From dependent type theory to higher algebraic structures

TL;DR

This dissertation develops a framework for dependently typed algebraic theories, characterizes their models as finitary monads, and explores their applications in higher algebraic structures and homotopy theory, extending classical algebraic and categorical results.

Contribution

It introduces dependently typed algebraic theories as a subclass of GATs, characterizes them via monads, and applies them to model complex algebraic and higher categorical structures, including homotopy models.

Findings

Dependently typed algebraic theories are characterized as finitary monads on presheaf categories.

These theories can model a wide range of algebraic structures like n-categories and operads.

Every locally finitely presentable category is the model category of some dependently typed algebraic theory.

Abstract

The first part of this dissertation defines "dependently typed algebraic theories", which are a strict subclass of the generalised algebraic theories (GATs) of Cartmell. We characterise dependently typed algebraic theories as finitary monads on certain presheaf categories, generalising a well-known result due to Lawvere, B\'enabou and Linton for ordinary multisorted algebraic theories. We use this to recognise dependently typed algebraic theories for a number of classes of algebraic structures, such as small categories, n-categories, strict and weak omega-categories, planar coloured operads and opetopic sets. We then show that every locally finitely presentable category is the category of models of some dependently typed algebraic theory. Thus, with respect to their Set-models, these theories are just as expressive as GATs, essentially algebraic theories and finite limit sketches. However, dependently typed algebraic theories admit a good definition of homotopy-models in spaces, via a left Bousfield localisation of a global model structure on simplicial presheaves. Some cases, such as certain "idempotent opetopic theories", have a rigidification theorem relating homotopy-models and (strict) simplicial models. The second part of this dissertation concerns localisations of presentable $(\infty,1)$-categories. We give a definition of "pre-modulator", and show that every accessible orthogonal factorisation system on a presentable $(\infty,1)$-category can be generated from a pre-modulator by iterating a plus-construction resembling that of sheafification. We give definitions of "modulator" and "left-exact modulator", and prove that they correspond to those factorisation systems that are modalities and left-exact modalities respectively. Thus every left-exact localisation of an $\infty$-topos is obtained by iterating the plus-construction associated to a left-exact modulator.