TITLE: Is the Fundamental Groupoid a Bicategory?
QUESTION [9 upvotes]: I am currently learning about fundamental group and I understand that there is really a fundamental groupoid. Given a topological space $X$, it is a category with

points in $X$ as objects
paths between points as arrows
homotopies between paths as 2-morphisms

You could continue by defining n-morphisms as homotopies between n-1 morphisms and this would give you something called an infinity category. My question is the following: If you were to stop at 2-morphisms would this give a bicategory like I suggested above? My concern is the following: usually in bicategories you have many types of 2-morphisms not all of them isomorphisms. However, in the fundamental groupoid, you seem to say that two paths are isomorphic if there is any homotopy between them at all. It seems like there are lot of missing 2- morphisms here.

REPLY [2 votes]: The other answer talks about the fundamental $2$-groupoid. I'll discuss the other possible correction to your construction: don't distinguish between parallel $2$-arrows.
Rather than having the $2$-morphisms be homotopies between homotopies, you can just have $2$-morphisms give the "is homotopic to" relation. That is, given paths $f$ and $g$, $\hom(f,g)$ is a one-point set if $f \simeq g$, and is empty otherwise.
The obvious functor from this construction to the fundamental groupoid is an equivalence, since for points $P$ and $Q$, the category $\hom(P,Q)$ is equivalent to the set of homotopy classes of paths from $P$ to $Q$.