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Minimality in diagrams of simplicial sets

Research paper by Carles Broto, Ramón Flores, Carlos Giraldo

Indexed on: 30 Oct '19Published on: 30 May '19Published in: Journal of Homotopy and Related Structures



Abstract

We formulate the concept of minimal fibration in the context of fibrations in the model category \({\mathbf {S}}^{\mathcal {C}}\) of \({\mathcal {C}}\)-diagrams of simplicial sets, for a small index category \({\mathcal {C}}\). When \({\mathcal {C}}\) is an EI-category satisfying some mild finiteness restrictions, we show that every fibration of \({\mathcal {C}}\)-diagrams admits a well-behaved minimal model. As a consequence, we establish a classification theorem for fibrations in \({\mathbf {S}}^{\mathcal {C}}\) over a constant diagram, generalizing the classification theorem of Barratt, Gugenheim, and Moore for simplicial fibrations (Barratt et al. in Am J Math 81:639–657, 1959).