>>12484331Let be the syntax category of a single-sorted finite product theory, and let be its category of models. There is an evident forgetful functor , sending a model to , the image of the generating object of . Moreover, this functor has a left adjoint defined on finite sets by the representables and more generally defined via Kan extension over the inclusion . Any -model thus has a simplicial resolution
induced from the comonad structure on . We define a polynomial on to be a connected component of the hom-complex between and treated as a discrete simplicial set; i.e. an element of .
