Realisability problem in arrow categories

Abstract

In this paper we raise the realisability problem in arrow categories. Namely, for a fixed category $\mathcal{C}$ and for arbitrary groups $H\le G_1\times G_2$, is there an object $\varphi :A_1\to A_2$ in $\mathrm{Arr}(\mathcal{C})$ such that ${\mathrm{Aut}}_{\mathrm{Arr}(\mathcal{C})}(\varphi )=H$, ${\mathrm{Aut}}_{\mathcal{C}}(A_1)=G_1$ and ${\mathrm{Aut}}_{\mathcal{C}}(A_2)=G_2$? We are interested in solving this problem when $\mathcal{C}=\mathcal{H}oTo{p}_{\ast }$, the homotopy category of pointed topological spaces. To that purpose, we first settle that question in the positive when $\mathcal{C}=\mathcal{G}raphs$. Then, we construct an almost fully faithful functor from $\mathcal{G}raphs$ to $\mathrm{CDGA}$, the category of commutative differential graded algebras, that provides among other things, a positive answer to our question when $\mathcal{C}=\mathrm{CDGA}$ and, as long as we work with finite groups, when $\mathcal{C}=\mathcal{H}oTo{p}_{\ast }$. Some results on representability of concrete categories are also obtained.

24 pages. Minor corrections. To appear in Collectanea Mathematica