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Diagonal fibrations are pointwise fibrations
#
Diagonal fibrations are pointwise fibrations

##
A.M. Cegarra and J. Remedios

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.

Journal of Homotopy and Related Structures, Vol. 2(2007), No. 2, pp. 81-92