nLab
Kan object

Idea

A Kan object $X$ in a category $C$ is a simplicial object in $C$ satisfying a generalized Kan condition.

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 $\mathrm{Set}$).

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. $C=\mathrm{Set}$) a simplicial set $X$ is a Kan complex if the following equivalent conditions are satisfied.

1. 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 $\begin{array}{ccc}{\Lambda }^k[n]& \stackrel{h_{k,n}}{\to }& X\\ {\downarrow }^{{\iota }_{k,n}}& {\nearrow }_{{\exists }_{h_{k,n}}}& & \\ \Delta [n]& & \end{array}$ commute where the ${\iota }_{k,n}$ are the horn inclusions.

2. ${\mathrm{hom}}_{sSet}(\Delta [n],X)\to {\mathrm{hom}}_{sSet}({\Lambda }^k[n],X)$ 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 $X:{\Delta }^{\mathrm{op}}\to C$ be a simplicial object in $C$.

The object of k-horns of n-simplices of $X$ is defined to be the weighted limit $X^{{\Lambda }_n^k}≔{\lim }_{{\Lambda }_n^k}X$

$X$ 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 $X^{{\Lambda }_n^k}=sSet({\Lambda }_n^k,X)$.

category: ∞-groupoid, simplicial object