Topological and algebraic pullback functors

TL;DR

This paper explores algebraic characterizations of properties of pullback functors in covering categories, linking them to subgroup concepts and product structures, with examples and open questions.

Contribution

It provides algebraic equivalents for properties of pullback functors, connecting them to subgroup notions and product constructions, and discusses related phenomena and open problems.

Findings

Nullity zero is equivalent to contranormal subgroups

Essential injectivity exhibits a Tannakian-like phenomenon

Essential surjectivity relates to Zappa-Szép products

Abstract

We give algebraic equivalents for certain desirable properties of pullback functors on categories of coverings and group sets, namely nullity zero, essential injectivity, and essential surjectivity. Nullity zero turns out to be equivalent to the notion of a contranormal subgroup. We observe a Tannakian-like phenomenon with essential injectivity. Essential surjectivity is intimately related to Zappa-Sz{\'e}p products. We include several examples, and some open questions.