finite jets or values of functions at finite sets of points. It is shown that closed ideals in these topologies are precisely the germ determined, near-point determined, and point determined ones, respectively. Also, the author proves that if \(\mathfrak T'\subset\mathfrak T\) is a sub-theory dense in \(\mathfrak T\) with respect to the topology \(\tau\), \(A\in\mathfrak T(\text{Set})\), and \(B\in\mathfrak T_\tau(\text{Set})\), then any continuous \(\mathfrak T'\)-morphism \(\phi:(A,\omega^\tau_A)\to(B,\omega^\tau_B)\) is a \(\mathfrak T\)-morphism, where \(\omega_A\) is the natural topology on \(A\).