Haruhisa Enomoto
We show that for a given exact category, there exists a bijection between semibricks (pairwise Hom-orthogonal set of bricks) and length wide subcategories (exact extension-closed length abelian subcategories). In particular, we show that a length exact category is abelian if and only if simple objects form a semibrick, that is, the Schur's lemma holds.
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We classify the derived tame Schur and infinitesimal Schur algebras and describe indecomposable objects in their derived categories.
We introduce the Frobenius-Schur indicator for categories with duality to give a category-theoretical understanding of various generalizations of the Frobenius-Schur theorem, including that for semisimple quasi-Hopf algebras, weak Hopf C*-algebras and association schemes. Our framework also clarifies a mechanism how the `twisted' theory arises from the ordinary case. As a demonstration, we give a twisted Frobenius-Schur theorem for semisimple quasi-Hopf algebras. We also give several applications to the quantum SL_2.
A Graded Schur Lemma and a graded-monoidal structure for induced modules over graded-commutative algebras
本文在有限阿贝尔群分次和辫化幺半范畴框架下推广舒尔引理，并证明当代数按分次群双特征标交换时，诱导模范畴具有由双特征标扭转的分次幺半结构，从而统一并推广了超代数情形。
Let $k$ be a commutative $\mathbb{Q}$-algebra. We study families of functors between categories of finitely generated $R$-modules which are defined for all commutative $k$-algebras $R$ simultaneously and are compatible with base changes. These operations turn out to be Schur functors associated to $k$-linear representations of symmetric groups. This result is closely related to Macdonald's classification of polynomial functors.
Ng and Schauenburg generalized higher Frobenius-Schur indicators to pivotal fusion categories and showed that these indicators may be computed utilizing the modular data of the Drinfel'd center of the given category. We consider two classes of fusion categories generated by a single non-invertible simple object: near groups, those fusion categories with one non-invertible simple object, and Haagerup-Izumi categories, those with one non-invertible simple object for every invertible object. Examples of both types arise as representations of finite or quantum groups or as Jones standard invariant…
The deformation cohomology of a tensor category controls deformations of its monoidal structure. Here we describe the deformation cohomology of tensor categories generated by one object (the so-called Schur-Weyl categories). Using this description we compute the deformation cohomology of free symmetric tensor categories generated by one object with an algebra of endomorphism free of zero-divisors. We compare the answers with the exterior invariants of the general linear Lie algebra.
We study the construction of arXiv:2111.11217 [math:AG] in more detail, especially in the case of Schur-finite rigid $\otimes$-categories. This leads to some groundwork on the ideal structure of rigid additive and abelian $\otimes$-categories.
We prove an injective version of Schanuel's lemma from homological algebra in the setting of exact categories.
A characterization of closed subfunctors through $3\times 3$-lemma property in extriangulated categories
本文在外三角范畴中引入子函子的3×3-引理性质，证明加法子函子闭当且仅当满足该性质，推广了Buan的结果，并给出饱和真类的新刻画。
We use category theory to propose a unified approach to the Schur-Weyl dualities involving the general linear Lie algebras, their polynomial extensions and associated quantum deformations. We define multiplicative sequences of algebras exemplified by the sequence of group algebras of the symmetric groups and use them to introduce associated monoidal categories. Universal properties of these categories lead to uniform constructions of the Drinfeld functor connecting representation theories of the degenerate affine Hecke algebras and the Yangians and of its q-analogue. Moreover, we construct act…
We define higher Frobenius-Schur indicators for objects in linear pivotal monoidal categories. We prove that they are category invariants, and take values in the cyclotomic integers. We also define a family of natural endomorphisms of the identity endofunctor on a $k$-linear semisimple rigid monoidal category, which we call the Frobenius-Schur endomorphisms. For a $k$-linear semisimple pivotal monoidal category -- where both notions are defined --, the Frobenius-Schur indicators can be computed as traces of the Frobenius-Schur endomorphisms.
We generalize the notion of an exact category and introduce weakly exact categories. A proof of the snake lemma in this general setting is given. Some applications are given to illustrate how one can do homological algebra in a weakly exact category.
We prove that a regular category $\mathcal C$ is a Mal'tsev category if and only if a strong form of the denormalised $3 \times 3$ Lemma holds true in $\mathcal C$. In this version of the $3 \times 3$ Lemma, the vertical exact forks are replaced by pullbacks of regular epimorphisms along arbitrary morphisms. The shape of the diagram it determines suggests to call it the Cuboid Lemma. This new characterisation of regular categories that are Mal'tsev categories (= $2$-permutable) is similar to the one previously obtained for Goursat categories (= $3$-permutable). We also analyse the "relative" v…
The Kaplansky-Walker Lemma for additive categories
本文针对带直和的幂等完全加法范畴建立Kaplansky-Walker型引理，将其应用于紧生成三角范畴，得到投射模相关定理及可数对象直和的同构细化结果。
We consider a subclass of the class of group-theoretical fusion categories: To every finite group $G$ and subgroup $H$ one can associate the category of $G$-graded vector spaces with a two-sided $H$-action compatible with the grading. We derive a formula that computes higher Frobenius-Schur indicators for the objects in such a category using the combinatorics and representation theory of the groups involved in their construction. We calculate some explicit examples for inclusions of symmetric groups.
We study ideal cotorsion pairs associated to weak proper classes of triangles in extension closed subcategories of triangulated categories. This approach allows us to extend the recent ideal approximations theory developed by Fu, Herzog et al. for exact categories in the above mentioned context, and to provide simplified proofs for the ideal versions of some standard results as Salce's Lemma, Wakamatsu's Lemma and Christensen's Ghost Lemma. In the last part of the paper we apply the theory in order to study connections between projective classes (in particular localization or smashing subcateg…
Surface Diagrams for Frobenius Algebras and Frobenius-Schur Indicators in Grothendieck-Verdier Categories
该研究为Grothendieck-Verdier范畴建立三维曲面图演算，推广刚性幺半范畴的Frobenius代数性质，引入高阶Frobenius-Schur示性数并证明其等价不变性，配套提供图形证明助手辅助文件。
A full subcategory of a Grothendieck category is called deconstructible if it consists of all transfinite extensions of some set of objects. This concept provides a handy framework for structure theory and construction of approximations for subcategories of Grothendieck categories. It also allows to construct model structures and t-structures on categories of complexes over a Grothendieck category. In this paper we aim to establish fundamental results on deconstructible classes and outline how to apply these in the areas mentioned above. This is related to recent work of Gillespie, Enochs, Est…
$2$-star-permutable categories were introduced in a joint work with Z. Janelidze and A. Ursini as a common generalisation of regular Mal'tsev categories and of normal subtractive categories. In the present article we first characterise these categories in terms of what we call star-regular pushouts. We then show that the $3 \times 3$ Lemma characterising normal subtractive categories and the Cuboid Lemma characterising regular Mal'tsev categories are special instances of a more general homological lemma for star-exact sequences. We prove that $2$-star-permutability is equivalent to the validit…
Group-theoretical fusion categories are defined by data concerning finite groups and their cohomology: A finite group $G$ endowed with a three-cocycle $ω$, and a subgroup $H\subset G$ endowed with a two-cochain whose coboundary is the restriction of $ω$. The objects of the category are $G$-graded vector spaces with suitably twisted $H$-actions; the associativity of tensor products is controlled by $ω$. Simple objects are parametrized in terms of projective representations of finite groups, namely of the stabilizers in $H$ of right $H$-cosets in $G$, with respect to two-cocycles defined by the …
We prove that Mal'tsev and Goursat categories may be characterised through stronger variations of the Shifting Lemma, that is classically expressed in terms of three congruences $R$, $S$ and $T$, and characterises congruence modular varieties. We first show that a regular category $\mathcal C$ is a Mal'tsev category if and only if the Shifting Lemma holds for reflexive relations on the same object in $\mathcal C$. Moreover, we prove that a regular category $\mathcal C$ is a Goursat category if and only if the Shifting Lemma holds for a reflexive relation $S$ and reflexive and positive relation…
We develop some basic concepts in the theory of higher categories internal to an arbitrary $\infty$-topos. We define internal left and right fibrations and prove a version of the Grothendieck construction and of Yoneda's lemma for internal categories.
We classify pointed fusion categories C(G, $ω$) up to Morita equivalence for 1 < |G| < 32. Among them, the cases |G| = 2 3 , 2 4 and 3 3 are emphasized. Although the equivalence classes of such categories are not distinguished by their Frobenius-Schur indicators, their categorical Morita equivalence classes are distinguished by the set of the indicators and ribbon twists of their Drinfeld centers. In particular, the modular data are a complete invariant for the modular categories Z(C(G, $ω$)) for |G[< 32. We use the computer algebra package GAP and present codes for treating complex-valued gro…
We obtain a generalization of the Two-Square Lemma proved for abelian categories by Fay, Hardie, and Hilton in 1989 and (in a special case) for preabelian categories by Generalov in 1994. We also prove the equivalence up to sign of two definitions of a connecting morphism of the Snake Lemma.
The Noether isomorphism theorems and the Zassenhaus Lemma from group theory have a non-pointed version in a suitable categorical context first considered by W. Tholen in his PhD thesis. This article leads to a unification of these results with the ones in the pointed categorical context considered by O. Wyler, by working in the framework of star-regular categories introduced by M. Gran, Z. Janelidze and A. Ursini. Some concrete examples of categories where these results hold are examined in detail.
Several results on exact sequences in categories of modules over trusses
本文在truss上模范畴中建立了蛇引理和九引理，推进了truss上模理论的范畴论研究。
Homological lemmas for (non-abelian) group-like structures by diagram chasing in a self-dual context
本文通过引入'诺特形式'的上下文，展示了如何在非阿贝尔群状结构范畴中，利用自对偶方法处理同调引理。
We study connections between additive and abelian 2-representations of fiat 2-categories, describe combinatorics of 2-categories in terms of multisemigroups and determine the annihilator of a cell 2-representation. We also describe, in detail, examples of fiat 2-categories associated to $\mathfrak{sl}_2$-categorification in the sense of Chuang and Rouquier, and 2-categorical analogues of Schur algebras.
The aim of this article is to better understand the correspondence between $n$-cubic extensions and $3^n$-diagrams, which may be seen as non-abelian Yoneda extensions, useful in (co)homology of non-abelian algebraic structures. We study a higher-dimensional version of the coequaliser/kernel pair adjunction, which relates $n$-fold reflexive graphs with $n$-fold arrows in any exact Mal'tsev category. We first ask ourselves how this adjunction restricts to an equivalence of categories. This leads to the concept of an effective $n$-fold equivalence relation, corresponding to the $n$-fold regular e…
Yoneda lemma and representation theorem for double categories
本文研究取值于 $\mathbb{C}\mathrm{at}$ 的正规松弛双预层，建立其米田引理、与离散双纤维化的 Grothendieck 2-等价，以及由双终对象刻画的表示定理。
Diagonal Lemma for Presheaves on Eilenberg-Zilber Categories
本文研究了Eilenberg-Zilber范畴上的对角引理，证明了在Kan-Quillen模型结构下，双层简复形的逐层弱等价可诱导对角简复形的弱等价。
We continue the study of enriched infinity categories, using a definition equivalent to that of Gepner and Haugseng. In our approach enriched infinity categories are associative monoids in an especially designed monoidal category of enriched quivers. We prove that, in case the monoidal structure in the basic category M comes from direct product, our definition is essentially equivalent to the approach via Segal objects. Furthermore, we compare our notion with the notion of category left-tensored over M, and prove a version of Yoneda lemma in this context. Version 2: An error in 2.6.2 corrected…
We give new characterisations of regular Mal'tsev categories with distributive lattice of equivalence relations through variations of the so-called Triangular Lemma and Trapezoid Lemma in universal algebra. We then give new characterisations of equivalence distributive Goursat categories (which extend $3$-permutable varieties) through variations of the Triangular and Trapezoid Lemmas involving reflexive and positive relations.
Invertible bimodule categories and generalized Schur orthogonality
该文将Schur正交性推广至弱Hopf代数，给出判定双模范畴可逆的新条件，用于构造Morita等价，并证明其等价于MPO注入性，解决弦网模型张量网络表示中的开放问题，还讨论了广义对称性及Wigner-Eckart定理。
We survey the basics of homological algebra in exact categories in the sense of Quillen. All diagram lemmas are proved directly from the axioms, notably the five lemma, the 3 x 3-lemma and the snake lemma. We briefly discuss exact functors, idempotent completion and weak idempotent completeness. We then show that it is possible to construct the derived category of an exact category without any embedding into abelian categories and we sketch Deligne's approach to derived functors. The construction of classical derived functors with values in an abelian category painlessly translates to exact ca…
Frobenius-Schur Indicators of Dual Fusion Categories and Semisimple Partially Dualized Quasi-Hopf Algebras
本文研究球面融合范畴的对偶范畴中对象的Frobenius-Schur指标，建立特定对象指标间的关系，并推导出半单Hopf代数及其部分对偶化quasi-Hopf代数的指标公式与指数等式。
Yoneda lemma
本文研究富集图范畴中的名称指派性质，并将米田引理推广到富集原图传输之间的连续变换上。
Key lemma and universal localization
本文研究了在任何基变换下仍保持局部化的无穷范畴局部化方法及其实现原理。
Homological Lemmas in a Non-pointed Context
本文证明经典同调引理的非点态版本在配备适当偏序单余反射子范畴的正则原模范畴中成立，并给出酉环、布尔代数、海廷代数、MV-代数及其拓扑模型和基本拓扑斯对偶范畴等实例。
For a small category $\mathcal{D}$ we define fibrations of simplicial presheaves on the category $\mathcal{D}\timesΔ$, which we call localized $\mathcal{D}$-left fibration. We show these fibrations can be seen as fibrant objects in a model structure, the localized $\mathcal{D}$-covariant model structure, that is Quillen equivalent to a category of functors valued in simplicial presheaves on $\mathcal{D}$, where the Quillen equivalence is given via a generalization of the Grothendieck construction. We use our understanding of this construction to give a detailed characterization of fibrations a…
The internal Yoneda lemma for locally Cartesian closed $\infty$-categories
在有限完备、局部笛卡尔闭的∞-范畴\(\C\)中阐述并证明约当引理等的内部版本，证明仅用有限极限等，不依赖外部约当引理，并应用于初等∞-拓扑斯，恢复相关定理。
In (B-Gran, 2004), was given a categorical formulation of the Shifting Lemma which is a characterization of the Congruence Modular Varieties among all the variety of Universal Algebra, introduced in (Gumm, 1983). Starting from a characterization of this Shifting Lemma by a property of the fibers of the fibration of points $¶\EE$ (B, 2005), on the model of what happens for Mal'tsev categories, we shall investigate three directions:\\ 1) a new one: in following the golden thread of abelian split epimorphisms naturally provided by this characterization;\\ 2) a more or less expected one: in measur…
We obtain two formulae for the higher Frobenius-Schur indicators: one for a spherical fusion category in terms of the twist of its center and the other one for a modular tensor category in terms of its twist. The first one is a categorical generalization of an analogous result by Kashina, Sommerhauser, and Zhu for Hopf algebras, and the second one extends Bantay's 2nd indicator formula for a conformal field theory to higher degree. These formulae imply the sequence of higher indicators of an object in these categories is periodic. We define the notion of Frobenius-Schur (FS-)exponent of a pivo…
We introduce subclasses of exact categories in terms of admissible intersections or admissible sums or both at the same time. These categories are recently studied by Brüstle, Hassoun, Shah, Tattar and Wegner to give characterisations of quasi-abelian and abelian categories. We also generalise the Schur lemma to the context of exact categories.
One-sided exact categories appear naturally as instances of Grothendieck pretopologies. In an additive setting they are given by considering the one-sided part of Keller's axioms defining Quillen exact categories. We study one-sided exact additive categories and a stronger version defined by adding the one-sided part of Quillen "obscure axiom". We show that some homological results, such as the Short Five Lemma and the 3 X 3 Lemma, can be proved in our context. We also note that the derived category of a one-sided exact additive category can be constructed.
In this note, we show that the long $n$-exact sequence theorem holds in any $n$-abelian category. As an application, the $n$-abelian version of Wakamatsu's Lemma is given.
We prove a stronger version of the octahedral axiom in a pre-triangulated category. The proof uses a new lemma about exact sequences in pointed additive categories which is based on a weak converse of the snake lemma.
A symmetric monoidal Comparison Lemma
本文证明比较引理的对称幺半版本，建立松弛与强对称幺半超层范畴间的等价，并用于获得紧支撑上同调理论结果的松弛对称幺半推广。
We introduce generalized Frobenius-Schur indicators for pivotal categories. In a spherical fusion category C, an equivariant indicator of an object in C is defined as a functional on the Grothendieck algebra of the quantum double Z(C) via generalized Frobenius-Schur indicators. The set of all equivariant indicators admits a natural action of the modular group. Using the properties of equivariant indicators, we prove a congruence subgroup theorem for modular categories. As a consequence, all modular representations of a modular category have finite images, and they satisfy a conjecture of Eholz…
We use the terms $\infty$-categories and $\infty$-functors to mean the objects and morphisms in an $\infty$-cosmos: a simplicially enriched category satisfying a few axioms, reminiscent of an enriched category of fibrant objects. Quasi-categories, Segal categories, complete Segal spaces, marked simplicial sets, iterated complete Segal spaces, $θ_n$-spaces, and fibered versions of each of these are all $\infty$-categories in this sense. Previous work in this series shows that the basic category theory of $\infty$-categories and $\infty$-functors can be developed only in reference to the axioms …
We give a computational approach to theorem proving in homological algebra. This approach is based on computations in the free abelian category of an additive category $\mathbf{A}$. We show that the free abelian category is amenable to explicit computations whenever we can decide homotopy equations in $\mathbf{A}$. As some consequences of our investigations, we recover Dowker's explicit formula for the connecting homomorphism $\partial$ in the snake lemma, we find a universal sense in which $\partial$ is unique, and we give a refined version of the 5-lemma.
Monobricks in extriangulated length categories
本文在extriangulated长度范畴中引入monobricks概念，证明其与左Schur子范畴的双射关系，并扩展了Enomoto关于abelian长度范畴的研究。
Everyone knows that if you have a bivariant homology theory satisfying a base change formula, you get an representation of a category of correspondences. For theories in which the covariant and contravariant transfer maps are in mutual adjunction, these data are actually equivalent. In other words, a 2-category of correspondences is the universal way to attach to a given 1-category a set of right adjoints that satisfy a base change formula. Through a bivariant version of the Yoneda paradigm, I give a definition of correspondences in higher category theory and prove an extension theorem for biv…
The theory of abelian categories proved very useful, providing an axiomatic framework for homology and cohomology of modules over a ring and, in particular, of abelian groups. For many years, a similar categorical framework has been lacking for non-abelian (co)homology, the subject of which includes the categories of non-abelian groups, of rings, Lie algebras, etc. The notion of semi-abelian category due to Janelidze, Marki and Tholen allows a unified study of many important homological properties of such categories. For instance, in any semi-abelian category, the classical diagram lemmas (the…
We consider a notion of exact sequences in any -not necessarily exact- pointed category relative to a given (E;M)-factorization structure. We apply this notion to introduce and investigate a new notion of exact sequences of semimodules over semirings relative to the canonical image factorization. Several homological results are proved using the new notion of exactness including some restricted versions of the Short Five Lemma and the Snake Lemma opening the door for introducing and investigating homology objects in such categories. Our results apply in particular to the variety of commutative …
Conservative functors to pointed categories
本文引入预指向范畴概念，研究指向范畴的保守函子，推广经典引理并建立普遍指向化构造。
In this article, we study the relationship among maximal green sequences, complete forward hom-orthogonal sequences and stability functions in abelian length categories. Mainly, we firstly give a one-to-one correspondence between maximal green sequences and complete forward hom-orthogonal sequences via mutual constructions, and then prove that a maximal green sequence can be induced by a central charge if and only if it satisfies crossing inequalities. As applications, we show that crossing inequalities can be computed by $c$-vectors for finite dimensional algebras; finally, we give the Rotati…
One-sided exact categories are obtained via a weakening of a Quillen exact category. Such one-sided exact categories are homologically similar to Quillen exact categories: a one-sided exact category $\mathcal{E}$ can be (essentially uniquely) embedded into its exact hull ${\mathcal{E}}^{\textrm{ex}}$; this embedding induces a derived equivalence $\textbf{D}^b(\mathcal{E}) \to \textbf{D}^b({\mathcal{E}}^{\textrm{ex}})$. Whereas it is well known that Quillen's obscure axioms are redundant for exact categories, some one-sided exact categories are known to not satisfy the corresponding obscure axi…
A Homological View of Categorical Algebra
该研究为范畴代数的同调与同伦方法构建基础，选取从z-正合范畴到阿贝尔范畴等多层公理框架并厘清其关系，开发含基本图引理的范畴计算工具，发现其可用于弱范畴环境。
In this paper, we study the ideal approximation theory associated to almost $n$-exact structures in the $n$-exangulated category. The notions of $n$-ideal cotorsion pairs and $n$-$\mathbb{F}$-phantom morphisms are introduced and studied. In particular, let $\mathscr{C}$ be an extriangulated category which satisfies the condition (WIC) and $\mathcal{T}$ be a nicely embedded $n$-cluster tilting subcategory of $\mathscr{C}$, we prove Salce's Lemma in $\mathcal{T}$.
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