Continuous (Efimov) K-theory

The right adjoint is the inclusion $k^{-1}(*) \hookrightarrow \mathsf{Ind}(\mathcal{C}^{\aleph_1})$, which is fully faithful, so we have a Bousfield localization.

For compact generation, it suffices to show the inclusion preserves filtered colimits. Let $(X_i)_{i \in I}$ be a filtered diagram in $k^{-1}(*)$. Since $k$ is a left adjoint, $k(\operatorname*{colim}_i X_i) \simeq \operatorname*{colim}_i k(X_i) \simeq \operatorname*{colim}_i * \simeq *$, so $\operatorname*{colim}_i X_i \in k^{-1}(*)$. The inclusion hence preserves filtered colimits, so the localization preserves compact objects, and their image generates the quotient. Since $\mathsf{Ind}(\mathcal{C}^{\aleph_1})$ is compactly generated, so is the quotient.

$\mathsf{Ind}(p)$ is the Bousfield localization of Proposition 1 , and it preserves compact objects, so by Gabriel–Ulmer we recover a right-exact $p$ on the small-category side.

For the pushout claim: $\mathsf{Ind}(p)^R$ is the inclusion $k^{-1}(*) \hookrightarrow \mathsf{Ind}(\mathcal{C}^{\aleph_1})$, which sits inside the pullback

Given a pushout diagram $A, B, C \in k^{-1}(*)$, the pushout $A \cup_C B$ taken in $\mathsf{Ind}(\mathcal{C}^{\aleph_1})$ satisfies $k(A \cup_C B) \simeq k(A) \cup_{k(C)} k(B) \simeq * \cup_* * \simeq *$, since $k$ preserves colimits. So $A \cup_C B \in k^{-1}(*)$, and the inclusion preserves pushouts.

Plugging the compactly generated $\mathsf{Ind}(\mathcal{C})$ into the previous cofibre sequence,

Compactly-generated makes $\hat{y} = \mathsf{Ind}(y)$ (idempotent completeness gives $\mathcal{C} \simeq \mathsf{Ind}(\mathcal{C})^{\aleph_0} \subset \mathsf{Ind}(\mathcal{C})^{\aleph_1}$). So the full sequence is obtained by applying $\mathsf{Ind}$ to $\mathcal{C} \xrightarrow{y} \mathsf{Ind}(\mathcal{C})^{\aleph_1} \xrightarrow{p} \mathsf{Calk}^{\mathrm{cont}}(\mathsf{Ind}(\mathcal{C}))$. Gabriel–Ulmer equivalence $\mathsf{Cat}^{\mathrm{rex,idem}} \simeq \mathsf{Pr}^L_{\aleph_0}$ lets us recover

the second equivalence because compact objects of a compactly generated category are automatically idempotent-complete.

See [sheaves-on-manifolds, Prop. 3.4.3] . The key non-trivial implication is $(4) \Rightarrow (5)$: for a type-$(4)$ sequence $\mathcal{C} \to \mathcal{D} \to \mathcal{E}$, natural equivalences

reduce the question to: tensoring with $\mathsf{Sp}$ turns the sequence into a Verdier cofibre sequence, $\mathsf{Calk}^{\mathrm{cont}}$ preserves these, and $\mathrm{K}$ sends Verdier cofibre sequences in $\mathsf{Cat}^{\mathrm{rex,idem}}$ to cofibre sequences.

Given Proposition 8 , we can extract each property of Theorem 7 :

(1) When $\mathcal{C}$ is compactly generated, $\mathcal{C} \simeq \mathsf{Ind}(\mathcal{C}^{\omega})$ and Proposition 4 gives $\mathsf{Calk}^{\mathrm{cont}}(\mathcal{C}) \simeq \mathsf{Calk}(\mathcal{C}^{\omega})$, so

(2) Filtered colimits and finite products: $\mathsf{Calk}^{\mathrm{cont}}$ preserves them ($\mathsf{Pr}^L_{\mathrm{ca}}$ is semi-additive and $\mathsf{Calk}^{\mathrm{cont}}$ is computed via $(-)^{\aleph_0}$ which preserves both operations on compactly generated quotients), and $\mathrm{K}$ does too.

(3) Product-preservation gives a map $\mathrm{K}^{\mathrm{cont}}(F)$ with $\mathrm{K}^{\mathrm{cont}}(F) + \operatorname{id} = \operatorname{id}$ in $\pi_0 \mathrm{End}(\mathrm{K}^{\mathrm{cont}}(\mathcal{C}))$; this group is abelian, so $\operatorname{id} = 0$.

(4) By Proposition 8 , $(4) \Rightarrow (5)$.

(5) By part of Proposition 8 , since $\mathcal{C} \to \mathsf{An}_* \otimes \mathcal{C}$ and $\mathcal{C} \to \mathsf{Sp} \otimes \mathcal{C}$ are inverted under $\mathrm{K}^{\mathrm{cont}}$.

Uniqueness. Given $\mathrm{K}$ and properties (1) and (2), $\mathrm{K}^{\mathrm{cont}}$ is determined. Indeed, every compactly assembled $\mathcal{C}$ is a filtered colimit of compactly generated categories (via $\mathsf{Ind}(-)^{\aleph_1}$-style approximations and cofinality), and (1) pins $\mathrm{K}^{\mathrm{cont}}$ on the compactly generated ones while (2) propagates the definition along the filtered colimit.

For $F \in \mathrm{Loc}(\mathcal{D})$, the natural cofibre sequence $\mathcal{C} \hookrightarrow \mathsf{Ind}(\mathcal{C}^{\aleph_1}) \to \mathsf{Ind}(\mathsf{Calk}^{\mathrm{cont}}(\mathcal{C}))$ and the Eilenberg swindle force

This gives $F^{\mathrm{cont}}(\mathsf{Ind}(-)) = F$ by Proposition 4 , and $F^{\mathrm{cont}}$ is determined by this. That $\mathsf{Calk}^{\mathrm{cont}}$ preserves Verdier cofibre sequences (part of Proposition 8 ) together with $F$’s localizing property gives localizingness of $F^{\mathrm{cont}}$. The finitary case is analogous, using that $\mathsf{Calk}^{\mathrm{cont}}$ preserves filtered colimits up to $\mathrm{k}$-equivalence.