Definition of a contractible category

Question

A simplicial set can be interpreted both as a generalized space as well as a generalized category. While reading about bisimplicial sets I came across the statement that the spatial $\Delta^{1}$ is contractible, whereas categorical $\Delta^{1}$ is not. So is there a notion of contractible category?

Answer 1

Something is contractible if it is (weakly) equivalent to a point. If we are thinking of simplicial sets as spaces then $\Delta^1$ is indeed equivalent to a point. On the other hand, if we are thinking of simplicial sets as categories then $\Delta^1$ is not equivalent to a point.