From: rscott@csws17.ic.sunysb.edu (Robert Scott)
Newsgroups: sci.math
Subject: Re: coskeleton
Date: 5 Sep 92 04:14:01 GMT
Distribution: sci
Organization: State University of New York at Stony Brook

ALLAN ADLER ASKED FOR INFORMATION ABOUT THE CONCEPT OF "CO-SKELETON".  
THE FOLLOWING IS PRETTY MUCH ALL I KNOW ABOUT IT; I HOPE IT HELPS.  AS 
USUAL, CORRECTIONS TO OR IMPROVEMENTS IN THE EXPOSITION ARE WELCOMED.


THE "CO-SKELETON" CONSTRUCTION IN THE THEORY OF SIMPLICIAL SETS IS 
BASICALLY JUST THE SIMPLICIAL VERSION OF THE CONCEPT OF "KILLING ALL 
HOMOTOPY GROUPS OF DIMENSION K OR HIGHER".  IN MORE DETAIL:


A SIMPLICIAL SET IS A CONTRAVARIANT FUNCTOR FROM THE CATEGORY D OF 
"MODEL SIMPLEXES" TO THE CATEGORY S OF SETS.  LET D(K) BE THE 
FULL SUBCATEGORY OF D CONTAINING JUST THE MODEL SIMPLEXES OF DIMENSION K 
OR LESS.  THEN THE INCLUSION FUNCTOR D(K)->D INDUCES A
FUNCTOR FROM THE CATEGORY OF SIMPLICIAL SETS TO THE CATEGORY OF 
"K-SIMPLICIAL SETS" BY THE OBVIOUS "RESTRICTION" PROCESS.  THAT IS, 
GIVEN A SIMPLICIAL SET X:D->S, WE OBTAIN A "K-SIMPLICIAL SET" 
X(K):D(K)->S SIMPLY BY RESTRICTING THE FUNCTOR X TO THE SUBCATEGORY 
D(K).  IN OTHER WORDS, X(K) IS OBTAINED FROM X BY JUST THROWING AWAY ALL 
OF THE SIMPLEXES OF DIMENSION HIGHER THAN K.

THIS RESTRICTION FUNCTOR FROM THE CATEGORY OF SIMPLICIAL SETS TO THE 
CATEGORY OF K-SIMPLICIAL SETS IS SOMETIMES CALLED "TRUNCATION", AND IT 
TURNS OUT TO HAVE BOTH A LEFT ADJOINT AND A RIGHT ADJOINT.  (AN 
"ADJOINT" TO A FUNCTOR IS A SORT OF "BEST APPROXIMATION TO AN INVERSE", 
AND THEY COME IN TWO FLAVORS, LEFT AND RIGHT.)  THUS WE HAVE THE 
FOLLOWING DIAGRAM:


                  <--------------------------------
                  R :=  RIGHT ADJOINT TO TRUNCATION

  SIMPLICIAL      -------------------------------->      K-SIMPLICIAL 
    SETS                  T :=  TRUNCATION                   SETS 

                  <--------------------------------
                  L :=  LEFT ADJOINT TO TRUNCATION



BY COMPOSING THE FUNCTORS IN THIS DIAGRAM WITH EACH OTHER, WE OBTAIN TWO 
DISTINCT ENDO-FUNCTORS OF THE CATEGORY OF SIMPLICIAL SETS:

                                  TL
                SIMPLICIAL     --------->      SIMPLICIAL
                  SETS         --------->        SETS
                                  TR

THE ENDO-FUNCTOR TL IS ALSO KNOWN AS "SKELETON", AND YOU PROBABLY 
ALREADY ARE FAMILIAR WITH IT.  THE ENDO-FUNCTOR TR, ON THE OTHER HAND, 
IS KNOWN AS "CO-SKELETON", AND IT TURNS OUT TO HAVE AN INTERESTING 
SIGNIFICANCE IN TERMS OF KILLING THE HIGHER HOMOTOPY GROUPS OF A 
SIMPLICIAL SET.

IF YOU LOOK CLOSELY AT THE RIGHT-ADJOINT PROPERTY OF THE FUNCTOR R, YOU
WILL SEE THAT WHAT IT REALLY AMOUNTS TO IS THAT "EVERY CONFIGURATION
SHAPED LIKE THE BOUNDARY OF A SIMPLEX OF DIMENSION HIGHER THAN K GETS
FILLED IN BY AN ACTUAL SIMPLEX".  THIS SOUNDS VERY MUCH LIKE THE SORT
OF THING YOU'D NEED IN ORDER TO KILL ALL OF THE HOMOTOPY GROUPS OF
DIMENSION K OR HIGHER, AND, SUBJECT TO ONE IMPORTANT TECHNICAL
RESTRICTION, THAT'S PRETTY MUCH WHAT HAPPENS.

THE TECHNICAL RESTRICTION HERE IS THAT THE SIMPLICIAL SET WHOSE
CO-SKELETON YOU TAKE SHOULD BE OF THE SPECIAL TYPE KNOWN AS A "KAN
COMPLEX".  THUS FOR KAN COMPLEXES, CO-SKELETON IS JUST A VERY NICE
CANONICAL WAY OF KILLING ALL OF THE KTH OR HIGHER HOMOTOPY GROUPS OF
THE COMPLEX.

(TO SEE WHAT GOES WRONG WHEN YOU TAKE THE CO-SKELETON OF A SIMPLICIAL 
SET WHICH IS NOT A KAN COMPLEX, CONSIDER THE SIMPLICIAL SET X PICTURED
AS FOLLOWS:

                             E
                         A  -->  B
                        / \      | F
                       H |      \ /
                         D  <--  C
                             G

THAT IS, THE ONLY NON-DEGENERATE SIMPLEXES IN X ARE THE 0-SIMPLEXES
A,B,C,D AND THE 1-SIMPLEXES E,F,G,H, AS PICTURED.  THE FIRST HOMOTOPY
GROUP OF X IS NON-TRIVIAL, BUT X IS ITS OWN 1-CO-SKELETON, SO CLEARLY
WE DID NOT SUCCEED IN THIS CASE IN KILLING THE FIRST HOMOTOPY GROUP. 
THE PROBLEM IS THAT THE HOMOTOPY CLASS THAT WE WANTED TO KILL, NAMELY
THE LOOP "EFGH", IS TOO "SPREAD OUT" TO BE KILLED BY THE INSERTION OF A
SINGLE 2-SIMPLEX.  THE SPECIALLY NICE SIMPLICIAL SETS KNOWN AS KAN
COMPLEXES, HOWEVER, NEVER EXHIBIT THIS SORT OF IMPOLITE BEHAVIOR.)

SO FAR I HAVE ONLY DISCUSSED THE CASE OF SIMPLICIAL SETS, WHEREAS ALLAN
ACTUALLY ASKED ABOUT SIMPLICIAL OBJECTS IN A CATEGORY C WHICH IS NOT
NECESSARILY THE CATEGORY OF SETS.  I IMAGINE THAT THE DEFINITION OF
CO-SKELETON AS "TRUNCATION, FOLLOWED BY THE RIGHT ADJOINT TO
TRUNCATION" IS TAKEN OVER DIRECTLY IN THIS CASE.  I AM NOT SURE WHAT
PROPERTIES THE CATEGORY C NEEDS IN ORDER FOR CO-SKELETON TO HAVE VERY
NICE PROPERTIES, BUT IN ORDER FOR IT MERELY TO EXIST I DON'T THINK
YOU'D NEED MUCH MORE THAN THAT C BE COMPLETE; LOOK UP "ADJOINT FUNCTOR
THEOREM" IN MACLANE'S BOOK "CATEGORIES FOR THE WORKING MATHEMATICIAN"
FOR INFORMATION ABOUT WHEN ADJOINTS TO FUNCTORS CAN BE EXPECTED TO
EXIST.



-JAMES DOLAN
Newsgroups: sci.math