以红楼梦&意象理论为主题的文化活动

 昨天单纯猫分享了一段他对数学作为一种文化消费品的思考。

• 与许多人的认识不同, 数学不只是推理法则驱动的机器, 而是思维模式的生动舞台; 这些思维模式, 在学术话语经过大众化, 通俗化, 进入了公共话语体系之后, 常常可以重塑人对世界和对生活的看法. 这需要既懂得数学行话, 又懂得如何与外行人交流的人去做

这段文字让我想到了红楼梦的一种 Morita 等价解读

读完之后我的大脑中生成了一个展览&Talk，在这个活动中，数学嵌入了一种文化消费体验。

这个活动的主题是红楼梦 \times Topos，形式是talk➕展览。

在一开始的talk里，单纯猫会阐述topos以及topos as bridge 的基本道理，接下来，我来阐述红楼梦里英雄屈死这一意象是如何连接闺阁与家国的。

而展览的部分可以展示许多红楼梦的谷子，周边，以及不同版本的红楼梦书籍，以及可以请朋友以书法写下一些红楼梦里的上好诗词。

例如林黛玉的《唐多令》

粉堕百花洲，香残燕子楼。一团团、逐队成毬。飘泊亦如人命薄，空缱绻，说风流。
草木也知愁，韶华竟白头。叹今生、谁舍谁收！嫁与东风春不管，凭尔去，忍淹留！

如果这个活动收费，那么cos 宝黛免票。

但无论收费与否，都应该鼓励参与者cos宝黛。

If，这是一个真正的艺术展览性质的活动，那么另外一种盈利方式是售卖展品同款谷子/周边，以及售卖红楼梦/Topos书籍。

对抽象与具体的思考，以及关于什么是自然的/Reflections on the abstract and the concrete, and on what it means to be natural.

 许多朋友和教授认为我思考数学的方式过于抽象，这体现为博客和文章，以及作业和考试中无处不在的范畴论。

比如我会用交换代数和范畴论来解线性常系数ode，会用Monoidal Category Theory来理解分析学（L^1 as Strong Monoidal Functor) 又或者我会用Hom函子的性质来定义什么是豪斯道夫空间，序列极限唯一空间，连通空间来证明一些命题。或者我不喜欢说Hom(X,R)是一个环，Hom(X,P)是一个偏序集，更不喜欢逐点验证它们满足那些axiom。我会用Lawvere Theory 以及更一般的sketch去刻画这些道理，并说明Hom(-,R),Hom(-,P)这两个Hom函子分别是内部的环和偏序集。又或者代数数论里我会用滤过余极限定义integral closure并借此说明它和群作用的不动点/局部化交换...这样的例子在我的博客里数不胜数。

而我本人和另外一些朋友认为这是自然的，优雅的，合理的。并且这一切都非常具体，并不抽象。我想两边的一大起码会有的共识是，使用范畴论的语言可以大规模降低证明的工作量，甚至会消解掉证明的必要性，因为在合适的语言下，这一切都是自然的。

不要将“初等”混同于“容易”：一个证明可以确定是初等的，但却不是容易的。实际上，存在着许多这样的定理：只要用一点点（现代数学的）高端方法就可以使定理的证明变得非常容易理解，并显示蕴涵于其中的思想，相反如果避免使用高端的概念与方法，而只是用初等的方法来证明，则会掩盖定理背后的思想内涵。另一方面，也要注意不要将高端等同于高质量，或者等同于“高级证明”（这是一个我很喜欢用的字眼，它会引起许多我以前学生们的哄笑）。在年轻的数学家们中间确实有一种（盲目）使用新奇的高端数学语言的倾向，以显示他们正在做的工作比较深刻。然而，只有真正理解了现代工具，并且与新的思想相结合，现代的工具才能发挥作用。那些在某些领域（例如数论）工作的人，如果不花时间和实质性的努力去学习掌握这些工具，就会使他们处于非常不利的境地。拒绝学习和掌握这些新工具，就好像只用凿子来拆一座建筑物。即使你使用凿子非常熟练，别人用推土机也比你有巨大的优越性，并且用不着掌握像你那样（使用凿子）的技巧。——《给青年数学家的建议》

一个可能是共识也可能不是的想法是，范畴论给出了更自然的理解。

这篇文章想讨论两个问题，什么是抽象和具体，以及什么是更自然的理解？并借此谈谈这个博客大概是什么（一种苏州古典园林）。

关于抽象和具体，我想通过一个“科幻”一样的小片段来阐述它。

假想这世界上有一种生活在巨大沼泽中的外星生物。在它们所生活的星球上，沼泽在陆地上泪光点点地分布着。这颗星球上的气象被称作风露清愁，沼泽在阳光和风的作用下不断地形变着。于是沼泽生物们天然地就熟悉同伦而不熟悉距离，面积。而地球人生活在坚硬的大地上，天然就熟悉“距离”，面积。沼泽生物们的文明早早就熟悉了同伦和范畴论的概念，其中范畴论教育在沼泽生物的文明里相当于地球上教小朋友认字。而关于距离和面积的概念需要在高中才开始有简单的接触。

当我们不再极端地人类中心主义地看待数学的时候，抽象和具体的概念似乎被消解了。在这个故事里我们可以看到，人类认为距离和面积是极为具体和直观的，但同伦与范畴是抽象的。而生活在风露清愁的沼泽生物们却恰恰相反。

当然你可能会说，我们并不认识人类以外的智慧生物（也许AI可以算一种智慧生物，但暂时不在本文的讨论范围内）。但即使人类内部，拥有不同基因和经历/文化背景的人之间的区别也是巨大的。抽象和具体依旧是一个非常主观的概念。一个2e人士和一个神经典型发育者对抽象和具体的看法可以天差地别。讨论一个数学理论对人类来说是抽象还是具体似乎并没有太大的意义。一个人营造的数学理论和对数学的理解应该被看作一座苏州古典园林，它是私人的，是园主审美，品味，以及观念...的体现。但它同时也具有一定的公共性质。苏州古典园林常常是文人雅集的场所，园林的主人时常会邀请品味相近的朋友们来此赏游。读者朋友们可以把我的博客看作我所营造的基于《园冶》的，“虽由人作，宛自天开”的为追求苏州古典园林。我给它起名为风露园（Zephyrdew Garden）。取自林黛玉的“风露清愁”。

下面我们谈谈什么是更自然的理解？我想说自然似乎本身就是理解的一部分。

“几星期前浮现在我心中的画面则又是另一番景象：那有待认知的未知之物，在我眼前犹如一片紧实坚硬的泥灰岩大地，抗拒着被穿透……大海在寂静中不知不觉地推进，似乎无事发生，毫无动静，水远得几乎听不到它的声音……然而它终有一刻包围了那顽固的物质，后者渐渐化为半岛，复又化为岛屿，继而化为小小的孤屿，最终轮到它被淹没，仿佛终于消溶于一望无际的汪洋之中。”——格罗滕迪克

为什么我认为范畴论给出了更自然的理解？

对我来说，数学是一种客观的现实——它独立于我们而存在，仿佛就在那里。我们就像站在壮丽风景前的户外画家，手里只有画笔和颜料，只能尽力捕捉自己所能触及的那一部分。 而传统的、以集合论为核心的数学，却很像是：想用几支暗淡的灰色和黑色铅笔去完成一幅印象派杰作。是的，技术上不是不可能，但这种方式显得粗糙又荒谬——它严重限制了我们能描绘什么、以及我们如何看见那种底层的美。你能想象只用这些铅笔去画莫奈的春日花园、或者他的睡莲吗？这简直不可想象！ 相比之下，意象理论为我们提供了更丰富的调色板和更有表现力的画笔——一种更扩展的语言，使我们能够照亮数学“现实”中那些在传统框架下会变得模糊甚至不可见的侧面。——引自我在Topos Seminar 的介绍里写下的文字

范畴论无疑能更好地捕捉数学结构以及关系，它提供了更好的画笔和颜料。没有这样的画笔和颜料，不到园林，我如何画下眼前的明媚？

如果以造园为类比，我希望这个博客是“自成天然之趣，不烦人事之功的。”好的造园要顺应地势、水流、植被的本然，不要用过度的人力去扭曲它。这就是为什么我喜欢用范畴的语言来叠山理水。

Many friends and professors think that my way of thinking about mathematics is too abstract, which shows up everywhere in my blog posts, articles, homework, and exams, all saturated with category theory.

For example, I use commutative algebra and category theory to solve linear constant-coefficient ODEs; I use monoidal category theory to understand analysis (L¹ as a strong monoidal functor); or I rely on properties of the Hom functor to define what it means to be a Hausdorff space, a space with unique sequential limits, or a connected space, and then prove some propositions. Or rather than saying that Hom(X, ℝ) is a ring and Hom(X, P) is a poset, and verifying the axioms point by point—something I dislike—I use Lawvere theories and, more generally, sketches to characterize these facts, showing that the Hom functors Hom(–, ℝ) and Hom(–, P) are internal ring and internal poset objects, respectively. Or in algebraic number theory, I define the integral closure via filtered colimits and use this to explain why it commutes with fixed points of group actions and with localizations… Countless such examples appear on my blog.

I myself, and some of my friends, find this natural, elegant, and reasonable. Moreover, it is all very concrete, not abstract at all. I think one consensus both sides can at least agree on is that using the language of category theory can massively reduce the workload of proofs—even dissolve the necessity of proving something altogether—because in the right language, everything becomes natural.

Do not confuse “elementary” with “easy”: a proof may certainly be elementary yet far from easy. In fact, there are many theorems for which just a little bit of (modern) high-level machinery can make the proof highly understandable and reveal the ideas within it, whereas avoiding those advanced concepts and methods and proving it only by elementary means would conceal the ideas behind the theorem. On the other hand, one must also be careful not to identify “advanced” with “high quality” or with a “higher proof” (a phrase I am fond of using, which used to provoke laughter among my former students). There is indeed a tendency among young mathematicians to blindly employ fancy new high-level mathematical language in order to show that what they are doing is profound. However, modern tools only work if one truly understands them and integrates them with new ideas. People working in certain fields (such as number theory) put themselves at a severe disadvantage if they do not spend the time and substantial effort to learn and master these tools. Refusing to learn and master such new tools is like tearing down a building using only a chisel. Even if you are highly skilled with the chisel, someone with a bulldozer has an enormous advantage over you and does not need the kind of finesse (in using the chisel) that you do. — *Advice for a Young Mathematician*

A thought that may or may not be a consensus: category theory provides a more natural understanding.

This essay aims to discuss two questions: What are “abstract” and “concrete”? And what constitutes a more natural understanding? Along the way, I shall also talk about what this blog is roughly like—a kind of Suzhou classical garden.

On the abstract and the concrete, I would like to illustrate with a little “science fiction” vignette.

Imagine there exist extraterrestrial beings living in vast marshes. On their planet, the marshes are scattered across the land like glittering teardrops. The climate of this planet is called “Zephyrdew Grief” (风露清愁); under the sunlight and wind, the marshes constantly deform. Thus, the marsh beings are innately familiar with homotopy, but not with distance or area. By contrast, we earthlings live on solid ground and are innately familiar with “distance” and “area.” The marsh civilization becomes acquainted early on with the concepts of homotopy and category theory; there, education in category theory is equivalent to teaching little children on Earth how to read. Concepts of distance and area, on the other hand, are only touched upon briefly in high school.

When we stop viewing mathematics from an extremely anthropocentric standpoint, the concepts of abstract and concrete seem to dissolve. In this story we can see that humans regard distance and area as supremely concrete and intuitive, but homotopy and categories as abstract. Yet for the marsh beings who live under the zephyrdew grief, the opposite is true.

Of course, you might say that we know no intelligent life other than humans (perhaps AI counts as a kind of intelligent being, but that is beyond the scope of this essay). Yet even within humanity, the differences among individuals with different genes, experiences, and cultural backgrounds are immense. Abstract and concrete remain highly subjective notions. A twice-exceptional (2e) person and a neurotypical person can have vastly divergent views of what is abstract and what is concrete. There seems to be little point in debating whether a given mathematical theory is abstract or concrete for humans. A person’s construction of a mathematical theory and their understanding of mathematics should be seen as a Suzhou classical garden—private, an embodiment of the garden owner’s aesthetics, tastes, ideas… Yet it also possesses a certain public character. Suzhou classical gardens were often venues for literati gatherings; the owner would frequently invite friends of similar tastes to visit and enjoy the garden. My dear readers may regard my blog as a Suzhou classical garden that I have constructed based on the principles of *The Craft of Gardens* (园冶), aspiring to the ideal of “though created by human hands, it appears as if born of heaven” (虽由人作，宛自天开). I have named it **Zephyrdew Garden** (风露园), taking its name from Lin Daiyu’s “zephyrdew grief” (风露清愁).

Now let us talk about what constitutes a more natural understanding. I want to say that naturalness seems to be a part of understanding itself.

> “The image that came to me a few weeks ago was yet another scene: the unknown thing to be known appeared before me like a stretch of compact, hard marl earth, resisting penetration… The sea advances imperceptibly and without sound; nothing seems to break, nothing moves, the water is so distant you can barely hear it… Yet eventually it surrounds the stubborn substance, which gradually becomes a peninsula, then an island, then a tiny islet, and finally is itself submerged, as if at last dissolving into the boundless ocean stretching out of sight.”  

> — Grothendieck

Why do I think category theory yields a more natural understanding?

To me, mathematics is an objective reality—it exists independently of us, as if just out there. We are like plein-air painters standing before a magnificent landscape, with only brushes and pigments at our disposal, trying our best to capture the portion we can reach.  

Traditional, set-theory–centered mathematics, in contrast, is much like attempting to complete an Impressionist masterpiece using only a few dull gray and black pencils. Yes, it is not technically impossible, but the approach appears crude and absurd—it seriously limits what we can depict and how we see the underlying beauty. Can you imagine using nothing but those pencils to paint Monet’s spring garden, or his *Water Lilies*? It is simply unthinkable!  

By comparison, topos theory provides us with a richer palette and more expressive brushes—a more extended language that allows us to illuminate aspects of mathematical “reality” that would become blurred or even invisible under the traditional framework. — quoted from my introduction to the Topos Seminar

Category theory undeniably captures mathematical structure and relationships better; it provides better brushes and pigments. Without such brushes and pigments, “if I do not enter the garden, how can I paint the radiant beauty before me?”

To use garden-making as an analogy, I hope this blog embodies the principle “by nature it achieves its own interest, without troubling human artifice” (自成天然之趣，不烦人事之功). Good garden-making follows the inherent nature of the terrain, the water, and the vegetation, and does not distort them through excessive human effort. That is why I enjoy using the language of categories to pile up mountains and lay out waterways.

Non Math

创建这个分区受单纯猫的单纯百科启发，其中非数学讨论了一些并非数学知识但对一些数学人而言非常有价值和启发的内容。我意识到了在今天这样一个时代，这样的栏目有它特别的意义。

在数学之外我有许多爱好以及许多好奇的领域。包括但不限于红楼梦，苏州古典园林，诗词，昆曲，越剧，音乐剧，汉服，欣赏印象派艺术，书法，古典音乐，民乐，进化美学与神经美学，神经科学，神经多样性（特别地，2e)，饮食/营养补充剂与well being，精神卫生...

而由于我的数学背景，我对其中一些领域的认识，理解，欣赏不可避免地受到数学的影响。而我的神经非典型发育的大脑总是喜欢在遥远的领域之间建立联系...

红楼梦作为一种Morita 等价

以红楼梦&意象理论为主题的文化活动

对抽象与具体的思考，以及关于什么是自然的/Reflections on the abstract and the concrete, and on what it means to be natural.

 

红楼梦作为一种Morita 等价 (Dream of the Red Chamber* embodies a kind of Morita equivalence.)

 我在想《红楼梦》是否体现出了一种 Morita 等价。 这里的依据来自红楼梦第一回的句子：

“施主，你把这有命无运，累及爹娘之物，抱在怀内作甚？”【甲戌眉批：八个字屈死多少英雄？屈死多少忠臣孝子？屈死多少仁人志士？屈死多少词客骚人？今又被作者将此一把眼泪洒与闺阁之中，见得裙钗尚遭逢此数，况天下之男子乎？看他所写开卷之第一个女子便用此二语以定终身，则知托言寓意之旨，谁谓独寄兴于一“情”字耶！武侯之三分，武穆之二帝，二贤之恨，及今不尽，况今之草芥乎？家国君父事有大小之殊，其理其运其数则略无差异。知运知数者则必谅而后叹也。

我想这里或许用 Olivia Caramello 的 Topos as Bridge 理论作类比是合适的。 闺阁草木是一个景（Site） 对象是女儿、诗社、草木；态射是眼泪、流言、葬花。这里的格罗滕迪克拓扑（Grothendieck topology）是家族伦理、嫡庶尊卑与随分从时的世俗规训。家国兴亡是另一个景（Site），对象是君臣、流寇、城池；态射是党争、构陷、杀戮。这里的格罗滕迪克拓扑是封建王权的倾轧、文人的气节与王朝的更迭。不过脂砚斋告诉我们 “家国君父事有大小之殊，其理其运其数则略无差异。知运知数者则必谅而后叹也。” 红楼梦里闺阁草木和家国兴亡这两个景上的层范畴构成的意象，或者说悲剧的结构都是英雄屈死。

闺阁女儿的英雄屈死贯穿了整本红楼梦。千红一哭万艳同悲，判词判曲，香菱的有命无运累及爹娘，王熙凤所谓脂粉堆里的英雄，葬花吟里，“天尽头，何处有香丘？未若锦囊收艳骨，一抷净土掩风流！质本洁来还洁去，强于污淖陷渠沟。”第七十八回老学士闲征姽婳词　痴公子杜撰芙蓉诔里林四娘这里，芙蓉女儿诔里宝玉把晴雯的屈死和贾谊，鲧并列。“高标见嫉，闺帏恨比长沙；直烈遭危，巾帼惨于羽野。”我甚至觉得晴雯之死有些像岳武穆屈死风波亭…而林四娘更是从闺阁女子转变为了姽婳将军。

在英雄屈死这一意象作为 bridge 的意义下，林妹妹在潇湘馆风刀霜剑下的挣扎，不仅是女子在闺阁中的自怜。当她的叹出质本洁来还洁去 强于污淖陷渠沟的时候，借由英雄屈死这座桥梁，她的眼泪在 Morita 等价的意义下与历史长河中所有不肯降志辱身的忠臣烈士、所有在文明黄昏中抱残守缺的遗民，发出了同频的悲鸣。（如屈原的“宁赴湘流，葬于江鱼之腹中”）

闺阁中的风刀霜剑和家国中的英雄屈死都在强调一种系统性的，无处不在的，看不见的对英雄的抹杀。这一悲剧的结构贯穿了风月宝鉴的正反两面…或许贯穿了更多。当我们批判学校教育的时候，当李归农谈论起自己的回忆的时候，当我们谈论2e人群1面临的困境的时候，我们何尝不是在谈论英雄屈死？ 更上一层，神州陆沉何尝不是一个伟大文明的英雄屈死？只是红楼梦又将一把眼泪洒于闺阁之中…

I have been wondering whether *Dream of the Red Chamber* embodies a kind of Morita equivalence. My point of departure is a sentence from the very first chapter:

“Patron, why do you hold in your arms that creature who has life but no fortune, and will only bring trouble to her parents?”  

[Jiaxu's eyebrow comment*: How many heroes have these eight words brought to a tragic end? How many loyal ministers and filial sons? How many benevolent men and lofty scholars? How many poets and men of feeling? And now the author takes this handful of tears and sprinkles it among the inner chambers, showing that even girls in skirts and hairpins meet such a fate—how much more must all the men under Heaven suffer it! Seeing that he uses these two phrases to seal the lifelong destiny of the very first girl he writes about, we understand the allegorical purport he entrusts to his work. Who says his inspiration rests solely on the word “love”? The Marquis of Wu’s Three Kingdoms, the Martial and Cultured Sovereigns’ two lost emperors, the eternal rancour of those two worthies—it still does not exhaust itself, to say nothing of us mere weeds and stalks. The affairs of family and state, sovereign and father, may differ in scale, but their principle (*li*), their operation, their allotted fate are precisely the same. Those who understand fate and allotted time will necessarily forgive and then sigh. ]

I think it is fitting to use Olivia Caramello’s “Topos as Bridge” theory as an analogy here.  

*Boudoir, plants and trees* form one **site**. Its objects: daughters, poetry clubs, plants and flowers; its morphisms: tears, rumours, burying flowers. The Grothendieck topology on this site consists of family ethics, the hierarchy of wife and concubine, the worldly discipline of “accepting one’s lot and following the season”.  

*The rise and fall of family and state* forms another **site**. Its objects: sovereign and minister, roving bandits, city walls; its morphisms: factional strife, false accusation, slaughter. The Grothendieck topology here is the crushing weight of feudal kingship, the integrity of the literatus, and the changing of dynasties.  

Yet the commentator Zhiyanzhai tells us: “The affairs of family and state, sovereign and father, may differ in scale, but their principle, their operation, their allotted fate are precisely the same.” The image constituted by the categories of sheaves on these two sites—the boudoir-vegetation site and the family-state site—or rather the very structure of tragedy that they share, is **the unjust death of heroes** (英雄屈死).  

The unjust death of the boudoir’s heroic daughters runs through the entire novel. A thousand reds weep, ten thousand beauties share one grief; the prophetic songs and verdicts; Xiangling’s “having life but no fortune, bringing trouble to her parents”; Wang Xifeng, called “a hero among the rouge and powder”; in the “Flower Burial Song”, “At the uttermost end of the sky / Where can I find a fragrant grave? / Better to wrap my fair bones in a brocade bag / And bury them in a handful of clean earth, / So that my pure essence may return to purity, / Rather than sink into the filthy ditches.” In Chapter 78, “The Old Scholar Idly Composes a Poem on the Warrior Maid, the Besotted Youth Fancifully Drafts an Elegy for the Hibiscus Maid”—here, in the story of Lin Siniang, and in the Elegy for the Hibiscus Maid, Baoyu places Qingwen’s unjust death alongside those of Jia Yi and Gun the Flood-Tamer: “Your lofty integrity met with envy; the hatred in the women’s quarters rivals that of Changsha. / Your upright fervour encountered danger; the young girl suffered a fate crueller than that on the wild plains of Feather Mountain.” I would even go so far as to say that Qingwen’s death is somewhat akin to Yue Fei’s unjust end at Fengbo Pavilion... And Lin Siniang, a boudoir girl, is transformed directly into a warrior-maid general.  

Understood through the image of the unjust death of heroes, acting as a bridge, the struggle of Sister Lin in the Bamboo Lodge, where “wind blades and frost swords” press in relentlessly, is not merely the self-pity of a girl in her inner chamber. When she sighs “My pure essence returns to purity—better this than sinking into the filthy ditches,” by means of the bridge of the heroic unjust death her tears, in the sense of Morita equivalence, resonate with the same frequency as every loyal minister and martyr across the long river of history who refused to compromise his integrity, and with every remnant subject clinging to the fragments in a civilization’s twilight. (For example, Qu Yuan: “I would rather cast myself into the Xiang stream and be buried in the bellies of fish.”)  

The wind blades and frost swords of the inner chambers, and the unjust deaths of heroes in the realm of family and state, both emphasize a systemic, omnipresent, invisible annihilation of the heroic. This tragic structure runs through both sides of the precious mirror of love and desire... and perhaps through far more. When we criticize school education today, when Li Guinong speaks of his memories, when we discuss the difficulties faced by twice-exceptional (2e) individualsⁱ, are we not, in just the same way, talking about the unjust death of heroes? And on an even larger scale, is not the “submerging of the Divine Land” (神州陆沉) itself the unjust death of a great civilization’s heroes? It is only that *Dream of the Red Chamber* once more takes that handful of tears and sprinkles it among the inner chambers...

Cayley–Hamilton Theorem via Yoneda Lemma

A Concise Proof of the Cayley–Hamilton Theorem

Once you have proved the Cayley–Hamilton theorem over algebraically closed fields (by AG),you know it holds for integral domains, and in particular for the integral domain

$$\mathcal{U}=\mathbb{Z}[x_1,\dots ,x_n,n]$$

This ring is the representing object for the functor $M_n$$M_n$, with universal element the generic matrix $X=(x_{i,j})$$X = (x_{i,j})$.
For any commutative ring $A$$A$, a matrix in $M_n(A)$$M_n(A)$ corresponds to a ring homomorphism $\phi :\mathcal{U}\to A$$\varphi : \mathcal{U} \to A$.

We already know that $p_X(X)=0$$p_X(X) = 0$; therefore

$$p_M(M)=\phi (p_X(X))=\phi (0)=0.$$

In abstract terms, the assignment ${\alpha }_R:M\mapsto p_M(M)$$\alpha_R : M \mapsto p_M(M)$ is actually a natural transformation, and it is the zero transformation.
Consider the following diagram; it is easy to verify that it commutes:

$$\begin{array}{ccc}{M}_n(R)& \stackrel{{\alpha }_R}{\to }& M_n(R)\\ \downarrow M_n(\varphi )& & \downarrow M_n(\varphi )\\ M_n(S)& \stackrel{{\alpha }_S}{\to }& M_n(S)\end{array}$$

Yoneda’s lemma tells us that

$$\mathrm{Nat}(M_n,M_n)\cong \mathrm{Nat}({\mathrm{Hom}}_{\mathsf{CRing}}(\mathcal{U},-),{\mathrm{Hom}}_{\mathsf{CRing}}(\mathcal{U},-))\cong M_n(\mathcal{U})$$

Thus $\alpha$$\alpha$ is completely determined by its action on the universal element. That is, we only need to consider...

Proof

Since $\mathrm{\Phi }$$\Phi$ is a natural transformation, we have the following commutative diagram:

This diagram shows that the natural transformation $\mathrm{\Phi }$$\Phi$ is completely determined by ${\mathrm{\Phi }}_A({\mathrm{id}}_A)=u$$\Phi_A(\mathrm{id}_A) = u$ since for each morphism $f:A\to X$$f: A \to X$ one has...

$${\alpha }_u(X)=0_u(X)⟹\alpha =0$$

Here $0$$0$ is the zero natural transformation, sending every matrix to the zero matrix.