Categories with Families: Unityped, Simply Typed, and Dependently Typed

Book chapter, 2021

We show how the categorical logic of the untyped, simply typed and dependently typed lambda calculus can be structured around the notion of category with families (cwf). To this end we introduce subcategories of simply typed cwfs (scwfs), where types do not depend on variables, and unityped cwfs (ucwfs), where there is only one type. We prove several equivalence and biequivalence theorems between cwf-based notions and basic notions of categorical logic, such as cartesian operads, Lawvere theories, categories with finite products and limits, cartesian closed categories, and locally cartesian closed categories. Some of these theorems depend on the restrictions of contextuality (in the sense of Cartmell) or democracy (in the sense of Clairambault and Dybjer). Some theorems are equivalences between notions with strict preservation of chosen structure. Others are biequivalences involving notions without chosen structure, and where properties are (necessarily) only preserved up to isomorphism. The cwf-based notions play the role of an abstract syntax of formal systems, and we discuss various constructions of initial ucwfs, scwfs, and cwfs with extra structure. As a corollary of our results we show that equality in the free locally cartesian closed category is undecidable.