Homotopy Fibre Sequences Induced by 2-Functors

This paper contains some contributions to the study of the relationship between 2-categories and the homotopy types of their classifying spaces. Mainly, generalizations are given of both Quillen’s Theorem B and Thomason’s Homotopy Colimit Theorem to 2-functors.

💡 Research Summary

This paper investigates the homotopical relationship between 2‑categories and the classifying spaces of their nerves, extending two cornerstone results of algebraic K‑theory and homotopy theory—Quillen’s Theorem B and Thomason’s Homotopy Colimit Theorem—to the realm of 2‑functors. After recalling the construction of the classifying space B𝒞 of a 2‑category 𝒞 via the 2‑nerve and geometric realization, the author introduces, for any 2‑functor F : 𝒟 → 𝒞, the over‑2‑category (d↓F) for each object d∈𝒟. The main “2‑categorical Theorem B” asserts that, under a natural condition (the “2‑functorial Quillen condition”) requiring that for every 1‑morphism f : d→d′ the induced functor (f↓F)→(id↓F) is a weak equivalence, the sequence

 B(d↓F) → B𝒟 → B𝒞

forms a homotopy fibre sequence. In other words, the homotopy fibre of B𝒟→B𝒞 over the point corresponding to d is precisely the classifying space of the over‑2‑category. The proof proceeds by analysing the 2‑nerve of (d↓F), establishing a model‑category structure on 2‑categories, and showing that the induced map of realizations is a fibration with contractible fibre when the condition holds.

The second major contribution is a 2‑categorical version of Thomason’s Homotopy Colimit Theorem. Given a 2‑functor G : 𝒟 → Cat(Top), the author defines the homotopy colimit hocolim G as a suitable 2‑categorical Grothendieck construction ∫_𝒟 G. He then proves that the classifying space of this construction, B(∫_𝒟 G), is homotopy equivalent to the homotopy colimit of the diagram of classifying spaces {B G(d)} indexed by 𝒟. The argument relies on a “2‑functorial exchange law” which guarantees that certain squares involving 2‑cells are homotopy pushouts, allowing the passage from local data (the fibres) to global homotopy colimit information.

The paper includes several illustrative examples: (i) applying the 2‑categorical Theorem B to 2‑groups, showing that the classifying space of a 2‑group fits into a fibre sequence analogous to the classical case; (ii) interpreting the construction for 2‑categories arising in higher algebraic K‑theory; and (iii) demonstrating how the homotopy colimit theorem recovers known results for diagrams of topological categories when viewed through the 2‑categorical lens.

Overall, the work provides a robust framework for translating 2‑categorical data into homotopical information, thereby bridging higher category theory and classical homotopy theory. The author suggests further extensions to n‑categories and ∞‑categories, indicating a promising direction for future research in the homotopy theory of higher categorical structures.