$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Warning The simplicial set $\operatorname{\raise {0.1ex}{\square }}^{I}$ and its normalized chain complex $\mathrm{N}_{\ast }( \operatorname{\raise {0.1ex}{\square }}^{I}; \operatorname{\mathbf{Z}})$ depend only on the choice of the finite set $I$. However, the fundamental chain $[ \operatorname{\raise {0.1ex}{\square }}^{I} ]$ of Construction is a priori ambiguous up to a sign. One can resolve this ambiguity by choosing a linear ordering on the set $I$ (as in Construction, which will be sufficient for our purposes in this section. However, less is needed: one needs only an orientation on the set $I$ (or equivalently an orientation of the topological manifold-with-boundary $| \operatorname{\raise {0.1ex}{\square }}^{I} | \simeq [0,1]^{I}$).