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Clausen predicted that Chevalley's id\`{e}le class group of a number field $F$ appears as the first $K$-group of the category of locally compact $F$-vector spaces. This has turned out to be true, and even generalizes to the higher $K$-groups in a suitable sense. We replace $F$ by a semisimple $\mathbb{Q}$-algebra, and obtain Fr\"{o}hlich's non-commutative id\`{e}le class group in an analogous fashion, modulo the reduced norm one elements. Even in the number field case our proof is simpler than the existing one, and based on the localization theorem for percolating subcategories. Finally, using class field theory as input, we interpret Hilbert's reciprocity law (as well as a noncommutative variant) in terms of our results., Comment: 36 pages. Comments welcome

It is well known that a resolving subcategory $\mathcal{A}$ of an abelian subcategory $\mathcal{E}$ induces several derived equivalences: a triangle equivalence $\mathbf{D}^-(\mathcal{A})\to \mathbf{D}^-(\mathcal{E})$ exists in general and furthermore restricts to a triangle equivalence $\mathbf{D}^{\mathsf{b}}(\mathcal{A})\to \mathbf{D}^{\mathsf{b}}(\mathcal{E})$ if $\operatorname{res.dim}_{\mathcal{A}}(E), Comment: 14 pages, comments welcome

Fragment and glider representations (introduced by F. Caenepeel, S. Nawal, and F. Van Oystaeyen) form a generalization of filtered modules over a filtered ring. Given a $\Gamma$-filtered ring $FR$ and a subset $\Lambda \subseteq \Gamma$, we provide a category $\operatorname{Glid}_\Lambda FR$ of glider representations, and show that it is a complete and cocomplete deflation quasi-abelian category. We discuss its derived category, and its subcategories of natural gliders and Noetherian gliders. If $R$ is a bialgebra over a field $k$ and $FR$ is a filtration by bialgebras, we show that $\operatorname{Glid}_\Lambda FR$ is a monoidal category which is derived equivalent to the category of representations of a semi-Hopf category (in the sense of E. Batista, S. Caenepeel, and J. Vercruysse). We show that the monoidal category of glider representations associated to the one-step filtration $k \cdot 1 \subseteq R$ of a bialgebra $R$ is sufficient to recover the bialgebra $R$ by recovering the usual fiber functor from $\operatorname{Glid}_\Lambda FR.$ When applied to a group algebra $kG$, this shows that the monoidal category $\operatorname{Glid}_\Lambda F(kG)$ alone is sufficient to distinguish even isocategorical groups., Comment: 33 pages, comments welcome

We consider the quotient of an exact or one-sided exact category $\mathcal{E}$ by a so-called percolating subcategory $\mathcal{A}$. For exact categories, such a quotient is constructed in two steps. Firstly, one localizes $\mathcal{E}$ at a suitable class $S_\mathcal{A} \subseteq \operatorname{Mor}(\mathcal{E})$ of morphisms. The localization $\mathcal{E}[S_\mathcal{A}^{-1}]$ need not be an exact category, but will be a one-sided exact category. Secondly, one constructs the exact hull $\mathcal{E}{/\mkern-6mu/} \mathcal{A}$ of $\mathcal{E}[S_\mathcal{A}^{-1}]$ and shows that this satisfies the 2-universal property of a quotient amongst exact categories. In this paper, we show that this quotient $\mathcal{E} \to \mathcal{E} {/\mkern-6mu/} \mathcal{A}$ induces a Verdier localization $\mathbf{D}^b(\mathcal{E}) \to \mathbf{D}^b(\mathcal{E} {/\mkern-6mu/} \mathcal{A})$ of bounded derived categories. Specifically, (i) we study the derived category of a one-sided exact category, (ii) we show that the localization $\mathcal{E} \to \mathcal{E}[S_\mathcal{A}^{-1}]$ induces a Verdier quotient $\mathbf{D}^b(\mathcal{E}) \to \mathbf{D}^b(\mathcal{E}[S^{-1}_\mathcal{A}])$, and (iii) we show that the natural embedding of a one-sided exact category $\mathcal{F}$ into its exact hull $\overline{\mathcal{F}}$ lifts to a derived equivalence $\mathbf{D}^b(\mathcal{F}) \to \mathbf{D}^b(\overline{\mathcal{F}})$. We furthermore show that the Verdier localization is compatible with several enhancements of the bounded derived category, so that the above Verdier localization can be used in the study of localizing invariants, such as non-connective $K$-theory., 37 pages. Improved exposition and the formulation of the main theorem. Added section 8. Comments welcome