P G Romeo, Riya Jose
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.
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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
Differential Bundles in Commutative Algebra and Algebraic Geometry
本文在切范畴框架下刻画交换代数与代数几何场景中的微分丛，证明交换环、仿射概形、概形上的微分丛范畴分别对应模、模反范畴、拟凝聚模层反范畴，拓展了微分丛的理论适用范围。
A tangent category is a category equipped with an endofunctor that satisfies certain axioms which capture the abstract properties of the tangent bundle functor from classical differential geometry. Cockett and Cruttwell introduced differential bundles in 2017 as an algebraic alternative to vector bundles in an arbitrary tangent category. In this paper, we prove that differential bundles in the category of smooth manifolds are precisely vector bundles. In particular, this means that we can give a characterisation of vector bundles that exhibits them as models of a tangent categorical essentiall…
We show that the derived category of a curve is embedded into the derived category of the moduli space of vector bundles on the curve of coprime rank and degree. We also generalize the semiorthogonal decomposition constructed by Narasimhan and Belmans-Mukhopadhyay. Finally, we produce a one-dimensional family of ACM bundles over the moduli space.
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…
Categories of bundles and categories of chains
本文重新审视纤维丛、向量丛和主G-丛等经典丛概念，讨论由丛范畴导出的链范畴，并验证这些范畴是具有子对象的范畴。
Differential bundles as functors from free modules
本文提出微分束与结构类别之间的等价关系，并通过函子构造方法证明其等价性。
From Abelianization to Tangent Categories
本文证明群范畴在阿贝尔化诱导下构成切范畴，通过引入线性指派推广该构造，证明有限余完备正则幺元范畴均为切范畴，并提供了多个新示例。
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…
On the normal functor in the category of smooth vector bundles
本文研究光滑实向量束范畴中的正常函子，探讨在迭代两次后出现的对称现象，通过发展双向量束及某些映射类的拉回与商理论，揭示正常函子的自然性。
The differential bundles of the geometric tangent category of an operad
本文通过operadic仿射概形的切范畴研究其几何，证明包络operad的仿射概形切范畴等于切片切范畴，并由此将微分丛分类为operadic意义下的A-模。
The formal theory of tangentads PART II
本文扩展了切线ads的正式理论，涵盖微分对象、微分流束及其连接，揭示了这些构造在切线范畴中的普遍性质及核心贡献。
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…
For a 2-category 2C we associate a notion of a principal 2C-bundle. In case of the 2-category of 2-vector spaces in the sense of M.M. Kapranov and V.A. Voevodsky this gives the 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 a lot of 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. A calculation bas…
The splitting of the Frobenius direct image of line bundles on toric varieties is used to explicitly construct an orthogonal basis of line bundles in the derived category D^b(X) where X is a Fano toric variety with (almost) maximal Picard number.
The Rosický Tangent Categories of Algebras over an Operad
本文证明任意 operad 的代数范畴及其反范畴均为切范畴，切丛分别由半直积与 Kähler 微分模构造且互为伴随，并由此从 operad 获得丰富的切范畴与 Cartesian 微分范畴例子。
Generic bundles over a localic category
本文构造了分类局部范畴和群胚，用于带有逻辑结构的各种丛，并证明了局部群胚比拓扑斯能分类更多种类的逻辑理论。
Reducing and Classifying Fiber Bundles over Small Categories
将小范畴上的纤维丛视为到小范畴范畴的局部常值函子，通过Grothendieck构造得到具有双纤维化投影的全范畴，并利用单值性分类纤维丛，证明规范群同构于单值子群的中心化子。
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 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…
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.
Stratifolds are considered from a categorical point of view. We show among others that the category of stratifolds fully faithfully embeds into the category of ${\mathbb R}$-algebras as does the category of smooth manifolds. We prove that a variant of the Serre-Swan theorem holds for stratifolds. In particular, the category of vector bundles over a stratifold is shown to be equivalent to the category of vector bundles over an associated affine scheme although the latter is in general larger than the stratifold itself.
A tensor model structure is constructed on the category of chain complexes of presheaves of R-modules for a sheaf of rings R in a Grothendieck topos. If the topos has enough points, then the homotopy category is equivalent to the derived category. Some t-structures, generalizing the perverse t-structures, are constructed on the derived category.
We describe the moduli space of extensions in the model category of simplicial presheaves. This article can be seen as a generalization of Blomgren-Chacholski results in the case of simplicial sets. Our description of the moduli space of extensions treat the equivariant and the nonequivariant case in the same setting. As a new result, we describe the moduli space of M-bundles over a fixed space X, when M is a simplicial monoid. Moreover, the moduli space of M-bundles is classified by the classifying space of the simplicial submonoid generated by homotopy invertible elements of M. We give a gen…
Principal 2-bundles and quotient 2-stacks
本文在双景的二范畴语境中推广主丛与商叠，引入主2-丛概念，构造商前2-叠，并证明在子典范双景及温和条件下其为2-叠。
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.
The goal of this article is to describe several presentations of the infinity category of algebras over some monad on the infinity category of chain complexes.
Etale spaces of residuated lattices
本文引入并研究了拓扑空间上剩余格的丛与étalé概念，证明了étalé范畴在丛范畴中的余反射性，提供了基于连续映射的étalé转移方法，并定义了反变截面函子。
We construct a category of flat vector bundles on an elliptic curve. It arises in the representation theory of quantum affine algebras and carries meromorphic braided structure with singularities on the diagonal of the square of the curve.
A Categorical Framework for the Direct Integration of Banach Spaces
该研究构建了抽象巴拿赫丛的拟阿贝尔范畴作为范畴框架，将直接积分定义为输入抽象巴拿赫丛对象、输出巴拿赫空间的函子，进而定义了层的直接积分概念。
Bundles of metric structures as left ultrafunctors
研究完备度量结构的超范畴，证明左超函子与连续理论模型丛的等价性，统一了Banach空间丛、C*-代数连续场等概念。
Smooth one-dimensional topological field theories are vector bundles with connection
本文证明光滑一维拓扑场论与带联络的向量丛等价，通过定义编码切割律的光滑边界范畴并利用下降与配边假设实现。
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