  The aim of this thesis is to give a concise introduction to homotopy type
theory, to Aczel's constructive set theory and to simplicial sets and their
homotopy theory in particular referring to their standard model structure,
showing some of their interactions. The original part of this thesis consists
in the final chapter where we introduce in the type theoretic context a
definition of weak Tarski universe motivated by categorical models like the one
given by simplicial sets. The weakening of this notion, although present in
some imprecise form in mathematical folklore was not published before, at the
best of our knowledge. Moreover, we show using the axiom of function
extensionality that the type theoretic interpretation of constructive set
theory generalises to homotopy type theory with a weak Tarski universe.
