Realisability problem in arrow categories
Realisability problem in arrow categories
In this paper we raise the realisability problem in arrow categories. Namely, for a fixed category C and for arbitrary groups H≤ G1× G2, is there an object ϕ: A1→ A2 in Arr (C) such that Aut Arr ( C )(ϕ) = H, Aut C(A1) = G1 and Aut C(A2) = G2? We are interested in solving this problem when C= HoTop∗, the homotopy category of simply-connected pointed topological spaces. To that purpose, we first settle that question in the positive when C= Graphs. Then, we construct an almost fully faithful functor from Graphs to CDGA , the category of commutative differential graded algebras, that provides among other things, a positive answer to our question when C= CDGA and, as long as we work with finite groups, when C= HoTop∗. Some results on representability of concrete categories are also obtained.
383–405
Costoya, Cristina
835f21d2-8417-43d4-8fda-79c32c6dbbd4
Méndez, David
082b86ce-ba78-4519-9713-ff1a38f65c96
Viruel, Antonio
e2cacd64-35e4-43de-9e5a-2df17c5775f8
2019
Costoya, Cristina
835f21d2-8417-43d4-8fda-79c32c6dbbd4
Méndez, David
082b86ce-ba78-4519-9713-ff1a38f65c96
Viruel, Antonio
e2cacd64-35e4-43de-9e5a-2df17c5775f8
Costoya, Cristina, Méndez, David and Viruel, Antonio
(2019)
Realisability problem in arrow categories.
Collectanea Mathematica, 71, .
(doi:10.1007/s13348-019-00265-2).
Abstract
In this paper we raise the realisability problem in arrow categories. Namely, for a fixed category C and for arbitrary groups H≤ G1× G2, is there an object ϕ: A1→ A2 in Arr (C) such that Aut Arr ( C )(ϕ) = H, Aut C(A1) = G1 and Aut C(A2) = G2? We are interested in solving this problem when C= HoTop∗, the homotopy category of simply-connected pointed topological spaces. To that purpose, we first settle that question in the positive when C= Graphs. Then, we construct an almost fully faithful functor from Graphs to CDGA , the category of commutative differential graded algebras, that provides among other things, a positive answer to our question when C= CDGA and, as long as we work with finite groups, when C= HoTop∗. Some results on representability of concrete categories are also obtained.
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Accepted/In Press date: 14 September 2019
e-pub ahead of print date: 24 September 2019
Published date: 2019
Identifiers
Local EPrints ID: 440699
URI: http://eprints.soton.ac.uk/id/eprint/440699
ISSN: 0010-0757
PURE UUID: 227d53ca-11a0-416c-afa6-9b2572131ff3
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Date deposited: 13 May 2020 17:06
Last modified: 16 Mar 2024 07:23
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Author:
Cristina Costoya
Author:
David Méndez
Author:
Antonio Viruel
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