Minimality in diagrams of simplicial sets

Carles Broto, Ramón Flores, Carlos Giraldo

Published 2018-04-17Version 1

If $\mathcal{C}$ is a small category, we use the simplicial cofibrantly generated model structure over the category $\mathbf{S}^\mathcal{C}$ of $\mathcal{C}$-diagrams of simplicial sets to formulate a definition of minimality for $\mathcal{C}$-diagrams which are free. When $\mathcal{C}$ is an artinian $EI$-category, we are able to show that every free diagram $X$ has a well-behaved minimal model. By generalizing the concepts of twisted cartesian product and fiber bundle to the category $\mathbf{S}^\mathcal{C}$ of $\mathcal{C}$-diagrams of simplicial sets, we also establish a classification result for fibrations in $\mathbf{S}^\mathcal{C}$ over a constant diagram, as it is done in the classical way for Kan fibrations in $\mathbf{S}$.