The point Rotman is trying to make is the following: if you have a singular qq-simplex sigma:DeltaqtoXsigma:DeltaqtoX, then for example the restriction sigma|[e0,e2,dots,eq]sigma|[e0,e2,dots,eq] is not a singular (q−1)(q−1)-simplex, simply because its domain [e0,e2,dots,eq][e0,e2,dots,eq] is not the standard simplex Deltaq−1Deltaq−1, which is instead [e0,e1,dots,eq−1][e0,e1,dots,eq−1].
He fixes this by composing with the face maps varepsilonvarepsilon, so as to get the domains right.
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