In classical model theory, Fraissé's theorem provides sufficient conditions on the finite (or finitely generated) models of a theory for the existence of an "ultrahomogeneous" model into which all of the finite models embed and such that any embedding of finite models extends to an automorphism of the structure. Work of Caramello explains how the classifying topos of such a model is equivalent to the continuous actions of its automorphism group, suitably topologized.
In this talk we will discuss how the conditions can be generalized when we relax "ultrahomogeneous" to "homogeneous". Equivalently, rather than considering mere embeddings of structures, we allow for more general homomorphisms, which preserve only the `positive' fragment of first-order logic. There is a corresponding relationship to actions of topological monoids. | 
