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.

中文翻译：

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箭头类别中的可实现性问题

在本文中，我们提出了箭头类别中的可实现性问题。即，对于固定类别\（\ mathcal {C} \）和任意组\（H \ le G_1 \ times G_2 \），在\（{中有一个对象\（\ phi：A_1 \ rightarrow A_2 \）\ text {Arr}}（\ mathcal {C}）\）使得\（{\ text {Aut}} _ {{\ text {Arr}}（\ mathcal {C}）}（\ phi）= H \ ），\（{\ text {Aut}} _ {\ mathcal {C}}（A_1）= G_1 \）和\（{\ text {Aut}} _ {\ mathcal {C}}（A_2）= G_2 \ ）？当\（\ mathcal {C} = \ mathcal {H} oTop _ * \）时，我们有兴趣解决这个问题，这是简单连接的尖形拓扑空间的同伦类别。为此，我们首先以积极的态度解决这个问题\（\ mathcal {C} = \ mathcal {G} raphs \）。然后，我们构造从\（\ mathcal {G} raphs \）到\（{\ text {CDGA}} \}的几乎完全忠实的仿函数，该类是可交换微分渐变代数的类别，除其他事项外，它提供了肯定的答案我们的问题，当\（\ mathcal {C} = {\文本{CDGA}} \）和，只要我们用有限的群体，工作的时候\（\ mathcal {C} = \ mathcal {H} OTOP _ * \）。还获得了有关具体类别可表示性的一些结果。