本文提出了箭头范畴的可实现性问题。即，对于固定范畴$\mathcal{C}$和任意群$H\le G_1\times G_2$，在$\operatorname{Arr}(\mathcal{C})$中是否存在对象$ϕ\colon A_1 \rightarrow A_2$，使得$\operatorname{Aut}_{\operatorname{Arr}(\mathcal{C})}(ϕ) = H$、$\operatorname{Aut}_{\mathcal{C}}(A_1) = G_1$和$\operatorname{Aut}_{\mathcal{C}}(A_2) = G_2$？当点拓扑空间的同伦范畴$\mathcal C =\mathcal{H}oTop_*$。为此目的，我们首先解决这个问题的积极时，$\mathcal C = \mathcal{G}raphs$。然后，我们构建一个几乎完全忠实的函子从$\mathcal{G}raphs$到$\operatorname{CDGA}$，交换微分分次代数的范畴，它提供了除其他事项外，一个积极的回答我们的问题时，$\mathcal C = \operatorname{CDGA}$，只要我们与有限群，当$\mathcal C =\mathcal{H}oTop_*$。还得到了具体范畴可表示性的一些结果。
英文摘要：
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.
