Hello,
  Roughly speaking, a quantum group is an algebraic structure which is
obtained by means of a deformation of a group. There rigorous definition is
here: https://ncatlab.org/nlab/show/quantum+group

Official reference to quantum groups: Kassel, Christian (1995), Quantum
groups, Graduate Texts in Mathematics, 155, Berlin, New York:
Springer-Verlag, doi:10.1007/978-1-4612-0783-2, ISBN 978-0-387-94370-1, MR
1321145

Deformations... homotopy type... Well, given a "well-behaved" family of
quantum groups, which are deformations of the same group, is it "natural"
to define this family as a homotopy type? Is HoTT, in some way, a natural
setting to work with quantum groups because types and homotopy types are
identified?

Kind Regards,
José M.

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