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

## TL;DR

This paper explores the categorical logic of various lambda calculus forms through categories with families, establishing equivalences and biequivalences with fundamental categorical structures and discussing initial object constructions.

## Contribution

It introduces subcategories of cwfs, such as scwfs and ucwfs, and proves several categorical equivalence and biequivalence theorems relating them to key categorical concepts.

## Key findings

- Established equivalences between cwfs and categorical logic structures
- Defined subcategories with simplified type dependencies
- Discussed constructions of initial cwfs with extra structure

## Abstract

We show how the categorical logic of untyped, simply typed and dependently typed lambda calculus can be structured around the notion of category with family (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 (used by Clairambault and Dybjer for their biequivalence theorems). Some theorems are equivalences between notions with strict preservation of chosen structure. Others are biequivalences between notions where properties are only preserved up to isomorphism. In addition to this we discuss various constructions of initial ucwfs, scwfs, and cwfs with extra structure.