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1. On the construction of limits and colimits in ∞-categories

   Riehl, Emily; Verity, Dominic (July 2020, Theory and applications of categories)

   null (Ed.)

   In previous work, we introduce an axiomatic framework within which to prove theorems about many varieties of infinite-dimensional categories simultaneously. In this paper, we establish criteria implying that an ∞-category --- for instance, a quasi-category, a complete Segal space, or a Segal category --- is complete and cocomplete, admitting limits and colimits indexed by any small simplicial set. Our strategy is to build (co)limits of diagrams indexed by a simplicial set inductively from (co)limits of restricted diagrams indexed by the pieces of its skeletal filtration. We show directly that the modules that express the universal properties of (co)limits of diagrams of these shapes are reconstructible as limits of the modules that express the universal properties of (co)limits of the restricted diagrams. We also prove that the Yoneda embedding preserves and reflects limits in a suitable sense, and deduce our main theorems as a consequence. 

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2. A Geometric Model for Syzygies Over 2-Calabi–Yau Tilted Algebras II

   https://doi.org/10.1093/imrn/rnad078  

   Schiffler, Ralf; Serhiyenko, Khrystyna (April 2023, International Mathematics Research Notices)

   Abstract In this article, we continue the study of a certain family of 2-Calabi–Yau tilted algebras, called dimer tree algebras. The terminology comes from the fact that these algebras can also be realized as quotients of dimer algebras on a disk. They are defined by a quiver with potential whose dual graph is a tree, and they are generally of wild representation type. Given such an algebra $$B$$, we construct a polygon $$\mathcal {S}$$ with a checkerboard pattern in its interior, which defines a category $$\text {Diag}(\mathcal {S})$$. The indecomposable objects of $$\text {Diag}(\mathcal {S})$$ are the 2-diagonals in $$\mathcal {S}$$, and its morphisms are certain pivoting moves between the 2-diagonals. We prove that the category $$\text {Diag}(\mathcal {S})$$ is equivalent to the stable syzygy category of the algebra $$B$$. This result was conjectured by the authors in an earlier paper, where it was proved in the special case where every chordless cycle is of length three. As a consequence, we conclude that the number of indecomposable syzygies is finite, and moreover the syzygy category is equivalent to the 2-cluster category of type $$\mathbb {A}$$. In addition, we obtain an explicit description of the projective resolutions, which are periodic. Finally, the number of vertices of the polygon $$\mathcal {S}$$ is a derived invariant and a singular invariant for dimer tree algebras, which can be easily computed form the quiver. 

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3. The Chow t-structure on the ∞-category of motivic spectra

   https://doi.org/10.4007/annals.2022.195.25  

   Kong, Hana J.; Wang, Guozhen; Xu, Zhouli (February 2022, Annals of mathematics)

   We define the Chow t-structure on the ∞-category of motivic spectra SH(k) over an arbitrary base field k. We identify the heart of this t-structure SH(k)c♡ when the exponential characteristic of k is inverted. Restricting to the cellular subcategory, we identify the Chow heart SH(k)cell,c♡ as the category of even graded MU2∗MU-comodules. Furthermore, we show that the ∞-category of modules over the Chow truncated sphere spectrum 1c=0 is algebraic. Our results generalize the ones in Gheorghe–Wang–Xu in three aspects: to integral results; to all base fields other than just C; to the entire ∞-category of motivic spectra SH(k), rather than a subcategory containing only certain cellular objects. We also discuss a strategy for computing motivic stable homotopy groups of (p-completed) spheres over an arbitrary base field k using the Postnikov–Whitehead tower associated to the Chow t-structure and the motivic Adams spectral sequences over k. 

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4. Accelerating Substructure Similarity Search for Formula Retrieval

   Zhong, Wei; Zanibbi, Richard (January 2020, Proc. European Conference on Information Retrieval)

   Formula retrieval systems using substructure matching are effective, but suffer from slow retrieval times caused by the complexity of structure matching. We present a specialized inverted index and rank-safe dynamic pruning algorithm for faster substructure retrieval. Formulas are indexed from their Operator Tree (OPT) representations. Our model is evaluated using the NTCIR-12 Wikipedia Formula Browsing Task and a new formula corpus produced from Math StackExchange posts. Our approach preserves the effectiveness of structure matching while allowing queries to be executed in real-time. 

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5. Consistent dimer models on surfaces with boundary

   https://doi.org/10.1017/fms.2025.18  

   Berggren, Jonah; Serhiyenko, Khrystyna (January 2025, Forum of Mathematics, Sigma)

   A dimer model is a quiver with faces embedded in a surface. We define and investigate notions of consistency for dimer models on general surfaces with boundary which restrict to well-studied consistency conditions in the disk and torus case. We define weak consistency in terms of the associated dimer algebra and show that it is equivalent to the absence of bad configurations on the strand diagram. In the disk and torus case, weakly consistent models are nondegenerate, meaning that every arrow is contained in a perfect matching; this is not true for general surfaces. Strong consistency is defined to require weak consistency as well as nondegeneracy. We prove that the completed as well as the noncompleted dimer algebra of a strongly consistent dimer model are bimodule internally 3-Calabi-Yau with respect to their boundary idempotents. As a consequence, the Gorenstein-projective module category of the completed boundary algebra of suitable dimer models categorifies the cluster algebra given by their underlying quiver. We provide additional consequences of weak and strong consistency, including that one may reduce a strongly consistent dimer model by removing digons and that consistency behaves well under taking dimer submodels. 

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