Segal-type algebraic models ofn–types

Abstract

Abstract. For each n ≥ 1, we introduce two new Segal-type models of ntypes of topological spaces: weakly globular n-fold groupoids, and a lax version of these. We show that any n-type can be represented up to homotopy by such models via an explicit algebraic fundamental n-fold groupoid functor. We compare these models to Tamsamani's weak n-groupoids, and extract from them a model for (k − 1)-connected n-types.

Pseudo-Functors Modelling Higher Structures

“…Further for all s ≥ 0 (p (j+1) ...p (n) X) s = p (j) ...p (n−1) X s = p (j,n−1) X s with X s = X 1 × X 0 s · · ·× X 0 X 1 for s ≥ 2. This proves (9). Remark 3.9.…”

“…In previous work the author developed the notion of weakly globular n-fold structure in all dimension n in a homotopical context: for the modeling of connected n-types with weakly globular cat n -groups [25], and for the modeling of general n-types with weakly globular n-fold groupoids [9]. This paper stretches far beyond a categorical generalization of the higher groupoidal case.…”

A survey of models for (∞, n)-categories

“…We have chosen to take this principle so seriously that we take it as our definition. One can just as well take a more categorical definition of what an ∞-groupoid should be and then try to prove the Homotopy Hypothesis for that particular definition; for example see [16] or [17].…”

Comonad cohomology of track categories

“…We here realize the first step of this program, for track categories -that is, categories enriched in groupoids. In the future we hope to extend this to the cohomology of n-track categories -that is, those enriched in the n-fold groupoidal models of n-types developed by the authors in [BP2] and [P2].…”

“…Instead, we use a version of André-Quillen cohomology, also known as comonad cohomology, since we use a comonad to produce a simplicial resolution of our track category (see [BBe]). We envisage a generalization to higher dimensions, using the n-fold nature of the models of n-types in [BP2] and [P2].…”