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We prove that any $\tau$-tilting finite incidence algebra of a finite poset is representation-finite, and that any $\mathbf{g}$-tame incidence algebra of a finite simply connected poset is tame. As the converse of these assertions are known to hold, we obtain characterizations of $\tau$-tilting finite incidence algebras and $\mathbf{g}$-tame simply connected incidence algebras. Both results are proved using the theory of concealed algebras. The former will be deduced from the fact that tame concealed algebras are $\tau$-tilting infinite, and to prove the latter, we show that wild concealed algebras are not $\mathbf{g}$-tame. We conjecture that any incidence algebra of a finite poset is wild if and only if it is not $\mathbf{g}$-tame, and prove a result showing that there are relatively few possible counterexamples. In the appendix, we determine the representation type of a $\tau$-tilting reduction of a concealed algebra of hyperbolic type., Comment: 31 pages. v2: Minor changes to structure and grammatical corrections

Let $\mathcal{C}$ be an extriangulated category and let $\mathcal{R}\subseteq \mathcal{C}$ be a rigid subcategory. Generalizing Iyama--Yang silting reduction, we devise a technical condition $\textbf{(gCP)}$ on $\mathcal{R}$ which is sufficient for the Verdier quotient $\mathcal{C}/\mathrm{thick}(\mathcal{R})$ to be equivalent to an ideal quotient. In particular, the Verdier quotient $\mathcal{C}/\mathrm{thick}(\mathcal{R})$ will admit an extriangulation in such a way that the localization functor $L_{\mathcal{R}}\colon \mathcal{C} \rightarrow \mathcal{C}/\mathrm{thick}(\mathcal{R})$ is extriangulated. When $\mathcal{C}$ is 0-Auslander, the condition $\textbf{(gCP)}$ holds for all rigid subcategories $\mathcal{R}$ admitting Bongartz completions. Furthermore, we prove that the Verdier quotient $\mathcal{C}/\mathrm{thick}(\mathcal{R})$ then remains 0-Auslander. As an application, we define the picture category of a connective $0$-Auslander exact dg category $\mathscr{A}$ with Bongartz completions, which generalizes the notion of $\tau$-cluster morphism category. We show that the picture category of $\mathscr{A}$ is a cubical category, in the sense of Igusa. The picture group of $\mathscr{A}$ is defined as the fundamental group of its picture category. When $H_0\mathscr{A}$ is $\mathbf{g}$-finite, the picture group of $\mathscr{A}$ is finitely presented., Comment: 33 pages. v2: Theorem 5.9 improved, minor changes. v3: Change of terminology (gHTCP to gCTCP), updated contact information and references. v4: k comm. ring -> field, minor corrections

We define extension $\infty$-categories for exact $\infty$-categories in terms of bifibrations. Extension $\infty$-categories are invariant when passing to the stable hull, and consequently we show that they form an $\Omega$-spectrum, generalizing a theorem of Retakh. Finally, we show that the homotopy groups of extension $\infty$-categories are naturally isomorphic to the higher extension groups of the extriangulated category given by the homotopy category., Comment: 28 pages v3: Updated contact information

We generalise $\tau$-cluster morphism categories to the setting of non-positive dg algebras with finite dimensional cohomology in all degrees. The compatibility of silting reduction with support $\tau$-tilting reduction will be an essential ingredient when linking our definition to those of Buan--Marsh and Buan--Hanson., Comment: 35 pages. v3: {\S}2, which previously included a flawed definition, has been removed, {\S}2 (previously {\S}3) considerably reshaped. v4: Thm. 5.3 has been removed. Contact information updated. v5: Setting generalised to non-positive dg algebras with finite dimensional cohomology in all degrees, Section 5 removed, substantial additions to some proofs

We discuss two different approaches for splitting the wavefunction of a single-particle-box (SPB) into two equal parts. Adiabatic insertion of a barrier in the center of a SPB in order to make two compartments which each have probability 1/2 to find the particle in it is one of the key steps for a Szilard engine. However, any asymmetry between the volume of the compartments due to an off-center insertion of the barrier results in a particle that is fully localized in the larger compartment, in the adiabatic limit. We show that rather than exactly splitting the eigenfunctions in half by a symmetric barrier, one can use a non-adiabatic insertion of an asymmetric barrier to induce excitations to the first excited state of the full box. As the barrier height goes to infinity the excited state of the full box becomes the ground state of one of the new boxes. Thus, we can achieve close to exact splitting of the probability between the two compartments using the more realistic non-adiabatic, not perfectly centered barrier, rather than the idealized adiabatic and central barrier normally assumed., Comment: 6 pages, 5 figures