Algebraic Models of Dependent Type Theory

It has been observed [Awo16, Fio12] that the rules governing the essentially algebraic notion of a category with families [Dyb96] precisely match those of a representable natural transformation between presheaves. This provides us with a natural, functorial description of essentially algebraic
objects which are used to model dependent type theory—following Steve Awodey, we call them natural models.
We can view natural models from several different viewpoints, of which we focus on three in this thesis. First, natural models are essentially algebraic, meaning that they can be described by specifying operations between sorts, subject to equational axioms—this allows us to assemble
natural models into a category with certain beneficial properties. Second, since natural models are natural transformations between presheaves, they are morphisms in a locally cartesian closed category, meaning that they can be regarded as polynomials [GK13]. Third, since natural models admit interpretations of dependent type theory, we can use them to provide a functorial semantics.
This thesis develops the theory of natural models in three new directions by viewing them in these three ways.
Natural models as essentially algebraic objects. The first development of the thesis is to bridge the gap between the presentation of natural models as models of an essentially algebraic theory, and the functorial characterisation of natural models as representable natural transformations. We
demonstrate that the functorial characterisations of natural models and morphisms thereof align as we hope with the essentially algebraic characterisations. Natural models as polynomials. The next development is to apply the theory of polynomials in locally cartesian closed categories to natural models. In doing so, we are able to characterise the
conditions under which a natural model admits certain type theoretic structure, and under which a natural transformation is representable, entirely in the internal language of a locally cartesian closed category. In particular, we prove that a natural model admits a unit type and dependent
sum types if and only if it is a polynomial pseudomonad, that it admits dependent product types if and only if it is a pseudoalgebra, and we prove various facts about the full internal subcategory associated with a natural model. Natural models as models of dependent type theory. The final development of the thesis is to demonstrate their suitability as a tool for the semantics of dependent type theory. We build the term model of a particularly simple dependent type theory and prove that it satisfies the appropriate universal property, and then we proceed by describing how to turn an arbitrary natural model into
one admitting additional type theoretic structure in an algebraically free way.