2021/06/10 by Conner, Gregory, Herfort, Wolfgang, Kent, Curtis +1
#Algebraic Topology (math.AT) #FOS: Mathematics #Group Theory (math.GR)
paper · doi:10.48550/arxiv.2106.06032
Classically, an abelian group G is said to be slender if every homomorphism from the countable product \mathbb Z\mathbb N to G factors through the projection to some finite product \mathbb Zn. Various authors have proposed generalizations to non-commutative groups, resulting in a plethora of similar but not completely equivalent concepts. In the first part of this work we present a unified treatment of these concepts and examine how are they related. In the second part of the paper we study slender groups in the context of co-small objects in certain categories, and give several new applications including the proof that certain homology groups of Barratt-Milnor spaces are cotorsion groups and a universal coefficients theorem for Čech cohomology with coefficients in a slender group.