  We fix a path model for the space of filters of the inverse semigroup
$\mathcal{S}_\Lambda$ associated to a left cancellative small category
$\Lambda$. Then, we compute its tight groupoid, thus giving a representation of
its $C^*$-algebra as a (full) groupoid algebra. Using it, we characterize when
these algebras are simple. Also, we determine amenability of the tight groupoid
under mild, reasonable hypotheses.
