5.2 Cohomology of simplicial topological spaces
To each simplicial topological space X_\bullet is associated a usual topological space |X_\bullet|, called its geometric realisation (see below). In the particular case where the X_n are discrete, we recover the usual notion of geometric realisation of a simplicial set.
In (Segal 1968), G. Segal defined the cohomology of X_\bullet with values in an abelian group A as \operatorname{H}^\bullet(|X_\bullet|,A). Under suitable hypotheses, the filtration of |X_\bullet| by successive skeletons gives a spectral sequence E_1^{pq} = \operatorname{H}^q(X_p,A) \Rightarrow\operatorname{H}^{p+q}(X_\bullet,A). \tag{5.2.1.1}
One of the interests in using this definition is that it also applies, for example, to the definition of the groups K(X_\bullet). We will adopt another definition, which is better adapted to sheaf-theoretic techniques.
Geometric realisation. For an integer n\geqslant0, we denote by |\Delta_n| the simplex in \mathbb{R}^{\Delta_n} whose vertices are the set of basis vectors. We identify \Delta_n with the set of vertices of |\Delta_n|. Every function f\colon\Delta_n\to\Delta_m extends by linearity to |f|\colon|\Delta_n|\to|\Delta_m|.
Let X_\bullet be a simplicial topological space. Let Y=\coprod_{n\geqslant0}(X_n\times|\Delta_n|). Let R be the finest equivalence relation on Y for which, for any f\colon\Delta_n\to\Delta_m in (\Delta), any x_m\in X_m, and any a\in|\Delta_n|, we have (x_m,|f|(a)) \equiv (f(x_m),a) \operatorname{mod}R. The geometric realisation of X_\bullet is by definition |X_\bullet| = Y/R.
Let X_\bullet be a simplicial topological space. The functors \operatorname{H}^i(X_\bullet,F^\bullet) of cohomology with coefficients in the abelian sheaf F^\bullet on X_\bullet are the derived functors of the functor \Gamma (5.1.13.1).
This definition is equivalent to the following, sometimes more convenient definition.
Let F^\bullet be an abelian sheaf on X_\bullet. We can show (by example, using (iii) of (5.1.9)) that F^\bullet always admits resolutions K on the right such that \operatorname{H}^r(X_q,K^{p,q}) =0 \quad\text{for } p,q\geqslant0 \text{ and } r>0. If K is such a resolution, then we canonically have that \operatorname{H}^n(X_\bullet,F^\bullet) \simeq \operatorname{H}^n(\mathrm{s}\Gamma^\bullet(X_\bullet,K)). \tag{5.2.3.1} It is easy to show directly that the right-hand side is independent (up to unique isomorphism) of the choice of K; for what follows, we are free to define \operatorname{H}^\bullet(X_\bullet,F^\bullet) by (5.2.3.1). We can further show that the spectral sequence of the complex \mathrm{s}\Gamma^\bullet(X_\bullet,K), filtered by L ((5.1.9.3) and (5.1.12.1) for S=P^t) E_1^{pq} = \operatorname{H}^q(X_p,F^p) \Rightarrow\operatorname{H}^{p+q}(X_\bullet,F^\bullet) \tag{5.2.3.2} is independent (up to unique isomorphism) of the choice of K (cf. (5.2.1.1)).
We will need to make precise the above construction when passing to derived categories, and to give a relative variant of it.
Let a\colon X_\bullet\to S, let S_\bullet be the constant simplicial space defined by S with augmentation map \varepsilon\colon S_\bullet\to S, and let a_\bullet\colon X_\bullet\to S_\bullet be the map induced by a. Then a_* = \varepsilon_* {a_\bullet}_*\colon (X_\bullet)^\sim \to (S)^\sim. \tag{5.2.4.1}
The derived version of this equation is \mathrm{R}a_* = \mathrm{R}\varepsilon_* \mathrm{R}{a_\bullet}_*\colon \mathrm{D}^+(X_\bullet) \to \mathrm{D}^+(S) \to (S)^\sim. \tag{5.2.4.2} where \mathrm{D}^+(X_\bullet) is the bounded-below derived category of the category of abelian sheaves on X_\bullet. We will calculate \mathrm{R}{a_\bullet}_* and \mathrm{R}\varepsilon_*.
Let u\colon X_\bullet\to Y_\bullet be a morphism of simplicial topological spaces. The functor \mathrm{R}u_*\colon\mathrm{D}^+(X_\bullet)\to\mathrm{D}^+(Y_\bullet) can be calculated “component by component”: if K is a complex of abelian sheaves on X_\bullet, then to calculate \mathrm{R}u_*K we take a resolution K\xrightarrow{\sim}K' such that the components F of K' satisfy \mathrm{R}^i {u_p}_*(F^p)=0 for i>0, p\geqslant0 (cf. (iii) of (5.1.9)); then \mathrm{R}u_*K\approx u_*K'.
The functor \mathrm{s}\colon\operatorname{C}^+(S_\bullet)\to\operatorname{C}^+(S) sends acyclic complexes to acyclic complexes. It thus trivially derives to \mathrm{s}\colon \mathrm{D}^+(S_\bullet) \to \mathrm{D}^+(S). If F is an injective sheaf on S_\bullet, then the chain complex \mathrm{s}F is a resolution of \varepsilon_*F. We thus obtain an isomorphism \mathrm{R}\varepsilon_*\xrightarrow{\sim}\mathrm{s}, and (5.2.4.2) becomes \mathrm{R}a_* = \mathrm{s}\mathrm{R}{a_\bullet}_* \tag{5.2.6.1}
Combined with (5.2.5) and specialised to the case where S=P^t, this equation proves (5.2.3.1). In concrete terms, this implies that, to calculate \mathrm{R}a_*K, we can proceed in two steps:
- We take a resolution K\xrightarrow{\sim}K' such that the components F of K' satisfy \mathrm{R}^i {a_p}_*(F^p)=0 for i>0. The complex {a_\bullet}_*K'\in\mathrm{D}^+(S_\bullet) (the derived category of the category of cosimplicial abelian sheaves on S) can be identified with \mathrm{R}{a_\bullet}_*K.
- \mathrm{R}a_*K is the simple complex \mathrm{s}{a_\bullet}_*K' associated to the double complex {a_\bullet}_*K'.
The spectral sequence (5.2.3.2) generalises to a spectral sequence E_1^{pq} = \mathrm{R}^q{a_p}_*(K|X_p) \Rightarrow\mathrm{R}^{p+q}a_*K \tag{5.2.7.1} induced by (5.1.12.2).