  We employ projective Fra\"iss\'e theory to define the "generic combinatorial
$n$-simplex" as the pro-finite, simplicial complex that is canonically
associated with a family of simply defined selection maps between finite
triangulations of the simplex. The generic combinatorial $n$-simplex is a
combinatorial object that can be used to define the geometric realization of a
simplicial complex without any reference to the Euclidean space. It also
reflects dynamical properties of its homeomorphism group down to finite
combinatorics.
  As part of our study of the generic combinatorial simplex, we define and
prove results on domination closure for Fra\"iss\'e classes, and we develop
further the theories of stellar moves and cellular maps. We prove that the
domination closure of selection maps contains the class of face-preserving
simplicial maps that are cellular on each face of the $n$-simplex and is
contained in the class of simplicial, face-preserving near-homeomorphisms.
Under the PL-Poincar\'e conjecture, this gives a characterization of the
domination closure of selections.
