Minimality in diagrams of simplicial sets

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.

33 pages