Realisability problem in arrow categories

Cristina Costoya, David Méndez, Antonio Viruel

Published: 24 Sept 2019, Last Modified: 29 Apr 2026Collectanea MathematicaEveryoneRevisionsBibTeXCC BY-SA 4.0
Abstract: In this paper we raise the realisability problem in arrow categories. Namely, for a fixed category C and for arbitrary groups H≤ G1× G2, is there an object ϕ: A1→ A2 in Arr (C) such that Aut Arr ( C )(ϕ) = H, Aut C(A1) = G1 and Aut C(A2) = G2? We are interested in solving this problem when C= HoTop∗, the homotopy category of simply-connected pointed topological spaces. To that purpose, we first settle that question in the positive when C= Graphs. Then, we construct an almost fully faithful functor from Graphs to CDGA , the category of commutative differential graded algebras, that provides among other things, a positive answer to our question when C= CDGA and, as long as we work with finite groups, when C= HoTop∗. Some results on representability of concrete categories are also obtained.