Some characterizations of acyclic maps

TL;DR

This paper explores two categorical characterizations of acyclic maps between path-connected spaces, linking them to higher categorical epimorphisms and balanced maps, and discusses the associated homotopy modality and generalized Blakers-Massey theorem.

Contribution

It introduces two new categorical characterizations of acyclic maps and identifies the related homotopy modality, expanding understanding of their properties in homotopy theory.

Findings

Acyclic maps characterized as higher categorical epimorphisms.

Acyclic maps characterized as balanced maps with specific homotopy properties.

Identification of the homotopy modality defined by acyclic maps.

Abstract

We discuss two categorical characterizations of the class of acyclic maps between (path-connected) spaces. The first one is in terms of the higher categorical notion of an epimorphism. The second one employs the notion of a balanced map, that is, a map whose homotopy pullbacks define also homotopy pushouts. We also identify the modality in the homotopy theory of spaces that is defined by the class of acyclic maps, and discuss the content of the generalized Blakers-Massey theorem for this modality.