一个相对米田引理（手稿）

双范畴的米田引理与表示定理

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。

Notes on Moduli theory, Stacks and 2-Yoneda's Lemma

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.

局部笛卡尔闭∞-范畴的内部约当引理

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。

Yoneda's lemma for internal higher categories

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 for enriched infinity categories

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…

从米田引理到拓扑斯态射

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。

A remark on Yoneda's Lemma

Yoneda'e Lemma is about the canonical isomorphism of all the natural transformations from a given representable covariant (contravariant, reps.) functor (from a locally small category to the category of sets) to a covariant (contravariant, reps.) functor. In this note we point out that given any representable functor and any functor we have the canonical natural transformation from the given representable functor to the "subset" functor of the given functor, "collecting all the natural transformations".

利用米田引理构造全同态加密方案

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。

Stable $\infty$-Operads and the multiplicative Yoneda lemma

We construct for every $\infty$-operad $\mathcal{O}^\otimes$ with certain finite limits new $\infty$-operads of spectrum objects and of commutative group objects in $\mathcal{O}$. We show that these are the universal stable resp. additive $\infty$-operads obtained from $\mathcal{O}^\otimes$. We deduce that for a stably (resp. additively) symmetric monoidal $\infty$-category $\mathcal{C}$ the Yoneda embedding factors through the $\infty$-category of exact, contravariant functors from $\mathcal{C}$ to the $\infty$-category of spectra (resp. connective spectra) and admits a certain multiplicative…

A bivariant Yoneda lemma and $(\infty,2)$-categories of correspondences

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…

形式化∞-范畴米田（Yoneda）引理

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。

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 …

紧卷积

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。

Higher extensions in exact Mal'tsev categories: distributivity of congruences and the $3^n$-Lemma

The aim of this article is to better understand the correspondence between $n$-cubic extensions and $3^n$-diagrams, which may be seen as non-abelian Yoneda extensions, useful in (co)homology of non-abelian algebraic structures. We study a higher-dimensional version of the coequaliser/kernel pair adjunction, which relates $n$-fold reflexive graphs with $n$-fold arrows in any exact Mal'tsev category. We first ask ourselves how this adjunction restricts to an equivalence of categories. This leads to the concept of an effective $n$-fold equivalence relation, corresponding to the $n$-fold regular e…

广义理论上的超映射

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。

沿2-纤维化的逐点Kan扩张与元素的2-范畴

Pointwise Kan extensions along 2-fibrations and the 2-category of elements

本研究将元素范畴的经典结果推广到2维，提出沿离散2-上纤维化的逐点Kan扩张定义，证明其恒为弱Kan扩张，推广了参数化Yoneda引理。

Kan extensions and the calculus of modules for $\infty$-categories

Various models of $(\infty,1)$-categories, including quasi-categories, complete Segal spaces, Segal categories, and naturally marked simplicial sets can be considered as the objects of an $\infty$-cosmos. In a generic $\infty$-cosmos, whose objects we call $\infty$-categories, we introduce modules (also called profunctors or correspondences) between $\infty$-categories, incarnated as as spans of suitably-defined fibrations with groupoidal fibers. As the name suggests, a module from $A$ to $B$ is an $\infty$-category equipped with a left action of $A$ and a right action of $B$, in a suitable se…