Abstract
We show by elementary means that every Kan fibration in simplicial sets can be embedded in a univalent Kan fibration.
Highlights
We show by elementary means that every Kan fibration in simplicial sets can be embedded in a univalent Kan fibration
A Kan fibration E → U in the category of simplicial sets is universal in case every Kan fibration Y → X satisfying certain size restrictions is a homotopy pullback of it
A Kan fibration E → U in the category of simplicial sets is univalent if the path space of U is equivalent to the space of equivalences between fibers of E → U ; see Sect. 4 below for a precise formulation
Summary
A Kan fibration E → U in the category of simplicial sets is universal in case every Kan fibration Y → X satisfying certain size restrictions is a homotopy pullback of it. In this context, a typical such size restriction is to require Y → X to have fibers of cardinality strictly less than a fixed regular cardinal. A typical such size restriction is to require Y → X to have fibers of cardinality strictly less than a fixed regular cardinal It was Voevodsky who first constructed such universal Kan fibrations and observed that the ones he constructed satisfied an additional property he dubbed univalence.
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