What is a pointed connected groupoid?

Question

Here it says a pointed connected groupoid is a connected groupoid that is pointed. This seems nonsensical: connected groupoids don't have a terminal object, except for the trivial case. The link in nlab about pointed objects only talks about categories with terminal objects so it doesn't help.

What is a pointed connected groupoid?

Answer 1

You've mixed up category levels; the definition of pointedness is about the terminal object of the $2$-category of connected groupoids itself, which is the point $1$. Then the $2$-category of pointed connected groupoids is the coslice $2$-category of the point; that is, it has

• objects given by functors $f : 1 \to X$ where $X$ is a connected groupoid (this just distinguishes an object of $X$, the "basepoint"),
• morphisms $(1 \xrightarrow{f} X) \to (1 \xrightarrow{g} Y)$ given by functors $h : X \to Y$ together with natural isomorphisms $\eta : h \circ f \cong g$ (this says that $h$ preserves the basepoint, but note that "preserving the basepoint" here is itself extra data, not just a property), and
• $2$-morphisms from $(h_1, \eta_1)$ to $(h_2, \eta_2)$ (both of which are morphisms as above) given by natural transformations $\eta : h_1 \to h_2$ such that $\eta_2 \circ (\eta f) = \eta_1$ (or something like that; I don't know a good convention for representing both vertical and horizontal composition, really I should be drawing a commutative diagram here).

Then the exercise is to show that this is actually just equivalent to the category of groups, with the equivalence given by taking the automorphism group of the basepoint; in particular there are no non-identity $2$-morphisms.