Minimality in diagrams of simplicial sets

TL;DR

This paper introduces the concept of minimal fibrations in diagrams of simplicial sets, proving that under certain conditions, every fibration has a minimal model, and extends classical classification results to this broader context.

Contribution

It generalizes the notion of minimal fibrations to diagrams of simplicial sets and establishes a classification theorem for fibrations over constant diagrams.

Findings

Existence of well-behaved minimal models for fibrations in certain diagram categories

Generalization of classical fibration classification theorems

Conditions under which minimal models can be constructed

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.