Epitopological and pseudotopological fundamental group functors

TL;DR

This paper introduces epitopological and pseudotopological fundamental group functors that extend the classical topological fundamental group, offering greater regularity and potential for advanced homotopy theoretical techniques.

Contribution

It defines new functors from pointed epitopological and pseudotopological spaces to their group objects, extending Biss's topological fundamental group with enhanced regularity and applicability.

Findings

Functors lift Biss's fundamental group to richer categories.

They retain the original topological fundamental group's information.

Potential to apply enriched homotopy theory techniques.

Abstract

In these notes the epitopological and pseudotopological fundamental group functors are introduced. These are functors from the category of pointed epitopological and pseudotopological spaces respectively, to the category of their respective group-objects. Their restrictions to the full subcategory of topological spaces are lifts of the topologized fundamental group functor introduced in [Daniel K. Biss, The topological fundamental group and generalized covering spaces, Topology and its Applications, 124(3):355-371, 2002] and thus retain its information. At the same time, they show greater regularity inherited from the convenient properties of EpiTop and PsTop. Moreover, the use of such convenient categories permits, in principle, to apply general techniques from enriched homotopy theory. Our approach should be compared with the alternative improvement of Biss's functor developed in [Jeremy Brazas, The fundamental group as a topological group, Topology and its Applications, 160(1):170-188, 2013] within the topological setting. Several open problems, including those aimed at understanding the precise relation to Brazas's approach, are scattered throughout the text.