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).