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 $\phi \colon A_1 \rightarrow A_2$ in $\operatorname{ Arr}(\mathcal{ C})$ such that $\operatorname{ Aut}_{ \operatorname{ Arr}(\mathcal{ C})}(\phi) = H$, $\operatorname{ Aut}_{ \mathcal{ C}}(A_1) = G_1$ 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.
Categories: math.AT math.CO math.RT