Does there exist a hardest DCFL? Greibach famously defined a language $H$, the so-called nondeterministic version of $D_2$, such that any CFL is an inverse morphic image of $H$.  Does there exist a similar statement with DCFL, possibly with some restriction on the morphisms allowed?

(See, e.g., M. Autebert, J. Berstel, and L. Boasson. Context-free languages and
pushdown automata. In R. Rozenberg and A. Salomaa, editors,
Handbook of Formal Languages, volume I, chapter 3. Springer Verlag, 1997.)
An identical homomorphism characterization of DCFL does not seem to be possible. The following is extracted from Greibach's original paper.


  We show that every context-free language can be expressed as $h^{-1}(L_0)$ or $h^{-1}(L_0-\{e\})$ for a homomorphism $h$. The algebraic statement is: the family of context-free languages is a principal AFDL; ... By way of contrast, the family of deterministic context-free languages is not a principal AFDL [7].


The paper 7 is the conference version of the paper. In the conference version, Theorem 4.2 states that "The family of deterministic context-free languages is not a principal AFDL". 

However some analogue characterization may still be possible. Okhotin provided homomorphic characterizations of conjunctive and Boolean grammars. For DCFL's the problem seems to be open. The following is the conclusion of Okhotin's paper (from 2013). 


  Every family of languages closed under inverse homomorphisms can potentially
  have an analogue of Greibach’s inverse homomorphic characterization.
  The question is, which families have it? Could it exist for linear, deterministic
  or unambiguous variants of ordinary (context-free) grammars? Could
  there be such a characterization for linear conjunctive grammars, unambiguous
  conjunctive grammars, etc.?