On the homotopy type of a (co)fibred category (Q2839800)

Segal's classical nerve of a (small) category has played an important role in homotopy theory [\textit{G. Segal}, Publ. Math., Inst. Hautes Étud. Sci. 34, 105--112 (1968; Zbl 0199.26404)], notably in Quillen's work on algebraic \(K\)-theory [\textit{D. Quillen}, Lect. Notes Math. 341, 85--147 (1973; Zbl 0292.18004)]. Grothendieck and others have studied fibred or cofibred categories, which are functors satisfying certain lifting properties. The paper under review studies the homotopy theory of cofibred categories, which the author calls (Grothendieck) fibrations to emphasize the analogy with fibrations in topology.NEWLINENEWLINEThe main results are as follows. In Section 3, the author constructs for every fibration \(\xi = (p : E \to B)\) a bisimplicial set \(N_f \xi\), called the fibred nerve of the fibration. For fibrations equipped with a cleavage, the author constructs a smaller variant \(N_c \xi\), called the cleaved nerve. If the fibration \(\xi\) is splitting (for example, a split extension of groups), then the inclusion of bisimplicial sets \(N_c \xi \to N_f \xi\) is a weak equivalence, i.e., induces a weak equivalence of diagonal simplicial sets \(d(N_c \xi) \to d(N_f \xi)\) (Theorem 3.2.3).NEWLINENEWLINEThe fibred nerve recovers the homotopy type of the total category. More precisely, there is a natural weak equivalence of simplicial sets \(d (N_f \xi) \to N E\) (Theorem 4.1.3), where \(N E\) denotes the classical nerve of the category \(E\). Using this, one recovers Quillen's Theorem A, providing sufficient conditions for a map of small categories to be a weak equivalence (Corollaries 4.2.2 and 4.2.3).NEWLINENEWLINEIn Section 5, various applications are described. The author recovers Thomason's theorem on homotopy colimits (Corollary 5.1.2). He constructs a homology Leray-Serre spectral sequence for fibrations of small categories (Theorem 5.2.1) and deduces a homology version of Quillen's Theorem A (Corollary 5.2.3). Given a fibration \(p : E \to B\) whose base-change functors between fibers are weak equivalences, the map induced on fibred classifying spaces \(|d(N_f E)| \to |d(N_f B)|\) is a quasifibration (Theorem 5.3.1). This implies Quillen's Theorem B, providing a long exact sequence of homotopy groups for certain maps of small categories (Corollary 5.3.2). Given a group \(G\) acting on a small category \(A\), one obtains an Eilenberg-Moore spectral sequence converging to the homology of the homotopy quotient \(A_{hG}\) (Proposition 5.4.2).