Segal-type algebraic models of n-types

TL;DR

This paper introduces new algebraic models for n-types of topological spaces using Segal-type structures, providing explicit algebraic representations and comparisons to existing models.

Contribution

It develops two novel Segal-type models for n-types and establishes their equivalence to topological n-types via an explicit algebraic fundamental functor.

Findings

Models can represent any n-type up to homotopy

Comparison with Tamsamani's models shows their relative strengths

Provides a model for (k-1)-connected n-types

Abstract

For each n\geq 1 we introduce two new Segal-type models of n-types 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