Being confused about differential cohomology III

Welcome to part III of being confused. We are going to construct differential cohomologies using the recollement machinary from the previous post. To start we will follow section 7 of this notes. Let $E$ be a (topological) cohomology theory:

Choose a pure sheaf $\hat{P}$, i.e., a sheaf $\hat{P}$ with $\hat{P}(*) = 0$.
Compute $$\Gamma_{#}(\hat{P}) = \hat{P}(\Delta_{\mathrm{alg}}^{\buleet})$$.
Define a map $f \colon E \to \Gamma_{\#}\hat{P}$. Then we simply construct $\hat{E}$ by the pullback procedure from last time.

Example: trivial case. take $\hat{P} = 0$ thus $\Gamma_{\#}(\hat{P}) = 0$. Then the differential refinement is just the constant sheaf $\Gamma^*(E)$.

Example: Ordinar differential cohomology Consider the de Rham complex $\Omega^*$, say viewed as a sheaf with valued in the derived $\infty$-category $D(\mbb{R})$. Note that $\Omega^k$ is in degree $-k$. By the de Rham theorem, the spectra $\Omega^*$ is equivalent to the constant discrete sheaf $\mbb{R}_{\disc}$, concentrated in degree $0$. Thus $\Omega^* \simeq \Gamma^*H\mbb{R}$. Therefore it is not pure and is a $\mbb{R}$-invariant sheaf. In particular, the purification $\Cyc(\Omega^*)$ is a $0$. For $k \in \mbb{N}$, let $\Omega^{\geq k}$ denote the stunted piece where we kill everything in degrees less than $k$. Note that $\Omega^{\geq k}(*) = 0$ thus $\Gamma_* \Omega^{\geq k}$ for $k \geq 1$, thus it is in fact a pure sheaf!

Therefore can compute $\Gamma_{\#}(\Omega^{\geq k})$: LemmaFor any $k \geq 0$, there is an equivalence $\Gamma_{\#}\Omega^{\geq k} \simeq H\mbb{R}$. ProofThis holds for $k = 0$. For the higher case, it suffices to show that $\Gamma_{\#}\Omega^k = 0$ for $k \geq 1$, which we will not prove here.

Now we can define $H\mbb{Z}(k)$ as the differential refinment of $H\mbb{Z}$ corresponding to the pair $(\Omega^{\geq k}, H\mbb{Z} \to H\mbb{R} = \Gamma_{\#}\Omega^{\geq k})$. It is straightforward to check this agrees with the definition that we gave in the first post.

Example: differential $K$-theory: Now we are finally ready to define some other differential cohomology theories haha. Let $V$ be a graded $\mbb{R}$ vector space, viewed as an element in $D(\mbb{R})$, hence spectra. Let $\Omega^{\geq k}(V)$ the sheave of de Rham forms with $V$ coefficients. Note that the total degree is the sum of the differential form degree plus the $V$ graiding. For $k \geq 0$, we have $$\Gamma_{\#}\Omega^{\geq k}(V) \simeq V$$; furthermore, for $k \geq 1$, the sheaf $\Omega^{\geq k}(V)$ is a pure sheaf.

Now we have to define a connecting morphism, in this case is the Chern character $\ch$: $\begin{equation} \ch \colon KU \to \mbb{R}[u^{\pm 1}], \end{equation}$ with $u$ in degree $2$. This allows us to define the differential refinement $\hat{KU}(k)$ of $KU$ defined by $(\Omega^{\geq k}[u^{\pm 1}], KU \to \mbb{R}[u^{\pm 1}] \simeq \Gamma_{\#}\Omega^{\geq k}[u^{\pm 1}])$.

Similarly, given the Pontryagin character $\ch_o \colon KO \to \mbb{R}[\beta^{\pm 1}]$ with $\beta$ in degree $4$. This defines a differetial refinments $\widehat{KO}(k)$ of $KO$.

More generally, given any spectrum $E$, then $E_{\mbb{R}} \coloneqq E \otimes H\mbb{R} \simeq H(E^* \otimes \mbb{R})$, where $E^* = \pi_{-*} E$ is the homotopy group, viewed as a graded abelian groups. Then $(\Omega^{\geq k})(E^*_{\mbb{R}})$ together with the map $E_{\mbb{R}} \to E_{\mbb{R}} \simeq \Gamma_{\#}(\Omega^{\geq k})(E^*_{\mbb{R}})$ defines a differential refinement $\hat{E}(k)$ of $E$.

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