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. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]