On the Existence of Categorical Universal Coverings

In this paper, we study necessary and sufficient conditions for the existence of categorical universal coverings using open covers of a given space $X$. As some applications, first we present a generalized version of the Shelah Theorem (Mycielski's conjecture: If $X$ is a Peano continuum, then $\pi_1(X,x)$ is uncountable or $X$ has a simply connected universal covering) which states that a first countable Peano space has a categorical universal covering or has an uncountable fundamental group. Second, we prove that the one point union $X_1\vee X_2=\frac{{X_1}\cup {X_2}}{{x_1}\sim {x_2}}$ has a categorical universal covering if and only if both $X_1$ and $X_2$ have categorical universal coverings.