Journal articles: '2-category theory'

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Bourke, John. "Accessible aspects of 2-category theory." Journal of Pure and Applied Algebra 225, no. 3 (March 2021): 106519. http://dx.doi.org/10.1016/j.jpaa.2020.106519.

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Bourke, John. "Skew structures in 2-category theory and homotopy theory." Journal of Homotopy and Related Structures 12, no. 1 (December 8, 2015): 31–81. http://dx.doi.org/10.1007/s40062-015-0121-z.

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Riehl, Emily, and Dominic Verity. "The 2-category theory of quasi-categories." Advances in Mathematics 280 (August 2015): 549–642. http://dx.doi.org/10.1016/j.aim.2015.04.021.

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del Hoyo, Matias L. "On the loop space of a 2-category." Journal of Pure and Applied Algebra 216, no. 1 (January 2012): 28–40. http://dx.doi.org/10.1016/j.jpaa.2011.05.001.

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Elgueta, Josep. "2-Cosemisimplicial objects in a 2-category, permutohedra and deformations of pseudofunctors." Journal of Pure and Applied Algebra 191, no. 3 (August 2004): 223–64. http://dx.doi.org/10.1016/j.jpaa.2003.12.007.

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Piacenza, Robert J. "Homotopy Theory of Diagrams and CW-Complexes Over a Category." Canadian Journal of Mathematics 43, no. 4 (August 1, 1991): 814–24. http://dx.doi.org/10.4153/cjm-1991-046-3.

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The purpose of this paper is to introduce the notion of a CW complex over a topological category. The main theorem of this paper gives an equivalence between the homotopy theory of diagrams of spaces based on a topological category and the homotopy theory of CW complexes over the same base category.A brief description of the paper goes as follows: in Section 1 we introduce the homotopy category of diagrams of spaces based on a fixed topological category. In Section 2 homotopy groups for diagrams are defined. These are used to define the concept of weak equivalence and J-n equivalence that generalize the classical definition. In Section 3 we adapt the classical theory of CW complexes to develop a cellular theory for diagrams. In Section 4 we use sheaf theory to define a reasonable cohomology theory of diagrams and compare it to previously defined theories. In Section 5 we define a closed model category structure for the homotopy theory of diagrams. We show this Quillen type homotopy theory is equivalent to the homotopy theory of J-CW complexes. In Section 6 we apply our constructions and results to prove a useful result in equivariant homotopy theory originally proved by Elmendorf by a different method.

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SOYLU YILMAZ, Elis. "4-Dimensional 2-Crossed Modules." Journal of New Theory, no. 40 (September 30, 2022): 46–53. http://dx.doi.org/10.53570/jnt.1148482.

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In this work, we defined a new category called 4-Dimensional 2-crossed modules. We identified the subobjects and ideals in this category. The notion of the subobject is a generalization of ideas like subsets from set theory, subspaces from topology, and subgroups from group theory. We then exemplified subobjects and ideals in the category of 4-Dimensional 2-crossed modules. A quotient object is the dual concept of a subobject. Concepts like quotient sets, spaces, groups, graphs, etc. are generalized with the notion of a quotient object. Using the ideal, we obtain the quotient of two subobjects and prove that the intersection of finite ideals is also an ideal in this category.

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Bachmann, Tom. "Motivic and real étale stable homotopy theory." Compositio Mathematica 154, no. 5 (March 20, 2018): 883–917. http://dx.doi.org/10.1112/s0010437x17007710.

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Let$S$be a Noetherian scheme of finite dimension and denote by$\unicode[STIX]{x1D70C}\in [\unicode[STIX]{x1D7D9},\mathbb{G}_{m}]_{\mathbf{SH}(S)}$the (additive inverse of the) morphism corresponding to$-1\in {\mathcal{O}}^{\times }(S)$. Here$\mathbf{SH}(S)$denotes the motivic stable homotopy category. We show that the category obtained by inverting$\unicode[STIX]{x1D70C}$in$\mathbf{SH}(S)$is canonically equivalent to the (simplicial) local stable homotopy category of the site$S_{\text{r}\acute{\text{e}}\text{t}}$, by which we mean thesmallreal étale site of$S$, comprised of étale schemes over$S$with the real étale topology. One immediate application is that$\mathbf{SH}(\mathbb{R})[\unicode[STIX]{x1D70C}^{-1}]$is equivalent to the classical stable homotopy category. In particular this computes all the stable homotopy sheaves of the$\unicode[STIX]{x1D70C}$-local sphere (over$\mathbb{R}$). As further applications we show that$D_{\mathbb{A}^{1}}(k,\mathbb{Z}[1/2])^{-}\simeq \mathbf{DM}_{W}(k)[1/2]$(improving a result of Ananyevskiy–Levine–Panin), reprove Röndigs’ result that$\text{}\underline{\unicode[STIX]{x1D70B}}_{i}(\unicode[STIX]{x1D7D9}[1/\unicode[STIX]{x1D702},1/2])=0$for$i=1,2$and establish some new rigidity results.

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Ozornova, Viktoriya, and Martina Rovelli. "The Duskin nerve of 2-categories in Joyal's cell category Θ2." Journal of Pure and Applied Algebra 225, no. 1 (January 2021): 106462. http://dx.doi.org/10.1016/j.jpaa.2020.106462.

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Treumann, David. "Exit paths and constructible stacks." Compositio Mathematica 145, no. 6 (September 21, 2009): 1504–32. http://dx.doi.org/10.1112/s0010437x09004229.

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AbstractFor a Whitney stratification S of a space X (or, more generally, a topological stratification in the sense of Goresky and MacPherson) we introduce the notion of an S-constructible stack of categories on X. The motivating example is the stack of S-constructible perverse sheaves. We introduce a 2-category EP≤2(X,S), called the exit-path 2-category, which is a natural stratified version of the fundamental 2-groupoid. Our main result is that the 2-category of S-constructible stacks on X is equivalent to the 2-category of 2-functors 2Funct(EP≤2(X,S),Cat) from the exit-path 2-category to the 2-category of small categories.

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Adachi, Takahide, Osamu Iyama, and Idun Reiten. "-tilting theory." Compositio Mathematica 150, no. 3 (December 3, 2013): 415–52. http://dx.doi.org/10.1112/s0010437x13007422.

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AbstractThe aim of this paper is to introduce $\tau $-tilting theory, which ‘completes’ (classical) tilting theory from the viewpoint of mutation. It is well known in tilting theory that an almost complete tilting module for any finite-dimensional algebra over a field $k$ is a direct summand of exactly one or two tilting modules. An important property in cluster-tilting theory is that an almost complete cluster-tilting object in a 2-CY triangulated category is a direct summand of exactly two cluster-tilting objects. Reformulated for path algebras $kQ$, this says that an almost complete support tilting module has exactly two complements. We generalize (support) tilting modules to what we call (support) $\tau $-tilting modules, and show that an almost complete support $\tau $-tilting module has exactly two complements for any finite-dimensional algebra. For a finite-dimensional $k$-algebra $\Lambda $, we establish bijections between functorially finite torsion classes in $ \mathsf{mod} \hspace{0.167em} \Lambda $, support $\tau $-tilting modules and two-term silting complexes in ${ \mathsf{K} }^{\mathrm{b} } ( \mathsf{proj} \hspace{0.167em} \Lambda )$. Moreover, these objects correspond bijectively to cluster-tilting objects in $ \mathcal{C} $ if $\Lambda $ is a 2-CY tilted algebra associated with a 2-CY triangulated category $ \mathcal{C} $. As an application, we show that the property of having two complements holds also for two-term silting complexes in ${ \mathsf{K} }^{\mathrm{b} } ( \mathsf{proj} \hspace{0.167em} \Lambda )$.

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Leslie, Spencer, and Gus Lonergan. "Parity sheaves and Smith theory." Journal für die reine und angewandte Mathematik (Crelles Journal) 2021, no. 777 (May 8, 2021): 49–87. http://dx.doi.org/10.1515/crelle-2021-0018.

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Abstract Let p be a prime number and let X be a complex algebraic variety with an action of ℤ / p ⁢ ℤ {\mathbb{Z}/p\mathbb{Z}} . We develop the theory of parity complexes in a certain 2-periodic localization of the equivariant constructible derived category D ℤ / p ⁢ ℤ b ⁢ ( X , ℤ p ) {D^{b}_{\mathbb{Z}/p\mathbb{Z}}(X,\mathbb{Z}_{p})} . Under certain assumptions, we use this to define a functor from the category of parity sheaves on X to the category of parity sheaves on the fixed-point locus X ℤ / p ⁢ ℤ {X^{\mathbb{Z}/p\mathbb{Z}}} . This may be thought of as a categorification of Smith theory. When X is the affine Grassmannian associated to some complex reductive group, our functor gives a geometric construction of the Frobenius-contraction functor recently defined by M. Gros and M. Kaneda via the geometric Satake equivalence.

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Ayala, R., and A. Quintero. "On the Ganea strong category in proper homotopy." Proceedings of the Edinburgh Mathematical Society 41, no. 2 (June 1998): 247–63. http://dx.doi.org/10.1017/s0013091500019623.

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This paper contains some basic relations between Ganea strong category and Lusternik Schnirelmann category in proper homotopy theory. We focus our interest on the case of category 2 in order to show that ℚn is the unique open n-manifold with proper Lusternik-Schnirelmann category 2 (n ≠ 3).

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Iwanari, Isamu. "The category of toric stacks." Compositio Mathematica 145, no. 03 (May 2009): 718–46. http://dx.doi.org/10.1112/s0010437x09003911.

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AbstractIn this paper, we show that there is an equivalence between the 2-category of smooth Deligne–Mumford stacks with torus embeddings and actions and the 1-category of stacky fans. To this end, we prove two main results. The first is related to a combinatorial aspect of the 2-category of toric algebraic stacks defined by I. Iwanari [Logarithmic geometry, minimal free resolutions and toric algebraic stacks, Preprint (2007)]; we establish an equivalence between the 2-category of toric algebraic stacks and the 1-category of stacky fans. The second result provides a geometric characterization of toric algebraic stacks. Logarithmic geometry in the sense of Fontaine–Illusie plays a central role in obtaining our results.

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Hoofman, Raymond. "The theory of semi-functors." Mathematical Structures in Computer Science 3, no. 1 (March 1993): 93–128. http://dx.doi.org/10.1017/s096012950000013x.

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The notion ofsemi-functorwas introduced in Hayashi (1985) in order to make possible a category-theoretical characterization of models of the non-extensional typed lambda calculus. Motivated by the further use of semi-functors in Martini (1987), Jacobs (1991) and Hoofman (1992a), (1992b) and (1992c), we consider the general theory of semi-functors in this paper. It turns out that the notion ofsemi natural transformationplays an important part in this theory, and that various categorical notions involving semi-functors can be viewed as 2-categorical notions in the 2-category of categories, semi-functors and semi natural transformations. In particular, we find that the notion ofnormal semi-adjunctionas defined in Hayashi (1985) is the canonical generalization of the notion of adjunction to the world of semi-functors. Further topics covered in this paper are the relation between semi-functors and splittings, the Karoubi envelope construction, semi-comonads, and a semi-adjoint functor theorem.

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Baas, Nils A., Marcel Bökstedt, and Tore August Kro. "Two-Categorical Bundles and their Classifying Spaces." Journal of K-Theory 10, no. 2 (February 23, 2012): 299–369. http://dx.doi.org/10.1017/is012001012jkt181.

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AbstractFor a 2-category 2C we associate a notion of a principal 2C-bundle. For the 2-category of 2-vector spaces, in the sense of M.M. Kapranov and V.A. Voevodsky, this gives the 2-vector bundles of N.A. Baas, B.I. Dundas and J. Rognes. Our main result says that the geometric nerve of a good 2-category is a classifying space for the associated principal 2-bundles. In the process of proving this we develop powerful machinery which may be useful in further studies of 2-categorical topology. As a corollary we get a new proof of the classification of principal bundles. Another 2-category of 2-vector spaces has been proposed by J.C. Baez and A.S. Crans. A calculation using our main theorem shows that in this case the theory of principal 2-bundles splits, up to concordance, as two copies of ordinary vector bundle theory. When 2C is a cobordism type 2-category we get a new notion of cobordism-bundles which turns out to be classified by the Madsen–Weiss spaces.

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SAWIN, STEPHEN F. "THREE-DIMENSIONAL 2-FRAMED TQFTS AND SURGERY." Journal of Knot Theory and Its Ramifications 13, no. 07 (November 2004): 947–63. http://dx.doi.org/10.1142/s0218216504003536.

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The notion of 2-framed three-manifolds is defined. The category of 2-framed cobordisms is described, and used to define a 2-framed three-dimensional TQFT. Using skeletonization and special features of this category, a small set of data and relations is given that suffice to construct a 2-framed three-dimensional TQFT. These data and relations are expressed in the language of surgery.

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Rennemo, Jørgen Vold. "The homological projective dual of." Compositio Mathematica 156, no. 3 (January 17, 2020): 476–525. http://dx.doi.org/10.1112/s0010437x19007772.

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We study the derived category of a complete intersection $X$ of bilinear divisors in the orbifold $\operatorname{Sym}^{2}\mathbb{P}(V)$. Our results are in the spirit of Kuznetsov’s theory of homological projective duality, and we describe a homological projective duality relation between $\operatorname{Sym}^{2}\mathbb{P}(V)$ and a category of modules over a sheaf of Clifford algebras on $\mathbb{P}(\operatorname{Sym}^{2}V^{\vee })$. The proof follows a recently developed strategy combining variation of geometric invariant theory (VGIT) stability and categories of global matrix factorisations. We begin by translating $D^{b}(X)$ into a derived category of factorisations on a Landau–Ginzburg (LG) model, and then apply VGIT to obtain a birational LG model. Finally, we interpret the derived factorisation category of the new LG model as a Clifford module category. In some cases we can compute this Clifford module category as the derived category of a variety. As a corollary we get a new proof of a result of Hosono and Takagi, which says that a certain pair of non-birational Calabi–Yau 3-folds have equivalent derived categories.

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Zheng, Qilian, and Jiaqun Wei. "(n + 2)-Angulated Quotient Categories." Algebra Colloquium 26, no. 04 (November 18, 2019): 689–720. http://dx.doi.org/10.1142/s1005386719000506.

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The notion of [Formula: see text]-mutation pairs of subcategories in an n-exangulated category is defined in this article. When (Ƶ, Ƶ) is a [Formula: see text]-mutation pair in an n-exangulated category (C, [Formula: see text]), the quotient category Ƶ/[Formula: see text] carries naturally an (n+2)-angulated structure. This result generalizes a theorem of Zhou and Zhu for extriangulated categories.

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Egorov, Dmitry, Yuriy Dyatlov, Maksim Bogdanov, Evgeni Shushpanov, and Angela Egorova. "VALUE: EMPIRICS AND THEORY." SOCIETY. INTEGRATION. EDUCATION. Proceedings of the International Scientific Conference 6 (May 25, 2018): 165–75. http://dx.doi.org/10.17770/sie2018vol1.3080.

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Disclosure of the essence of the category "value" is not a prerequisite for its use: the introduction of this concept is correct and simple a priori. At the same time, it is obviously desirable. Objectives: 1) to show the connection between the concept of "value" and the empirical level; 2)to justify the interpretation of value from the perspective of information theory. Results. Manifestations of the "value" category in economic practice: the notions of "fair price", "reasonable profit", statistics of input-output balances, the world practice of planning of long-term energy production projects, etc. An adequate interpretation of value is the information concept: at the basis of value as the result of labor, and the rarity of a thing is information. Value is an information measure of the object’s worth. The complexity of operationalization does not ensue the unscientific (metaphysical) character of the concept of "value" as such. From the recognition of value as an objective basis for the observable price phenomenon, there appear very specific consequences: 1) feedback through the market must have an objective basis as a starting measure, that is, money must have a standard; 2) in some cases direct pricing (and / or their directive definition) is justified and appropriate.

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Hu, Po, Igor Kriz, and Kyle Ormsby. "Remarks on motivic homotopy theory over algebraically closed fields." Journal of K-Theory 7, no. 1 (January 21, 2010): 55–89. http://dx.doi.org/10.1017/is010001012jkt098.

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AbstractWe discuss certain calculations in the 2-complete motivic stable homotopy category over an algebraically closed field of characteristic 0. Specifically, we prove the convergence of motivic analogues of the Adams and Adams-Novikov spectral sequences, and as one application, discuss the 2-complete version of the complex motivic J -homomorphism.

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Wen, Xiao-Gang. "A theory of 2+1D bosonic topological orders." National Science Review 3, no. 1 (November 24, 2015): 68–106. http://dx.doi.org/10.1093/nsr/nwv077.

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Abstract In primary school, we were told that there are four phases of matter: solid, liquid, gas, and plasma. In college, we learned that there are much more than four phases of matter, such as hundreds of crystal phases, liquid crystal phases, ferromagnet, anti-ferromagnet, superfluid, etc. Those phases of matter are so rich, it is amazing that they can be understood systematically by the symmetry breaking theory of Landau. However, there are even more interesting phases of matter that are beyond Landau symmetry breaking theory. In this paper, we review new ‘topological’ phenomena, such as topological degeneracy, that reveal the existence of those new zero-temperature phase—topologically ordered phases. Microscopically, topologically orders are originated from the patterns of long-range entanglement in the ground states. As a truly new type of order and a truly new kind of phenomena, topological order and long-range entanglement require a new language and a new mathematical framework, such as unitary fusion category and modular tensor category to describe them. In this paper, we will describe a simple mathematical framework based on measurable quantities of topological orders (S, T, c) proposed around 1989. The framework allows us to systematically describe all 2+1D bosonic topological orders (i.e. topological orders in local bosonic/spin/qubit systems).

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Kusumayanti, Andi, and Nur Jannah. "ITEM ANALYSIS OF UIN ALAUDDIN MAKASSAR INDEPENDENT ENTRANCE EXAMINATION QUESTIONS WITH MODERN TEST THEORY." MaPan 10, no. 1 (June 10, 2022): 159–74. http://dx.doi.org/10.24252/mapan.2022v10n1a11.

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This study aims to find out how the quality of UMM UIN Alauddin Makassar questions in 2021 in the field of mathematics with the modern test method of the 2-PL model with indicators of difficulty level and differentiating power. This study is a quantitative analysis that emphasizes the analysis of the characteristics of the items empirically. The subjects of the research were the participants of the UIN Alauddin Makassar Independent Entrance Exam in 2021. The objects selected were multiple-choice math questions made by the UMM Alauddin Makassar UMM Question Making Team which consisted of 8 question packages with 5 items for each package. The method used in data collection is the documentation method, by collecting student responses to the 2021 UMM UIN Alauddin Makassar problem in mathematics made by the UMM Alauddin Makassar UMM Question Making Team, which is then analyzed quantitatively based on modern test methods. Based on the results of data analysis using the RStudio application, it is known that the items for the 2021 UIN Alauddin Makassar Independent Entrance Exam, based on the level of difficulty, obtained: (1) easy category items as many as 2 items, namely items number 4 and 5, (2) items very easy category questions as many as 3 items, namely items number 1, 2 and 3. Furthermore, in terms of discriminating power, obtained: (1) very bad category items as many as 3 items, namely item numbers 3, 4 and 5, (2 ) items in the bad category as many as 2 items, namely items number 1 and 2. Thus, it can be stated that overall the questions for the 2021 Alauddin UIN Makassar Mandiri Entrance Exam are not good.

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Lekili, Yankı, and Alexander Polishchuk. "Homological mirror symmetry for higher-dimensional pairs of pants." Compositio Mathematica 156, no. 7 (June 18, 2020): 1310–47. http://dx.doi.org/10.1112/s0010437x20007150.

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Using Auroux’s description of Fukaya categories of symmetric products of punctured surfaces, we compute the partially wrapped Fukaya category of the complement of $k+1$ generic hyperplanes in $\mathbb{CP}^{n}$, for $k\geqslant n$, with respect to certain stops in terms of the endomorphism algebra of a generating set of objects. The stops are chosen so that the resulting algebra is formal. In the case of the complement of $n+2$ generic hyperplanes in $\mathbb{C}P^{n}$ ($n$-dimensional pair of pants), we show that our partial wrapped Fukaya category is equivalent to a certain categorical resolution of the derived category of the singular affine variety $x_{1}x_{2}\ldots x_{n+1}=0$. By localizing, we deduce that the (fully) wrapped Fukaya category of the $n$-dimensional pair of pants is equivalent to the derived category of $x_{1}x_{2}\ldots x_{n+1}=0$. We also prove similar equivalences for finite abelian covers of the $n$-dimensional pair of pants.

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Caprau, Carmen, and Joel Smith. "The Singular Temperley-Lieb Category." ISRN Geometry 2014 (April 24, 2014): 1–9. http://dx.doi.org/10.1155/2014/321509.

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We introduce and study the singular Temperley-Lieb category over ℤ[q,q-1], which is a free pivotal category over two self-dual generators and is an extension of the (classical) Temperley-Lieb category. Our construction is motivated by a state model for the sl(2) polynomial of an oriented link and provides a categorical perspective to this link invariant. We also construct a couple of polynomial invariants for oriented tangles from category theory point of view.

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Elgueta, Josep. "The groupoid of finite sets is biinitial in the 2-category of rig categories." Journal of Pure and Applied Algebra 225, no. 11 (November 2021): 106738. http://dx.doi.org/10.1016/j.jpaa.2021.106738.

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Elmendorf, A. D. "Function spectra." Mathematical Proceedings of the Cambridge Philosophical Society 108, no. 1 (July 1990): 31–34. http://dx.doi.org/10.1017/s0305004100068924.

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Boardman's stable category (see [5]) is a closed category ([4], VII·7), and in the best of all possible worlds, the category of spectra underlying the stable category would be closed as well; this would make life considerably easier for those doing calculations in stable homotopy theory. Unfortunately none of the categories of spectra introduced to date are closed; only S, the category introduced in [2], is even symmetric monoidal. The problem with making S closed is that it comes equipped with an augmentation to I, the category of universes and linear isometries (called Un in [2]), which preserves the symmetric monoidal structure. Since I is not closed, this makes it difficult to see how S might be closed.

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Ana-Maria Anghel, Cristina, and Nathan Geer. "Modified Turaev-Viro invariants from quantum 𝔰𝔩(2|1)." Journal of Knot Theory and Its Ramifications 29, no. 04 (March 23, 2020): 2050018. http://dx.doi.org/10.1142/s0218216520500182.

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The category of finite dimensional modules over the quantum superalgebra [Formula: see text] is not semi-simple and the quantum dimension of a generic [Formula: see text]-module vanishes. This vanishing happens for any value of [Formula: see text] (even when [Formula: see text] is not a root of unity). These properties make it difficult to create a fusion or modular category. Loosely speaking, the standard way to obtain such a category from a quantum group is: (1) specialize [Formula: see text] to a root of unity; this forces some modules to have zero quantum dimension, (2) quotient by morphisms of modules with zero quantum dimension, (3) show the resulting category is finite and semi-simple. In this paper, we show an analogous construction works in the context of [Formula: see text] by replacing the vanishing quantum dimension with a modified quantum dimension. In particular, we specialize [Formula: see text] to a root of unity, quotient by morphisms of modules with zero modified quantum dimension and show the resulting category is generically finite semi-simple. Moreover, we show the categories of this paper are relative [Formula: see text]-spherical categories. As a consequence, we obtain invariants of 3-manifold with additional structures.

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Cannon, Clare, Regardt J. Ferreira, and Fred Buttell. "Critical Race Theory, Parenting, and Intimate Partner Violence: Analyzing Race and Gender." Research on Social Work Practice 30, no. 1 (April 29, 2018): 122–34. http://dx.doi.org/10.1177/1049731518772151.

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Purpose: This study sought to investigate similarities and differences among race, gender, parenting attitudes, and conflict negotiation tactics of perpetrators of intimate partner violence in a batterer intervention program. Method: This research utilized a nonequivalent, control group secondary analysis of 238 women and men. Results: Logistic regression indicated the following: (1) An increased likelihood for scoring higher on the Conflict Tactics Scale-2 (CTS-2), Physical Assault subscale, and high-risk Adult–Adolescent Parenting Inventory-2 (AAPI-2) parenting group for those in the African American category compared to the White category; (2) African American women are more likely to be unemployed, score higher on the CTS-2 Physical Assault subscale, and in the high-risk AAPI-2 parenting group than African American men; and (3) White women, compared to White men, are more likely to experience injury and to score in the high-risk AAPI-2 group. Conclusions: Critical race theory provides a necessary understanding of these findings within structural inequality in the United States. Further results and implications are discussed.

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Caprau, Carmen. "Twin TQFTs and Frobenius Algebras." Journal of Mathematics 2013 (2013): 1–25. http://dx.doi.org/10.1155/2013/407068.

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We introduce the category of singular 2-dimensional cobordisms and show that it admits a completely algebraic description as the free symmetric monoidal category on atwin Frobenius algebra, by providing a description of this category in terms of generators and relations. A twin Frobenius algebra(C,W,z,z∗)consists of a commutative Frobenius algebraC, a symmetric Frobenius algebraW, and an algebra homomorphismz:C→Wwith dualz∗:W→C, satisfying some extra conditions. We also introduce a generalized 2-dimensional Topological Quantum Field Theory defined on singular 2-dimensional cobordisms and show that it is equivalent to a twin Frobenius algebra in a symmetric monoidal category.

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Whelan, E. A. "An Infinite Construction in Ring Theory." Glasgow Mathematical Journal 33, no. 1 (January 1991): 121–23. http://dx.doi.org/10.1017/s0017089500008119.

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1. Point (3) of the main theorem of our paper [3, Theorem 1.1] is incorrect: this note corrects the main and consequential errors, and shows that (after minor adjustments) almost all the other results of [3], including the remaining seven points of Theorem 1.1, remain correct.2. The theme of [3] was a family of functors G,(–), defined on the category of rings with unity for each cardinal t. For t = 0, 1, the results of [3] are unchanged, but, for 2≤t<∞, major, and, for t infinite, less major, corrections are necessary; we therefore assume 2≤t. Terminology and notation are standard or as in [3], and I would like to thank A. W. Chatters and an anonymous referee for comments which prompted this correction.

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Kurniawan, Nur Ichsanuddin Achmad, and Sudji Munadi. "Analysis of the quality of test instrument and students’ accounting learning competencies at vocational school." Jurnal Penelitian dan Evaluasi Pendidikan 23, no. 1 (June 29, 2019): 68–75. http://dx.doi.org/10.21831/pep.v23i1.22484.

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The study is aimed at describing: (1) characteristics of the items of the national examination try-out test of the accounting subject matter in the 2015/2016 academic year on classical test theory and modern test theory; and (2) classification of students’ masteries in the learning of accounting. The study is explorative research. Analyses are conducted using the classical and modern test theories for item characteristics and descriptive quantitative for students’ masteries in accounting using the test set for the national examination try-out in the 2015/2016 academic year. A total of 414 students do the Package A test. Results show that (1) based on the classical test analyses, a number of 11 items (27.5%) belong to the “easy” category, 22 items (55%) “medium” category, and 7 items (17.5%) “difficult” category allowing a total of 19 (47.5%) to be categorized as good items; meanwhile, on the modern-theory analyses, a total of 34 items (85%) belong to the “good” category. (2) Around 38% of the students have competencies of the medium and low categories. Most students have difficulty in answering questions of the higher-order thinking levels.

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Schmitt, Vincent. "Completions of Non-Symmetric Metric Spaces Via Enriched Categories." gmj 16, no. 1 (March 2009): 157–82. http://dx.doi.org/10.1515/gmj.2009.157.

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Abstract It is known from [Lawvere, Repr. Theory Appl. Categ. 1: 1–37 2002] that nonsymmetric metric spaces correspond to enrichments over the monoidal closed category [0, ∞]. We use enriched category theory and in particular a generic notion of flatness to describe various completions for these spaces. We characterise the weights of colimits commuting in the base category [0, ∞] with the conical terminal object and cotensors. Those can be interpreted in metric terms as very general filters, which we call filters of type 1. This correspondence extends the one between minimal Cauchy filters and weights which are adjoint as modules. Translating elements of enriched category theory into the metric context, one obtains a notion of convergence for filters of type 1 with a related completeness notion for spaces, for which there exists a universal completion. Another smaller class of flat presheaves is also considered both in the context of both metric spaces and preorders. (The latter being enrichments over the monoidal closed category 2.) The corresponding completion for preorders is the so-called dcpo completion.

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HUANG, YI-ZHI, and ANTUN MILAS. "INTERTWINING OPERATOR SUPERALGEBRAS AND VERTEX TENSOR CATEGORIES FOR SUPERCONFORMAL ALGEBRAS, I." Communications in Contemporary Mathematics 04, no. 02 (May 2002): 327–55. http://dx.doi.org/10.1142/s0219199702000622.

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We apply the general theory of tensor products of modules for a vertex operator algebra (developed by Lepowsky and the first author) and the general theory of intertwining operator algebras (developed by the first author) to the case of the N=1 superconformal minimal models and related models in superconformal field theory. We show that for the category of modules for a vertex operator algebra containing a subalgebra isomorphic to a tensor product of rational vertex operator superalgebras associated to the N =1 Neveu–Schwarz Lie superalgebra, the intertwining operators among the modules have the associativity property, the category has a natural structure of vertex tensor category, and a number of related results hold. We obtain, as a corollary and special case, a construction of a braided tensor category structure on the category of finite direct sums of minimal modules of central charge [Formula: see text] for the N = 1 Neveu–Schwarz Lie superalgebra for any fixed integers p, q larger than 1 such that p - q ∈ 2ℤ and (p - q)/2 and q relatively prime to each other.

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Purnomo, Muchammad Khafith Octafian, and Pradnyo Wijayanti. "The Exploration of Mathematical Objects in Anime Series Reviewed from Onto-Semiotic Approach (OSA) Theory." MATHEdunesa 10, no. 3 (October 14, 2021): 497–506. http://dx.doi.org/10.26740/mathedunesa.v10n3.p497-506.

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Mathematics is often associated with a boring subject in the classroom. There’s a need for innovation, especially in the learning process, so students don’t feel bored learning mathematics. One of the innovations available is to connect mathematics with the student's interests. One of the things the students are interested in is animation works, especially anime series. This research aims to explore the mathematical objects in the anime series reviewed from the Onto-Semiotic Approach (OSA) theory. This research is descriptive exploratory research. Data were collected through observation of 5 anime series titles using a research instrument in the form of an observation sheet. The analysis process was carried out on the identified mathematical objects from the 5 anime series titles, then categorizing the identified mathematical objects according to the characteristics of the 6 categories of primary mathematical objects in OSA theory. The results indicate that there are 6 mathematical objects identified as the language category, such as the quadratic equation and trigonometry, 5 mathematical objects identified as the situation category, no mathematical object identified as the concept category, 1 mathematical object identified as the proposition category, 1 mathematical object identified as the procedure category, and 2 mathematical objects as the argument category. The result of this research can be used as an innovation in mathematics learning and as additional research on the use of popular culture in the classroom.

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Sudarma, I. Komang, Dewa Gede Agus Putra Prabawa, and I. Kadek Suartama. "The Application of Information Processing Theory to Design Digital Content in Learning Message Design Course." International Journal of Information and Education Technology 12, no. 10 (2022): 1043–49. http://dx.doi.org/10.18178/ijiet.2022.12.10.1718.

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The development research being carried out has the aim of producing digital content developed based on information processing theory for the message design course in Educational Technology Study Program in Education Science Faculty of Universitas Pendidikan Ganesha. This is a development research in which the Hannafin & Peck model is used. The developed digital content is evaluated using formative evaluation techniques, including 1) expert validation, 2) one-to-one evaluation, and 3) small group evaluation. The subjects involved in this study were 2 experts, namely media experts and instructional design experts, 3 students in one-to-one evaluation, and 9 students in small group evaluation. The methods and instruments used to collect data in this study were observation and questionnaires. Based on the expert’s judgment, the design aspect is in the good category, the media aspect is in the very good category. Students’ responses at the one-to-one and small group evaluation stages are in the good categories. Thus, it can be concluded that the attractiveness of digital content is in the good category.

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Ramadani, Elia Maryam, Aripin Aripin, and Rifaatul Maulidah. "PENGEMBANGAN MEDIA PEMBELAJARAN BERBASIS APLIKASI ANDROID MENGGUNAKAN POWERPOINT ISPRING PADA MATERI TEORI KINETIK GAS." JURNAL EDUSCIENCE 9, no. 1 (April 1, 2022): 243–54. http://dx.doi.org/10.36987/jes.v9i1.2594.

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This study aims to develop an android application-based learning media using powerpoint ispring on gas kinetic theory material to describe the level of validity and practicality. This study uses research and development methods with the ADDIE development model but only carried out 4 stages, namely Analysis, Design, Development, and Implementation. The results of the validation of learning media that have been developed to develop learning media based on media experts are 4.25 with a very good category, material experts are 4.38 with very good categories, digital experts are 4.23 with very good categories, and experts are 4 ,2 with good category. The learning media that has been developed was tested on 101 students and 2 educators of XI MIPA SMAN 1 Taraju as research subjects and produced product practicality data based on the results of student responses, namely 4.52 with a very good category and the teacher's response being 4.5 with a category very good. Thus, android application-based learning media is suitable for use in physics learning.Keywords: Learning Media; Android; Powerpoint Ispring; Kinetic Theory of Gas

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Mackaay, Marco, Volodymyr Mazorchuk, Vanessa Miemietz, Daniel Tubbenhauer, and Xiaoting Zhang. "Finitary birepresentations of finitary bicategories." Forum Mathematicum 33, no. 5 (August 7, 2021): 1261–320. http://dx.doi.org/10.1515/forum-2021-0021.

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Abstract In this paper, we discuss the generalization of finitary 2-representation theory of finitary 2-categories to finitary birepresentation theory of finitary bicategories. In previous papers on the subject, the classification of simple transitive 2-representations of a given 2-category was reduced to that for certain subquotients. These reduction results were all formulated as bijections between equivalence classes of 2-representations. In this paper, we generalize them to biequivalences between certain 2-categories of birepresentations. Furthermore, we prove an analog of the double centralizer theorem in finitary birepresentation theory.

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Brochier, Adrien, David Jordan, and Noah Snyder. "On dualizability of braided tensor categories." Compositio Mathematica 157, no. 3 (March 2021): 435–83. http://dx.doi.org/10.1112/s0010437x20007630.

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We study the question of dualizability in higher Morita categories of locally presentable tensor categories and braided tensor categories. Our main results are that the 3-category of rigid tensor categories with enough compact projectives is 2-dualizable, that the 4-category of rigid braided tensor categories with enough compact projectives is 3-dualizable, and that (in characteristic zero) the 4-category of braided multi-fusion categories is 4-dualizable. Via the cobordism hypothesis, this produces respectively two-, three- and four-dimensional framed local topological field theories. In particular, we produce a framed three-dimensional local topological field theory attached to the category of representations of a quantum group at any value of $q$.

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Vogel, Pierre. "Functoriality of Khovanov homology." Journal of Knot Theory and Its Ramifications 29, no. 04 (March 18, 2020): 2050020. http://dx.doi.org/10.1142/s0218216520500200.

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In this paper, we prove that every Khovanov homology associated to a Frobenius algebra of rank 2 can be modified in such a way as to produce a TQFT on oriented links, that is a monoidal functor from the category of cobordisms of oriented links to the homotopy category of complexes.

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Orlicki, Andrzej. "On Enumerated Algebras and Some Monads in the Category of Enumerated Sets." Fundamenta Informaticae 8, no. 3-4 (July 1, 1985): 285–307. http://dx.doi.org/10.3233/fi-1985-83-403.

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In this paper we investigate some properties of categories of enumerated algebras. In section 1 we prove that in case of the category of enumerated algebras satisfying some fixed set of identities, the forgetful functor into the category of enumerated sets is monadic. Then in sections 2 and 3 some applications of this result are presented. In particular, we show (see Theorem 3.2, section 3) that each finitary monad in the category of denumerable sets has a lifting to a monad in the category of enumerated sets.

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Bowles, Terry. "The Brief Attachment Adjective Checklist: A Measure of the Fourfold Definition of the Theory of Attachment." Journal of Relationships Research 1, no. 1 (August 1, 2010): 17–30. http://dx.doi.org/10.1375/jrr.1.1.17.

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AbstractThe aims of this research were to develop and validate a Brief Attachment Adjective Checklist (BAAC) to represent the four category model of attachment, compare it with a current measure of attachment, and use both to predict relationship satisfaction. A 32-item operationalisation of a hypothesised four-category model was analysed using a principal component analysis. Results of Study 1 (n = 174) indicated the items of the four-category model reflected good factor structure. Comparison with the four-paragraph Relationship Questionnaire (RQ) showed low correlation between the two operationalisations, suggesting that they were measuring different aspects of attachment. Analyses showed that the BAAC was a weak but better predictor of relationship satisfaction than the RQ. A confirmatory factor analysis in Study 2 (n = 131) refined the structure of the BAAC. The pattern of correlations showing relative independence of the BAAC and the RQ in Study 1 was also shown in Study 2. The frequency of respondents in dominant attachment categories of the RQ was consistent with previous research but the frequency of respondents in dominant attachment categories differed for RQ compared with the BAAC. The replication of the prediction of relationship satisfaction in Study 2 showed that both measures were weak predictors of relationship satisfaction but the BAAC was a better predictor than the RQ.

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Ardana, Made I., Wisna I. Putu Ariawan, and Gusti Ayu Dessy Sugiharni. "The expansion of sociocultural theory-oriented mathematical learning model." Cypriot Journal of Educational Sciences 16, no. 6 (December 31, 2021): 3016–32. http://dx.doi.org/10.18844/cjes.v16i6.6493.

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The research aim was to obtain a Sociocultural Theory-oriented Mathematics Learning Model with Tri Hita Karana insight that is valid, practical, and effective in developing the good character of students in learning mathematics. This research was a research and development with reference to the development of Plomp. The data collection techniques used tests and questionnaires. The results showed that: (1) the Sociocultural Theory-oriented Mathematics Learning Model with Tri Hita Karana insight is valid, practical, and effective to use to develop the good character of students; (2) there was an increase in the good character of students from the category ‘sometimes shows good behavior according to the Tri Hita Karana aspect and often behaves not in accordance with the Tri Hita Karana aspect’ to the category ‘often and consistently shows good behavior according to the Tri Hita Karana aspect’; and (3) positive student responses to learning. Keywords: Mathematical learning, sociocultural theory, tri hita karana, pawongan, tri pramana

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Azizah, Nur, and Muhbib Abdul Wahab. "IMPLEMENTASI TEORI VERBAL LINGUISTIC INTELLIGENCE DAN INTERPERSONAL INTELLIGENCE DALAM PEMBELAJARAN MAHÂRAT AL-KALÂM DAN MAHÂRAT AL-QIRÂ’AH." Arabi : Journal of Arabic Studies 7, no. 2 (December 21, 2022): 208–24. http://dx.doi.org/10.24865/ajas.v7i2.510.

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This research aims to analyse the implementation of method, media, and evaluation of speaking skill and reading skill learning in Arabic learning based on the theory verbal linguistic intelligence and interpersonal intelligence. The method of this research was mixed method with triangulation concuren strategy by collecting quantitative data and qualitative data. The finding indicate that the learning of speaking skill and reading skill in learning Arabic at state Islamic Senior High School 2 and 4 Tanah Datar West Sumatera become very effective with applying the linguistic of theory verbal intelligence and interpersonal intellligence in good impact. Developed with that theory make test result Arabic subject the students are in the high category. The benefits of developing this theory in learning can also be concluded from the result of the questionnaire that have shared to students of states Islamic Senior High School 2 and 4 Tanah Datar West Sumatera. It was shown that maharat al-kalam and maharat al-qira’ah in the high category.

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Keller, Bernhard, and Idun Reiten. "Acyclic Calabi–Yau categories." Compositio Mathematica 144, no. 5 (September 2008): 1332–48. http://dx.doi.org/10.1112/s0010437x08003540.

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AbstractWe prove a structure theorem for triangulated Calabi–Yau categories: an algebraic 2-Calabi–Yau triangulated category over an algebraically closed field is a cluster category if and only if it contains a cluster-tilting subcategory whose quiver has no oriented cycles. We prove a similar characterization for higher cluster categories. As an application to commutative algebra, we show that the stable category of maximal Cohen–Macaulay modules over a certain isolated singularity of dimension 3 is a cluster category. This implies the classification of the rigid Cohen–Macaulay modules first obtained by Iyama and Yoshino. As an application to the combinatorics of quiver mutation, we prove the non-acyclicity of the quivers of endomorphism algebras of cluster-tilting objects in the stable categories of representation-infinite preprojective algebras. No direct combinatorial proof is known as yet. In the appendix, Michel Van den Bergh gives an alternative proof of the main theorem by appealing to the universal property of the triangulated orbit category.

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Li, Yizheng, and Dingguo Wang. "Lie algebras with differential operators of any weights." Electronic Research Archive 31, no. 3 (2022): 1195–211. http://dx.doi.org/10.3934/era.2023061.

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<abstract><p>In this paper, we define a cohomology theory for differential Lie algebras of any weight. As applications of the cohomology, we study abelian extensions and formal deformations of differential Lie algebras of any weight. Finally, we consider homotopy differential operators on $ \mathrm{L}_{\infty} $ algebras and 2-differential operators of any weight on Lie 2-algebras, and we prove that the category of 2-term $ \mathrm{L}_{\infty} $ algebras with homotopy differential operators of any weight is same as the category of Lie 2-algebras with 2-differential operators of any weight.</p></abstract>

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Chen, Qiaoyun, and Guiying Jiang. "Why are you amused: Unveiling multimodal humor from the prototype theoretical perspective." European Journal of Humour Research 6, no. 1 (June 13, 2018): 62. http://dx.doi.org/10.7592/ejhr2018.6.1.chen.

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This paper looks at multimodal humour through the lens of prototype theory in the framework of conventional incongruity theory of humour, aiming for a unified linguistic and semiotic approach to humour. From this perspective, humour can be achieved through the following three aspects of linguistic and non-linguistic categories: 1) prototypicality versus non-prototypicality of category members; 2) the family resemblance shared by category members; 3) vague inter-categorical boundary. The cognitive mechanisms behind this type of multimodal humour and its comprehension are discussed. The intermodal relationships involved are examined and categorised into two major types: complementary and non-complementary ones.

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Morrison, Scott, and David Penneys. "Monoidal Categories Enriched in Braided Monoidal Categories." International Mathematics Research Notices 2019, no. 11 (October 3, 2017): 3527–79. http://dx.doi.org/10.1093/imrn/rnx217.

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Abstract We introduce the notion of a monoidal category enriched in a braided monoidal category $\mathcal{V}$. We set up the basic theory, and prove a classification result in terms of braided oplax monoidal functors to the Drinfeld centre of some monoidal category $\mathcal{T}$. Even the basic theory is interesting; it shares many characteristics with the theory of monoidal categories enriched in a symmetric monoidal category, but lacks some features. Of particular note, there is no cartesian product of braided-enriched categories, and the natural transformations do not form a 2-category, but rather satisfy a braided interchange relation. Strikingly, our classification is slightly more general than what one might have anticipated in terms of strong monoidal functors $\mathcal{V}\to Z(\mathcal{T})$. We would like to understand this further; in a future article, we show that the functor is strong if and only if the enriched category is ‘complete’ in a certain sense. Nevertheless it remains to understand what non-complete enriched categories may look like. One should think of our construction as a generalization of de-equivariantization, which takes a strong monoidal functor ${\mathsf {Rep}}(G) \to Z(\mathcal{T})$ for some finite group $G$ and a monoidal category $\mathcal{T}$, and produces a new monoidal category $\mathcal{T} _{{/\hspace{-2px}/}G}$. In our setting, given any braided oplax monoidal functor $\mathcal{V} \to Z(\mathcal{T})$, for any braided $\mathcal{V}$, we produce $\mathcal{T} _{{/\hspace{-2px}/}\mathcal{V}}$: this is not usually an ‘honest’ monoidal category, but is instead $\mathcal{V}$-enriched. If $\mathcal{V}$ has a braided lax monoidal functor to ${\mathsf {Vec}}$, we can use this to reduce the enrichment to ${\mathsf {Vec}}$, and this recovers de-equivariantization as a special case. This is the published version of arXiv:1701.00567.

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CHENG, SHUN-JEN, and WEIQIANG WANG. "CHARACTER FORMULAE IN CATEGORY O $$ \mathcal{O} $$ FOR EXCEPTIONAL LIE SUPERALGEBRAS D(2|1; ζ)." Transformation Groups 24, no. 3 (November 9, 2018): 781–821. http://dx.doi.org/10.1007/s00031-018-9506-5.

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Liu, Tong. "On lattices in semi-stable representations: a proof of a conjecture of Breuil." Compositio Mathematica 144, no. 1 (January 2008): 61–88. http://dx.doi.org/10.1112/s0010437x0700317x.

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AbstractFor p≥3 an odd prime and a nonnegative integer r≤p−2, we prove a conjecture of Breuil on lattices in semi-stable representations, that is, the anti-equivalence of categories between the category of strongly divisible lattices of weight r and the category of Galois stable $\mathbb {Z}_p$-lattices in semi-stable p-adic Galois representations with Hodge–Tate weights in {0,…,r}.

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Journal articles on the topic '2-category theory'

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