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\centerline {\bf Simplicial}
\medskip
The standard $n${\it -simplex} is a point for $n=0$, the interval $[0,1]$ for n=1, the
triangle for $n=2$, the tetrahedron for $n=3$, and more generally the toplogical space
$$\Delta^n = \{(x_0,\ldots, x_n)\  \vert\  x_i\in \R ,\  0\leq x_i\leq 1,\ x_0+\cdots
+ x_n = 1\}.
$$
It is the convex envelope in $\R ^{n+1}$ of its $n+1$ vertices with coordinates
$(0,\ldots, 1, \ldots, 0)$.
The $n$-simplex has $n+1$ {\it faces} which are $n-1$-simplices. There are the images
of the following injective maps (for
$i=0,\ldots, n$):   \par
 $\delta_i : \Delta^{n-1}\to \Delta^n,\
\delta_i(x_0,\ldots, x_{n-1})= (x_0,\ldots, x_{i-1}, 0, x_i,\ldots , x_{n-1})$.\par
 On the other hand one can smash an $n+1$-simplex onto an
$n$-simplex by the following surjective maps (for
$i=0,\ldots, n$):\par
 $\sigma_i : \Delta^{n+1}\to \Delta^n,\
\sigma_i(x_0,\ldots, x_{n+1})= (x_0,\ldots, x_i + x_{i+1},\ldots , x_{n+1})$. 

These maps satisfy relations dual to the ones shown below, for instance $\delta
_j\delta _i = \delta _i\delta _{j-1}\  {\rm for}\  i<j$.

By definition a {\it simplicial set} $X_.$ (resp. {\it simplicial module} $M_.$) is a
family of sets $X_n$ (resp. modules $M_n$) for $n\geq 0$ and maps called {\it faces}
$d_i: X_{n-1}\to X_n$, $0\leq i\leq n$,
and {\it degeneracies} $s_j: X_{n+1}\to X_n$, $0\leq j\leq n$ 
(resp. linear maps $d_i: M_{n-1}\to M_n$ and 
$s_j: M_{n+1}\to M_n$), satisfying the following
relations: $$\eqalign{ d_id_j &= d_{j-1}d_i\  {\rm for}\  i<j,\cr
s_is_j&=s_{j+1}s_i \  {\rm for}\   i\leq j,\cr
d_is_j &= \cases {s_{j-1}d_i& for $i<j$,\cr
{\rm id}&for $i=j,\ i=j+1$,\cr
s_jd_{i-1}& for $i>j+1$.\cr}\cr}
$$
A simplicial set admits a {\it geometric realization}, which is a topological set
constructed as follows: one takes one $n$-simplex for each element in $X_n$, and then
one glue them together according to the rules given by the faces and degeneracies:
\smallskip

$\vert X_.\vert := \bigcup_{n\geq 0} X_n\times \Delta^n / \approx.$
\smallskip

A simplicial set is a contravariant functor from a certain category
$\Delta$ to the category of sets. The objects of $\Delta$ are the integers $[n],\ n\geq
0$, and the morphisms are generated by $\delta_i:[n]\to [n-1]$ and
$\sigma_j:[n+1]\to [n]$, subject to the above relations. So the family of standard
$n$-simplices is a covariant functor from $\Delta$ to the category of
topological spaces. The category $\Delta$ has a nice interpretation in terms of ordered
finite sets and non-decreasing maps.

Simplicial modules are important since they give rise to {\it chain complexes} as follows.
The module of $n$-chains is $M_n$ and the boundary map is $d= \sum _{i=0}^{i=n}(-1)^id_i$.
The first relation ensures that $d\circ d= 0$. 

Starting with a simplicial set $X_.$, one gets a simplicial module ${\bf Z}[X_.]$, then a
chain complex and, finally, homology groups $H_n({\bf Z}[X_*])$. It turns out that they
are precisely the singular homology groups of the geometric realization of $X_.$.
 \smallskip
\noindent {\bf Ariane's thread:}
J.P. May, {\it Simplicial objects in algebraic topology},  Reprint of the 1967 original. Chicago Lectures in
Mathematics. University of Chicago Press, Chicago, IL, 1992. viii+161 pp. 


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