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 $\phi :A_1 \rightarrow A_2$ in ${\text {Arr}}(\mathcal {C})$ such that ${\text {Aut}}_{{\text {Arr}}(\mathcal {C})}(\phi ) = H$, ${\text {Aut}}_{\mathcal {C}}(A_1) = G_1$ and ${\text {Aut}}_{\mathcal {C}}(A_2) = G_2$? We are interested in solving this problem when $\mathcal {C} =\mathcal {H}oTop_*$, the homotopy category of simply-connected 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 ${\text {CDGA}}$, the category of commutative differential graded algebras, that provides among other things, a positive answer to our question when $\mathcal {C} = {\text {CDGA}}$ 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.