Homotopy theory of Grothendieck ∞-groupoids and ∞-categories

This work investigates the homotopy theory of globular models for higher categorical structures. In particular, we focus on weak ∞-groupoids, but most of the constructions can be performed also for weak ∞-categories, and we explicitly mention this when appropriate. Motivated by Grothendieck’s homotopy hypothesis, we study algebraic models of homotopy types in the form of ∞-groupoids, and we address the problem of constructing a path object for these structures, after having introduced their homotopy theory. In detail, we define (trivial) cofibrations and (trivial) fibrations of ∞-groupoids, and prove some basic facts about the induced factorization systems. The construction of a path-object endofunctor is a highly non-trivial task, and the first step we take is to characterize those globular theories whose category of models can be endowed with cofibrantly generated semi-model structure of a precise form, and we also give a sufficient condition for this to happen, based on the existence of a path object endofunctor. We then construct the underlying globular set of this path object based on the notion of cylinders, and show how to endow it with systems of structures, involving compositions, identities and inverses. Using the combinatorics of finite planar rooted trees we construct an approximation of the algebraic structure needed for the construction of the path object and we introduce modifications to “correct” this approximation in low dimensions, and we thus interpret all operations of dimension less than or equal to 2. Finally, we manage to complete this construction for the finite-dimensional case of Grothendieck 3-groupoids, thanks to the introduction of a bicategory of cylinders and modifications. We thus establish a semi-model structure on this category, which is conjectured to model homotopy 3-types.