This generalization of the notion of Kan complex takes into account that the category in which is a simplicial object (namely ) is now not equal to, but only enriched over, the category in which the horns on which the horn-filler condition (the Kan condition) is imposed are simplicial objects (namely ).
Note that a Kan object in a category other than is not per se (a model for) an internal ∞-groupoid (discussed there).
Recall that in the unenriched case (i.e. ) a simplicial set is a Kan complex if the following equivalent conditions are satisfied.
The object of k-horns of n-simplices of is defined to be the weighted limit
is called a Kan object (in ) if is an epimorphism for all and all . We obtain a family of related notions by requiring these maps to be different kinds of epimorphisms (regular, split, etc.).
Note that this condition—called the internal horn-filler condition—coincides with the usual horn-filler condition (i.e. the Kan condition) if , since for -enriched functors and we have in the case that the weighted limit coincides with the hom object; so in particular .