Exponential domination in function spaces

Given a Tychonoff space X and an infinite cardinal kappa, we prove that exponential kappa-domination in X is equivalent to exponential kappa-cofinality of C-p(X). On the other hand, exponential kappa-cofinality of X is equivalent to exponential kappa-domination in C-p(X). We show that every exponentially kappa-cofinal space X has a kappa(+)-small diagonal; besides, if X is kappa-stable, then nw(X) <= kappa. In particular, any compact exponentially kappa-cofinal space has weight not exceeding kappa. We also establish that any exponentially kappa-cofinal space X with l(X) <= kappa and t(X) <= kappa has i-weight not exceeding kappa while for any cardinal kappa, there exists an exponentially omega-cofinal space X such that l(X) >= kappa.