Given a topological space $X$, we have the notion of the fundamental groupoid $\pi_1(X)$.

Here, the fundamental groupoid $\pi_1(X)$ is made into a topological groupoid giving a topology on the morphism set.

One can then talk about $\pi_1(\pi_1(X))$. Is this related to (same as) the fundamental $2$-groupoid?

Given a topological space $X$, we have the notion of the fundamental groupoid $\Pi_1(X)$.

Here, the fundamental groupoid $\Pi_1(X)$ is made into a topological groupoid giving a topology on the morphism set.

One can then talk about $\Pi_1(\Pi_1(X))$. Is this related to (same as) the fundamental $2$-groupoid?

Given a topological space $X$, we have the notion of the fundamental groupoid $\pi_1(X)$.

Here, the fundamental groupoid $\pi_1(X)$ is made into a topological grouopid giving a topology on the morphism set.

One can then talk about $\pi_1(\pi_1(X))$. Is this related to (same as) the fundamental $2$-groupoid?

Given a topological space $X$, we have the notion of the fundamental groupoid $\pi_1(X)$.

Here, the fundamental groupoid $\pi_1(X)$ is made into a topological groupoid giving a topology on the morphism set.

One can then talk about $\pi_1(\pi_1(X))$. Is this related to (same as) the fundamental $2$-groupoid?