2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/125358It is shown that the quantized Teichm"uller spaces have factorization properties like those required in the definition of a modular functor.Material from math.QA/0405332, substantially reorganized. V2: ref. added; V3: Part III rewritten to make it more accessible, some arguments in Part IV simplified, V4: few last corrections, to appear in "Handbook of Teichmueller theory"Quantum AlgebraHigh Energy Physics - Theorytext