Diagonal fibrations are pointwise fibrations

TL;DR

This paper proves that in the category of bisimplicial sets, every diagonal fibration also qualifies as a pointwise fibration, unifying two different notions of fibrations within model structures.

Contribution

It establishes that diagonal fibrations are necessarily pointwise fibrations, clarifying the relationship between these two concepts in bisimplicial set model structures.

Findings

Diagonal fibrations are pointwise fibrations.

Unification of different fibrations in bisimplicial sets.

Clarification of model structure relationships.

Abstract

On the category of bisimplicial sets there are different Quillen closed model structures associated to various definitions of fibrations. In one of them, which is due to Bousfield and Kan and that consists of seeing a bisimplicial set as a simplicial object in the category of simplicial sets, fibrations are those bisimplicial set maps such that each of the induced simplicial set maps is a Kan fibration, that is, the pointwise fibrations. In another of them, introduced by Moerdijk, a bisimplicial map is a fibration if it induces a Kan fibration of associated diagonal simplicial sets, that is, the diagonal fibrations. In this note, we prove that every diagonal fibration is a pointwise fibration.