Bundle functors and fibrations

Fibrations as Eilenberg-Moore algebras

We give an elementary exposition of some fundamental facts about fibered (or rather opfibered) categories, in terms of monads and 2-categories. The account avoids any mention of category-valued functors and pseudofunctors.

Monoidal Grothendieck construction

We lift the standard equivalence between fibrations and indexed categories to an equivalence between monoidal fibrations and monoidal indexed categories, namely weak monoidal pseudofunctors to the 2-category of categories. In doing so, we investigate the relation between this `global' monoidal structure where the total category is monoidal and the fibration strictly preserves the structure, and a `fibrewise' one where the fibres are monoidal and the reindexing functors strongly preserve the structure, first hinted by Shulman. In particular, when the domain is cocartesian monoidal, lax monoidal…

微分束作为自由模的函子

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。

Lax colimits and free fibrations in $\infty$-categories

We define and discuss lax and weighted colimits of diagrams in $\infty$-categories and show that the coCartesian fibration associated to a functor is given by its lax colimit. A key ingredient, of independent interest, is a simple characterization of the free Cartesian fibration associated to a a functor of $\infty$-categories. As an application of these results, we prove that lax representable functors are preserved under exponentiation, and also that the total space of a presentable Cartesian fibration between $\infty$-categories is presentable, generalizing a theorem of Makkai and Paré to t…

函子的弱和强纤维化

Weak and Strong Fibrations of Functors

本文提出了一种针对小范畴的同调框架，扩展了代数拓扑的经典不变量到范畴设置。通过构造真正的路径范畴，定义了函子的强和弱纤维化，并引入了纤维化替代和扩展了如Svarc genus和sectional category等同调不变量到小范畴。最后，将该框架应用于小范畴中的运动规划，提供了Farber拓扑复杂度的范畴类比，并消除了现有方法中常见的有限性约束。

分裂双侧2-纤维的虚拟双范畴

Virtual double categories of split two-sided 2-fibrations

本文提出通过虚拟双范畴研究分裂双侧2-纤维，并展示其与2-函子之间的双侧Grothendieck对应。

Unified Functorial Signal Representation I: From Grothendieck fibration to Base structured categories

In this paper we study categories $(F,\mathbf{C},\mathbf{D})$ and $(\mathbb{F},\mathbf{C},\mathbf{Set})$ and prove them to be fibred on $\mathbf{C}$. Then we examine Grothendieck construction in the context of an ordinary functor $F: \mathbf{C} \rightarrow \mathbf{D}$ through the concept of trivial categorification, using an appropriate functor $\mathbf{F}: \mathbf{C} \xrightarrow{F} \mathbf{D} \xrightarrow{I} \mathbf{Cat}$ to construct $\int_{\mathbf{C}^{op}} \bar{\mathbf{F}}$. This category characterizes a functor as an abstract right category action while its dual $\mathcal{X} \rtimes_{\mat…

Framed bicategories and monoidal fibrations

In some bicategories, the 1-cells are `morphisms' between the 0-cells, such as functors between categories, but in others they are `objects' over the 0-cells, such as bimodules, spans, distributors, or parametrized spectra. Many bicategorical notions do not work well in these cases, because the `morphisms between 0-cells', such as ring homomorphisms, are missing. We can include them by using a pseudo double category, but usually these morphisms also induce base change functors acting on the 1-cells. We avoid complicated coherence problems by describing base change `nonalgebraically', using cat…

Lax 幺半伴随、二变量纤维化与配偶演算

Lax monoidal adjunctions, two-variable fibrations and the calculus of mates

本文建立面向 ∞-范畴函子的配偶演算，推广 unstraightening 等价，并通过二变量纤维化刻画 lax/oplax 幺半伴随及 Gray 张量积函子。

A Categorical Model for the Hopf Fibration

We give a description up to homeomorphism of $S^3$ and $S^2$ as classifying spaces of small categories, such that the Hopf map $S^3\to{}S^2$ is the realization of a functor.

Differential bundles and fibrations for tangent 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…

Derived sections of Grothendieck fibrations and the problems of homotopical algebra

The description of algebraic structure of n-fold loop spaces can be done either using the formalism of topological operads, or using variations of Segal's $Γ$-spaces. The formalism of topological operads generalises well to different categories yielding such notions as $\mathbb E_n$-algebras in chain complexes, while the $Γ$-space approach faces difficulties. In this paper we discuss how, by attempting to extend the Segal approach to arbitrary categoires, one arrives to the problem of understanding "weak" sections of a homotopical Grothendieck fibration. We propose a model for such sections, c…

Fibrations and Yoneda's lemma in an $\infty$-cosmos

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 …

双变量纤维化、分解系统与张成空间的$\infty$-范畴

Two-variable fibrations, factorisation systems and $\infty$-categories of spans

本文在Barwick适当三元组框架下证明张成空间$\infty$-范畴的万有性质，引入正交适当三元组并证明其自等价性，进而统一推广了展开定理、正交分解系统、双变量纤维化及参数化伴随等多个范畴论结果。

在 Wolfram 语言中使用 Categorica I：图表、函子与纤维化

Applied Category Theory in the Wolfram Language using Categorica I: Diagrams, Functors and Fibrations

本文介绍了基于 Wolfram 语言的开源计算范畴论框架 Categorica，支持范畴结构操作、泛性质推理及自动化图表定理证明，首篇聚焦图表、函子与广群的处理及证明能力。

关于光滑向量束范畴中正常函子的研究

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。

操作素纤维与 unary 操作素 2-范畴

Operadic Fibrations and Unary Operadic 2-categories

本文提出 unary 操作素 2-范畴作为操作素 Grothendieck 构造的框架，并描述了其与机器学习中 'Para' 构造的关联。

Tangent bundle and Gauss functor of a combinatorial manifold

Pure combinatorial models for BPL_n and Gauss map of a combinatorial manifold are described.

Left fibrations and homotopy colimits II

For a small simplicial category A, we prove that the homotopy colimit functor from the category of simplicial diagrams on A to the category of simplicial sets over the homotopy-coherent nerve of A provides a left Quillen equivalence between the projective model structure on the former category and the covariant model structure on the latter. We compare this Quillen equivalence to the straightening-unstraightening equivalence previously established by Lurie, where the left adjoint goes in the opposite direction. The existence of left Quillen functors in both directions considerably simplifies t…

广义余Cartesian纤维的合成构造

A synthetic construction of universal cocartesian fibrations

本文提出一种模型无关的方法，构造了$(\infty,1)$-范畴的定向单值余Cartesian纤维，并证明了其直拉等价性。核心步骤是证明余Cartesian纤维沿局部化降维，通过分析局部化的映射空间实现。

柯西完备性、松弛满态射与分裂纤维化的有效下降

Cauchy Completeness, Lax Epimorphisms and Effective Descent for Split Fibrations

本文刻画了分裂纤维化语境下的有效下降态射：证明松弛满态射等价于柯西完备化间的等价，并由此将有效下降态射归结为离散余纤维化情形，附有Beck-Chevalley条件评论。

Locally (co)Cartesian fibrations as realisation fibrations and the classifying space of cospans

We show that the conditions in Steimle's 'additivity theorem for cobordism categories' can be weakened to only require \emph{locally} (co)Cartesian fibrations, making it applicable to a larger class of functors. As an application we compute the difference in classifying spaces between the infinity category of cospans of finite sets and its homotopy category.

关于 $L_\infty$-bundles 的 étale 纤维丛结构

On the structure of étale fibrations of $L_\infty$-bundles

本文研究了 $L_\infty$-bundles 的 étale 纤维丛结构，证明其可由简单形态组成，并应用逆函数定理描述了同调范畴的性质。

自由纤维化、懒惰余限和Kan扩展对于$(\infty,2)$-范畴

Free fibrations, lax colimits and Kan extensions for $(\infty,2)$-categories

本文研究了$(\infty,2)$-范畴的纤维化、余限和Kan扩展，提出了自由纤维化的构造，并给出了部分懒惰余限和Kan扩展的纤维化描述。

Duality functors for triple vector bundles

We calculate the group of dualization operations for triple vector bundles, showing that it has order 96 and not 72 as given in Mackenzie's original treatment. The group is a nonsplit extension of S4 by the Klein group. Dualization operations are interpreted as functors on appropriate categories and are said to be equal if they are naturally isomorphic. The method set out here will be applied in a subsequent paper to the case of n-fold vector bundles.

Quillen伴随的一个简单刻画

A simple characterization of Quillen adjunctions

本文研究了模型范畴间的富化右伴随函子，指出保持无环纤维化和纤维对象的函子通常即为右Quillen函子，提供了Quillen伴随的简单刻画。

Hofmann-Streicher 抬升的纤维范畴

Hofmann-Streicher lifting of fibred categories

本文基于Awodey的分析，定义了Hofmann-Streicher提升的相对版本，并构建了新的2-纤维化，其中opcartesian和cartesian提升源于伪伴随。

严格4-范畴的弦图与映射4-群胚的纤维化

String diagrams for $4$-categories and fibrations of mapping $4$-groupoids

本文为严格4-范畴建立弦图演算，并证明在4-范畴表示的余有限包含下，映射空间限制函子构成严格4-群胚纤维化。

$\infty$-operads 的伸直-非伸直等价

A straightening-unstraightening equivalence for $\infty$-operads

本文为 $\infty$-operads 建立伸直-非伸直伴随等价，并证明 Hinich-Moerdijk 比较函子诱导 operadic 与 dendroidal 左纤维化等价，同时刻画幺半非伸直函子的本质像。

Moduli space of fibrations in the category of simplicial presheaves

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…

通过六函子形式体系的同调流形

Homology manifolds via six functor formalisms

利用谱层六函子形式体系研究同调流形，刻画上同调光滑对象，证明ANR同调流形是Poincaré对偶复形，推广Wilder单调映射定理，引入同伦流形并证明不稳定版本的Wilder可定向猜想。

Model categorical Koszul-Tate resolution for algebras over differential operators

Derived D-Geometry is considered as a convenient language for a coordinate-free investigation of nonlinear partial differential equations up to symmetries. One of the first issues one meets in the functor of points approach to derived D-Geometry, is the question of a model structure on the category C of differential non-negatively graded quasi-coherent commutative algebras over the sheaf D of differential operators of an appropriate underlying variety. In [BPP15a], we described a cofibrantly generated model structure on C via the definition of its weak equivalences and its fibrations. In the p…

2-向量丛的平行截面函子

A Parallel Section Functor for 2-Vector Bundles

本文将本质有限广群上的2-向量丛关联到作为同伦不动点的平行截面2-向量空间，并将其扩张为对称幺半2-函子以提供拉回-前推映射，动机源于扩展等变拓扑场论的轨形化。