Simplicial  sets,   Postnikov  systems,
and  bounded  cohomology

Contents
A.1.   Introduction 2  
A.2.   Simplicial  sets and $\Delta$-sets 5  
A.3.   Postnikov  systems  and  minimality 9  
A.4.   Classifying  spaces of  categories and  groups 11  
A.5.   Bundles  with  Eilenberg–MacLane  fibers 14  
A.6.   Unraveling  simplicial  sets 22  
A.7.   Isometric  isomorphisms  in  bounded cohomology 28  
Appendices     
A.A.1.   The constructions of  Milnor  and  Segal 33  
A.A.2.   Few  technical  lemmas 35  
References   39

1. Introduction

Bounded cohomology  of  topological  spaces.   The bounded cohomology  groups  $\widehat{H}^{\hskip 0.35002pt*}(\hskip 1.49994ptX\hskip 1.49994pt)$ of  a  topological  space  $X$  were  introduced  by  Gromov  [Gr].   The definition of  $\widehat{H}^{\hskip 0.35002pt*}(\hskip 1.49994ptX\hskip 1.49994pt)$ is  almost  the same as  the definition of  the singular cohomology  $H^{\hskip 0.35002pt*}(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 1.99997pt\mathbf{R}\hskip 1.49994pt)$ of  $X$  with  real  coefficients.   Namely,   in order  to define  $\widehat{H}^{\hskip 0.35002pt*}(\hskip 1.49994ptX\hskip 1.49994pt)$  one needs only  to replace arbitrary  singular $n$-cochains by  singular $n$-cochains which are  bounded as real-valued  functions on  the set  of  singular $n$-simplices.   The effect  of  this change  is  rather dramatic.   It  turns out  that  $\widehat{H}^{\hskip 0.35002pt*}(\hskip 1.49994ptX\hskip 1.49994pt)$ depends only  on  the fundamental  group of  $X$,   or ,   what  is  the same,  $\widehat{H}^{\hskip 0.35002pt*}(\hskip 1.49994ptX\hskip 1.49994pt)$ does not  depend on  the higher  homotopy  groups of  $X$.   At  the same  time $\widehat{H}^{\hskip 0.35002pt*}(\hskip 1.49994ptX\hskip 1.49994pt)$ carries an additional  structure,  a canonical  semi-norm,   and  this semi-norm  is  the  raison  d’être  of  the  theory.

Gromov’s  exposition  [Gr]  of  the bounded cohomology  theory  is  rather cryptic.   For more  than  three decades  the only  available detailed  proofs of  the main  results of  the bounded cohomology  theory  were author’s  proofs  [$I_{\hskip 0.35002pt1}$],   under a  technical  assumption  removed  in  [$I_{\hskip 0.35002pt2}$].   Only  recently  R.  Frigerio  and  M.  Moraschini  [F M]  reconstructed  Gromov’s  proofs and  provided a detailed  exposition of  the  theory  following  Gromov’s  outline.

Simplicial  sets and  Postnikov  systems.   The modern singular  homology and cohomology  theory  was created  by  S.  Eilenberg  in  his paper  [E].   Later on  Eilenberg,   in collaboration  with  S.  MacLane  and  J.  Zilber,   undertook a detailed analysis of  this  theory.   In  particular ,   Eilenberg  and  MacLane  [$E{\hskip 0.50003pt}M_{\hskip 0.35002pt1}$],  [$E{\hskip 0.50003pt}M_{\hskip 0.35002pt2}$]  studied  the influence of  the homotopy  groups of  spaces on  their  homology  and cohomology  groups.   The second  paper  [$E{\hskip 0.50003pt}M_{\hskip 0.35002pt2}$]  in  this series relied  on  the notion of  complete semi-simplicial  complexes  just  introduced  by  Eilenberg  and  Zilber  [E Z].   Nowadays  complete semi-simplicial  complexes are known as  simplicial  sets.

The problem of  influence of  homotopy  groups on  homology  and cohomology  groups was addressed  by  Eilenberg  and  MacLane  only  in  fairly  special  cases.   Their  results were subsumed  by  the  theory of  natural  systems  of  M.M.  Postnikov  [$P_{\hskip 0.35002pt1}$],  [$P_{\hskip 0.35002pt2}$].   Natural  systems were quickly  renamed  into  Postnikov  systems.   The  theory  of  Postnikov  systems allows,   at  least  in  principle,   to determine  the homology  and cohomology  groups of  a space starting  with  its  homotopy  groups and some additional  invariants,   known as  Postnikov  invariants.   Simplicial  sets served as  the natural  framework  for  Postnikov’s  theory  [$P_{\hskip 0.35002pt2}$].

Simplicial  sets and  bounded cohomology  theory.   The bounded cohomology  groups  $\widehat{H}^{\hskip 0.35002pt*}(\hskip 1.49994ptK\hskip 1.49994pt)$ of  a simplicial  set  $K$  are defined  in an obvious manner .   Namely,   the simplicial $n$-cochains of  $K$ with  real  coefficients are  the real-valued  functions on  the set $K_{\hskip 0.70004ptn}$ of  $n$-simplices of  $K$.   A simplicial $n$-cochain  is  bounded  if  it  is  bounded as  a real-valued  function on $K_{\hskip 0.70004ptn}$,   and  the coboundary of  a bounded cochain  is  obviously  bounded.   The bounded cohomology  $\widehat{H}^{\hskip 0.35002pt*}(\hskip 1.49994ptK\hskip 1.49994pt)$  are  defined as  the cohomology  of  the cochain complex of  bounded cochains.   Homotopies of  simplicial  maps  lead  to cochain  homotopies of  complexes of  bounded cochains and,   similarly  to  the case of  spaces,  $\widehat{H}^{\hskip 0.35002pt*}(\hskip 1.49994ptK\hskip 1.49994pt)$ depends only  on  the homotopy  type of  $K$.

It  is  only  natural  to expect  that  the  theories of  simplicial  sets and  Postnikov  systems can  be adapted  to  the bounded version of  the cohomology  theory  and,   in  particular ,   used  to prove  that  the bounded cohomology  groups  $\widehat{H}^{\hskip 0.35002pt*}(\hskip 1.49994ptK\hskip 1.49994pt)$  do not  depend on  the higher  homotopy  groups of  $K$.   The goal  of  the present  paper  is  to show  that  this  is  indeed  the case and,   moreover ,   that  the  tools provided  by  the  theories  of  simplicial  sets and  Postnikov  systems are nearly  ready  for using  in  the bounded cohomology  theory.

Kan  extension  property.   It  is  well  known  that  in  the  the  theory of  simplicial  sets  the internal  notions of  homotopy,  homotopy  type,  and  homotopy  groups are reasonable only  for simplicial  sets satisfying  an additional  condition  known as  the  Kan  extension  property.   Such simplicial  sets are called  Kan  simplicial  sets,  or ,  more recently,    fibrant  simplicial  sets.   The need  to restrict  the class of  considered simplicial  sets  is  clear  in  the bounded cohomology  theory.   Indeed,   the bounded cohomology  groups of  simplicial  sets arising  from  finite simplicial  complexes are  the same as  the usual  real  cohomology  groups.   It  turns out  the  Kan  extensions condition  is  exactly  what  is  needed  for  the bounded cohomology  theory.

The main  theorems.   The first  main  theorem  of  this paper  is  concerned  with  locally  trivial  bundles of  simplicial  sets.   The above notwithstanding,   the base  is  allowed  to be an arbitrary  simplicial  set.   In  this  theorem $K\hskip 1.00006pt(\hskip 1.00006pt\pi\hskip 0.50003pt,\hskip 1.99997ptn\hskip 1.49994pt)$ denotes  the  Eilenberg–MacLane  simplicial  set,   as  in  the classical  papers of  Eilenberg  and  MacLane.   See  Section  Simplicial  sets,   Postnikov  systems, and  bounded  cohomology  for  the details.

Theorem  A.   Let  $E\hskip 0.50003pt,\hskip 3.00003ptB$  be simplicial  sets,   and  let  $p\hskip 1.00006pt\colon\hskip 1.00006ptE\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptB$  be a  locally  trivial  bundle with  the  Eilenberg–MacLane  simplicial  set  $K\hskip 1.00006pt(\hskip 1.00006pt\pi\hskip 0.50003pt,\hskip 1.99997ptn\hskip 1.49994pt)$  as  the fiber .   If  $n\hskip 1.99997pt>\hskip 1.99997pt1$,   then  the map induced  by $p$  in  bounded cohomology  is  an  isometric  isomorphism.   

See  Theorem  Simplicial  sets,   Postnikov  systems, and  bounded  cohomology.   This  result  should  be considered as a  relative property  of  $E$  with  respect  to  $B$.   Since $K\hskip 1.00006pt(\hskip 1.00006pt\pi\hskip 0.50003pt,\hskip 1.99997ptn\hskip 1.49994pt)$ has  the  Kan  extension property,   the  latter  is  present,   albeit  implicitly.

The second  main  theorem  is  concerned  with  the  “classifying  spaces”  $\mathit{B}\hskip 1.49994pt\pi$  of  discrete groups $\pi$,   which are actually  not  spaces,   but  another  incarnation of  simplicial  sets  $K\hskip 1.00006pt(\hskip 1.00006pt\pi\hskip 0.50003pt,\hskip 1.99997pt1\hskip 1.49994pt)$.   See  Section  Simplicial  sets,   Postnikov  systems, and  bounded  cohomology  for  a discussion of  classifying  spaces.

Theorem  B.   Let $\pi$ be a discrete group and  $\kappa\hskip 1.99997pt\subset\hskip 1.99997pt\pi$ be a normal  amenable subgroup of  $\pi$.   Let  $p\hskip 1.00006pt\colon\hskip 1.00006pt\pi\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997pt\pi/\kappa$  be  the quotient  homomorphism.   Then  $\mathit{B}\hskip 0.50003ptp\hskip 1.00006pt\colon\hskip 1.00006pt\mathit{B}\hskip 1.49994pt\pi\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997pt\mathit{B}\hskip 1.49994pt(\hskip 1.00006pt\pi/\kappa\hskip 1.00006pt)$  induces  isometric  isomorphism  in  bounded cohomology.   

See  Theorem  Simplicial  sets,   Postnikov  systems, and  bounded  cohomology.   Note  that  the classifying spaces  $\mathit{B}\hskip 1.49994pt\pi$  are  Kan  simplicial  sets.

In addition  to  the classical  theory of  simplicial  sets,   the proofs of  these  theorems  involve  the notion of  $\Delta$-sets,   introduced  under  the name  semi-simplicial  complexes  by  Eilenberg  and  Zilber  in  the same paper  [E Z]  in  which  they  introduced simplicial  sets.   The $\Delta$-sets differ  from simplicial  sets by  the  lack  of  degeneracy  operators.   See  Section  Simplicial  sets,   Postnikov  systems, and  bounded  cohomology  for  the details.   The modern  term  “$\Delta$-set ”  goes back  to  C.  Rourke  and  B.  Sanderson  [RS].   Our  main example of  a $\Delta$-set  is  the infinitely  dimensional  simplex $\Delta\hskip 0.50003pt[\hskip 1.00006pt\infty\hskip 1.00006pt]$,   the union over $n\hskip 1.99997pt\in\hskip 3.00003pt\mathbf{N}$  of  the standard $n$-dimensional  simplices $\Delta\hskip 0.50003pt[\hskip 0.24994ptn\hskip 1.00006pt]$  considered as $\Delta$-sets.   Again,   see  Section  Simplicial  sets,   Postnikov  systems, and  bounded  cohomology  for  the details.   The $\Delta$-sets are used  to construct  unravelings  of  simplicial  sets.   The idea of  the unraveling  goes back  to  G.  Segal  [S]  and  J.  Milnor’s  construction of  classifying  spaces  [Mi].   The  unraveling  of  a simplicial  set  $K$  is  simply  the $\Delta$-set  $K\hskip 1.00006pt\times\hskip 1.00006pt\Delta\hskip 0.50003pt[\hskip 1.00006pt\infty\hskip 1.00006pt]$.   In  this product  the degeneracy operators  of  $K$ are ignored and  the product  is  taken  dimension-wise,   exactly  as  the products of  simplicial  sets.   The main  result  about  unravelings  is  the following.

Theorem  C.   Let  $K$  be a simplicial  set.   The projection  $p\hskip 1.00006pt\colon\hskip 1.00006ptK\hskip 1.00006pt\times\hskip 1.00006pt\Delta\hskip 0.50003pt[\hskip 1.00006pt\infty\hskip 1.00006pt]\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptK$  induces  isometric  isomorphisms  in  the bounded cohomology  groups.   

See  Theorem  Simplicial  sets,   Postnikov  systems, and  bounded  cohomology.   The proof  is  based on a modification of  the method of  acyclic models.

Applications.   When combined  with  the basic facts of  the  theory  of  Postnikov  systems,   Theorem  A  easily  implies  the following  theorem.

Theorem  D.   Let  $K$  be  a connected  Kan  simplicial  set  and  $f\hskip 1.00006pt\colon\hskip 1.00006ptK\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997pt\mathit{B}\hskip 1.99997pt\pi_{\hskip 0.35002pt1}\hskip 1.00006pt(\hskip 1.49994ptK\hskip 0.50003pt,\hskip 1.99997ptv\hskip 1.49994pt)$,   where $v$  is  a vertex of  $K$,   be a simplicial  map  inducing isomorphism of  fundamental  groups.   Then  $f$  induces an  isomorphism  in  bounded cohomology.   

See  Theorem  Simplicial  sets,   Postnikov  systems, and  bounded  cohomology.   This  theorem  together  with  Theorem  B  easily  implies  the following.

Theorem  E.   Let  $K\hskip 0.50003pt,\hskip 3.00003ptL$  be connected  Kan  simplicial  sets and  let  $v$ be a vertex of  $K$.   Let  $f\hskip 1.00006pt\colon\hskip 1.00006ptK\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptL$  be a simplicial  map.   If  $f_{\hskip 0.70004pt*}\hskip 1.00006pt\colon\hskip 1.00006pt\pi_{\hskip 0.35002pt1}\hskip 1.00006pt(\hskip 1.49994ptK\hskip 0.50003pt,\hskip 1.99997ptv\hskip 1.49994pt)\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997pt\pi_{\hskip 0.35002pt1}\hskip 1.00006pt(\hskip 1.49994ptL\hskip 0.50003pt,\hskip 1.99997ptf\hskip 1.00006pt(\hskip 1.00006ptv\hskip 1.49994pt)\hskip 1.49994pt)$  is  surjective and  has amenable kernel,   then  $f$  induces an  isometric  isomorphisms  in  bounded cohomology.   

See  Theorem  Simplicial  sets,   Postnikov  systems, and  bounded  cohomology.   When applied  to  the singular  simplicial  sets of  topological  spaces,   Theorems  E  and  D  turn  into  Gromov’s  Mapping  theorem  and  its  Corollary  (A)  respectively.   See  [Gr],   p.  40.   In  fact,   Gromov  deduces  his  Mapping  theorem  from  his  Corollary  (A).   Observing  some similarity  between  this deduction  (see  [Gr],   the  top of  p.  47)  and  Segal’s  unraveling  of  categories  [S]  was  the starting  point  of  the present  paper .

The structure of  the  paper .   Sections  Simplicial  sets,   Postnikov  systems, and  bounded  cohomology,  Simplicial  sets,   Postnikov  systems, and  bounded  cohomology,  and  the first  half  of  Section  Simplicial  sets,   Postnikov  systems, and  bounded  cohomology  are devoted  to  the basic definitions and  a  review of  theories used  in  the paper .   The second  half  of  Section  Simplicial  sets,   Postnikov  systems, and  bounded  cohomology  introduces some ideas behind  the proof  of  Theorem  B  and  the definition of  unravelings.   Appendix  Simplicial  sets,   Postnikov  systems, and  bounded  cohomology  provides additional  motivation,   but  is  not  used  in  the main  part  of  the  paper .   Section  Simplicial  sets,   Postnikov  systems, and  bounded  cohomology  is  the  technical  heart  of  the paper ,   laying the groundwork  for  the proof  of  Theorem  A.   Theorem  C  is  proved  in  Section  Simplicial  sets,   Postnikov  systems, and  bounded  cohomology,   which does not  depends on  the rest  of  the paper.   Section  Simplicial  sets,   Postnikov  systems, and  bounded  cohomology  is  devoted  to  the proofs of  Theorems  A  and  B  and deducing  Theorems  D  and  E  from  them.   Appendix  Simplicial  sets,   Postnikov  systems, and  bounded  cohomology  is  devoted  to  the proofs of  several  technical  lemmas.

2. Simplicial  sets  and  $\Delta$-sets

The categories  $\bm{\Delta}$  and  $\Delta$.   We will  include  $0$  in  the set  $\mathbf{N}$  of  natural  numbers.   For every  $n\hskip 1.99997pt\in\hskip 3.00003pt\mathbf{N}$  let  $[\hskip 0.24994ptn\hskip 1.00006pt]$  be  the set  $\{\hskip 1.49994pt0\hskip 0.50003pt,\hskip 3.00003pt1\hskip 0.50003pt,\hskip 3.00003pt\ldots\hskip 0.50003pt,\hskip 3.00003ptn\hskip 1.49994pt\}$.   The category  $\bm{\Delta}$  has sets  $[\hskip 0.24994ptn\hskip 1.00006pt]$  as objects and  non-decreasing  maps  $[\hskip 0.24994ptm\hskip 1.00006pt]\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997pt[\hskip 0.24994ptn\hskip 1.00006pt]$  as morphisms from  $[\hskip 0.24994ptm\hskip 1.00006pt]$  to  $[\hskip 0.24994ptn\hskip 1.00006pt]$,   with  the composition  being  the composition of  maps.   The category  $\Delta$  is  the subcategory  of  $\bm{\Delta}$  having  the same objects and  strictly  increasing  maps  $[\hskip 0.24994ptm\hskip 1.00006pt]\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997pt[\hskip 0.24994ptn\hskip 1.00006pt]$  as morphisms from  $[\hskip 0.24994ptm\hskip 1.00006pt]$  to  $[\hskip 0.24994ptn\hskip 1.00006pt]$.

Simplicial  sets and $\Delta$-sets.   A  simplicial  set  is  a contravariant  functor  from  $\bm{\Delta}$  to  the category  of  sets.   Similarly ,  a  $\Delta$-set  is  a contravariant  functor  from  $\Delta$  to  the category  of  sets.   So,   a simplicial  set  $K$  consists of  a  set  $K_{\hskip 0.70004ptn}$  for every  $n\hskip 1.99997pt\in\hskip 3.00003pt\mathbf{N}$  and a map  $\theta^{\hskip 0.35002pt*}\hskip 1.00006pt\colon\hskip 1.00006ptK_{\hskip 0.70004ptn}\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptK_{\hskip 0.70004ptm}$  for every  non-decreasing  map  $\theta\hskip 1.00006pt\colon\hskip 1.00006pt[\hskip 0.24994ptm\hskip 1.00006pt]\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997pt[\hskip 0.24994ptn\hskip 1.00006pt]$.   For $\Delta$-set  $K$  the map  $\theta^{\hskip 0.35002pt*}$ is  defined only  if  $\theta$  is  strictly  increasing.   If  $K$  is  a  simplicial  set ,   then  the restriction of  the functor  $K$  to  the subcategory  $\Delta$  of  $\bm{\Delta}$  is  a $\Delta$-set ,   which  we will  denote by  $\Delta\hskip 1.00006ptK$  or  simply  by  $K$.

The elements of  $K_{\hskip 0.70004ptn}$  are called  $n$-simplices,   or  simplices  of  dimension  $n$  of  $K$,   and  the maps  $\theta^{\hskip 0.35002pt*}$  the  structure maps  of  $K$.   The $0$-simplices are also called  vertices  and  if  $\sigma\hskip 1.99997pt\in\hskip 3.00003ptK_{\hskip 0.70004ptn}$,   then  the  vertices  of  $\sigma$ are  $0$-simplices of  the form $\theta^{\hskip 0.35002pt*}\hskip 1.00006pt(\hskip 1.00006pt\sigma\hskip 1.49994pt)$ with  $\theta$  being a map  $[\hskip 1.00006pt0\hskip 1.00006pt]\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997pt[\hskip 0.24994ptn\hskip 1.00006pt]$.

If  $K\hskip 0.50003pt,\hskip 3.00003ptL$  are either simplicial  or  $\Delta$-sets,   then a  simplicial  map  $K\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptL$  is  a natural  transformation of  functors,   i.e.  as a sequence of  maps  $K_{\hskip 0.70004ptn}\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptL_{\hskip 0.70004ptn}$  such  that

is  a commutative diagram  for every  morphism  $\theta\hskip 1.00006pt\colon\hskip 1.00006pt[\hskip 0.24994ptm\hskip 1.00006pt]\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997pt[\hskip 0.24994ptn\hskip 1.00006pt]$  of  $\bm{\Delta}$  or  $\Delta$  respectively.

The face and degeneracy  operators.   For every  $n\hskip 1.99997pt\in\hskip 3.00003pt\mathbf{N}$  and  $i\hskip 1.99997pt\in\hskip 1.99997pt[\hskip 0.24994ptn\hskip 1.00006pt]$  there  is  a unique surjective non-decreasing  map $s\hskip 1.00006pt(\hskip 1.00006pti\hskip 1.49994pt)\hskip 1.00006pt\colon\hskip 1.00006pt[\hskip 0.24994ptn\hskip 1.99997pt+\hskip 1.99997pt1\hskip 1.00006pt]\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997pt[\hskip 0.24994ptn\hskip 1.00006pt]$  taking  the value $i$ twice.   If  $n\hskip 1.99997pt>\hskip 1.99997pt0$,   then then  there is  a unique strictly  increasing  map $d\hskip 1.00006pt(\hskip 1.00006pti\hskip 1.49994pt)\hskip 1.00006pt\colon\hskip 1.00006pt[\hskip 0.24994ptn\hskip 1.99997pt-\hskip 1.99997pt1\hskip 1.00006pt]\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997pt[\hskip 0.24994ptn\hskip 1.00006pt]$  not  taking  the value  $i$.   The structure maps  $s_{\hskip 0.70004pti}\hskip 3.99994pt=\hskip 3.99994pts\hskip 1.00006pt(\hskip 1.00006pti\hskip 1.49994pt)^{\hskip 0.35002pt*}$  and  $\partial_{\hskip 0.35002pti}\hskip 3.99994pt=\hskip 3.99994ptd\hskip 1.00006pt(\hskip 1.00006pti\hskip 1.49994pt)^{\hskip 0.35002pt*}$  are  called  the $i$th  degeneracy  and  face  operators  respectively .   If  $\sigma$  is  a simplex  of  $K$,   then  $\partial_{\hskip 0.70004pti}\hskip 1.00006pt\sigma$  is  called  the  $i$th  face  of  $\sigma$.

Clearly,   every  non-decreasing  map  $[\hskip 0.24994ptm\hskip 1.00006pt]\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997pt[\hskip 0.24994ptn\hskip 1.00006pt]$  admits a unique presentation as a composition  $[\hskip 0.24994ptm\hskip 1.00006pt]\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997pt[\hskip 0.50003ptk\hskip 1.00006pt]\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997pt[\hskip 0.24994ptn\hskip 1.00006pt]$  of  a surjective non-decreasing map  $[\hskip 0.24994ptm\hskip 1.00006pt]\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997pt[\hskip 0.50003ptk\hskip 1.00006pt]$  and  a strictly  increasing map  $[\hskip 0.50003ptk\hskip 1.00006pt]\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997pt[\hskip 0.24994ptn\hskip 1.00006pt]$.   On  the other  hand,   every  strictly  increasing  map  is  a composition of  several  maps of  the form  $d\hskip 1.00006pt(\hskip 1.00006pti\hskip 1.49994pt)$,   and every  surjective non-decreasing map  is  a composition of  several  maps of  the form  $s\hskip 1.00006pt(\hskip 1.00006pti\hskip 1.49994pt)$.

It  follows  that  every  structure map  is  a composition of  several  face  and  degeneracy  operators.   These operators satisfy  some simple and  well  known  relations  implied  by  relations between  maps  $s\hskip 1.00006pt(\hskip 1.00006pti\hskip 1.49994pt)\hskip 1.00006pt,\hskip 3.99994ptd\hskip 1.00006pt(\hskip 1.49994ptj\hskip 1.49994pt)$,   which  we do  not  reproduce here.   Conversely ,   the face and degeneracy operators  $\partial_{\hskip 0.35002ptj}\hskip 1.99997pt,\hskip 3.99994pts_{\hskip 0.70004pti}$  satisfying  these relations can  be extended  to a contravariant  functor  from  $\bm{\Delta}$  to  the category  of  sets,   i.e.  to a simplicial  set .   Similarly,   face operators  $\partial_{\hskip 0.35002ptj}$  satisfying  the relations involving only  face operators can  be extended  to a functor  from  $\Delta$  to  the category  of  sets,   i.e.  to a $\Delta$-set .   If  $K$  is  a  simplicial  set ,   then  the $\Delta$-set  $\Delta\hskip 1.00006ptK$  is  the result  of  ignoring  the degeneracy  operators of  $K$.

Non-degenerate simplices.   A simplex $\sigma$ of  a simplicial  set $K$ is  said  to be  degenerate  if  it  belongs  to  the image of  some $s_{\hskip 0.70004pti}$,   and  non-degenerate  otherwise.   An $n$-simplex $\sigma$ is  degenerate  if  and  only  if  $\sigma\hskip 3.99994pt=\hskip 3.99994pt\theta^{\hskip 0.35002pt*}\hskip 1.00006pt(\hskip 1.00006pt\tau\hskip 1.49994pt)$  for an $m$-simplex $\tau$ with $m\hskip 1.99997pt<\hskip 1.99997ptn$ and a surjective non-decreasing  $\theta\hskip 1.00006pt\colon\hskip 1.00006pt[\hskip 0.24994ptn\hskip 1.00006pt]\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997pt[\hskip 0.24994ptm\hskip 1.00006pt]$.   By  a  lemma of  Eilenberg  and  Zilber  [E Z],   if  $\tau$  is  required  to be non-degenerate,   then  the presentation  $\sigma\hskip 3.99994pt=\hskip 3.99994pt\theta^{\hskip 0.35002pt*}\hskip 1.00006pt(\hskip 1.00006pt\tau\hskip 1.49994pt)$  is  unique.   See  Lemma  Simplicial  sets,   Postnikov  systems, and  bounded  cohomology.

Simplicial  sets  from $\Delta$-sets.   Let  $D$  be a $\Delta$-set .   It  gives rise  to simplicial  set $\bm{\Delta}\hskip 1.00006ptD$  defined as  follows.   The  $n$-simplices of  $\bm{\Delta}\hskip 1.00006ptD$  are  the pairs  $(\hskip 1.00006pt\sigma\hskip 0.50003pt,\hskip 3.00003pt\rho\hskip 1.49994pt)$  such  that  $\sigma$  is  an $l$-simplex of  $D$  for some  $l\hskip 1.99997pt\leqslant\hskip 1.99997ptn$  and  $\rho\hskip 1.00006pt\colon\hskip 1.00006pt[\hskip 0.24994ptn\hskip 1.00006pt]\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997pt[\hskip 1.00006ptl\hskip 1.99997pt]$  is  a surjective non-decreasing  map.   In order  to define $\theta^{\hskip 0.70004pt*}$  for a non-decreasing  map  $\theta\hskip 1.00006pt\colon\hskip 1.00006pt[\hskip 0.24994ptm\hskip 1.00006pt]\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997pt[\hskip 0.24994ptn\hskip 1.00006pt]$,   we represent  $\theta$  as  the composition  $\theta\hskip 3.99994pt=\hskip 3.99994pt\tau\hskip 1.00006pt\circ\hskip 1.49994pt\varphi$,   where  $\tau$  is  a  strictly  increasing  map and  $\varphi$  is  a  surjective non-decreasing  map,  and  set  $\theta^{\hskip 0.35002pt*}\hskip 1.00006pt(\hskip 1.00006pt\sigma\hskip 0.50003pt,\hskip 3.00003pt\rho\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994pt(\hskip 1.49994pt\tau^{\hskip 0.70004pt*}\hskip 1.00006pt(\hskip 1.00006pt\sigma\hskip 1.49994pt)\hskip 0.50003pt,\hskip 3.00003pt\varphi\hskip 1.49994pt)$.   One can easily  check  that  $(\hskip 1.00006pt\theta\hskip 1.00006pt\circ\hskip 1.49994pt\eta\hskip 1.49994pt)^{\hskip 0.70004pt*}\hskip 3.99994pt=\hskip 3.99994pt\theta^{\hskip 0.70004pt*}\hskip 0.50003pt\circ\hskip 1.99997pt\hskip 0.50003pt\eta^{\hskip 0.70004pt*}$  and  hence  $\bm{\Delta}\hskip 0.50003ptK$  is  indeed a simplicial  set .   The correspondence  $D\hskip 3.99994pt\longmapsto\hskip 3.99994pt\bm{\Delta}\hskip 1.00006ptD$  naturally  extends  to simplicial  maps,   i.e.  leads  to a functor from  the category  of  $\Delta$-sets  to  the category  of  simplicial  sets.

Simplicial  sets and $\Delta$-sets from simplicial  complexes.   Recall  that  a  simplicial  complex  $S$  is  a set  of  vertices  $V\hskip 3.99994pt=\hskip 3.99994ptV_{\hskip 0.70004ptS}$  together  with a collection of  finite subsets of  $V$,   called  simplices  of  $S$,   subject  to  the condition  that  a subset  of  a simplex  is  also a simplex.   Elements of  a simplex  $\sigma\hskip 1.99997pt\subset\hskip 1.99997ptV$  are called  the  vertices of  $\sigma$.   A simplex $\tau$  is  said  to be a  face  of  a simplex $\sigma$  if  $\tau\hskip 1.99997pt\subset\hskip 1.99997pt\sigma$.   A  local  order  on a simplicial  complex  $S$  is  an assignment  of  a  linear order  $<_{\hskip 0.70004pt\sigma}$  on $\sigma$ for each simplex $\sigma$.   These orders are required  to agree in  the sense  that  $<_{\hskip 0.70004pt\tau}$  is  the restriction of  $<_{\hskip 0.70004pt\sigma}$  if  $\tau$  is  a  face of  $\sigma$.   For example,   if  $<$  is  a  linear order on  $V_{\hskip 0.70004ptS}$,   then  the restrictions of  $<$  to simplices form a  local  order on  $S$.   The simplest  examples of  simplicial  sets and $\Delta$-sets  are provided  by  the following construction.

A  locally  ordered simplicial  complex  $S$  gives rise  to a $\Delta$-set  $\Delta\hskip 0.50003ptS$  and  a simplicial  set $\bm{\Delta}\hskip 0.50003ptS$.   The $n$-simplices of  $\Delta\hskip 0.50003ptS$  are injective maps  $\sigma\hskip 1.00006pt\colon\hskip 1.00006pt[\hskip 0.24994ptn\hskip 1.00006pt]\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptV$  such  that  the image  $\sigma\hskip 1.00006pt(\hskip 1.49994pt[\hskip 0.24994ptn\hskip 1.00006pt]\hskip 1.49994pt)$  is  a  simplex  of  $S$  and $\sigma$ is  order-preserving.   Of  course,   the sets  $[\hskip 0.24994ptn\hskip 1.00006pt]$  are  considered  with  their  natural  order  induced  from  $\mathbf{N}$.   The $n$-simplices of  $\bm{\Delta}\hskip 0.50003ptS$  are maps  $\sigma\hskip 1.00006pt\colon\hskip 1.00006pt[\hskip 0.24994ptn\hskip 1.00006pt]\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptV$  such  that  $\sigma\hskip 1.00006pt(\hskip 1.49994pt[\hskip 0.24994ptn\hskip 1.00006pt]\hskip 1.49994pt)$  is  a  simplex  of  $S$  and $\sigma$  in  non-decreasing  with  respect  to  the orders on $[\hskip 0.24994ptn\hskip 1.00006pt]$ and  this simplex.   In  both cases  the structure maps $\theta^{\hskip 0.35002pt*}$  are defined  by  $\theta^{\hskip 0.35002pt*}\hskip 1.00006pt(\hskip 1.00006pt\sigma\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994pt\sigma\hskip 1.00006pt\circ\hskip 1.49994pt\theta$.   An  easy  check  shows  that  $\bm{\Delta}\hskip 0.50003ptS\hskip 3.99994pt=\hskip 3.99994pt\bm{\Delta}\hskip 0.50003pt\Delta\hskip 0.50003ptS$.   The  local  order  involved  in  this construction almost  never  matters and  is  rarely  mentioned.

Basic examples.   For  $n\hskip 1.99997pt\in\hskip 3.00003pt\mathbf{N}$  the set $[\hskip 0.24994ptn\hskip 1.00006pt]$ can  be considered as a simplicial  complex  having $[\hskip 0.24994ptn\hskip 1.00006pt]$ as its set  of  vertices and all  subsets of  $[\hskip 0.24994ptn\hskip 1.00006pt]$ as simplices.   The usual  order on $\mathbf{N}$  turns $[\hskip 0.24994ptn\hskip 1.00006pt]$  into a  locally  ordered simplicial  complex.   The $\Delta$-set $\Delta\hskip 0.50003pt[\hskip 0.24994ptn\hskip 1.00006pt]$  has as $k$-simplices strictly  increasing  maps  $[\hskip 1.00006ptk\hskip 1.49994pt]\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997pt[\hskip 0.24994ptn\hskip 1.00006pt]$,   and  the simplicial  set $\bm{\Delta}\hskip 0.50003pt[\hskip 0.24994ptn\hskip 1.00006pt]$  has as $k$-simplices  non-decreasing  maps  $[\hskip 1.00006ptk\hskip 1.49994pt]\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997pt[\hskip 0.24994ptn\hskip 1.00006pt]$.   Clearly,  $\bm{\Delta}\hskip 0.50003pt[\hskip 0.24994ptn\hskip 1.00006pt]\hskip 3.99994pt=\hskip 3.99994pt\bm{\Delta}\hskip 1.00006pt\Delta\hskip 0.50003pt[\hskip 0.24994ptn\hskip 1.00006pt]$.   We will  need also  the simplicial  complex  $[\hskip 1.00006pt\infty\hskip 1.00006pt]$  having  $\mathbf{N}$  as its set  of  vertices and all  finite subsets of  $\mathbf{N}$  as simplices,   as also  the $\Delta$-set  $\Delta\hskip 0.50003pt[\hskip 1.00006pt\infty\hskip 1.00006pt]$  and  the simplicial  set  $\bm{\Delta}\hskip 0.50003pt[\hskip 1.00006pt\infty\hskip 1.00006pt]\hskip 3.99994pt=\hskip 3.99994pt\bm{\Delta}\hskip 1.00006pt\Delta\hskip 0.50003pt[\hskip 1.00006pt\infty\hskip 1.00006pt]$.

Every  non-decreasing  map  $\theta\hskip 1.00006pt\colon\hskip 1.00006pt[\hskip 0.24994ptm\hskip 1.00006pt]\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997pt[\hskip 0.24994ptn\hskip 1.00006pt]$  defines a simplicial  map  $\theta_{\hskip 0.70004pt*}\hskip 1.00006pt\colon\hskip 1.00006pt\bm{\Delta}\hskip 0.50003pt[\hskip 0.24994ptm\hskip 1.00006pt]\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997pt\bm{\Delta}\hskip 0.50003pt[\hskip 0.24994ptn\hskip 1.00006pt]$  by  the rule  $\theta_{\hskip 0.70004pt*}\hskip 1.00006pt(\hskip 1.00006pt\sigma\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994pt\theta\hskip 1.00006pt\circ\hskip 1.00006pt\sigma$.   Clearly,  $(\hskip 1.49994pt\theta\hskip 1.00006pt\circ\hskip 1.00006pt\eta\hskip 1.99997pt)_{\hskip 0.70004pt*}\hskip 3.99994pt=\hskip 3.99994pt\theta_{\hskip 0.70004pt*}\hskip 1.00006pt\circ\hskip 1.99997pt\eta_{\hskip 0.70004pt*}$  and  the assignments $n\hskip 3.99994pt\longmapsto\hskip 3.99994pt\bm{\Delta}\hskip 0.50003pt[\hskip 0.24994ptn\hskip 1.00006pt]$  and  $\theta\hskip 3.99994pt\longmapsto\hskip 3.99994pt\theta_{\hskip 0.70004pt*}$  define a covariant  functor from $\bm{\Delta}$ to  the category  of  simplicial  sets.   Similarly,   strictly  increasing maps  $\theta\hskip 1.00006pt\colon\hskip 1.00006pt[\hskip 0.24994ptm\hskip 1.00006pt]\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997pt[\hskip 0.24994ptn\hskip 1.00006pt]$  define simplicial  maps  $\theta_{\hskip 0.70004pt*}\hskip 1.00006pt\colon\hskip 1.00006pt\Delta\hskip 0.50003pt[\hskip 0.24994ptm\hskip 1.00006pt]\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997pt\Delta\hskip 0.50003pt[\hskip 0.24994ptn\hskip 1.00006pt]$  and  lead  to a functor  from  $\Delta$  to $\Delta$-sets.

Kan  extension and  lifting  properties.   Let $n\hskip 1.99997pt\in\hskip 3.00003pt\mathbf{N}$ and  $k\hskip 1.99997pt\in\hskip 1.99997pt[\hskip 0.24994ptn\hskip 1.00006pt]$.   The  $k$-horn  of  $[\hskip 0.24994ptn\hskip 1.00006pt]$  is  the simplicial  complex  $[\hskip 0.24994ptn\hskip 1.00006pt]_{\hskip 0.35002ptk}$ having $[\hskip 0.24994ptn\hskip 1.00006pt]$ as  the set  of  vertices and subsets of  $[\hskip 0.24994ptn\hskip 1.00006pt]$ not  containing  $[\hskip 0.24994ptn\hskip 1.00006pt]\hskip 1.99997pt\smallsetminus\hskip 1.99997pt\{\hskip 1.49994ptk\hskip 1.99997pt\}$ as simplices.   Equivalently ,  $[\hskip 0.24994ptn\hskip 1.00006pt]_{\hskip 0.70004ptk}$ is  obtained  from $[\hskip 0.24994ptn\hskip 1.00006pt]$  by  removing  simplices $[\hskip 0.24994ptn\hskip 1.00006pt]$ and $[\hskip 0.24994ptn\hskip 1.00006pt]\hskip 1.99997pt\smallsetminus\hskip 1.99997pt\{\hskip 1.49994ptk\hskip 1.99997pt\}$.   The $k$-horn $[\hskip 0.24994ptn\hskip 1.00006pt]_{\hskip 0.35002ptk}$  of  $[\hskip 0.24994ptn\hskip 1.00006pt]$  leads  to  the $k$-horn  $\bm{\Lambda}_{\hskip 0.35002ptk}\hskip 0.50003pt[\hskip 0.24994ptn\hskip 1.00006pt]\hskip 3.99994pt=\hskip 3.99994pt\bm{\Delta}\hskip 0.50003pt[\hskip 0.24994ptn\hskip 1.00006pt]_{\hskip 0.35002ptk}$  of  $\bm{\Delta}\hskip 0.50003pt[\hskip 0.24994ptn\hskip 1.00006pt]$.

A simplicial  set  $K$  is  said  to have  the  Kan  extension  property,  or  to be a  Kan  simplicial  set ,   if  every  simplicial  map  $\bm{\Lambda}_{\hskip 0.35002ptk}\hskip 0.50003pt[\hskip 0.24994ptn\hskip 1.00006pt]\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptK$  can  be extended  to a simplicial  map  $\bm{\Delta}\hskip 1.00006pt[\hskip 0.24994ptn\hskip 1.00006pt]\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptK$.   Let  $E\hskip 0.50003pt,\hskip 3.00003ptB$  be simplicial  sets.   A simplicial  map  $p\hskip 1.00006pt\colon\hskip 1.00006ptE\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptB$  is  said  to  have  Kan  lifting  property,  or  to be a  Kan  fibration  if  every commutative diagram of  solid  arrows of  the form

where  $i$  is  the inclusion,   can  be completed  by  a dashed arrow  to a commutative diagram.   A  simplicial  set  $K$  is  Kan  if  and  only  if  the unique map  $K\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997pt\bm{\Delta}\hskip 0.50003pt[\hskip 1.00006pt0\hskip 1.00006pt]$  is  a  Kan  fibration.

Simplices and simplicial  maps.   Let  $\bm{\iota}_{\hskip 0.70004ptn}$  be  the identity  map  $[\hskip 0.24994ptn\hskip 1.00006pt]\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997pt[\hskip 0.24994ptn\hskip 1.00006pt]$  considered as an $n$-simplex of  $\bm{\Delta}\hskip 0.50003pt[\hskip 0.24994ptn\hskip 1.00006pt]$.   Then every $m$-simplex of  $\bm{\Delta}\hskip 0.50003pt[\hskip 0.24994ptn\hskip 1.00006pt]$  is  equal  to  $\theta^{\hskip 0.35002pt*}\hskip 1.00006pt(\hskip 1.99997pt\bm{\iota}_{\hskip 0.70004ptn}\hskip 1.49994pt)$  for a unique non-decreasing  map  $\theta\hskip 1.00006pt\colon\hskip 1.00006pt[\hskip 0.24994ptm\hskip 1.00006pt]\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997pt[\hskip 0.24994ptn\hskip 1.00006pt]$.   It  follows  that  for every simplicial  set  $K$  simplicial  maps  $f\hskip 1.00006pt\colon\hskip 1.00006pt\bm{\Delta}\hskip 0.50003pt[\hskip 0.24994ptn\hskip 1.00006pt]\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptK$ are uniquely  determined  by  the images $f\hskip 1.00006pt(\hskip 1.99997pt\bm{\iota}_{\hskip 0.70004ptn}\hskip 1.00006pt)$.   Conversely,   if  $\sigma\hskip 1.99997pt\in\hskip 3.00003ptK_{\hskip 0.70004ptn}$,   then  there  is  a unique simplicial  map

$$\quad i_{\hskip 1.04996pt\sigma}\hskip 1.00006pt\colon\hskip 1.00006pt\bm{\Delta}\hskip 0.50003pt[\hskip 0.24994ptn\hskip 1.00006pt]\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptK$$

such  that  $\sigma\hskip 3.99994pt=\hskip 3.99994pti_{\hskip 1.04996pt\sigma}\hskip 1.00006pt(\hskip 1.99997pt\bm{\iota}_{\hskip 0.70004ptn}\hskip 1.00006pt)$.

Skeletons.   Let  $n\hskip 1.99997pt\in\hskip 3.00003pt\mathbf{N}$.   If  $D$  is  a $\Delta$-set,   then  the $n$th  skeleton  $\operatorname{sk}_{\hskip 0.70004ptn}\hskip 0.50003ptD$  of  $D$  is  the $\Delta$-subset  of  $D$  consisting of  all $k$-simplices with  $k\hskip 1.99997pt\leqslant\hskip 1.99997ptn$.   If  $K$  is  a simplicial  set,   then  $\operatorname{sk}_{\hskip 0.70004ptn}\hskip 0.50003ptK$  consists of  all  simplices of  the form  $\theta^{\hskip 0.70004pt*}\hskip 1.00006pt(\hskip 1.00006pt\sigma\hskip 1.49994pt)$,   where $\sigma$  is  a $k$-simplex  for some  $k\hskip 1.99997pt\leqslant\hskip 1.99997ptn$.   The  boundary  $\partial{\hskip 0.50003pt}\bm{\Delta}\hskip 0.50003pt[\hskip 0.24994ptn\hskip 1.00006pt]$ of  $\bm{\Delta}\hskip 0.50003pt[\hskip 0.24994ptn\hskip 1.00006pt]$ is  defined as  the skeleton $\operatorname{sk}_{\hskip 0.70004ptn\hskip 0.70004pt-\hskip 0.70004pt1}\hskip 0.50003pt\bm{\Delta}\hskip 0.50003pt[\hskip 0.24994ptn\hskip 1.00006pt]$.   The simplicial  set $\partial{\hskip 0.50003pt}\bm{\Delta}\hskip 0.50003pt[\hskip 0.24994ptn\hskip 1.00006pt]$ has as simplices non-decreasing  maps $[\hskip 0.24994ptm\hskip 1.00006pt]\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997pt[\hskip 0.24994ptn\hskip 1.00006pt]$ with  the image  $\neq\hskip 3.99994pt[\hskip 0.24994ptn\hskip 1.00006pt]$.

Products.   The  product  $K\hskip 1.00006pt\times\hskip 1.00006ptL$ of  simplicial  sets and $\Delta$-sets  $K\hskip 0.50003pt,\hskip 3.00003ptL$  is  defined dimension-wise.   In  more details,  $(\hskip 1.49994ptK\hskip 1.00006pt\times\hskip 1.00006ptL\hskip 1.49994pt)_{\hskip 0.70004ptn}\hskip 3.99994pt=\hskip 3.99994pt\hskip 1.00006ptK_{\hskip 0.70004ptn}\hskip 1.00006pt\times\hskip 1.49994ptL_{\hskip 0.70004ptn}$  and  the structure maps of  $K\hskip 1.00006pt\times\hskip 1.00006ptL$  are  the products of  the structure maps of  $K$  and  $L$.   This  dimension-wise  product  is  hardly  natural  for  finite $\Delta$-sets,   but  products  with  $\Delta\hskip 0.50003pt[\hskip 1.00006pt\infty\hskip 1.00006pt]$ play a  key  role in  our  theory.

Homotopies.   The maps  $d\hskip 1.00006pt(\hskip 1.00006pt0\hskip 1.00006pt),\hskip 3.00003ptd\hskip 1.00006pt(\hskip 1.00006pt1\hskip 1.00006pt)\hskip 1.99997pt\colon\hskip 1.99997pt[\hskip 1.00006pt0\hskip 1.00006pt]\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997pt[\hskip 1.00006pt1\hskip 1.00006pt]$  take $0$ to $1$ and $0$ respectively.   Let  $i\hskip 1.00006pt(\hskip 1.00006pt0\hskip 1.00006pt)\hskip 3.99994pt=\hskip 3.99994ptd\hskip 1.00006pt(\hskip 1.00006pt1\hskip 1.00006pt)_{\hskip 0.70004pt*}$  and  $i\hskip 1.00006pt(\hskip 1.00006pt1\hskip 1.00006pt)\hskip 3.99994pt=\hskip 3.99994ptd\hskip 1.00006pt(\hskip 1.00006pt0\hskip 1.00006pt)_{\hskip 0.70004pt*}$  be  the corresponding  maps  $\bm{\Delta}\hskip 0.50003pt[\hskip 1.00006pt0\hskip 1.00006pt]\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997pt\bm{\Delta}\hskip 0.50003pt[\hskip 1.00006pt1\hskip 1.00006pt]$.   Suppose  that  $K\hskip 0.50003pt,\hskip 3.00003ptL$  are simplicial  sets and  $f,\hskip 3.00003ptg\hskip 1.99997pt\colon\hskip 1.49994ptK\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptL$  are simplicial  maps.   A  homotopy  between $f$ and  $g$  is  a simplicial  map  $h\hskip 1.00006pt\colon\hskip 1.00006ptK\hskip 1.00006pt\times\hskip 1.00006pt\bm{\Delta}\hskip 0.50003pt[\hskip 1.00006pt1\hskip 1.00006pt]\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptL$  such  that

$$\quad f\hskip 3.99994pt=\hskip 3.99994ptk\hskip 1.00006pt\circ\hskip 1.00006pt\bigl{(}\hskip 1.99997pt\operatorname{id}_{\hskip 1.04996ptK}\hskip 1.00006pt\times\hskip 1.00006pti\hskip 1.00006pt(\hskip 1.00006pt0\hskip 1.00006pt)\hskip 1.99997pt\bigr{)}\quad\ \mbox{and}\quad\ g\hskip 3.99994pt=\hskip 3.99994ptk\hskip 1.00006pt\circ\hskip 1.00006pt\bigl{(}\hskip 1.99997pt\operatorname{id}_{\hskip 1.04996ptK}\hskip 1.00006pt\times\hskip 1.00006pti\hskip 1.00006pt(\hskip 1.00006pt1\hskip 1.00006pt)\hskip 1.99997pt\bigr{)}\hskip 3.00003pt,$$

where  $K$  is  identified  with  $K\hskip 1.00006pt\times\hskip 1.00006pt\bm{\Delta}\hskip 0.50003pt[\hskip 1.00006pt0\hskip 1.00006pt]$.   Homotopy  equivalences  are defined  in  terms of  homotopies in  the usual  manner .   A simplicial  set  $K$  is  said  to be  contractible  if  $K$  is  homotopy  equivalent  to $\bm{\Delta}\hskip 0.50003pt[\hskip 1.00006pt0\hskip 1.00006pt]$.   In  this case  $\operatorname{id}_{\hskip 1.04996ptK}$  is  homotopic  to  the composition  $K\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997pt\bm{\Delta}\hskip 0.50003pt[\hskip 1.00006pt0\hskip 1.00006pt]\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptK$  of  the unique map  $K\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997pt\bm{\Delta}\hskip 0.50003pt[\hskip 1.00006pt0\hskip 1.00006pt]$  with  $i_{\hskip 0.70004ptv}\hskip 1.00006pt\colon\hskip 1.00006pt\bm{\Delta}\hskip 0.50003pt[\hskip 0.24994ptn\hskip 1.00006pt]\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptK$  for some  $v\hskip 1.99997pt\in\hskip 3.00003ptK_{\hskip 1.04996pt0}$.   If  $K$  is  contractible,   then  the projection  $K\hskip 1.00006pt\times\hskip 1.00006ptL\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptL$  is  a  homotopy  equivalence  for every  $L$.

Local  systems of  coefficients.   Let  $K$  be either a simplicial  or  a $\Delta$-set.   Let  $\varepsilon$  be a $1$-simplex of  $K$  and  $v\hskip 0.50003pt,\hskip 3.00003ptw$  are vertices of  $K$.   We say  that  $\varepsilon$  connects $v$ with $w$  if  $\partial_{\hskip 0.70004pt1}\hskip 1.00006pt\varepsilon\hskip 3.99994pt=\hskip 3.99994ptv$  and  $\partial_{\hskip 0.70004pt0}\hskip 1.00006pt\varepsilon\hskip 3.99994pt=\hskip 3.99994ptw$.   A  local  system of  coefficients,   or  simply  a  local  system  $\pi$ on  $K$  is  an assignment  of  a  group $\pi_{\hskip 0.70004ptv}$  to every  vertex  $v$ of  $K$  and  an  isomorphism  $\varepsilon^{\hskip 0.70004pt*}\hskip 1.00006pt\colon\hskip 1.00006pt\pi_{\hskip 0.70004ptw}\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997pt\pi_{\hskip 0.70004ptv}$  to every $1$-simplex $\varepsilon$ connecting $v$ with $w$.   These groups and  isomorphisms are subject  to  the following  condition :   if  $\omega\hskip 1.99997pt\in\hskip 3.00003ptK_{\hskip 1.04996pt2}$  and  $\rho\hskip 3.99994pt=\hskip 3.99994pt\partial_{\hskip 1.04996pt2}\hskip 1.00006pt\omega\hskip 0.50003pt,\hskip 3.99994pt\ \sigma\hskip 3.99994pt=\hskip 3.99994pt\partial_{\hskip 1.04996pt0}\hskip 1.00006pt\omega\hskip 0.50003pt,\hskip 3.99994pt\ \tau\hskip 3.99994pt=\hskip 3.99994pt\partial_{\hskip 0.70004pt1}\hskip 1.00006pt\omega$,   then  $\tau^{\hskip 0.70004pt*}\hskip 3.99994pt=\hskip 3.99994pt\rho^{\hskip 0.70004pt*}\hskip 1.00006pt\circ\hskip 1.99997pt\sigma^{\hskip 0.70004pt*}$.   A  local  system of  coefficients $\pi$  is  said  to be  abelian  if  all  groups $\pi_{\hskip 0.70004ptv}$  are abelian.

The  leading  vertex  of  an $n$-simplex $\sigma$  is  the vertex  $v_{\hskip 0.70004pt\sigma}\hskip 3.99994pt=\hskip 3.99994pt\theta^{\hskip 0.70004pt*}\hskip 0.50003pt(\hskip 1.00006pt\sigma\hskip 1.00006pt)$,   where  $\theta\hskip 1.00006pt\colon\hskip 1.00006pt[\hskip 1.00006pt0\hskip 1.00006pt]\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997pt[\hskip 0.24994ptn\hskip 1.00006pt]$  is  the inclusion,   and  the  leading  edge  of  an $n$-simplex $\sigma$  is  the $1$-simplex  $\varepsilon_{\hskip 0.70004pt\sigma}\hskip 3.99994pt=\hskip 3.99994pt\eta^{\hskip 0.70004pt*}\hskip 0.50003pt(\hskip 1.00006pt\sigma\hskip 1.00006pt)$,   where  $\eta\hskip 1.00006pt\colon\hskip 1.00006pt[\hskip 1.00006pt1\hskip 1.00006pt]\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997pt[\hskip 0.24994ptn\hskip 1.00006pt]$  is  the inclusion.   An $n$-cochain of  $K$  with coefficients in  the  local  system $\pi$  is  a map $c$  assigning  every  $\sigma\hskip 1.99997pt\in\hskip 3.00003ptK_{\hskip 0.70004ptn}$  an element  $c\hskip 1.49994pt(\hskip 1.00006ptv\hskip 1.49994pt)\hskip 1.99997pt\in\hskip 3.00003pt\pi_{\hskip 0.70004ptv}$,   where  $v\hskip 3.99994pt=\hskip 3.99994ptv_{\hskip 0.70004pt\sigma}$  is  the  leading  vertex of  $\sigma$.   The group of  such cochains  is  denoted  by  $C^{\hskip 0.70004ptn}\hskip 1.00006pt(\hskip 1.49994ptK\hskip 0.50003pt,\hskip 1.99997pt\pi\hskip 1.49994pt)$.   For abelian $\pi$  the  coboundary  operators  $\partial^{\hskip 0.70004pt*}\hskip 1.00006pt\colon\hskip 1.00006ptC^{\hskip 0.70004ptn}\hskip 1.00006pt(\hskip 1.49994ptK\hskip 0.50003pt,\hskip 1.99997pt\pi\hskip 1.49994pt)\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptC^{\hskip 0.70004ptn\hskip 0.70004pt+\hskip 0.70004pt1}\hskip 1.00006pt(\hskip 1.49994ptK\hskip 0.50003pt,\hskip 1.99997pt\pi\hskip 1.49994pt)$  are defined  by  the formula

$$\quad\partial^{\hskip 0.70004pt*}\hskip 1.00006ptc\hskip 1.49994pt(\hskip 1.00006pt\sigma\hskip 1.00006pt)\hskip 3.99994pt=\hskip 3.99994pt\hskip 1.00006pt\varepsilon_{\hskip 0.70004pt\sigma}^{\hskip 0.70004pt*}\hskip 1.00006pt\bigl{(}\hskip 1.49994ptc\hskip 1.49994pt\bigl{(}\hskip 1.99997pt\partial_{\hskip 1.04996pt0}\hskip 1.00006pt\sigma\hskip 1.49994pt\bigr{)}\hskip 1.99997pt\bigr{)}\hskip 3.99994pt+\hskip 3.99994pt\sum\nolimits_{\hskip 0.70004pti\hskip 1.39998pt=\hskip 1.39998pt1}^{\hskip 0.70004ptn\hskip 0.70004pt+\hskip 0.70004pt1}\hskip 3.99994pt(\hskip 1.99997pt-\hskip 1.99997pt1\hskip 1.49994pt)^{\hskip 0.70004pti}\hskip 3.00003ptc\hskip 1.49994pt\bigl{(}\hskip 1.99997pt\partial_{\hskip 0.70004pti}\hskip 1.00006pt\sigma\hskip 1.49994pt\bigr{)}\hskip 3.99994pt\in\hskip 3.99994pt\pi_{\hskip 0.70004ptv_{\hskip 0.50003pt\sigma}}\hskip 3.99994pt.$$

The  cocycles  and  coboundaries  are defined  in  terms  $\partial^{\hskip 0.70004pt*}$  in  the usual  manner .

3. Postnikov  systems  and  minimality

Comparing  simplices.   Let  $K$  be a simplicial  set  and  $q\hskip 1.99997pt\in\hskip 3.00003pt\mathbf{N}$.   Recall  that  for every $\sigma\hskip 1.99997pt\in\hskip 3.00003ptK_{\hskip 0.70004ptq}$ there exists a unique simplicial  map $i_{\hskip 1.04996pt\sigma}\hskip 1.00006pt\colon\hskip 1.00006pt\bm{\Delta}\hskip 0.50003pt[\hskip 0.50003ptq\hskip 1.00006pt]\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptK$ such  that  $i_{\hskip 1.04996pt\sigma}\hskip 1.00006pt(\hskip 1.99997pt\bm{\iota}_{\hskip 0.70004ptq}\hskip 1.00006pt)\hskip 3.99994pt=\hskip 3.99994pt\sigma$.   Two simplices  $\sigma\hskip 0.50003pt,\hskip 3.00003pt\tau\hskip 1.99997pt\in\hskip 3.00003ptK_{\hskip 0.70004ptq}$  are said  to be  $n$-equivalent   if  the restrictions of  the maps

$$\quad i_{\hskip 1.04996pt\sigma}\hskip 1.49994pt,\hskip 3.99994pti_{\hskip 1.04996pt\tau}\hskip 1.99997pt\colon\hskip 1.99997pt\bm{\Delta}\hskip 0.50003pt[\hskip 0.50003ptq\hskip 1.00006pt]\hskip 3.99994pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 3.99994ptK$$

to  $\operatorname{sk}_{\hskip 0.70004ptn}\hskip 0.50003pt\bm{\Delta}\hskip 0.50003pt[\hskip 0.50003ptq\hskip 1.00006pt]$  are  equal,   or ,   equivalently ,   if  $\theta^{\hskip 0.70004pt*}\hskip 1.00006pt(\hskip 1.00006pt\sigma\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994pt\theta^{\hskip 0.70004pt*}\hskip 1.00006pt(\hskip 1.00006pt\tau\hskip 1.49994pt)$  for every  non-decreasing  map  $\theta\hskip 1.00006pt\colon\hskip 1.00006pt[\hskip 0.24994ptn\hskip 1.00006pt]\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997pt[\hskip 0.50003ptq\hskip 1.00006pt]$.   We write  $\sigma\hskip 3.99994pt\sim_{\hskip 0.70004ptn}\hskip 3.99994pt\tau$  if  $\sigma\hskip 0.50003pt,\hskip 3.00003pt\tau$  are $n$-equivalent.   Obviously,   if  $q\hskip 1.99997pt\leqslant\hskip 1.99997ptn$,   then  $\sigma\hskip 3.99994pt\sim_{\hskip 0.70004ptn}\hskip 3.99994pt\tau$  if  and  only  if  $\sigma\hskip 3.99994pt=\hskip 3.99994pt\tau$.

Clearly,  $\sim_{\hskip 0.70004ptn}$  is  an equivalence relation on  the set  of  simplices.   If  $\sigma\hskip 0.50003pt,\hskip 3.00003pt\tau\hskip 1.99997pt\in\hskip 3.00003ptK_{\hskip 0.70004ptq}$  and  $\sigma\hskip 1.99997pt\sim_{\hskip 0.70004ptn}\hskip 1.99997pt\tau$,   then  $\theta^{\hskip 0.70004pt*}\hskip 1.00006pt(\hskip 1.00006pt\sigma\hskip 1.49994pt)\hskip 1.99997pt\sim_{\hskip 0.70004ptn}\hskip 1.99997pt\theta^{\hskip 0.70004pt*}\hskip 1.00006pt(\hskip 1.00006pt\tau\hskip 1.49994pt)$  for every  non-decreasing  map  $\theta\hskip 1.00006pt\colon\hskip 1.00006pt[\hskip 0.24994ptm\hskip 1.00006pt]\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997pt[\hskip 0.50003ptq\hskip 1.00006pt]$.   Therefore  the structural  maps $\theta^{\hskip 0.70004pt*}$ of  $K$  induce maps between sets of  equivalence classes of  $\sim_{\hskip 0.70004ptn}$.   These induced  maps are  the structure maps of  a canonical  structure of  a simplicial  set  on  the set  $K\hskip 1.00006pt(\hskip 1.00006ptn\hskip 1.49994pt)$  of  equivalence classes with  respect  to  $\sim_{\hskip 0.70004ptn}$.   Clearly,   there  is  a canonical  simplicial  map

$$\quad p_{\hskip 0.35002ptn}\hskip 1.00006pt\colon\hskip 1.00006ptK\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptK\hskip 1.00006pt(\hskip 1.00006ptn\hskip 1.49994pt)\hskip 3.00003pt.$$

Also,   if  $n\hskip 1.99997pt\leqslant\hskip 1.99997ptm$  and  $\sigma\hskip 1.99997pt\sim_{\hskip 0.70004ptm}\hskip 1.99997pt\tau$,   then  $\sigma\hskip 1.99997pt\sim_{\hskip 0.70004ptn}\hskip 1.99997pt\tau$,   and  hence  there  is  a canonical  simplicial  map

$$\quad p_{\hskip 0.35002ptm,\hskip 0.70004ptn}\hskip 1.00006pt\colon\hskip 1.00006ptK\hskip 1.00006pt(\hskip 1.00006ptm\hskip 1.49994pt)\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptK\hskip 1.00006pt(\hskip 1.00006ptn\hskip 1.49994pt)\hskip 3.00003pt.$$

Clearly,   the maps  $p_{\hskip 0.35002ptn}$  and  $p_{\hskip 0.35002ptm,\hskip 0.70004ptn}$  induce isomorphisms of  $n$th  skeletons.   When  there  is  no danger of  confusion,   we denote maps  $p_{\hskip 0.35002ptn}$  and  $p_{\hskip 0.35002ptm,\hskip 0.70004ptn}$  simply  by  $p$.

Let  $M$  be a simplicial  subset  of  $K$.   Clearly,   for simplices  $\sigma\hskip 0.50003pt,\hskip 3.00003pt\tau$  of  $M$  the relation  $\sigma\hskip 3.99994pt\sim_{\hskip 0.70004ptn}\hskip 3.99994pt\tau$  holds in  $M$  if  and  only  if  it  holds in  $K$.   Therefore  there  is  a canonical  injective  map  $M\hskip 1.00006pt(\hskip 1.00006ptn\hskip 1.49994pt)\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptK\hskip 1.00006pt(\hskip 1.00006ptn\hskip 1.49994pt)$.   In such a situation  we will  identify  $M\hskip 1.00006pt(\hskip 1.00006ptn\hskip 1.49994pt)$ with  its image in  $K\hskip 1.00006pt(\hskip 1.00006ptn\hskip 1.49994pt)$.

Two simplices  $\sigma\hskip 0.50003pt,\hskip 3.00003pt\tau\hskip 1.99997pt\in\hskip 3.00003ptK_{\hskip 0.70004ptq}$  are said  to be  homotopic  if  the maps  $i_{\hskip 1.04996pt\sigma}$  and  $i_{\hskip 1.04996pt\tau}$  are  homotopic relatively  to  $\partial\hskip 1.00006pt\bm{\Delta}\hskip 0.50003pt[\hskip 0.50003ptq\hskip 1.00006pt]$.   We write  $\sigma\hskip 3.99994pt\sim\hskip 3.99994pt\tau$  if  $\sigma\hskip 0.50003pt,\hskip 3.00003pt\tau$  are homotopic.   If  $K$  is  a  Kan  simplicial  set ,   then  $\sim$  is  an equivalence relation on  the set  of  simplices.

Postnikov  systems.   The sequence of  simplicial  sets  $K\hskip 1.00006pt(\hskip 1.00006pt0\hskip 1.00006pt)\hskip 0.50003pt,\hskip 3.00003ptK\hskip 1.00006pt(\hskip 1.00006pt1\hskip 1.00006pt)\hskip 0.50003pt,\hskip 3.00003pt\ldots\hskip 0.50003pt,\hskip 3.00003ptK\hskip 1.00006pt(\hskip 1.00006ptn\hskip 1.49994pt)\hskip 0.50003pt,\hskip 3.00003pt\ldots$  together  with  the maps  $p_{\hskip 0.35002ptn}$  and  $p_{\hskip 0.35002ptm,\hskip 0.70004ptn}$  is  called  Postnikov  or  Moore-Postnikov  system  of  $K$.   More precisely,   this definition  is  the version of  the original  construction of  Postnikov  [$P_{\hskip 0.35002pt1}$],   [$P_{\hskip 0.35002pt2}$]  due  to  Moore  [Mo].   This construction and  the notion of  homotopic simplices are  useful  only  for  Kan  simplicial  sets.   If  $K$  is  Kan,   then every  term  $K\hskip 1.00006pt(\hskip 1.00006ptn\hskip 1.49994pt)$  of  the  Postnikov  system  is  also  Kan,   and all  maps  $p_{\hskip 0.35002ptn}$  and  $p_{\hskip 0.35002ptm,\hskip 0.70004ptn}$  are  Kan  fibrations.   See  [Ma],   Proposition  8.2.

Minimality.   Postnikov  systems are especially  powerful  when  $K$  is  minimal  in  the following sense.   A  Kan  simplicial  set  $K$  is  said  to be  minimal  if  every  two homotopic simplices of  $K$  are equal,   i.e.  that  $\sigma\hskip 3.99994pt\sim\hskip 3.99994pt\tau$  implies  $\sigma\hskip 3.99994pt=\hskip 3.99994pt\tau$  for every  two simplices  $\sigma\hskip 0.50003pt,\hskip 3.00003pt\tau$  of  $M$.   This notion  is  going  back  to  Eilenberg  and  Zilber  [E Z]  and  Postnikov  [$P_{\hskip 0.35002pt1}$].   Every  Kan  simplicial  set  $K$  contains a  minimal  Kan  simplicial  subset  $M$  as a strong  deformation  retract.   Moreover ,   every  two such simplicial  subsets  $M$  are  isomorphic.   See  [Ma],   Theorems  9.5  and  9.8.

Let  $E\hskip 0.50003pt,\hskip 3.00003ptB$  be simplicial  sets and  $p\hskip 1.00006pt\colon\hskip 1.00006ptE\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptB$  be  a  Kan  fibration.   Two simplices  $\sigma\hskip 0.50003pt,\hskip 3.00003pt\tau\hskip 1.99997pt\in\hskip 3.00003ptE_{\hskip 0.70004ptq}$  are  fiberwise  homotopic   if  $p\hskip 1.00006pt(\hskip 1.00006pt\sigma\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994ptp\hskip 1.00006pt(\hskip 1.00006pt\tau\hskip 1.49994pt)$  and  there exists  a  relative  to  $\partial\hskip 1.00006pt\bm{\Delta}\hskip 0.50003pt[\hskip 0.50003ptq\hskip 1.00006pt]$  homotopy

$$\quad h\hskip 1.00006pt\colon\hskip 1.00006pt\bm{\Delta}\hskip 0.50003pt[\hskip 0.50003ptq\hskip 1.00006pt]\hskip 1.00006pt\times\hskip 1.00006pt\bm{\Delta}\hskip 0.50003pt[\hskip 1.00006pt1\hskip 1.00006pt]\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptE$$

between  the maps  $i_{\hskip 1.04996pt\sigma}$  and  $i_{\hskip 1.04996pt\tau}$  such  that  the diagram

where  $\rho\hskip 3.99994pt=\hskip 3.99994ptp\hskip 1.00006pt(\hskip 1.00006pt\sigma\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994ptp\hskip 1.00006pt(\hskip 1.00006pt\tau\hskip 1.49994pt)$  and  $\operatorname{p{\hskip 0.50003pt}r}$  is  the projection,   is  commutative.   In  particular ,   if  $\sigma\hskip 0.50003pt,\hskip 3.00003pt\tau$  are fiberwise homotopic,   then  $\sigma\hskip 0.50003pt,\hskip 3.00003pt\tau$  are homotopic.   A  Kan  fibration  $p\hskip 1.00006pt\colon\hskip 1.00006ptE\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptB$  is  said  to be  minimal  if  every  two fiberwise homotopic simplices are equal.   For every  Kan  fibration  $p\hskip 1.00006pt\colon\hskip 1.00006ptE\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptB$  there exists a simplicial  subset  $M$  of  $E$  such  that  $p\hskip 1.00006pt|\hskip 1.49994ptM\hskip 1.00006pt\colon\hskip 1.00006ptE\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptB$  is  a  minimal  Kan  fibration and  $M$  is  a  fiberwise  strong  deformation  retract  of  $E$  in  a  natural  sense.   See  [G  J],   Chapter  I,   Proposition  10.3,   or  [Ma],   Theorem  10.9.

3.1. Theorem.   Every  minimal  Kan  fibration  with connected  base  is  a  locally  trivial  bundle.   

Proof.   See  [G Z],   Section  VI.5.4,   or  [Ma],   Theorem  11.11.    $\blacksquare$

Postnikov  systems and  locally  trivial  bundles.   If  $K$  is  a  minimal  Kan  simplicial  set,   then all  maps  $p_{\hskip 0.35002ptn}$  and  $p_{\hskip 0.35002ptm,\hskip 0.70004ptn}$  are  minimal  Kan  fibrations.   See  [Ma],   Lemma  12.1.   Therefore,   Theorem  Simplicial  sets,   Postnikov  systems, and  bounded  cohomology  implies  that  these maps are  locally  trivial  bundles.   In  particular ,

$$\quad p\hskip 3.99994pt=\hskip 3.99994ptp_{\hskip 0.35002ptn,\hskip 0.70004ptn\hskip 0.70004pt-\hskip 0.70004pt1}\hskip 1.00006pt\colon\hskip 1.00006ptK\hskip 1.00006pt(\hskip 1.00006ptn\hskip 1.49994pt)\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptK\hskip 1.00006pt(\hskip 1.00006ptn\hskip 1.99997pt-\hskip 1.99997pt1\hskip 1.00006pt)$$

is  a  locally  trivial  bundle.   Its  fiber  is  the  Eilenberg–MacLane  simplicial  set  $K\hskip 1.00006pt(\hskip 1.00006pt\pi\hskip 0.50003pt,\hskip 1.99997ptn\hskip 1.49994pt)$,   where  $\pi\hskip 3.99994pt=\hskip 3.99994pt\pi_{\hskip 0.35002ptn}\hskip 1.00006pt(\hskip 1.49994ptK\hskip 0.50003pt,\hskip 1.99997ptv\hskip 1.49994pt)$  is  the $n$th  homotopy  groups of  $K$.   See  [Ma],   the  beginning  of  Section  25.   For our  purposes  it  is  sufficient  to know  that  for  $n\hskip 1.99997pt>\hskip 1.99997pt1$  the fiber  is  an  Eilenberg–MacLane  simplicial  set  $K\hskip 1.00006pt(\hskip 1.00006pt\pi\hskip 0.50003pt,\hskip 1.99997ptn\hskip 1.49994pt)$  with an abelian  group $\pi$,   and  to identify  the simplicial  set  $K\hskip 1.00006pt(\hskip 1.00006pt1\hskip 1.00006pt)$.   See  Lemma  Simplicial  sets,   Postnikov  systems, and  bounded  cohomology   for  the  latter .

4. Classifying  spaces  of  categories  and  groups

Classifying spaces of  categories.   This section  is  devoted  to some classical  constructions of  Milnor  [Mi]  and  Segal  [S]  (who attributed some of  the ideas of  [S]  to  Grothendieck ).

A set  $S$  with a partial  order  $\leqslant$  defines a category  having  $S$  as its set  of  objects.   For  $a\hskip 0.50003pt,\hskip 3.00003ptb\hskip 1.99997pt\in\hskip 1.99997ptS$  there  is  exactly  one morphisms  $a\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptb$  if  $a\hskip 1.99997pt\leqslant\hskip 1.99997ptb$,   an none otherwise.   In  particular ,   sets  $[\hskip 0.24994ptn\hskip 1.00006pt]$  together  with  their natural  order can  be considered as categories.   From  this point  of  view  non-decreasing  maps  $\theta\hskip 1.00006pt\colon\hskip 1.00006pt[\hskip 0.24994ptm\hskip 1.00006pt]\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997pt[\hskip 0.24994ptn\hskip 1.00006pt]$  are nothing else but  functors  $[\hskip 0.24994ptm\hskip 1.00006pt]\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997pt[\hskip 0.24994ptn\hskip 1.00006pt]$.

Every  small  category  $\mathcal{C}$ defines a simplicial  set $\mathit{B}\hskip 1.99997pt\mathcal{C}$,   its  nerve  in  the sense of  G.  Segal  [S],   often called also  the  classifying  space  of  $\mathcal{C}$.   The vertices of  $\mathit{B}\hskip 1.99997pt\mathcal{C}$  are  the objects of  $\mathcal{C}$,   and  the $n$-simplices are functors  $\sigma\hskip 1.00006pt\colon\hskip 1.00006pt[\hskip 0.24994ptn\hskip 1.00006pt]\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997pt\mathcal{C}$.   As usual,   the structure maps are defined as compositions.   Namely,   if  $\theta\hskip 1.00006pt\colon\hskip 1.00006pt[\hskip 0.24994ptm\hskip 1.00006pt]\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997pt[\hskip 0.24994ptn\hskip 1.00006pt]$  is  a non-decreasing  map,   then  $\theta^{\hskip 0.35002pt*}\hskip 1.00006pt(\hskip 1.00006pt\sigma\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994pt\sigma\hskip 1.00006pt\circ\hskip 1.49994pt\theta$,   where in  the right  hand side  $\theta$  is  considered as a functor.   Clearly,   a functor  $[\hskip 0.24994ptn\hskip 1.00006pt]\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997pt\mathcal{C}$  is  determined  by  its values on  objects and on morphisms  $i\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997pti\hskip 1.99997pt+\hskip 1.99997pt1$,   where  $i\hskip 1.99997pt\in\hskip 1.99997pt[\hskip 0.24994ptn\hskip 1.99997pt-\hskip 1.99997pt1\hskip 1.00006pt]$.   Therefore $n$-simplices of  $\mathit{B}\hskip 1.99997pt\mathcal{C}$ correspond  to sequences of  morphisms of  the form

(4.1)

where each $v_{\hskip 0.70004pti}$  is  an object  of  $\mathcal{C}$  and each  $p_{\hskip 0.70004pti}$  is  a morphism  $v_{\hskip 0.70004pti\hskip 0.70004pt-\hskip 0.70004pt1}\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptv_{\hskip 0.70004pti}$.   Of  course,   the objects  $v_{\hskip 0.70004pti}$  are determined  by  the morphisms  $p_{\hskip 0.70004ptk}$  and  hence $n$-simplices correspond  to sequences  $(\hskip 1.49994ptp_{\hskip 0.70004pt1}\hskip 1.00006pt,\hskip 3.99994ptp_{\hskip 1.04996pt2}\hskip 1.00006pt,\hskip 3.99994pt\ldots\hskip 1.00006pt,\hskip 3.99994ptp_{\hskip 0.70004ptn}\hskip 1.49994pt)$  of  morphisms such  that  the composition  $p_{\hskip 0.70004pti\hskip 0.70004pt+\hskip 0.70004pt1}\hskip 1.00006pt\circ\hskip 1.00006ptp_{\hskip 0.70004pti}$  is  defined  for each  $i$  between  $1$  and  $n\hskip 1.99997pt-\hskip 1.99997pt1$.   For  $0\hskip 1.99997pt<\hskip 1.99997pti\hskip 1.99997pt<\hskip 1.99997ptn$  the boundary  operators  $\partial_{\hskip 0.70004pti}$  acts by  replacing  $v_{\hskip 0.70004pti}$  and  morphisms  $p_{\hskip 0.70004pti}\hskip 0.50003pt,\hskip 3.00003ptp_{\hskip 0.70004pti\hskip 0.70004pt+\hskip 0.70004pt1}$  by  the composition  $p_{\hskip 0.70004pti\hskip 0.70004pt+\hskip 0.70004pt1}\hskip 1.00006pt\circ\hskip 1.00006ptp_{\hskip 0.70004pti}$.   The boundary  operators  $\partial_{\hskip 1.04996pt0}$  and  $\partial_{\hskip 0.70004ptn}$  act  by  simply  removing  $v_{\hskip 1.04996pt0}\hskip 0.50003pt,\hskip 3.00003ptp_{\hskip 0.70004pt1}$  and  $p_{\hskip 0.70004ptn\hskip 0.70004pt-\hskip 0.70004pt1}\hskip 0.50003pt,\hskip 3.00003ptv_{\hskip 0.70004ptn}$  respectively.   The degeneracy  operator  $s_{\hskip 0.70004pti}$  acts by  inserting  the identity  morphism  $v_{\hskip 0.70004pti}\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptv_{\hskip 0.70004pti}$.   Cf.  [G  J],   Example  I.1.4.

Let  $\mathcal{C}\hskip 0.50003pt,\hskip 3.00003pt\mathcal{D}$  be  two categories.   A functor  $f\hskip 1.00006pt\colon\hskip 1.00006pt\mathcal{C}\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997pt\mathcal{D}$  defines,   in an obvious  way,   a simplicial  map  $\mathit{B}\hskip 0.50003ptf\hskip 1.00006pt\colon\hskip 1.00006pt\mathit{B}\hskip 1.99997pt\mathcal{C}\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997pt\mathit{B}\hskip 1.99997pt\mathcal{D}$.   Given  two functors  $f,\hskip 3.00003ptg\hskip 1.00006pt\colon\hskip 1.00006pt\mathcal{C}\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997pt\mathcal{D}$,   a natural  transformation  $f\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptg$  defines a homotopy  between  $\mathit{B}\hskip 0.50003ptf$  and  $\mathit{B}\hskip 1.00006ptg$.   Indeed,   a natural  transformation  $t\hskip 1.00006pt\colon\hskip 1.00006ptf\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptg$  can  be considered as  a  functor  $\mathcal{C}\hskip 1.00006pt\times\hskip 1.00006pt[\hskip 1.00006pt1\hskip 1.00006pt]\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997pt\mathcal{D}$,   where  $[\hskip 1.00006pt1\hskip 1.00006pt]$  is  considered as a category.   One can easily  see  that  the operation  $\mathcal{C}\hskip 3.99994pt\longmapsto\hskip 3.99994pt\mathit{B}\hskip 1.99997pt\mathcal{C}$  commutes with  the products.   Since,   obviously,  $\mathit{B}\hskip 1.99997pt[\hskip 1.00006pt1\hskip 1.00006pt]\hskip 3.99994pt=\hskip 3.99994pt\bm{\Delta}\hskip 0.50003pt[\hskip 1.00006pt1\hskip 1.00006pt]$,   the natural  transformation  $t$  defines a simplicial  map  $\mathit{B}\hskip 1.00006ptt\hskip 1.00006pt\colon\hskip 1.00006pt\mathit{B}\hskip 1.99997pt\mathcal{C}\hskip 1.00006pt\times\hskip 1.00006pt\bm{\Delta}\hskip 0.50003pt[\hskip 1.00006pt1\hskip 1.00006pt]\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997pt\mathit{B}\hskip 1.99997pt\mathcal{D}$,   i.e.  a  homotopy.   We  leave  to  the reader  the verification  that  this  is  a homotopy  between  $\mathit{B}\hskip 0.50003ptf$  and  $\mathit{B}\hskip 1.00006ptg$.

A discrete group $\pi$ can  be considered as a category  with a single object  and  $\pi$  being  the set  of  morphisms from  this object  to  itself ,   with  the composition  being  the group multiplication.   The classifying space  $\mathit{B}\hskip 1.49994pt\pi$  is  a  Kan  simplicial  set.   See  [G  J],   Lemma  I.3.5.   Comparing  the definitions shows  that  the usual  and  the bounded cohomology of  the group  $\pi$  are,   in  fact,   cohomology of  the classifying space  $\mathit{B}\hskip 1.49994pt\pi$.

Milnor’s  classifying  spaces.   Another classical  construction of  classifying  spaces of  groups  is  due  to  Milnor  [Mi].   While we will  not  use  it  directly,   it  serves as a motivation  for  the definitions of  unravelings  of  classifying spaces and simplicial  sets  below and  in  Section  Simplicial  sets,   Postnikov  systems, and  bounded  cohomology.

For a discrete group  $\pi$  let  $\mathcal{E}\hskip 0.50003pt\pi$  be  the simplicial  complex  having  the product  $\pi\hskip 1.00006pt\times\hskip 1.00006pt\mathbf{N}$  as  the set  of  vertices and as simplices finite subsets  $\sigma\hskip 1.99997pt\subset\hskip 1.99997pt\hskip 0.50003pt\pi\hskip 1.00006pt\times\hskip 1.00006pt\mathbf{N}$  such  that  the projection  $\pi\hskip 1.00006pt\times\hskip 1.00006pt\mathbf{N}\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997pt\mathbf{N}$  is  injective on $\sigma$.   There  is  a  left  action  $\pi\hskip 1.00006pt\times\hskip 1.00006pt\mathcal{E}\hskip 0.50003pt\pi\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997pt\mathcal{E}\hskip 0.50003pt\pi$  of  the group $\pi$ on  $\mathcal{E}\hskip 0.50003pt\pi$  by  the rule

$$\quad h\hskip 1.00006pt\cdot\hskip 1.49994pt(\hskip 1.49994ptg\hskip 0.50003pt,\hskip 1.99997ptk\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994pt(\hskip 1.99997pth\hskip 0.50003pt\cdot\hskip 0.50003ptg\hskip 0.50003pt,\hskip 1.99997ptk\hskip 1.49994pt)\hskip 3.00003pt,$$

where  $h\hskip 1.99997pt\in\hskip 1.99997pt\pi$  and  $(\hskip 1.99997ptg\hskip 0.50003pt,\hskip 1.99997ptk\hskip 1.49994pt)\hskip 1.99997pt\in\hskip 1.99997pt\pi\hskip 1.00006pt\times\hskip 1.00006pt\mathbf{N}$  is  a vertex of  $\mathcal{E}\hskip 0.50003pt\pi$.   The quotient  $\mathcal{B}\hskip 1.00006pt\pi\hskip 3.99994pt=\hskip 3.99994pt\pi\hskip 0.50003pt\backslash\hskip 1.00006pt\mathcal{E}\hskip 0.50003pt\pi$  of  the simplicial  complex  $\mathcal{E}\hskip 0.50003pt\pi$  by  this action  is  a  well-defined simplicial  complex.   We will  call  $\mathcal{B}\hskip 1.00006pt\pi$  the  Milnor’  classifying  space  of  $\pi$.   Milnor  defined directly  the geometric realization $|\hskip 1.99997pt\mathcal{B}\hskip 1.00006pt\pi\hskip 1.99997pt|$ for arbitrary  topological  group $\pi$.   By  this reason  his construction  is  different  from  ours one.

It  is  convenient  to enhance  the structure of  Milnor’s  classifying  space  to a $\Delta$-set.   The natural  order on  $\mathbf{N}$  defines  local  orders on  the simplicial  set  $\mathcal{E}\hskip 0.50003pt\pi$ and  $\mathcal{B}\hskip 1.00006pt\pi$  and  allows  to  turn  them  into $\Delta$-sets,   which  we will  still  denote by  $\mathcal{E}\hskip 1.00006pt\pi$  and  $\mathcal{B}\hskip 1.00006pt\pi$.   The  local  order on  $\mathcal{E}\hskip 1.00006pt\pi$  is  invariant  under  the  left  action of  $\pi$,   and  $\mathcal{B}\hskip 1.00006pt\pi\hskip 3.99994pt=\hskip 3.99994pt\pi\hskip 0.50003pt\backslash\hskip 1.00006pt\mathcal{E}\hskip 0.50003pt\pi$  as $\Delta$-sets also.

One of  advantages of  $\mathcal{B}\hskip 1.00006pt\pi$  is  the existence of  many  automorphisms.   Let  $C^{\hskip 0.70004pt0}\hskip 1.00006pt(\hskip 1.49994pt\mathbf{N}\hskip 0.50003pt,\hskip 3.00003pt\pi\hskip 1.49994pt)$  be  the group of  all  maps  $\mathbf{N}\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997pt\pi$.   Let  us  define a right  action of  $C^{\hskip 0.70004pt0}\hskip 1.00006pt(\hskip 1.49994pt\mathbf{N}\hskip 0.50003pt,\hskip 3.00003pt\pi\hskip 1.49994pt)$  on $\mathcal{E}\hskip 0.50003pt\pi$  by  the rule

$$\quad(\hskip 1.49994ptg\hskip 0.50003pt,\hskip 1.99997ptk\hskip 1.49994pt)\hskip 3.99994pt\longmapsto\hskip 3.99994pt\left(\hskip 1.49994ptg\hskip 1.00006pt\cdot\hskip 1.00006ptc\hskip 1.49994pt(\hskip 1.00006ptk\hskip 1.49994pt)\hskip 0.50003pt,\hskip 1.99997ptk\hskip 1.49994pt\right)\hskip 1.00006pt,$$

where  $c\hskip 1.99997pt\in\hskip 1.99997ptC^{\hskip 0.70004pt0}\hskip 1.00006pt(\hskip 1.49994pt\mathbf{N}\hskip 0.50003pt,\hskip 3.00003pt\pi\hskip 1.49994pt)$  and  $(\hskip 1.49994ptg\hskip 0.50003pt,\hskip 1.99997ptk\hskip 1.49994pt)\hskip 1.99997pt\in\hskip 1.99997pt\pi\hskip 1.00006pt\times\hskip 1.00006pt\mathbf{N}$  is  a vertex of  $\mathcal{E}\hskip 0.50003pt\pi$.   Clearly,   the right  action of  $C^{\hskip 0.70004pt0}\hskip 1.00006pt(\hskip 1.49994pt\mathbf{N}\hskip 0.50003pt,\hskip 3.00003pt\pi\hskip 1.49994pt)$  on $\mathcal{E}\hskip 0.50003pt\pi$  preserves  the  order of  vertices and commutes with  the  left  action of  $\pi$.   Hence  this action  leads  to a right  action of  $C^{\hskip 0.70004pt0}\hskip 1.00006pt(\hskip 1.49994pt\mathbf{N}\hskip 0.50003pt,\hskip 3.00003pt\pi\hskip 1.49994pt)$ on  $\mathcal{B}\hskip 1.00006pt\pi$.   If  $\kappa\hskip 1.99997pt\subset\hskip 1.99997pt\pi$  is  a subgroup of  $\pi$,   then  $C^{\hskip 0.70004pt0}\hskip 1.00006pt(\hskip 1.49994pt\mathbf{N}\hskip 0.50003pt,\hskip 3.00003pt\kappa\hskip 1.49994pt)$  is  a subgroup of  $C^{\hskip 0.70004pt0}\hskip 1.00006pt(\hskip 1.49994pt\mathbf{N}\hskip 0.50003pt,\hskip 3.00003pt\pi\hskip 1.49994pt)$,   and  if  $\kappa$  is  a  normal  subgroup,   then

(4.2)

$$\quad\mathcal{B}\hskip 1.00006pt(\hskip 1.00006pt\pi/\kappa\hskip 1.00006pt)\hskip 3.99994pt=\hskip 3.99994pt\mathcal{B}\hskip 1.00006pt\pi\left/\hskip 1.00006ptC^{\hskip 0.70004pt0}\hskip 1.00006pt(\hskip 1.49994pt\mathbf{N}\hskip 0.50003pt,\hskip 3.00003pt\kappa\hskip 1.49994pt)\right.\hskip 1.00006pt.$$

This obvious property  is  the main  reason of  our  interest  in  $\mathcal{B}\hskip 1.00006pt\pi$.   This property  strongly  contracts with  the properties of  the classifying  spaces  $\mathit{B}\hskip 1.49994pt\pi$.   Namely,   the classifying  space  $\mathit{B}\hskip 1.49994pt(\hskip 1.00006pt\pi/\kappa\hskip 1.00006pt)$  is  not  a quotient  of  $\mathit{B}\hskip 1.49994pt\pi$,   at  least  not  in any  natural  way.

Unravelings  of  classifying spaces of  groups.   The $\Delta$-set  $\mathcal{B}\hskip 1.00006pt\pi$  is  isomorphic  to  $\mathit{B}\hskip 1.49994pt\pi\hskip 1.00006pt\times\hskip 1.00006pt\Delta\hskip 1.00006pt[\hskip 1.00006pt\infty\hskip 1.00006pt]$.   See  Lemma  Simplicial  sets,   Postnikov  systems, and  bounded  cohomology.   While we are not  going  to use  this result,   it  motivates our  interest  to  the $\Delta$-set  $\mathit{B}\hskip 1.49994pt\pi\hskip 1.00006pt\times\hskip 1.00006pt\Delta\hskip 1.00006pt[\hskip 1.00006pt\infty\hskip 1.00006pt]$,   which  we will  call  the  unraveling  of  $\mathit{B}\hskip 1.49994pt\pi$.

As we will  see in a  moment,   the action of  $C^{\hskip 0.70004pt0}\hskip 1.00006pt(\hskip 1.49994pt\mathbf{N}\hskip 0.50003pt,\hskip 3.00003pt\pi\hskip 1.49994pt)$  can  be defined  directly  for  $\mathit{B}\hskip 1.49994pt\pi\hskip 1.00006pt\times\hskip 1.00006pt\Delta\hskip 1.00006pt[\hskip 1.00006pt\infty\hskip 1.00006pt]$.   In  fact,   it  is  easier and  more useful  to define  an action of  $C^{\hskip 0.70004pt0}\hskip 1.00006pt(\hskip 1.49994pt\mathbf{N}\hskip 0.50003pt,\hskip 3.00003pt\pi\hskip 1.49994pt)$  on  the simplicial  set  $\mathit{B}\hskip 1.49994pt\pi\hskip 1.00006pt\times\hskip 1.00006pt\bm{\Delta}\hskip 1.00006pt[\hskip 1.00006pt\infty\hskip 1.00006pt]$  first.   The  latter  is  the classifying  space of  a category.   Indeed,   the simplicial  set $\bm{\Delta}\hskip 1.00006pt[\hskip 1.00006pt\infty\hskip 1.00006pt]$ has  non-decreasing  maps  $[\hskip 0.24994ptn\hskip 1.00006pt]\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997pt\mathbf{N}$  as $n$-simplices,   with  the usual  structure maps.   Let  $\bm{n}$  be  the category  having  $\mathbf{N}$ as  its  set  of  objects,   exactly  one morphism  $n\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptm$  when  $n\hskip 1.99997pt\leqslant\hskip 1.99997ptm$,   and no morphisms  $n\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptm$  when  $n\hskip 1.99997pt>\hskip 1.99997ptm$.   Clearly,  $\bm{\Delta}\hskip 1.00006pt[\hskip 1.00006pt\infty\hskip 1.00006pt]\hskip 3.99994pt=\hskip 3.99994pt\mathit{B}\hskip 1.00006pt\bm{n}$.   It  follows  that

$$\quad\mathit{B}\hskip 1.49994pt\pi\hskip 1.00006pt\times\hskip 1.00006pt\bm{\Delta}\hskip 1.00006pt[\hskip 1.00006pt\infty\hskip 1.00006pt]\hskip 3.99994pt=\hskip 3.99994pt\mathit{B}\hskip 1.49994pt\pi\hskip 1.00006pt\times\hskip 1.00006pt\mathit{B}\hskip 1.00006pt\bm{n}\hskip 3.99994pt=\hskip 3.99994pt\mathit{B}\hskip 1.49994pt(\hskip 1.49994pt\pi\hskip 1.00006pt\times\hskip 1.00006pt\bm{n}\hskip 1.49994pt)\hskip 3.00003pt.$$

The category  $\pi\hskip 1.00006pt\times\hskip 1.00006pt\bm{n}$  has  $\mathbf{N}$  as  the set  of  objects.   The set  of  morphisms  $n\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptm$  is  a copy of  $\pi$  if  $n\hskip 1.99997pt\leqslant\hskip 1.99997ptm$,   and  is  empty  if  $n\hskip 1.99997pt>\hskip 1.99997ptm$.   Given  $c\hskip 1.99997pt\in\hskip 1.99997ptC^{\hskip 0.70004pt0}\hskip 1.00006pt(\hskip 1.49994pt\mathbf{N}\hskip 0.50003pt,\hskip 3.00003pt\pi\hskip 1.49994pt)$,   let  $a\hskip 1.00006pt(\hskip 1.00006ptc\hskip 1.49994pt)\hskip 1.00006pt\colon\hskip 1.00006pt\pi\hskip 1.00006pt\times\hskip 1.00006pt\bm{n}\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997pt\pi\hskip 1.00006pt\times\hskip 1.00006pt\bm{n}$  be  the functor  equal  to  the identity  on objects and and acting  on  morphisms  $n\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptm$  identified  with elements of  the group $\pi$  by  the rule

$$\quad g\hskip 3.99994pt\longmapsto\hskip 3.99994ptc\hskip 1.49994pt(\hskip 1.00006ptn\hskip 1.49994pt)^{\hskip 0.70004pt-\hskip 0.70004pt1}\hskip 1.00006pt\cdot\hskip 1.00006ptg\hskip 1.00006pt\cdot\hskip 1.00006ptc\hskip 1.49994pt(\hskip 1.00006ptm\hskip 1.49994pt)\hskip 3.00003pt.$$

Clearly,  $a\hskip 1.00006pt(\hskip 1.00006ptc\hskip 1.49994pt)$ is  an automorphism of  $\pi\hskip 1.00006pt\times\hskip 1.00006pt\bm{n}$ and even an automorphism over $\bm{n}$,   in  the sense  that  $\operatorname{p{\hskip 0.50003pt}r}\hskip 1.00006pt\circ\hskip 1.99997pta\hskip 1.00006pt(\hskip 1.00006ptc\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994pt\operatorname{p{\hskip 0.50003pt}r}$,   where  $\operatorname{p{\hskip 0.50003pt}r}\hskip 1.00006pt\colon\hskip 1.00006pt\pi\hskip 1.00006pt\times\hskip 1.00006pt\bm{n}\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997pt\bm{n}$  is  the projection.   Also,   all  diagrams

are commutative and  hence  morphisms  $c\hskip 1.49994pt(\hskip 1.00006ptn\hskip 1.49994pt)\hskip 1.00006pt\colon\hskip 1.00006ptn\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptn$  form a natural  transformation  from  the identity  functor  to  $a\hskip 1.00006pt(\hskip 1.00006ptc\hskip 1.49994pt)$.   It  follows  that  the simplicial  map

$$\quad\mathit{B}\hskip 1.00006pta\hskip 1.00006pt(\hskip 1.00006ptc\hskip 1.49994pt)\hskip 1.00006pt\colon\hskip 1.00006pt\mathit{B}\hskip 1.49994pt(\hskip 1.49994pt\pi\hskip 1.00006pt\times\hskip 1.00006pt\bm{n}\hskip 1.49994pt)\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997pt\mathit{B}\hskip 1.49994pt(\hskip 1.49994pt\pi\hskip 1.00006pt\times\hskip 1.00006pt\bm{n}\hskip 1.49994pt)$$

is  an automorphism of  $\mathit{B}\hskip 1.49994pt(\hskip 1.49994pt\pi\hskip 1.00006pt\times\hskip 1.00006pt\bm{n}\hskip 1.49994pt)$  homotopic  to  the identity.   The map  $c\hskip 3.99994pt\longmapsto\hskip 3.99994pt\mathit{B}\hskip 1.00006pta\hskip 1.00006pt(\hskip 1.00006ptc\hskip 1.49994pt)$  defines an action of  $C^{\hskip 0.70004pt0}\hskip 1.00006pt(\hskip 1.49994pt\mathbf{N}\hskip 0.50003pt,\hskip 3.00003pt\pi\hskip 1.49994pt)$ on  $\mathit{B}\hskip 1.49994pt(\hskip 1.49994pt\pi\hskip 1.00006pt\times\hskip 1.00006pt\bm{n}\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994pt\mathit{B}\hskip 1.49994pt\pi\hskip 1.00006pt\times\hskip 1.00006pt\bm{\Delta}\hskip 1.00006pt[\hskip 1.00006pt\infty\hskip 1.00006pt]$.   Since  the functors $a\hskip 1.00006pt(\hskip 1.00006ptc\hskip 1.49994pt)$ are automorphisms over $\bm{n}$,   the simplicial  maps  $\mathit{B}\hskip 1.00006pta\hskip 1.00006pt(\hskip 1.00006ptc\hskip 1.49994pt)$  are automorphisms of  $\mathit{B}\hskip 1.49994pt\pi\hskip 1.00006pt\times\hskip 1.00006pt\bm{\Delta}\hskip 1.00006pt[\hskip 1.00006pt\infty\hskip 1.00006pt]$  over  $\bm{\Delta}\hskip 1.00006pt[\hskip 1.00006pt\infty\hskip 1.00006pt]$.   It  follows  that  maps  $\mathit{B}\hskip 1.00006pta\hskip 1.00006pt(\hskip 1.00006ptc\hskip 1.49994pt)$  leave  the $\Delta$-subset  $\mathit{B}\hskip 1.49994pt\pi\hskip 1.00006pt\times\hskip 1.00006pt\Delta\hskip 1.00006pt[\hskip 1.00006pt\infty\hskip 1.00006pt]$  of  $\mathit{B}\hskip 1.49994pt\pi\hskip 1.00006pt\times\hskip 1.00006pt\bm{\Delta}\hskip 1.00006pt[\hskip 1.00006pt\infty\hskip 1.00006pt]$  invariant.   By  restricting  these maps  to  $\mathit{B}\hskip 1.49994pt\pi\hskip 1.00006pt\times\hskip 1.00006pt\Delta\hskip 1.00006pt[\hskip 1.00006pt\infty\hskip 1.00006pt]$  we get  an action of  $C^{\hskip 0.70004pt0}\hskip 1.00006pt(\hskip 1.49994pt\mathbf{N}\hskip 0.50003pt,\hskip 3.00003pt\pi\hskip 1.49994pt)$ on  $\mathit{B}\hskip 1.49994pt\pi\hskip 1.00006pt\times\hskip 1.00006pt\Delta\hskip 1.00006pt[\hskip 1.00006pt\infty\hskip 1.00006pt]$.   If  $\kappa\hskip 1.99997pt\subset\hskip 1.99997pt\pi$  is  a normal  subgroup of  $\pi$,   then a direct  verification shows  that

$$\quad\mathit{B}\hskip 1.49994pt(\hskip 1.00006pt\pi/\kappa\hskip 1.00006pt)\hskip 1.00006pt\times\hskip 1.00006pt\Delta\hskip 1.00006pt[\hskip 1.00006pt\infty\hskip 1.00006pt]\hskip 3.99994pt=\hskip 3.99994pt\mathit{B}\hskip 1.49994pt\pi\hskip 1.00006pt\times\hskip 1.00006pt\Delta\hskip 1.00006pt[\hskip 1.00006pt\infty\hskip 1.00006pt]\left/\hskip 1.00006ptC^{\hskip 0.70004pt0}\hskip 1.00006pt(\hskip 1.49994pt\mathbf{N}\hskip 0.50003pt,\hskip 3.00003pt\kappa\hskip 1.49994pt)\right.\hskip 1.00006pt.$$

Of  course,   this  is  simply  another  form of  the property  (4.2).   We will  need a slightly  stronger ,   but  still  obvious,   form of  this property.   Let  $1\hskip 1.99997pt\in\hskip 1.99997pt\pi$  be  the unit  of  $\pi$,   and  let  $C_{\hskip 0.70004pt0}\hskip 1.00006pt(\hskip 1.49994pt\mathbf{N}\hskip 0.50003pt,\hskip 3.00003pt\pi\hskip 1.49994pt)$  be  the group of  maps  $c\hskip 1.00006pt\colon\hskip 1.00006pt\mathbf{N}\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997pt\pi$  such  that  $c\hskip 1.49994pt(\hskip 1.00006ptn\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994pt1$  for almost  every  $n$.   Then

(4.3)

$$\quad\mathit{B}\hskip 1.49994pt(\hskip 1.00006pt\pi/\kappa\hskip 1.00006pt)\hskip 1.00006pt\times\hskip 1.00006pt\Delta\hskip 1.00006pt[\hskip 1.00006pt\infty\hskip 1.00006pt]\hskip 3.99994pt=\hskip 3.99994pt\mathit{B}\hskip 1.49994pt\pi\hskip 1.00006pt\times\hskip 1.00006pt\Delta\hskip 1.00006pt[\hskip 1.00006pt\infty\hskip 1.00006pt]\left/\hskip 1.00006ptC_{\hskip 1.04996pt0}\hskip 1.00006pt(\hskip 1.49994pt\mathbf{N}\hskip 0.50003pt,\hskip 3.00003pt\kappa\hskip 1.49994pt)\right.\hskip 1.00006pt.$$

5. Bundles  with  Eilenberg–MacLane  fibers

Locally  trivial  bundles.   Let  $p\hskip 1.00006pt\colon\hskip 1.00006ptE\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptB$  be a simplicial  map  thought  as a bundle,   and  let  $i\hskip 1.00006pt\colon\hskip 1.00006ptA\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptB$  a simplicial  map.   Let  $i^{\hskip 0.70004pt*}\hskip 0.50003ptE\hskip 1.99997pt\subset\hskip 3.00003ptE\hskip 1.00006pt\times\hskip 1.00006ptA$  be  the simplicial  subset  of  $E\hskip 1.00006pt\times\hskip 1.00006ptA$  having as $n$-simplices pairs  $(\hskip 1.00006pt\sigma\hskip 0.50003pt,\hskip 3.00003pt\tau\hskip 1.49994pt)$  such  that  $\sigma\hskip 1.99997pt\in\hskip 3.00003ptE_{\hskip 0.70004ptn}$,  $\tau\hskip 1.99997pt\in\hskip 1.99997ptA_{\hskip 0.70004ptn}$,   and  $p\hskip 1.00006pt(\hskip 1.00006pt\sigma\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994pti\hskip 1.00006pt(\hskip 1.00006pt\tau\hskip 1.49994pt)$.   In other  terms,

$$\quad(\hskip 1.49994pti^{\hskip 0.70004pt*}\hskip 0.50003ptE\hskip 1.49994pt)_{\hskip 0.70004ptn}\hskip 3.99994pt=\hskip 3.99994pt\left\{\hskip 3.00003pt(\hskip 1.00006pt\sigma\hskip 0.50003pt,\hskip 3.00003pt\tau\hskip 1.49994pt)\hskip 1.99997pt\in\hskip 3.00003ptE_{\hskip 0.70004ptn}\hskip 1.00006pt\times\hskip 1.00006ptA_{\hskip 0.70004ptn}\hskip 3.99994pt\bigl{|}\hskip 3.99994ptp\hskip 1.00006pt(\hskip 1.00006pt\sigma\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994pti\hskip 1.00006pt(\hskip 1.00006pt\tau\hskip 1.49994pt)\hskip 3.00003pt\right\}\hskip 3.00003pt.$$

The restriction  $i^{\hskip 0.70004pt*}p\hskip 1.00006pt\colon\hskip 1.00006pti^{\hskip 0.70004pt*}\hskip 0.50003ptE\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptA$  of  the projection  $E\hskip 1.00006pt\times\hskip 1.00006ptA\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptA$  to  $i^{\hskip 0.70004pt*}\hskip 0.50003ptE$  is  called  the  pull-back  of  $p$  by  $i$,   or  the  bundle  induced  from  the bundle  $p\hskip 1.00006pt\colon\hskip 1.00006ptE\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptB$  by  $i$.   The bundle  $i^{\hskip 0.70004pt*}p$  has  the usual  universal  properties of  pull-backs.   A simplicial  map  $p\hskip 1.00006pt\colon\hskip 1.00006ptE\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptB$  is  said  to  be a  trivial  bundle  with  the  fiber $F$  if  there exists a commutative diagram

such  that  $t\hskip 1.00006pt\colon\hskip 1.00006pt\ B\hskip 1.00006pt\times\hskip 1.00006ptF\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptE$  is  an  isomorphism.   Such  $t$  is  called a  trivialization  of  $p$.   A map  $p\hskip 1.00006pt\colon\hskip 1.00006ptE\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptB$  is  a  locally  trivial  bundle  with  the fiber  $F$  if  for every  simplex $\sigma$ of  $B$  the pull-back  $i_{\hskip 0.70004pt\sigma}^{\hskip 0.70004pt*}\hskip 1.00006ptp$  is  a  trivial  bundle with  the  fiber $F$.   In  this case  $E$  is  called  the  total  space  and  $B$  the  base  of  $p$.   Clearly,   if  $p$  is  a  locally  trivial  bundle,   then  $p$  is  surjective.

Normalized  and  non-abelian cochains.   Let  $n\hskip 1.99997pt\in\hskip 3.00003pt\mathbf{N}$,  $n\hskip 1.99997pt>\hskip 1.99997pt1$,   and  let  $\pi$  be an abelian  group.   A cochain of  a simplicial  set  $K$  is  said  to be  normalized  if  it  is  equal  to $0$ on degenerate simplices.   We will  denote  by  $\mathcal{C}^{\hskip 0.70004ptn}\hskip 1.00006pt(\hskip 1.00006ptK\hskip 0.50003pt,\hskip 1.99997pt\pi\hskip 1.49994pt)$  the  group of  normalized $n$-cochains of  $K$  with  coefficients  in  $\pi$ and  by  $\mathcal{Z}^{\hskip 0.70004ptn}\hskip 1.00006pt(\hskip 1.00006ptK\hskip 0.50003pt,\hskip 1.99997pt\pi\hskip 1.49994pt)$  the subgroup of  normalized cocycles.