homotopy group of a poset.\
The definition of a homotopy of paths ([@Rob3], p.322) needs some preliminaries. An *ampliation* of a 1-simplex $b$ is a 2-simplex $c$ such that $\partial_1c=b$. We denote by ${\mathsf{A}}(b)$ the set of the ampliations of $b$. An *elementary ampliation* of a path $p=\{b_n,\ldots, b_1\}$, is a path $q$ of the form $$\label{Ab:1}
q = \{b_n, \ldots, b_{j+1},\partial_0c,\partial_2c, b_{j-1},\ldots b_2,b_1\}
 \qquad c\in  {\mathsf{A}}(b_j).$$ Consider now a pair $\{b_2, b_1\}$ of 1-simplices satisfying $\partial_1b_2=\partial_0b_1$. A *contraction* of $\{b_2,b_1\}$ is a 2-simplex $c$ satisfying $\partial_0c=b_2$, $\partial_2c=b_1$. We denote by ${\mathsf{C}}(b_2,b_1)$ the set of the contractions of $\{b_2,b_1\}$. An *elementary contraction* of a path $p=\{b_n,\ldots,b_1\}$ is a path $q$ of the form $$\label{Ab:2}
q = \{b_n,\ldots, b_{j+2}, \partial_1c,
b_{j-1},\ldots, b_1\} \qquad c\in {\mathsf{C}}(b_{j+1},b_{j}).$$ An *elementary deformation* of a path $p$ is a path $q$ which is either an elementary ampliation or an elementary contraction of $p$.\

![$q$ is an elementary ampliation of the path $p$. $q_1$ is an elementary contraction of $p$. The symbol $\delta$ stands for $\partial$.](defor.eps)

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Note that a path $q$ is an elementary ampliation of a path $p$ if, and only if, $p$ is an elementary contraction of $q$. This can be easily seen by 