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Warning In the formulation of Proposition, the ordering on the set $I = \{ 1, 2, \cdots , n \} $ is dictated by the “prefix” convention that the composition of a string of morphisms

\[ X_0 \xrightarrow { f_{1} } X_1 \xrightarrow { f_{2} } X_2 \xrightarrow { f_{3} } \cdots \xrightarrow { f_{n} } X_ n \]

is denoted by $f_{n} \circ \cdots \circ f_{1}$, in which the indices appear (from left to right) in the opposite of their numerical order. Note that reversing the order on $I$ changes the definition of the fundamental chain $[ \operatorname{\raise {0.1ex}{\square }}^{I} ]$ by a factor of $(-1)^{n(n-1)/2}$ (see Warning