Terms and derivatives of polynomial functors via negation

Abstract

Given a locally cartesian closed category E, a polynomial (s,p,t) may be defined as a diagram consisting of three arrows in E of a certain shape. In this paper we define the homogeneous and monomial terms comprising a polynomial (s,p,t) and give sufficient conditions on E such that the homogeneous terms of polynomials exist. We use these homogeneous terms to exhibit an infinite family of coreflection adjunctions between polynomials and homogeneous polynomials of order n. We show that every locally cartesian closed category E with a strict initial object admits a negation operator and a (dense,closed) orthogonal factorization system. We see that both terms and derivatives of polynomial functors are constructed from this negation operator, and that if one takes the localization of E by the class W of dense monomorphisms, then derivatives of all polynomial functors exist. All results are shown formally using only the theory of extensive categories and distributivity pullbacks.