Gluing of infinitesimal models of algebraic theories

Categories of models of algebraic theories have good categorical properties except for gluing. Building upon insights and examples from Synthetic Differential Geometry, we introduce a generalisation of models of algebraic theories to infinitesimal models. We demonstrate that the category of infinitesimal models retains most of the good categorical properties, but with a stark improvement in the behaviour of gluing. This makes infinitesimal models an interesting natural construction with the ability to interpolate between algebra and geometry.