K. S. S. Nambooripad 引入了一类有趣的范畴，称为正规范畴，即具有子对象、态射可分解且具有足够多锥的范畴。这些正规范畴在正则半群结构的研究中起着基础性作用。在文献[6]中，我们讨论了链丛范畴和链范畴。本文重新审视了丛的经典概念，包括纤维丛、向量丛和主G-丛，并讨论了由丛范畴导出的链范畴。此外，验证了这些范畴是具有子对象的范畴。
英文摘要：
K. S. S. Nambooripad introduced an interesting class of categories known as normal categories, which are categories with subobjects, morphisms admitting factorization and having sufficiently many cones. These normal categories plays fundamental role in the study of structure of regular semigroups. In [6] we discussed the category of chain bundles and category of chains. In the present paper revisits classical notions of bundles, including fibre bundles, vector bundles, and principal G-bundles, and discuss the chain categories arising from the category of bundles. Moreover, its is verified that these categories are categories with subobjects.
Differential lambda-categories were introduced by Bucciarelli et al. as models for the simply typed version of the differential lambda-calculus of Ehrhard and Regnier. A differential lambda-category is a cartesian closed differential category of Blute et al. in which the differential operator is compatible with the closed structure. We prove that any differential lambda-category is equipped with a canonical strong commutative monad whose construction resembles that of the tangent bundle in the category of smooth manifolds. Most of the results of this note remain valid in an arbitrary cartesian…
In [7], we introduced the category of chain bundles with examples and established its significance. Here we shall describe certain categories which we call the category of chains in the chain bundle category and discuss some interesting categorical properties of this chain categories
For a category with subobjects and factorization, here we describe a new category which we call category of chain bundles and it is shown that this new category is also a category with subobjects and admits factorization under certain restriction. Further we provide several examples of bundle categories and discuss some interesting properties of these categories.
Tangent categories are categories equipped with a tangent functor: an endofunctor with certain natural transformations which make it behave like the tangent bundle functor on the category of smooth manifolds. They provide an abstract setting for differential geometry by axiomatizing key aspects of the subject which allow the basic theory of these geometric settings to be captured. Importantly, they have models not only in classical differential geometry and its extensions, but also in algebraic geometry, combinatorics, computer science, and physics. This paper develops the theory of "different…
We introduce the notions of multiplier C*-category and continuous bundle of C*-categories, as the categorical analogues of the corresponding C*-algebraic notions. Every symmetric tensor C*-category with conjugates is a continuous bundle of C*-categories, with base space the spectrum of the C*-algebra associated with the identity object. We classify tensor C*-categories with fibre the dual of a compact Lie group in terms of suitable principal bundles. This also provides a classification for certain C*-algebra bundles, with fibres fixed-point algebras of O_d.
We give an account, in terms of fibered categories and their fibrewise duals, of aspects of the theory of bundle functors and star-bundle functors in differential geometry.
I categorify the definition of fibre bundle, replacing smooth manifolds with differentiable categories, Lie groups with coherent Lie 2-groups, and bundles with a suitable notion of 2-bundle. To link this with previous work, I show that certain 2-categories of principal 2-bundles are equivalent to certain 2-categories of (nonabelian) gerbes. This relationship can be (and has been) extended to connections on 2-bundles and gerbes. The main theorem, from a perspective internal to this paper, is that the 2-category of 2-bundles over a given 2-space under a given 2-group is (up to equivalence) indep…
We construct the categories of standard vector bundles over schemes and define direct sum and tensor product. These categories are equivalent to the usual categories of vector bundles with additional properties. The tensor product is strictly associative, strictly commutative with line bundles, and strictly functorial on base change.
There are several ways to construct omega-categories from combinatorial objects such as pasting schemes or parity complexes. We make these constructions into a functor on a category of chain complexes with additional structure, which we call augmented directed complexes. This functor from augmented directed complexes to omega-categories has a left adjoint, and the adjunction restricts to an equivalence on a category of augmented directed complexes with good bases. The omega-categories equivalent to augmented directed complexes with good bases include the omega-categories associated to globes, …
The category of strict omega-categories has an important full subcategory whose objects are the simple omega-categories freely generated by planar trees or by globular cardinals. We give a simple description of this subcategory in terms of chain complexes, and we give a similar description of the opposite category, the category of finite discs, in terms of cochain complexes. Berger has shown that the category of simple omega-categories has a filtration by iterated wreath products of the simplex category. We generalise his result by considering wreath products of categories of chain complexes o…
K. S. S. Nambooripad intoduced nornal categories to enable to describe the structure of regular semigroups fully. In this paper we describe the ideal categories of the regular semigroup $OX_n,$ of non-invertible order-preserving transformations on a finite chain $X_n=\{1\leq 2\leq \cdots \leq n \}$ which are normal categories. Further it is shown that the principal left ideal category of $OX_n$ as the power set category $P_o(X_n)$ of $OX_n$ and the principal right ideal category as $\prod_o(X_n)$ category of ordered partitions of $X_n$ and described the cone semigroup $T\mathscr L(OX_n)$ and p…
We show that the projectivization of the exceptional rank 2 vector bundle on an arbitrary smooth V14 Fano threefold after a certain natural flop turns into the projectivization of an instanton vector bundle on a smooth cubic threefold. And vice versa, starting from a smooth cubic threefold with an instanton vector bundle of charge 2 on it we reconstruct V14 threefold. Relying on the geometric properties of the above correspondence we prove that the orthogonals to the exceptional pairs in the bounded derived categories of coherent sheaves on a smooth V14 threefold and on the corresponding cubic…
We introduce a cohomological invariant arising from a class in nonabelian cohomology. This invariant generalizes the Dixmier-Douady class and encodes the obstruction to a C*-algebra bundle being the fixed-point algebra of a gauge action. As an application, the duality breaking for group bundles vs. tensor C*-categories with non-simple unit is discussed in the setting of Nistor-Troitsky gauge-equivariant K-theory: there is a map assigning a nonabelian gerbe to a tensor category, and "triviality" of the gerbe is equivalent to the existence of a dual group bundle. At the C*-algebraic level, this …
Stellenbosch University（斯泰伦博斯大学）；National Institute for Theoretical and Computational Sciences（国家理论与计算科学研究所）；Université Paris Cité, CNRS, IRIF（巴黎西岱大学，法国国家科学研究中心，IRIF）；National Research Foundation of South Africa（南非国家研究基金会）
We construct an equivalence of categories from a strong categorical sl(2) action, following the work of Chuang-Rouquier. As an application, we give an explicit, natural equivalence between the derived categories of coherent sheaves on cotangent bundles to complementary Grassmannians.
We present a construction of a 2-Hilbert space of sections of a bundle gerbe, a suitable candidate for a prequantum 2-Hilbert space in higher geometric quantisation. We introduce a direct sum on the morphism categories in the 2-category of bundle gerbes and show that these categories are cartesian monoidal and abelian. Endomorphisms of the trivial bundle gerbe, or higher functions, carry the structure of a rig-category, which acts on generic morphism categories of bundle gerbes. We continue by presenting a categorification of the hermitean metric on a hermitean line bundle. This is achieved by…
We describe dual notions of tangent bundle for an infinity-topos, each underlying a tangent infinity-category in the sense of Bauer, Burke and the author. One of those notions is Lurie's tangent bundle functor for presentable infinity-categories, and the other is its adjoint. We calculate that adjoint for injective infinity-toposes, where it is given by applying Lurie's tangent bundle on infinity-categories of points.
A categorical principal bundle is a structure comprised of categories that is analogous to a classical principal bundle; examples arise from geometric contexts involving bundles over path spaces. We show how a categorical principal bundle can be constructed from local data specified through transition functors and natural transformations.
We study the general structure of 2-C*-categories closed under conjugation, projections and direct sums. We do not assume units to be simple, i.e. for i_A the 1-unit corresponding to an object A, the space Hom(i_A, i_A) is a commutative unital C*-algebra. We show that 2-arrows can be viewed as continuous sections in Hilbert bundles and describe the behaviour of the fibres with respect to the categorical structure. We give an example of a 2-C*-Category giving rise to bundles of finite Hopf-algebras in duality. We make some remarks concerning Frobenius algebras and Q-systems in the general conte…
Let $F:\mathcal{A}\to \mathcal{B}$ be a left adjoint between abelian categories and let $Ch(F)$ be the induced left adjoint on chain complexes. If the abelian categories $\mathcal{A}$ and $\mathcal{B}$ are equipped with sufficiently nice cotorsion pairs, then we find model structures on their categories of chain complexes. This paper gives novel criteria under which the induced adjunction becomes a Quillen adjunction. In particular, our criteria can be checked on the level of the underlying abelian categories instead of chain complexes. We do the same for adjunctions of $n$ variables and we al…
We construct a new cylinder object for semifree differential graded (dg) categories in the category of dg categories. Using this, we give a practical formula computing homotopy colimits of semifree dg categories. Combining it with the result of Ganatra, Pardon, and Shende, we get a formula computing wrapped Fukaya categories of Weinstein manifolds using their sectorial coverings. This formula has lots of applications including a practical computation of the wrapped Fukaya category of any cotangent bundle or plumbing space. In this paper, we compute wrapped Fukaya categories of cotangent bundle…
We propose a definition of involutive categorical bundle (Fell bundle) enriched in an involutive monoidal category and we argue that such a structure is a possible suitable environment for the formalization of different equivalent versions of spectral data for commutative C*-categories.
Tangent categories provide an axiomatic approach to key structural aspects of differential geometry that exist not only in the classical category of smooth manifolds but also in algebraic geometry, homological algebra, computer science, and combinatorics. Generalizing the notion of (linear) connection on a smooth vector bundle, Cockett and Cruttwell have defined a notion of connection on a differential bundle in an arbitrary tangent category. Herein, we establish equivalent formulations of this notion of connection that reduce the amount of specified structure required. Further, one of our equ…
Chains of model structures arising from cotorsion pairs on extriangulated categories
School of Mathematical Sciences, Zhejiang University of Technology（浙江工业大学数学科学学院）；School of Science, Zhejiang University of Science and Technology（浙江科技大学理学院）；School of Mathematical Sciences, Zhejiang Normal University（浙江师范大学数学与计算机科学学院）；School of Mathematics and Statistics, Changsha University of Science and Technology（长沙理工大学数学与统计学院）
In this paper we show that the Baues-Wirsching complex used to define cohomology of categories is a 2-functor from a certain 2-category of natural systems of abelian groups to the 2-category of chain complexes, chain homomorphism and relative homotopy classes of chain homotopies. As a consequence we derive (co)localization theorems for this cohomology.
Tangent categories provide an axiomatic framework for understanding various tangent bundles and differential operations that occur in differential geometry, algebraic geometry, abstract homotopy theory, and computer science. Previous work has shown that one can formulate and prove a wide variety of definitions and results from differential geometry in an arbitrary tangent category, including generalizations of vector fields and their Lie bracket, vector bundles, and connections. In this paper we investigate differential and sector forms in tangent categories. We show that sector forms in any t…
Parallel transport of a connection in a smooth fibre bundle yields a functor from the path groupoid of the base manifold into a category that describes the fibres of the bundle. We characterize functors obtained like this by two notions we introduce: local trivializations and smooth descent data. This provides a way to substitute categories of functors for categories of smooth fibre bundles with connection. We indicate that this concept can be generalized to connections in categorified bundles, and how this generalization improves the understanding of higher dimensional parallel transport.
We construct a prequantum 2-Hilbert space for any line bundle gerbe whose Dixmier-Douady class is torsion. Analogously to usual prequantisation, this 2-Hilbert space has the category of sections of the line bundle gerbe as its underlying 2-vector space. These sections are obtained as certain morphism categories in Waldorf's version of the 2-category of line bundle gerbes. We show that these morphism categories carry a monoidal structure under which they are semisimple and abelian. We introduce a dual functor on the sections, which yields a closed structure on the morphisms between bundle gerbe…
Tate objects have been studied by many authors. They allow us to deal with infinite dimensional spaces by identifying some more structure. In this article, we set up the theory of Tate objects in stable $(\infty,1)$-categories, while the literature only treats with exact categories. We will prove the main properties expected from Tate objects. This new setting includes several useful examples: Tate objects in the category of spectra for instance, or in the derived category of a derived algebraic object -- which can be thought as structured infinite dimensional vector bundle in derived setting.
We prove that the bounded derived category of the surface S constructed by Barlow admits a length 11 exceptional sequence consisting of (explicit) line bundles. Moreover, we show that in a small neighbourhood of S in the moduli space of determinantal Barlow surfaces, the generic surface has a semiorthogonal decomposition of its derived category into a length 11 exceptional sequence of line bundles and a category with trivial Grothendieck group and Hochschild homology, called a phantom category. This is done using a deformation argument and the fact that the derived endomorphism algebra of the …
We construct exceptional collections of line bundles of maximal length 4 on $S=(C \times D)/G$ which is a surface isogenous to a higher product with $p_g=q=0$ where $G=G(32,27)$ is a finite group of order 32 having number 27 in the list of Magma library. From these exceptional collections, we obtain new examples of quasiphantom categories as their orthogonal complements.
The classical Dold-Kan correspondence is known to admit a categorification in the form of an equivalence between the $\infty$-categories of $2$-simplicial stable $\infty$-categories and connective chain complexes of stable $\infty$-categories. In this work, we extend these concepts to give a categorification of the classical Dwyer-Kan correspondence by showing that the $\infty$-category of $2$-duplicial stable $\infty$-categories is equivalent to the $\infty$-category of conective chain complexes with right adjoints.
