John W. Barrett, Marco Mackaay
The representation theory for categorical groups is constructed. Each categorical group determines a monoidal bicategory of representations. Typically, these categories contain representations which are indecomposable but not irreducible. A simple example is computed in explicit detail.
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A key invariant of a braided categorical group is its quadratic form, introduced by Joyal and Street. We show that the categorical group is braided equivalent to a simultaneously skeletal and strictly associative one if and only if the polarization of this quadratic form is the symmetrization of a bilinear form. This generalizes the result of Johnson-Osorno that all Picard groupoids can simultaneously be strictified and skeletalized, except that in the braided case there is a genuine obstruction.
This is intended as a self-contained introduction to the representation theory developed in order to create a Poincare 2-category state sum model for Quantum Gravity in 4 dimensions. We review the structure of a new representation 2-category appropriate to Lie 2-group symmetries and discuss its application to the problem of finding a state sum model for Quantum Gravity. There is a remarkable richness in its details, reflecting some desirable characteristics of physical 4-dimensionality. We begin with a review of the method of orbits in Geometric Quantization, as an aid to the intuition that th…
We analyze the action of the Brauer-Picard group of a pointed fusion category on the set of Lagrangian subcategories of its center. Using this action we compute the Brauer-Picard groups of pointed fusion categories associated to several classical finite groups. As an application, we construct new examples of weakly group-theoretical fusion categories.
We consider compatible group structures on a $V$-category, where $V$ is a quantale, and we study the topological and algebraic properties of such groups. Examples of such structures are preordered groups, metric and ultrametric groups, probabilistic (ultra)metric groups. In particular, we show that, when $V$ is a frame, symmetric $V$-groups satisfy very strong categorical-algebraic properties, typical of the category of groups. In particular, symmetric $V$-groups form a protomodular category.
We prove that the 2-category of symmetric categorical groups have enough projective and injective objects. This was conjectured by Bourn and Vitale in 2002 and was anounced recently by Fang Huang, Shao-Han Chen, Wei Chen and Zhu-Jun Zheng in arXiv:1006.4677 with wrong proof.
To $2$-categorify the theory of group representations, we introduce the notions of the $3$-representation of a group in a strict $3$-category and the strict $2$-categorical action of a group on a strict $2$-category. We also $2$-categorify the concept of the trace by introducing the $2$-categorical trace of a $1$-endomorphism in a strict $3$-category. For a $3$-representation $ρ$ of a group $G$ and an element $f$ of $G$, the $2$-categorical trace $\mathbb{T}r_2 ρ_f $ is a category. Moreover, the centralizer of $f$ in $G$ acts categorically on this $2$-categorical trace. We construct the induce…
We show that either of the two reasonable choices for the category of compact quantum groups is nice enough to allow for a plethora of universal constructions, all obtained "by abstract nonsense" via the adjoint functor theorem. This approach both recovers constructions which have appeared in the literature, such as the quantum Bohr compactification of a locally compact semigroup, and provides new ones, such as the coproduct of a family of compact quantum groups, and the compact quantum group freely generated by a locally compact quantum space. In addition, we characterize epimorphisms and mon…
Equivalence of the categories of categorical groups and cssc-crossed modules
本文基于前期研究中范畴群与cssc-交叉模的双向构造，证明二者对应关系可定义函子，建立了范畴群范畴与cssc-交叉模范畴的等价性。
A finite tensor category is called pointed if all its simple objects are invertible. We find necessary and sufficient conditions for two pointed semisimple categories to be dual to each other with respect to a module category. Whenever the dual of a pointed semisimple category with respect to a module category is pointed, we give explicit formulas for the Grothendieck ring and for the associator of the dual. This leads to the definition of categorical Morita equivalence on the set of all finite groups and on the set of all pairs (G, w), where G is a finite group and w is a 3-cocycle on G. A gr…
Prolongations of a group extension can be studied in a more general situation that we call group extensions of the co-type of a crossed module. Cohomology classification of such extensions is obtained by applying the obstruction theory of monoidal functors.
This is an expanded version of the lectures given by the author on the 3rd school "Lie algebras, algebraic groups and invariant theory" in Togliatti, Russia. In these notes we explain the concept of a categorical Kac-Moody action by studying an example of the category of rational representations of a general linear group in positive characteristic. We also deal with some more advanced topics: a categorical action on the polynomial representations and crystals of categorical actions.
From cssc-crossed modules to categorical groups
本文为任意 cssc-交叉模构造相干范畴群，结合前文结果，将在续篇中证明范畴群范畴与 cssc-交叉模范畴之间的等价性。
We introduce a notion of c-group, which is a group up to congruence relation and consider the corresponding category. Extensions, actions and crossed modules (c-crossed modules) are defined in this category and the semi-direct product is constructed. We prove that each categorical group gives rise to c-groups and to a c-crossed module, which is a connected, special and strict c-crossed module in the sense defined by us. The results obtained here will be applied in the proof of an equivalence of the categories of categorical groups and connected, special and strict c-crossed modules.
We present a categorical point of view on dynamical quantum groups in terms of categories of Harish-Chandra bimodules. We prove Tannaka duality theorems for forgetful functors into the monoidal category of Harish-Chandra bimodules in terms of a slight modification of the notion of a bialgebroid. Moreover, we show that the standard dynamical quantum groups $F(G)$ and $F_q(G)$ are related to parabolic restriction functors for classical and quantum Harish-Chandra bimodules. Finally, we exhibit a natural Weyl symmetry of the parabolic restriction functor using Zhelobenko operators and show that it…
By regarding the classical non abelian cohomology of groups from a 2-dimensional categorical viewpoint, we are led to a non abelian cohomology of groupoids which continues to satisfy classification, interpretation and representation theorems generalizing the classical ones. This categorical approach is based on the fact that if groups are regarded as categories, then, on the one hand, crossed modules are 2-groupoids and, cocycles are lax 2-functors and the cocycle conditions are precisely the coherence axioms for lax 2-functors, and, on the other hand group extensions are fibrations of categor…
In this paper we introduce the notion of a categorical Mackey functor. This categorical notion allows us to obtain new Mackey functors by passing to Quillen's $K$-theory of the corresponding abelian categories. In the case of an action by monoidal autoequivalences on a monoidal category the Mackey functor obtained at the level of Grothendieck rings has in fact a Green functor structure.
Right-preordered groups from a categorical perspective
本文从范畴论角度研究右预序群，刻画其极限、余极限、正合性与分裂扩张，并指出其范畴与幺半群范畴的相似性及有序群半直积的自然右预序。
We study the gerbal representations of a finite group $G$ or, equivalently, module categories over Ostrik's category $Vec_G^α$ for a 3-cocycle $α$. We adapt Bartlett's string diagram formalism to this situation to prove that the categorical character of a gerbal representation is a module over the twisted Drinfeld double $D^α(G)$. We interpret this twisted Drinfeld double in terms of the inertia groupoid of a categorical group.
If $X$ is a topological group, then its fundamental groupoid $π_1X$ is a group-groupoid which is a group object in the category of groupoids. Further if $X$ is a path connected topological group which has a simply connected cover, then the category of covering spaces of $X$ and the category of covering groupoids of $π_1X$ are equivalent. In this paper we prove that if $(X,x_0)$ is an $H$-group, then the fundamental groupoid $π_1X$ is a categorical group. This enable us to prove that the category of the covering spaces of an $H$-group $(X,x_0)$ is equivalent to the category of covering groupoid…
A Mackey type decomposition for group actions on abelian categories is described. This allows us to define new Mackey functors which associates to any subgroup the $K$-theory of the corresponding equivariantized abelian category. In the case of an action by tensor autoequivalences the Mackey functor at the level of Grothendieck rings has a Green functor structure. As an application we give a description of the Grothendieck rings of equivariantized fusion categories under group actions by tensor autoequivalences on graded fusion categories.
We extend the notion of the nerve of a category for a small class of crossed simplicial groups, explicitly describing them using generators and relations. We do this by first considering a generalised bar construction of a group before looking at twisted versions of some of these nerves. As an application we show how we can use the twisted nerves to give equivariant versions of certain derived stacks.
We classify various types of graded extensions of a finite braided tensor category $\cal B$ in terms of its $2$-categorical Picard groups. In particular, we prove that braided extensions of $\cal B$ by a finite group $A$ correspond to braided monoidal $2$-functors from $A$ to the braided $2$-categorical Picard group of $\cal B$ (consisting of invertible central $\cal B$-module categories). Such functors can be expressed in terms of the Eilnberg-Mac~Lane cohomology. We describe in detail braided $2$-categorical Picard groups of symmetric fusion categories and of pointed braided fusion categorie…
We give explicit and elementary constructions of the categorical extensions of a torus by the circle and discuss an application to loop group extensions. Examples include maximal tori of simple and simply connected compact Lie groups and the tori associated to the Leech and Niemeyer lattices. We obtain the extraspecial 2-groups as the isomorphism classes of categorical fixed points under an involution action.
In this paper we introduce a description of ordered groupoids as a particular type of double categories. This enables us to turn Lawson's correspondence between ordered groupoids and left-cancellative categories into a biequivalence. We use this to identify which ordered functors are maps of sites in the sense that they give rise to geometric morphisms between the induced sheaf categories, and establish a Comparison Lemma for maps between Ehresmann sites.
We introduce and develop a categorification of the theory of Real representations of finite groups. In particular, we generalize the categorical character theory of Ganter--Kapranov and Bartlett to the Real setting. Given a Real representation of a group $\mathsf{G}$, or more generally a finite categorical group, on a linear category, we associate a number, the modified secondary trace, to each graded commuting pair $(g, ω) \in \mathsf{G} \times \hat{\mathsf{G}}$, where $\hat{\mathsf{G}}$ is the background Real structure on $\mathsf{G}$. This collection of numbers defines the Real $2$-characte…
We develop a theory of twisted actions of categorical groups using a notion of semidirect product of categories. We work through numerous examples to demonstrate the power of these notions. Turning to representations, which are actions that respect vector space structures, we establish an analog of Schur's lemma in this context. Keeping new terminology to a minumum, we concentrate on examples exploring the essential new notions introduced.
We set up a fibred categorical theory of obstruction and classification of morphisms that specializes to the one of monoidal functors between categorical groups and also to the Schreier-Mac Lane theory of group extensions. Further applications are provided, as for example a classification of unital associative algebra extensions with non-abelian kernel in terms of Hochschild cohomology.
We prove that the KLR algebra associated with the cyclic quiver of length $e$ is a subquotient of the KLR algebra associated with the cyclic quiver of length $e+1$. We also give a geometric interpretation of this fact. This result has an important application in the theory of categorical representations. We prove that a category with an action of $\widetilde{\mathfrak{sl}}_{e+1}$ contains a subcategory with an action of $\widetilde{\mathfrak{sl}}_{e}$. We also give generalizations of these results to more general quivers and Lie types.
Categorical-Algebraic Properties of Lattice-ordered Groups
本文研究了格序群的范畴-代数性质，证明其具有纤维-wise代数上闭、算术和强原型模的特性，并指出该范畴不具作用可及性，同时展示了格序阿贝尔群子变种的代数一致性。
We discuss the relationship between (co)homology groups and categorical diagonalization. We consider the category of chain complexes in the category of finitely generated free modules on a commutative ring. For a fixed chain complex with zero maps as an object, a chain map from the object to another chain complex is defined, and the chain map introduce a mapping cone. We found that the fixed object is isomorphic to the (co)homology groups of the codomain of the chain map if and only if the chain map is injective to the cokernel of differentials of the codomain chain complex and the mapping con…
Model theoretic internality provides conditions under which the group of automorphisms of a model over a reduct is itself a definable group. In this paper we formulate a categorical analogue of the condition of internality, and prove an analogous result on the categorical level. The model theoretic statement is recovered by considering the category of definable sets.
We give a definition of an operad with general groups of equivariance suitable for use in any symmetric monoidal category with appropriate colimits. We then apply this notion to study the 2-category of algebras over an operad in Cat. We show that any operad is finitary, that an operad is cartesian if and only if the group actions are nearly free (in a precise fashion), and that the existence of a pseudo-commutative structure largely depends on the groups of equivariance. We conclude by showing that the operad for strict braided monoidal categories has two canonical pseudo-commutative structure…
Categorical bundles provide a natural framework for gauge theories involving multiple gauge groups. Unlike the case of traditional bundles there are distinct notions of triviality, and hence also of local triviality, for categorical bundles. We study categorical principal bundles that are product bundles in the categorical sense, developing the relationship between functorial sections of such bundles and trivializations. We construct functorial cocycles with values in categorical groups using a suitable family of locally defined functions on the object space of the base category. Categorical p…
Consciousness in a Higher Categorical Context
本文在更高范畴背景下，通过微观可逆性和意识概念，提出两种对偶的Segal范畴表示方法。
The Right Angled Artin Group Functor as a Categorical Embedding
本文证明可通过纯代数方式在群中刻画直角 Artin 群，并在群同态中刻画图同态，进而给出从 raag 恢复原图的新算法。
Categorical forms of the rigid direct-system representation for finite local-unit-aligned totally ordered monoids
本文研究有限局部单位对齐全序幺半群的刚性直接系统表示的范畴性，证明其对严格块态射和固有$\ au$-相容同态均为自函子，并给出相应等价性。
A categorical representation of games
本文用可达性与偏好的多重图表示博弈，构造完备且余完备的博弈范畴，并推广 Nash 均衡及保持均衡的子范畴。
Any $Γ$-graded categorical group is determined by a factor set of a categorical group. This paper studies the factor set of the group $Γ$ with coefficients in the categorical group of the type $(Π,A).$ Then, an interpretation of the notion of $Γ-$operator $3-$cocycle is presented and the proof of cohomological classification theorem for the a $Γ-$graded Gr-category is also presented.
Representation theory for categorical symmetries
本文研究范畴对称性对量子场论扩展算符的作用，引入高阶管范畴与高阶管代数，证明扭变扇区扩展算符按高阶表示变换，并在三维情形下通过可逆与非可逆对称范畴的多种例子推广了二维已知构造。
Categorical Bass-Serre Theory for Tree Quotients
提出内部范畴论框架统一经典和 profinite Bass-Serre 理论，通过 Bass-Serre 范畴中群作用图的基本群恢复原群。
Categorical representation of DRC-semigroups
本文证明DRC-半群范畴与特定双序范畴范畴同构，并构造左伴随函子以确立自由DRC-半群的存在性。
We provide a construction for categorical representation learning and introduce the foundations of "$\textit{categorifier}$". The central theme in representation learning is the idea of $\textbf{everything to vector}$. Every object in a dataset $\mathcal{S}$ can be represented as a vector in $\mathbb{R}^n$ by an $\textit{encoding map}$ $E: \mathcal{O}bj(\mathcal{S})\to\mathbb{R}^n$. More importantly, every morphism can be represented as a matrix $E: \mathcal{H}om(\mathcal{S})\to\mathbb{R}^{n}_{n}$. The encoding map $E$ is generally modeled by a $\textit{deep neural network}$. The goal of repre…
It has been proven by Schupp and Bergman that the inner automorphisms of groups can be characterized purely categorically as those group automorphisms that can be coherently extended along any outgoing homomorphism. One is thus motivated to define a notion of (categorical) inner automorphism in an arbitrary category, as an automorphism that can be coherently extended along any outgoing morphism, and the theory of such automorphisms forms part of the theory of covariant isotropy. In this paper, we prove that the categorical inner automorphisms in any category $\mathsf{Group}^{\mathcal{J}}$ of p…
Categorical (Co)Limits of Quantum Graphs
本文通过量子图作为左理想的新表征，提出不依赖表示的量子图映射概念，并引入量子图的范畴（余）极限，同时探讨图的另一种量子化方式。
In this paper we study loops, neardomains and nearfields from a categorical point of view. By choosing the right kind of morphisms, we can show that the category of neardomains is equivalent to the category of sharply 2-transitive groups. The other categories are also shown to be equivalent with categories whose objects are sets of permutations with suitable extra properties. Up to now the equivalence between neardomains and sharply 2-transitive groups was only known when both categories were equipped with the obvious isomorphisms as morphisms. We thank Hubert Kiechle for this observation.
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