  Tanaka introduced a notion of Lusternik Schnirelmann category, denoted
$\mathrm{ccat}\, \mathcal{C}$, of a small category $\mathcal{C}$. Among other
properties, he proved an analog of Varadarajan's theorem for fibrations,
relating the LS-categories of the total space, the base and the fiber.
  In this paper we recall the notion of homotopic distance $\mathrm{D}(F,G)$
between two functors $F,G\colon \mathcal{C} \to \mathcal{D}$, later introduced
by us, which has $\mathrm{ccat}
\mathcal{C}=\mathrm{D}(\mathrm{id}_{\mathcal{C}},\bullet)$ as a particular
case. We consider another particular case, the distance $\mathrm{D}(p_1,p_2)$
between the two projections $p_1,p_2\colon \mathcal{C}\times \mathcal{C} \to
\mathcal{C}$, which we call the categorical complexity of the small category
$\mathcal{C}$. Moreover, we define the higher categorical complexity of a small
category and we show that it can be characterized as a higher distance.
  We prove the main properties of those invariants. As a final result we prove
a Varadarajan's theorem for the homotopic distance for Grothendieck
bi-fibrations between small categories.
