Is the geometric realization of a pointed category contractible?

Question

Given a pointed category $X$, is $|N(X)|$ contractible ($N(X)$ is the nerve of $X$)? Or are there counter-examples?

Answer 1

Any category that has either a terminal object or an initial object has contractible nerve. This follows from the fact that the nerve functor turns natural transformations into homotopies, and thus turns adjoint functors into inverse homotopy equivalences (since the unit and counit of the adjunction become homotopies to the identity). A terminal object is exactly a right adjoint to the unique functor $X\to 1$, and so gives a homotopy equivalence $|N(X)|\simeq |N(1)|$ and hence shows $|N(X)|$ is contractible since $|N(1)|$ is a one-point space. Similarly, an initial object is a left adjoint to $X\to 1$.