Yoneda lemma for complete Segal spaces
David Kazhdan, Yakov Varshavsky
本文用完备Segal空间的语言给出了无穷范畴的Yoneda引理的一个完备证明。
In this note we formulate and give a self-contained proof of the Yoneda lemma for infinity categories in the language of complete Segal spaces.
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Yoneda Lemma for Elementary Higher Toposes
我们证明了Yoneda引理内的一个小学高拓扑，推广Yonda引理的空间。
A remark on Yoneda's Lemma
Yoneda 'e引理是关于给定可表示协变（逆变，reps.）的所有自然变换的典范同构函子（从局部小范畴到集范畴）到协变（反变，代表）函子本文指出，给定任意可表示函子和任意函子，我们有从给定可表示函子到给定函子的“子集”函子的规范自然变换，“集合了所有的自然变换“。
We study the Yoneda lemma for arbitrary simplicial spaces. We do that by introducing left fibrations of simplicial spaces and and studying its associated model structure, the covariant model structure. In particular, we prove a recognition principle for covariant equivalences over an arbitrary simplicial space and invariance of the covariant model structure with respect to complete Segal space equivalences.
A relative Yoneda Lemma (manuscript)
我们通过一个相对的Yoneda引理构造了集值右Kan-扩张。
This note is a survey on the basic aspects of moduli theory along with some examples. In that respect, one of the purposes of this current document is to understand how the introduction of stacks circumvents the non-representability problem of the corresponding moduli functor $\mathcal{F}$ by using the 2-category of stacks. To this end, we shall briefly revisit the basics of 2-category theory and present a 2-categorical version of Yoneda's lemma.
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。
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.
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。
Enriched Yoneda Lemma
我们提出了一个版本的丰富Yoneda引理的常规（而不是无穷大）类别。我们要求基幺半范畴有余极限，但不要求它是闭的或对称的。
We use the complete Segal approach to the theory of Cartesian fibrations to define and study representable Cartesian fibrations, generalizing representable right fibrations which have played a key role in $\infty$-category theory. In particular, we give a construction of representable Cartesian fibrations using over-categories and prove the Yoneda lemma for representable Cartesian fibration, which generalizes the established Yoneda lemma for right fibrations. We then use the theory of Cartesian fibrations to study complete Segal objects internal to an $\infty$-category. Concretely, we prove th…
Stable $\infty$-Operads and the multiplicative Yoneda lemma
我们为每个$\infty$ -运算$\mathcal{O}^\otimes$构造了具有一定有限极限的新的$\infty$ -运算的谱对象和交换群对象在$\mathcal{O}$。我们表明，这些都是普遍稳定的响应。添加剂$\infty$ -从$\mathcal{O}^\otimes$获得的操作物。我们推断，对于一个稳定的（resp。加法）对称monoidal $\infty$ -范畴$\mathcal{C}$ Yoneda嵌入因子通过从$\mathcal{C}$到谱的$\infty$ -范畴的精确、反变函子的$\infty$ -范畴（分别为.连接谱），并承认一定的乘法细化。作为应用，我们证明了具有Smash积的谱的对称幺半群$\infty$ -范畴Sp的正合松弛对称幺半群内函子中的单位函子Sp $\to$ Sp是初始的.
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…
Fibrations and Yoneda's lemma in an $\infty$-cosmos
我们使用术语$\infty$ -范畴和$\infty$ -函子来表示$\infty$ -宇宙中的对象和态射：一个满足一些公理的简单丰富范畴，让人想起一个丰富的范畴的对象。拟范畴、Segal范畴、完备Segal空间、标记单纯集、迭代完备Segal空间、$θ_n$ -空间以及它们各自的纤维化版本都是这个意义上的$\infty$ -范畴。本系列以前的工作表明，$\infty$ -范畴和$\infty$ -函子的基本范畴论只能参照$\infty$ -宇宙的公理来发展;实际上，大部分工作是同伦2-范畴、$\infty$ -范畴、$\infty$ -函子和自然变换的严格2-范畴的内部工作。在准范畴的$\infty$ -宇宙中，我们精确地重新获得了由Joyal和Lurie发展的相同的范畴论，尽管我们的定义自然是2-范畴的，没有利用区分每个模型的组合细节。本文介绍了一类$\infty$ -函子--carnivibration及其群胚变体。笛卡尔纤维化形成了一个基石，在抽象处理的“类范畴”结构的一个洛杉矶街，并发挥了重要作用，在卢里的工作准范畴。在建立了它们的基本理论之后，我们陈述并证明了Yoneda引理，它具有可表示纤维化映射的拟范畴与纤维在其表示元上的拟范畴之间的等价形式。一个配套文件将应用这些结果建立一个演算的模块之间的$\infty$类别，这将被用来定义和研究点态Kan扩展沿$\infty$…
Constructing a fully homomorphic encryption scheme with the Yoneda Lemma
本文利用米田引理统一解释多种经典同态加密系统，提出 Yoneda Encryption Scheme 框架，并据此设计无需自举即可支持任意加密乘加序列的全同态加密方案 ACES。
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…
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。
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…
Formalizing the $\infty$-Categorical Yoneda Lemma
本研究借助支持合成∞-范畴理论的新型证明助手Rzk，首次实现∞-范畴理论结论的形式化，核心成果包括范畴论基本定理米田引理的形式化，且框架内构造自动具备自然性或函子性。
From Yoneda to Topoi morphisms
本文仅用米田引理、自然变换技巧与基础范畴论，重新证明拓扑斯理论中site态射分类、平坦性与左正合性等价两个核心结果，推导过程初等且主线清晰。
Compact convolution
我们陈述了一个Yoneda型引理，它导致各种函子范畴是紧闭的。
Supermaps on generalised theories
本文基于超映射理论，建立了一个Yoneda引理，用于描述任意电路理论中的高阶量子操作，并应用于boxworld的高阶过程和高阶真实量子理论的定义。
In previous work, we introduce an axiomatic framework within which to prove theorems about many varieties of infinite-dimensional categories simultaneously. In this paper, we establish criteria implying that an $\infty$-category - for instance, a quasi-category, a complete Segal space, or a Segal category - is complete and cocomplete, admitting limits and colimits indexed by any small simplicial set. Our strategy is to build (co)limits of diagrams indexed by a simplicial set inductively from (co)limits of restricted diagrams indexed by the pieces of its skeletal filtration. We show directly th…
Infinity category theory from scratch
我们使用术语“$\infty$ -categories”和“$\infty$ -functors”来表示“$\infty$ -cosmos”中的对象和态射。“拟范畴、Segal范畴、完备Segal空间、自然标记单纯集、迭代完备Segal空间、$θ_n$ -空间以及它们各自的纤维化版本在这个意义上都是$\infty$ -范畴。我们表明，基本的范畴论的$\infty$ -范畴和$\infty$ -函子可以从一个$\infty$ -宇宙的公理发展;事实上，大部分的工作是内部的一个严格的2-范畴的$\infty$ -范畴，$\infty$ -函子，和自然变换。在准范畴的$\infty$ -宇宙中，我们精确地重新获得了Joyal和Lurie所发展的同一理论，尽管在大多数情况下，我们的定义是2-范畴的而不是组合的，是标准概念的新的化身。在第一讲中，我们定义了一个$\infty$ -宇宙，并引入了它的“同伦2-范畴”，利用形式范畴论定义和研究了$\infty$ -范畴之间的等价和互逆。在第二讲中，我们研究了取值于$\infty$ -范畴的图的（上）极限以及（上）极限与极限之间的关系。在第三讲中，我们引入了逗号$\infty$ -范畴，它们被用来编码（余）极限和伴随性的泛性质，并证明了“模型独立性”结果。在第四讲中，我们介绍了（上）卡氏纤维化，描述了$\infty$ -范畴之间的“模”演算，并利用…
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