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Notes on condensed mathematics

Kiran S. Kedlaya

Section 8 Condensed cohomology

In this section, we explain some links between condensed abelian groups and classical notions of algebraic topology, e.g., the singular cohomology of CW complexes.
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Reference.

This section is based on [8], Lecture 4, except that we take advantage of the material from [8], Lecture 5 to reverse the flow of some information. See also [6], Lecture 4 (with the same proviso).
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Subsection 8.1 Condensed cohomology on compact spaces

Definition 8.1.1.

For \(X \in \CSet\) and \(M\) a topological abelian group, we define the condensed cohomology groups of \(X\) with values in \(M\) to be
\begin{equation*} H^i(X, M) := \Ext^i_{\CAb}(\ZZ[X], \underline{M}). \end{equation*}
When \(X \in \Top\) we may also write \(H^i_{\cond}(X, M)\) to mean \(H^i(\underline{X}, M)\text{.}\)
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Proof.

This is included in Lemma 7.3.7: we have \(\ZZ[\underline{X}] \cong \prod_i \underline{\ZZ} \cong P_\solid\) and the latter is internally projective in \(\CAb_\solid\) by Proposition 5.4.8.
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Proof.

Proof.

We loosely follow [6], Theorem 3.2. Using Remark 4.2.2, we may construct a simplicial hypercovering \(X_\bullet \to X\) by objects of \(\Prof\text{.}\) By Corollary 8.1.3, the terms of this hypercovering are acyclic and so we may compute \(R\Gamma_{\cond}(X, M)\) by the Čech complex
\begin{equation*} 0 \to \Gamma(X_0, M) \to \Gamma(X_1, M) \to \cdots\text{.} \end{equation*}
Since continuous functions from any \(X_n\) to the discrete group \(M\) are locally constant, the augmented version of this complex is itself a colimit of complexes associated to simplicial hypercoverings of finite sets, which are acyclic by Corollary 8.1.3 again.
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Remark 8.1.5.

For \(X \in \CHaus\text{,}\) we can also interpret \(H^i_{\sheaf}(X,S)\) as Čech cohomology ([11], Théorème 5.10.1). In some cases we can also interpret it as singular cohomology, see for example Subsection 8.2.
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We next consider coefficients in the topological (not discrete) group \(\RR\text{.}\)
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Proof.

Suppose first that \(X\) and the \(X_n\) are all finite. Then the hypercover splits; this means that the augmented Čech complex admits a chain homotopy which is “combinatorial” in the sense that every map is given by pullback along a map of underlying sets. (This amounts to a very basic case of the Dold–Kan correspondence.) In this case the claim evidently holds even for \(\epsilon = 0\text{.}\)
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By the same token, if \(X\) and the \(X_n\) are all profinite, then we can write the augmented Cech complex as a completed colimit of sequences of the previous form. By some elementary analysis (left to the reader), this implies the claim but now with \(\epsilon \gt 0\text{.}\)
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In the general case, given \(f \in \Cts(X_n, \RR)\) nonzero with \(df = 0\text{,}\) for any \(x \in X\) we can apply the previous case to the restriction of \(f\) to \(X_n \times_X \{x\}\text{.}\) We may then deduce the general case by spreading the results out using Tietze’s extension theorem, invoking compactness to see that the spread out functions cover \(X\text{,}\) and constructing a suitable partition of unity argument to patch together the local choices. See [6], Theorem 3.3 for the detailed argument.
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Subsection 8.2 Cohomology of CW complexes

Definition 8.2.1.

A CW complex is a topological space obtained as a colimit \(X_0 \subseteq X_1 \subseteq \cdots\) in which each \(X_k\) is obtained from \(X_{k-1}\) by glueing in some copies of the \(k\)-dimensional ball, with the boundaries contained in \(X_{k-1}\text{.}\)
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Proof.

For a CW complex, we have \(H^i_{\sing}(X, M) \cong H^i_{\sheaf}(X, M)\text{.}\) We can thus deduce this from Theorem 8.1.4.
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Remark 8.2.3.

Theorem 8.2.2 can be thought of as a reinterpretation of the Dold–Thom theorem in algebraic topology.
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Subsection 8.3 Locally compact abelian groups

See [15] for a much more detailed development of this topic.
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Proof.

See [14], specifically Theorem 24.30 for the first point, Theorem 24.8 for the second, and Theorem 23.17 for the third.
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Proof.

Since \(\underline{M}\) is solid by Theorem 7.4.1, we have \(R\Hom_{\CAb}\left(\prod_I \underline{\RR}, \underline{M}\right) = 0\) by Corollary 6.3.3. Meanwhile, via Theorem 7.4.1 we have
\begin{equation*} R\Hom_{\CAb}\left(\prod_I \underline{\ZZ}, \underline{M}\right) = R\Hom_{\ZZ_\solid}\left(\prod_I \ZZ_\solid, M_\solid\right) = \bigoplus_I M[0] \end{equation*}
because \(\prod_I \underline{\ZZ}\) is projective in \(\Mod_{\ZZ_\solid}\text{.}\) This yields the claim.
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Proof.

In light of point 1 of Theorem 8.3.1, it suffices to check that \(R\Hom_{\LCAb}(A,B) \cong R\Hom_{\CAb}(\underline{A}, \underline{B})\) when each of \(A,B\) is either compact, discrete, or isomorphic to \(\RR\text{.}\) We break down the argument as follows.
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Suppose first that \(A\) is discrete. Using resolutions and filtered colimits we may reduce to the case \(A = \ZZ\text{.}\) In this case, \(A\) and \(\underline{A}\) are projective in their respective categories and the result is clear.
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Suppose next that \(A\) is compact. By taking a resolution of the dual group, we may reduce to the case where \(A = \prod_I \TT\) with \(I\) at most countable. At this point, when \(B\) is discrete we may deduce the desired isomorphism by comparing Lemma 8.3.2 with [15], Example 4.11; when \(B = \RR\) we may instead compare Theorem 8.1.6 with [15], Proposition 4.15 for the same effect. Combining these and using the exact sequence (8.1), we deduce the claim also when \(B = \TT\text{,}\) and consequently when \(B = \prod_I \TT\text{;}\) again we may formally promote this to the general case of \(B\) compact.
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Finally, to resolve the case \(A = \RR\) we may use (8.1) to reduce to the previously treated cases \(A = \ZZ\) (discrete) and \(A = \TT\) (compact).
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