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 $ϕ\colon A_1 \rightarrow A_2$ in $\operatorname{Arr}(\mathcal{C})$ such that $\operatorname{Aut}_{\operatorname{Arr}(\mathcal{C})}(ϕ) = H$, $\operatorname{Aut}_{\mathcal{C}}(A_1) = G_1$ and $\operatorname{Aut}_{\mathcal{C}}(A_2) = G_2$? We are interested in solving this problem when $\mathcal C =\mathcal{H}oTop_*$, 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 $\operatorname{CDGA}$, the category of commutative differential graded algebras, that provides among other things, a positive answer to our question when $\mathcal C = \operatorname{CDGA}$ and, as long as we work with finite groups, when $\mathcal C =\mathcal{H}oTop_*$. Some results on representability of concrete categories are also obtained.