What is a contractible simplicial object?

[Michael Barr <barr@math.mcgill.ca>]

A few weeks ago, I asked on this group how one defines contractibility
of simplicial objects.  Later I tried Math Overflow and discovered that
the question had been asked but not satisfactorily answered several
years ago.  Below is a \TeX file that gives an entirely satisfactory
answer.

\def\dot{{_\bullet}}
  \def\to{\mathop{\longrightarrow}\limits}
For an augmented simplicial set, the answer is clear.  Given $X_\dot\to
X_{-1}$, a contraction is a sequence of maps $t=t_n:X_n\to X_{n+1}$ for
all $n\ge -1$ such that $d^0t=1$, $d^it=td^{i-1}$ for $i>0$, $s^0t=tt$,
and $s^it=ts^{i-1}$ for $i>0$.  These are the equations a degeneracy
$s^{-1}$ would satisfy.

Suppose $X\dot$ is a simplicial object in a category with split
idempotents and there
are maps $t=t_n:X_n\to X_{n+1}$ for all $n\ge0$ satisfying all the above
equations as far as they are defined.  We
have, in $X_0$
  $$d^1td^1t=d^1d^2tt=d^1d^1tt=d^1td^0t=d^1t$$
  so that $d^1t$ is idempotent.  If we factor it as $X_0\to^{d^0}X_{-1}
\to^{t_{-1}}X_0$ we nearly have a contractible augmented simplicial
object. There two things to verify: first that $d_0^0d_1^0=d_0^0d_1^1$
and second that $s_0^0t_{-1}=t_0t_{-1}$. I have used the normally
omitted lower indices to clarify that the low dimension indentities
follow from the higher ones. For the first,
  $$t_{-1}d_1^0d_2^0=d_0^1t_0d_1^0=d_1^1d_2^1t_2=d_1^1d_2^2t_1=
d_1^1t_0d^1 =t_{-1}d_0^0d_1^1$$
  and $t_{-1}$ being a (split) monic can be canceled to give
the result. For the second,
  $$s_0^0t_{-1}d_0^0=s_0^0d_1^1t_0=d_2^2s_1^0t_0=d_2^2t_1t_0=
t_0d_1^1t_0=t_0t_{-1}d_0^0$$
  and $d_0^0$ being a (split) epic can be canceled to give
  the result.


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