Kan complex

In algebraic topology , a branch of mathematics , Kan complexes are an aid for the combinatorial definition of homotopy groups.

definition

The violet two-Simplex has black edges as borders: . ${\displaystyle \sigma }$ ${\displaystyle \tau _{0},\tau _{2}}$ ${\displaystyle \partial _{0}\sigma =\tau _{0},\partial _{2}\sigma =\tau _{2}}$

A simplicial set is a Kan complex if it satisfies the Kan extension property :

for all and every -elemental set of -Simplices with for all there is a -Simplex with for . ${\displaystyle n\in \mathbb {N} ,0\leq k\leq n+1}$ ${\displaystyle (n+1)}$ ${\displaystyle \left\{\tau _{0},\ldots ,\tau _{k-1},\tau _{k+1},\ldots \tau _{n+1}\right\}}$ ${\displaystyle n}$ ${\displaystyle \partial _{i}\tau _{j}=\partial _{j-1}\tau _{i}}$ ${\displaystyle i<j}$ ${\displaystyle (n+1)}$ ${\displaystyle \sigma }$ ${\displaystyle \partial _{i}\sigma =\tau _{i}}$ ${\displaystyle i=0,\ldots ,k-1,k+1,\ldots ,n+1}$

Homotopy groups

DM Kan gave a combinatorial definition of homotopy groups for Kan complexes.

Example: the singular chain complex

Let be a topological space and its singular chain complex : the n-simplices of are the continuous mappings of the standard n-simplex after . ${\displaystyle X}$ ${\displaystyle S_{*}(X)}$ ${\displaystyle S_{*}(X)}$ ${\displaystyle X}$

${\displaystyle S_{*}(X)}$is a Kan complex, its homotopy groups (in the sense of Kan) match the homotopy groups of . ${\displaystyle X}$

literature

• J. Peter May : Simplicial objects in algebraic topology. Reprint of the 1967 original. Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL 1992, ISBN 0-226-51181-2 .

Web links

• G. Friedman: an elementary illustrated introduction to simplicial sets

Individual evidence

1. ^ Daniel Marinus Kan: A combinatorial definition of homotopy groups. In: Ann. of Math. (2) 67 1958, pp. 282-312.