On weak epimorphisms in homotopy theory

Joseph Roitberg

Vol. 121 (1986), No. 1, 183–187

DOI: 10.2140/pjm.1986.121.183

Abstract

Weakened versions of the categorical notions of epimorphism and monomorphism have proved to be of some interest in pointed homotopy theory. A weak epimorphism, for instance, is a morphism e (in any category with 0 objects) such that g ∘ e = 0 implies g = 0.

In 1967, Ganea utilized extensive homotopy-theoretic calculations to exhibit examples, in the pointed homotopy category, of weak monomorphisms which are not monomorphisms. In this note, we exploit the properties of a remarkable group discovered by Higman in 1951 to exhibit examples, again in the pointed homotopy category, of weak epimorphisms which are not epimorphisms, thereby confirming a suspicion enunciated by Hilton in the early 1960’s.

Mathematical Subject Classification 2000

Primary: 55P99

Milestones

Received: 29 May 1984

Revised: 6 August 1984

Published: 1 January 1986