What is the relationship between self-duality and groupoid-ness? Does any condition imply the other? Is there an example which helps understand the difference?


To go from a self-duality $F$ on a category to a groupoid, I guess I have to check for $f: A \to B$ that $F(f) \circ f$ and $f \circ F(f)$ are the identities on $A$ and $B$ respectively.
I also guess it is safe to assume that all groupoid are isomorphic (not just equivalent) to its dual.


Am I right in my guesswork?



A category being self-dual vs. it being a groupoid
The category $\mathbf{2}$ with two objects ($0$ and $1$, say) and just one map between them ($0 \to 1$) is self-dual and clearly not a groupoid. To see that it is self-dual, just consider the functor that exchanges $0$ and $1$. It is clearly its own quasi-inverse. If it were a groupoid, then the objects $0$ and $1$ would be isomorphic but there is no map from $1$ to $0$.


