nLab
Kan object

Kan objects

Idea

A Kan object XX in a category CC is a simplicial object in CC satisfying a generalized Kan condition.

This generalization of the notion of Kan complex takes into account that the category in which XX is a simplicial object (namely CC) 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 SetSet).

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

Motivation

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

  1. The map X*X\to * is a Kan fibration. This means for all nn\in\mathbb{N} and for all 0kn0\le k\le n and for all h k,nh_{k , n} the is a morphism h k,n\exists_{h_{k, n}} making Λ k[n] h k,n X ι k,n h k,n Δ[n] \array{\Lambda^k[n]&\xrightarrow{h_{k , n}}&X\\\downarrow^{\iota_{k,n}}&\nearrow_{\exists_{h_{k,n}}}&&\\\Delta[n]&&} commute where the ι k,n\iota_{k,n} are the horn inclusions.

  2. hom sSet(Δ[n],X)hom sSet(Λ k[n],X)hom_\sSet(\Delta[n],X)\to \hom_\sSet(\Lambda^k[n],X) is an epimorphism in Set\Set for all nn and all 0kn0\le k\le n where sSet\sSet is regarded as a Set\Set-enriched category.

Definition

Definition

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

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

XX is called a Kan object (in CC) if X[n]X Λ n kX[n]\to \X^{\Lambda_n^k} is an epimorphism for all nn and all 0kn0\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=SetC=\Set, since for VV-enriched functors F:KCF:K\to C and W:KVW:K\to V we have in the case V=CV=C that the weighted limit lim WF=[K,V](W,F)\lim_W F=[K,V](W,F) coincides with the hom object; so in particular X Λ n k=sSet(Λ n k,X)X^{\Lambda_n^k}=\sSet(\Lambda_n^k,X).

Revised on September 16, 2012 13:05:39 by Tim Porter (95.147.237.41)