The classifying space of a (braided) monoidal category induces a (double) loop space up to homotopy. The collection of some geometric groups gives rise to a braided monoidal category, therefore generates a double loop space structure. The braid structures inside mapping class groups can be explicitly expressed in terms of self-homomorphisms on the fundamental groups and also in terms of Dehn twists. There are various embeddings of braid groups into mapping class groups. We explain why those embeddings induce trivial homomorphim on homology groups. Here, the construction of a monoidal 2-category plays a key role. In this talk I will also introduce further embedding problems of Artin groups into mapping class groups.