In mathematics, especially in
homotopy theory, a left fibration of
simplicial set
In mathematics, a simplicial set is an object composed of ''simplices'' in a specific way. Simplicial sets are higher-dimensional generalizations of directed graphs, partially ordered sets and categories. Formally, a simplicial set may be defined ...
s is a map that has the right lifting property with respect to the horn inclusions
.
A right fibration is one with the right lifting property with respect to the horn inclusions
.
A
Kan fibration is one with the right lifting property with respect to every horn inclusion; hence, a Kan fibration is both a left and right fibration.
On the other hand, a left fibration is a
coCartesian fibration and a right fibration a
Cartesian fibration. In particular,
category fibered in groupoids over another category is a special case of a right fibration of simplicial sets in the
∞-category
In mathematics, more specifically category theory, a quasi-category (also called quasicategory, weak Kan complex, inner Kan complex, infinity category, ∞-category, Boardman complex, quategory) is a generalization of the notion of a category. T ...
setup.
References
*Ch. 2 of Lurie's ''
Higher Topos Theory''.
*
Simplicial sets
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