This generalization of the notion of Kan complex takes into account that the category in which $X$ is a simplicial object (namely $C$) 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 $Set$).

Note that a Kan object in a category other than $\Set$ is not per se (a model for) an internal ∞-groupoid (discussed there).

Motivation

Recall that in the unenriched case (i.e. $C=Set$) a simplicial set$X$ is a Kan complex if the following equivalent conditions are satisfied.

The map $X\to *$ is a Kan fibration. This means for all $n\in\mathbb{N}$ and for all $0\le k\le n$ and for all $h_{k , n}$ the is a morphism $\exists_{h_{k, n}}$ making $\array{\Lambda^k[n]&\xrightarrow{h_{k , n}}&X\\\downarrow^{\iota_{k,n}}&\nearrow_{\exists_{h_{k,n}}}&&\\\Delta[n]&&}$ commute where the $\iota_{k,n}$ are the horn inclusions.

$hom_\sSet(\Delta[n],X)\to \hom_\sSet(\Lambda^k[n],X)$ is an epimorphism in $\Set$ for all $n$ and all $0\le k\le n$ where $\sSet$ is regarded as a $\Set$-enriched category.

Definition

Definition

Let $C$ be a category, and let $X:\Delta^{op}\to C$ be a simplicial object in $C$.

The object of k-horns of n-simplices of$X$ is defined to be the weighted limit$\X^{\Lambda_n^k}\coloneqq \lim_{\Lambda_n^k}X$

$X$ is called a Kan object (in $C$) if $X[n]\to \X^{\Lambda_n^k}$ is an epimorphism for all $n$ and all $0\le k\le n$. 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 $C=\Set$, since for $V$-enriched functors $F:K\to C$ and $W:K\to V$ we have in the case $V=C$ that the weighted limit$\lim_W F=[K,V](W,F)$ coincides with the hom object; so in particular $X^{\Lambda_n^k}=\sSet(\Lambda_n^k,X)$.