Realisability problem in arrow categories

• Autores: Cristina Costoya , David Méndez Martínez, Antonio Angel Viruel Arbaizar
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 71, Fasc. 3, 2020, págs. 383-405
• Idioma: inglés
• DOI: 10.1007/s13348-019-00265-2
• Texto completo no disponible (Saber más ...)
• Resumen

  • 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.

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