Homotopy invariants in small categories

TL;DR

This paper explores homotopy invariants in small categories, introducing categorical complexity and higher complexity, and establishes properties and a Varadarajan's theorem analog for these invariants.

Contribution

It introduces the homotopic distance and categorical complexity as new invariants, extending classical results to small categories and fibrations.

Findings

Defined the homotopic distance between functors

Introduced categorical and higher categorical complexity

Proved properties and a Varadarajan's theorem for these invariants

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 $F,G\colon \mathcal{C} \to \mathcal{D}$, 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.