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Gabriel-Zisman cohomology and spectral sequences. (English) Zbl 1469.18015

The paper is devoted to the (co)homology of simplicial sets introduced by Gabriel and Zisman for the case of homology \(H^{GZ}_n(X, T)\) with coefficients in functors taking values in an abelian category \(\mathcal M\) with exact coproducts. In the work [P. Gabriel and M. Zisman, Calculus of fractions and homotopy theory. Berlin-Heidelberg-New York: Springer-Verlag (1967; Zbl 0186.56802), Appendix II, Proposition 4.2], it was proved that for a fixed simplicial set the functors \(H^{GZ}_n(X, -)\) are isomorphic to the left satellites of the colimit functor \(\mathrm{colim}^{(\Delta/X)^{op}}_n: {\mathcal M}^{(\Delta/X)^{op}} \to {\mathcal M}\). If instead of the abelian category \(\mathcal M\) we substitute the opposite category \({\mathcal A}^{op}\), then we obtain the Gabriel-Zismann cohomology \(H^n_{GZ}(X, T)\cong \lim^n_{\Delta/X}T\) for any functor \(T: \Delta/X\to {\mathcal A}\) where \(\mathcal A\) is the abelian category with exact products.
The following definition is introduced, which is useful for analyzing the functorial properties of the Gabriel-Zismann cohomology.
Definition 1.8 Let \(\mathscr C\) be a small category and \(\mathscr A\) a complete abelian category. Given two functors \(P : {\mathscr C} \to \Delta\) and \(T : {\mathscr C} \to {\mathscr A}\) the nth cohomology of \(P\) with coefficients in \(T\) is defined as \(H^n(P, T ) := H^n(Ran_P(T ))\).
Homology \(H_n(P,T)\) is dually defined (Definition 1.9).
An additive category \(\mathscr A\) is called cotensored over the category of Abelian groups \(Ab\) if for each \(b\in Ob {\mathscr A}\) the functor \(Hom_{\mathscr A} (-, b): {\mathscr A}^{op}\to Ab \) has a left adjoint, denoted by \(Hom(-, b): Ab \to {\mathscr A}^{op}\). \(\mathscr A\) is called tensored over \(Ab\) if for each \(a\in Ob{\mathscr A}\) the functor \(Hom_{\mathscr A} (a, -)\) has a left adjoint \(-\otimes a: Ab \to {\mathscr A}\).
Theorem 1.11 Let \({\mathscr A}\) be an additive category. For any functor \(P: {\mathscr C}\to \Delta\) there exists a resolution \(B^P_*\) of the constant functor \(\underline{\mathbb Z}\) such that
1. If \({\mathscr A}\) is complete and cotensored over \(Ab\), then
\[ Ran_P(T)\cong \overline{Hom}_{\mathscr C}(B^P_*, T): \Delta \to {\mathscr A} \] natural in \(T: {\mathscr C}\to {\mathscr A}\).
2. If \({\mathscr A}\) is cocomplete and tensored over \(Ab\), then \[ Lan^{P}(T) \cong B^P_*\underline{\otimes}_{\mathscr C}T: \Delta \to {\mathscr A} \] natural in \(T: {\mathscr C}^{op}\to {\mathscr A}\).
3. If \(P: {\mathscr C}\to \Delta\) is a discrete fibration over \(\Delta\), then there is a natural isomorphism \[ B^P_n \cong \bigoplus_{c\in {\mathscr C}: P(c)=n} {\mathbb Z}Hom_{\mathscr C}(c,d) \] and hence \(B^P_*\) is a free resolution of \(\underline{\mathbb Z}\).
Hence it follows that if \(P: {\mathscr C}\to \Delta\) is a discrete fibration, then for any complete abelian category \({\mathscr A}\) with exact products, the cohomology groups of \(P\) with coefficients in \(T\) are derived functors, \(H^n(P,T)\cong Ext^n_{\mathscr C}(\underline{\mathbb Z}, T) \cong \lim^n_T=H^n({\mathscr C}, T)\) and the dual statement is also true (Corollary 1.12).
Let \(X\) be a simplicial set. The forgetful functor \(P_X: \Delta/X \to \Delta\) is the discrete fibration. Consequently, \(C^*(P_X, T)\cong C^*_{GZ}(X, T)\) and \( H^*(P_X, T)\cong H^*_{GZ}(X, T)\cong H^*(\Delta/X, T)\) (Theorem 1.13). The dual is also true (Theorem 1.14).
Applications of the Gabriel-Zismann (co)homology theory are given: Thomason (co)homology of categories (Example 1.15) in [I. Gálvez-Carrillo, F. Neumann and A. Tonks, Thomason cohomology of categories. J. Pure Appl. Algebra 217, No. 11, 2163-2179 (2013; Zbl 1285.18020)], sheaves on topological spaces (Example 1.16) in [T. Fimmel, Math. Nachr. 190, 51–122 (1998; Zbl 0903.55008)], Parshin-Beilinson adeles of schemes (Example 1.17) in [A. A. Beilinson, Funct. Anal. Appl. 14, 34–35 (1980); translation from Funkts. Anal. Prilozh. 14, No. 1, 44–45 (1980; Zbl 0509.14018); A. N. Parshin, Math. USSR, Izv. 10(1976), 695–729 (1977; Zbl 0366.14003)], buildings of reductive algebraic groups (Example 1.18) in [K.S. Brown, Buildings. New York etc.: Springer-Verlag (1989; Zbl 0715.20017)] and [P. Schneider and U. Stuhler, Publ. Math., Inst. Hautes Étud. Sci. 85, 97–191 (1997; Zbl 0892.22012)].
The second section contains results on the cohomology introduced in Definitions 1.8 and 1.9.
Theorem 2.1. Let \({\mathscr C}\) and \({\mathscr D}\) be small categories and \(T: {\mathscr C}\to {\mathscr A}\) be a functor to a complete abelian category. Let \(u: {\mathscr C}\to {\mathscr D}\) be a functor together with functors \(P: {\mathscr C}\to \Delta\) and \(Q: {\mathscr D}\to \Delta\) such that \(P= Q\circ u\). Then there is a spectral sequence: \[ E^{p,q}_2\cong H^p(Q, Ran^q_u(T))\Rightarrow H^{p+q}(P,T), \] which is natural in \(u\) and \(T\) and where \(Ran^q_u(T)\) denotes \(q\)th right satellite of \(Ran_u(T)\).
The dual statement is also true, for homology (Theorem 2.2).
The final part of the article is devoted to the spectral sequence of the Kan fibration described by P. Gabriel and M. Zisman in [Calculus of fractions and homotopy theory. Berlin-Heidelberg-New York: Springer-Verlag (1967; Zbl 0186.56802), Appendix II, Theorem 4.5]. A similar but different method for constructing the spectral sequence of the Kan fibration was developed in [A. A. Khusainov, Sib. Math. J. 32, No. 1, 116–122 (1991; Zbl 0741.55009), Subsections 4.1–4.3].

MSC:

18G40 Spectral sequences, hypercohomology
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References:

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