If our topological space is connected, locally connected and semi-locally simply-connected, then we know that a universal cover exists. Knowing the existence, my question is how to find universal cover explicitly? Any help in this regard will be appreciated. Thank you.
This question is related to this stackexchange question and answer on lifted topologies.
An algebraic model of a covering map is a covering groupoid morphism, $q: H \to G $, namely a groupoid morphism such that for each $x \in Ob(H)$ and $g$ from $q(x)$ to some $y$ there is a unique $h$ in $H$ starting at $x$ such that $q(h)=g$.
If $p: X \to B$ is a covering map, then the induced morphism of groupoids $\pi_1(X) \to \pi_1(B)$ is a covering morphism of groupoids.
If $G$ is a connected groupoid, then an easy construction of a universal covering groupoid of $G$ is to choose $x \in Ob(G)$ and let $Ob(H)$ be the set of elements of $G$ starting at $x$, with $Ob(q)$ being the end point map. An element of $H(g,g')$ is to be a pair $(h,g)$ of elements of $G$ such that $hg=g'$.
Note that this construction if $G=\pi_1(B)$ requires no conditions on $B$. It is the construction of the lifted topology which requires the local conditions.