Infinitesimal Models of Algebraic Theories
Smooth manifolds have been always understood intuitively as spaces that are infinitesimally linear at each point, and thus infinitesimally affine when forgetting about the base point. The aim of this thesis is to develop a general theory of infinitesimal models of algebraic theories that provides us with a formalisation of these notions, and which is in accordance with the intuition when applied in the context of Synthetic Differential Geometry. This allows us to study well-known geometric structures and concepts from the viewpoint of infinitesimal geometric algebra. Infinitesimal models of algebraic theories generalise the notion of a model by allowing the operations of the theory to be interpreted as partial operations rather than total operations. The structures specifying the domains of definition are the infinitesimal structures. We study and compare two definitions of infinitesimal models: actions of a clone on infinitesimal structures and models of the infinitesimalisation of an algebraic theory in cartesian logic. The last construction can be extended to first-order theories, which allows us to define infinitesimally euclidean and projective spaces, in principle. As regards the category of infinitesimal models of an algebraic theory in a Grothendieck topos we prove that it is regular and locally presentable. Taking a Grothendieck topos as a base we study lifts of colimits along the forgetful functor with a focus on the properties of the category of infinitesimally affine spaces. We conclude with applications to Synthetic Differential Geometry. Firstly, with the help of syntactic categories we show that the formal dual of every smooth ring is an infinitesimally affine space with respect to an infinitesimal structure based on nil-square infinitesimals. This gives us a good supply of infinitesimally affine spaces in every well-adapted model of Synthetic Differential Geometry. In particular, it shows that every smooth manifold is infinitesimally affine and that every smooth map preserves this structure. In the second application we develop some basic theory of smooth loci and formal manifolds in naive Synthetic Differential Geometry using infinitesimal geometric algebra.