Yoneda Lemma for Elementary Higher Toposes

一个相对米田引理（手稿）

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。

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…

米田引理

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。

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".

Enriched Yoneda Lemma

We present a version of enriched Yoneda lemma for conventional (not infinity-) categories. We require the base monoidal category to have colimits, but do not require it to be closed or symmetric monoidal.

Yoneda Lemma for $\mathcal{D}$-Simplicial Spaces

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…

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.

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…

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 …

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。

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。

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。

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。

通过Tripos到Topos构造对局部层化的代数抽象

An Algebraic Abstraction of the Localic Sheafification via the Tripos-to-Topos Construction

本文通过代数抽象局部预层与层化，并借助比较引理联系locale超紧化，为局部拓扑斯与可实现性拓扑斯建立统一的几何框架。

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。

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。

Enriched infinity categories I: enriched presheaves

This is the first of a series of papers on enriched infinity categories, seeking to reduce enriched higher category theory to the higher algebra of presentable infinity categories, which is better understood and can be approached via universal properties. In this paper, we introduce enriched presheaves on an enriched infinity category. We prove analogues of most familiar properties of presheaves. For example, we compute limits and colimits of presheaves, prove that all presheaves are colimits of representable presheaves, and prove a version of the Yoneda lemma.

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…

高阶希尔伯特空间的正交基

Orthonormal bases for higher Hilbert spaces

本文通过范畴化Baez的高阶线性代数，为3-希尔伯特空间提供正交基等工具，赋予其C*-3-范畴自富集结构，证明酉Yoneda引理与Deligne积的酉版本性质，推进高阶希尔伯特空间理论研究。