Homotopy invariants in small categories

Abstract

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 $F,G:\mathcal{C}\to \mathcal{D}$, 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 $p_1,p_2:\mathcal{C}\times \mathcal{C}\to \mathcal{C}$, 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.