数学意义上的相等 | 等同性的范畴论诠释

注：英文引文，机翻未校。

如有内容异常，请看原文。

When is one thing equal to some other thing?

一物何时等同于另一物？

Barry Mazur

June 12, 2007

巴里·马祖尔

2007 年 6 月 12 日

In memory of Saunders MacLane

谨以此文纪念桑德斯·麦克莱恩

1 The awkwardness of equality

1 相等性的尴尬之处

One can't do mathematics for more than ten minutes without grappling, in some way or other, with the slippery notion of equality. Slippery, because the way in which objects are presented to us hardly ever, perhaps never, immediately tells us--without further commentary--when two of them are to be considered equal. We even see this, for example, if we try to define real numbers as decimals, and then have to mention aliases like 20 = 19.999 ... 20=19.999\dots 20=19.999..., a fact not unknown to the merchants who price their items 19.99 19.99 19.99.

任何人研究数学超十分钟，就必会以这样或那样的方式，直面"相等性"这一难以捉摸的概念。它之所以难以把握，是因为对象呈现在我们面前的方式，几乎从未------或许永远不会------无需额外说明，就能让我们立刻判断出两个对象何时应被视作相等。即便我们尝试将实数定义为小数时，也能发现这一点，比如我们不得不提及诸如 20 = 19.999 ⋯ 20=19.999\cdots 20=19.999⋯ 这样的等价形式，那些将商品标价为 19.99 美元的商家，对此事实其实心知肚明。

The heart and soul of much mathematics consists of the fact that the "same" object can be presented to us in different ways. Even if we are faced with the simple-seeming task of "giving" a large number, there is no way of doing this without also, at the same time, "giving" a hefty amount of extra structure that comes as a result of the way we pin down-or the way we present-our large number. If we write our number as 1729 we are, sotto voce, offering a preferred way of "computing it" (add one thousand to seven hundreds to two tens to nine). If we present it as 1 + 12 3 1 + 12^3 1+123 we are recommending another mode of computation, and if we pin it down-as Ramanujan did- as the first number expressible as a sum of two cubes in two different ways, we are being less specific about how to compute our number, but have underscored a characterizing property of it within a subtle diophantine arena.

诸多数学理论的内容，在于"同一个"对象能以不同方式呈现在我们面前。即便只是完成"给出"一个大数这一看似简单的任务，我们也必然会同时"赋予"这个大数大量额外的结构------而这一结构，源于我们确定或呈现这个大数的方式。若我们将某个数写作 1729，实则是隐晦地给出了一种"计算它"的方式（1000 加 700 加 20 加 9）；若将其表示为 1 + 12 3 1 + 12^3 1+123，则是推荐了另一种计算方式；而如拉马努金那般，将其定义为第一个能以两种不同方式表示为两个立方数之和的数时，我们虽未明确其计算方法，却凸显了它在深奥的丢番图理论范畴中的一个特征性质。

The issue of "presentation" sometimes comes up as a small pedagogical hurdle- no more than a pebble in the road, perhaps, but it is there-- when one teaches young people the idea of congruence mod N. How should we think of 1, 2, 3, . . . mod 691? Are these ciphers just members of a new number system that happens to have similar notation as some of our integers? Are we to think of them as equivalence classes of integers, where the equivalence relation is congruence mod 691? Or are we happy to deal with them as the good old integers, but subjected to that equivalence relation? The eventual answer, of course, is: all three ways-having the flexibility to adjust our viewpoint to the needs of the moment serves us well. But that may be too stiff a dose of flexibility to impose on our students all at once.

向初学者讲授模 N N N 同余的概念时，"呈现方式"的问题有时会成为一个小小的教学障碍------或许只是路上的一颗小石子，但它真实存在。我们该如何理解 1、2、3、... 模 691 的含义？这些数字仅仅是一个新数系的元素，只是恰好与我们的一些整数符号相同吗？我们应将其视作整数在模 691 同余这一等价关系下的等价类吗？还是说，我们可以将其仍当作熟悉的整数，只是需遵循这一等价关系来处理？当然，最终的答案是：三种理解方式皆可------根据当下需求调整视角的灵活性，能为我们提供便利。但要让学生一下子接受这种灵活性，或许有些过于困难。

To define the mathematical objects we intend to study, we often-perhaps always-first make it understood, more often implicitly than explicitly, how we intend these objects to be presented to us, thereby delineating a kind of superobject; that is, a species of mathematical objects garnished with a repertoire of modes of presentation. Only once this is done do we try to erase the scaffolding of the presentation, to say when two of these super-objects-possibly presented to us in wildly different ways- are to be considered equal. In this oblique way, the objects that we truly want enter the scene only defined as equivalence classes of explicitly presented objects. That is, as specifically presented objects with the specific presentation ignored, in the spirit of "ham and eggs, but hold the ham."

在定义我们想要研究的数学对象时，我们往往------或许总是------会先让人们理解这些对象将以何种方式呈现，这种理解大多是隐含的，而非明确的，由此划定一种"超对象"；也就是说，这类数学对象附带了一系列的呈现方式。只有完成这一步，我们才会尝试抛开呈现的"脚手架"，去界定两个可能以截然不同方式呈现的超对象何时应被视作相等。通过这种间接的方式，我们真正想要研究的对象，最终仅被定义为显式呈现对象的等价类而登场。换言之，这些对象是被具体呈现的，但我们会忽略其具体的呈现方式，就如同"要火腿煎蛋，却去掉火腿"的逻辑一般。

This issue has been with us, of course, forever: the general question of abstraction, as separating what we want from what we are presented with. It is neatly packaged in the Greek verb aphairein, as interpreted by Aristotle 1 ^1 1 in the later books of the Metaphysics to mean simply separation: if it is whiteness we want to think about, we must somehow separate it from white horse, white house, white hose, and all the other white things that it invariably must come along with, in order for us to experience it at all.

当然，这一问题自始至终都存在：这就是抽象的一般性问题------将我们想要研究的东西，与它所依附的呈现载体分离开来。古希腊动词"aphairein"恰好概括了这一概念，亚里士多德在《形而上学》的后几卷中将其解释为简单的"分离"：若我们想要思考"白色"这一概念，就必须将其与白马、白房子、白水管，以及所有它必然依附的白色事物分离开来，否则我们根本无法对其进行思考。

1 Aristotle first uses this term in Book XI Chap 3 1061a line 29 of the Metaphysics; his discussion

in Book XIII, Chap 2 begins to confront some of the puzzles the term poses.

亚里士多德在《形而上学》第十一卷第三章 1061a 第 29 行首次使用了这个术语；而在第十三卷第二章中，他的讨论开始面对这个术语所带来的某些难题。

The little trireme of possibilities we encounter in teaching congruence mod 691 (i.e., is 5 mod 691 to be thought of as a symbol, or a stand-in for any number that has remainder 5 when divided by 691, or should we take the tack that it (i.e., "5 mod 691") is the (equivalence) class of all integers that are congruent to 5 mod 691?) has its analogue elsewhere-perhaps everywhere-in mathematics. Familiarity with a concept will allow us to finesse, or ignore, this, as when we are happy to deal with a fraction a / b a/b a/b ambiguously as an equivalence class of pairs of integers ( a , b ) (a, b) (a,b) with b ≠ 0 b ≠0 b=0, where the equivalence relationship is given by the rule ( a , b ) ∼ ( a ′ , b ′ ) (a, b) \sim(a', b') (a,b)∼(a′,b′) if and only if a b ′ = a ′ b a b'=a' b ab′=a′b, or as a particular member of this class. Few mathematical concepts enter our repertoire in a manner other than ambiguously a single object and at the same time an equivalence class of objects. This is especially true for the concept of natural number, as we shall see in the next section where we examine the three possible ways we have of coming to terms with the number 5.

我们在讲授模 691 同余时遇到的这三种可能性（即，5 模 691 应被视作一个符号？还是任何被 691 除余 5 的数的代表？亦或是将"5 模 691"定义为所有与 5 模 691 同余的整数构成的（等价）类？），在数学的其他领域------或许是所有领域------都能找到类似的情况。对一个概念的熟悉，会让我们得以巧妙处理或忽略这一问题，比如我们会轻松地将分数 a / b a/b a/b 模糊地理解为：要么是满足 b ≠ 0 b≠0 b=0 的整数对 ( a , b ) (a,b) (a,b) 在等价关系（ ( a , b ) ∼ ( a ′ , b ′ ) (a,b)\sim(a',b') (a,b)∼(a′,b′) 当且仅当 a b ′ = a ′ b ab'=a'b ab′=a′b）下的等价类，要么是该等价类中的某个特定元素。几乎所有数学概念进入我们的认知时，都是这样既被当作单个对象，又同时被当作对象的等价类，这种模糊性是普遍存在的。自然数的概念尤其如此，在下一节中，我们将通过分析理解数字 5 的三种可能方式来印证这一点。

One of the templates of modern mathematics, category theory, offers its own formulation of equivalence as opposed to equality; the spirit of category theory allows us to be content to determine a mathematical object, as one says in the language of that theory, up to canonical isomorphism. Category theory goes further than merely "content" with the inevitability that any particular mathematical object tends to come to us along with the contingent scaffolding of the specific way in which it is presented to us, but has this inevitability built in to its very vocabulary, and in an elegant way, makes profound use of this. It will allow itself the further flexibility of viewing any mathematical object "as" a representation of the theory in which the object is contained to the proto-theory of modern mathematics, namely, to set theory.

范畴论作为现代数学的一种范式，对"等价"而非"相等"给出了独有的定义；用范畴论的语言来说，其理念是，我们只需将数学对象确定到"典范同构"的程度，便足矣。范畴论的视角不止是对"任何特定数学对象必然会随其具体呈现方式这一偶然的脚手架一同出现在我们面前"这一事实的被动接受，它还将这一必然性融入了自身的基本术语体系，并以精妙的方式对其加以运用。范畴论还赋予了我们更大的灵活性：我们可以将任意数学对象，"视作"其所属理论到现代数学的原初理论------即集合论------的一种表示。

My aim in this article is to address a few points about mathematical objects and equality of mathematical objects following the line of thought of the preceding paragraph. I see these "points" borne out by the doings of working mathematicians as they go about their daily business thinking about, developing, and communicating mathematics, but I haven't found them specifically formulated anywhere 2 ^2 2. I don't even see how questions about these issues can even be raised within the framework of vocabulary that people employ when they talk about the foundations of mathematics for so much of the literature on philosophy of mathematics continues to keep to certain staples: formal systems, consistency, undecidability, provability and unprovability, and rigor in its various manifestations.

本文的目的，是循着上一段的思路，探讨关于数学对象及其相等性的几个问题。我发现，一线数学家在日常思考、发展和交流数学理论的过程中，实则一直在践行这些观点，但我从未在任何文献中看到对这些观点的明确阐述。甚至，在人们探讨数学基础时所使用的术语框架下，我都不知道该如何提出这些问题------因为绝大多数数学哲学文献，始终围绕着某些固定主题展开：形式系统、一致性、不可判定性、可证性与不可证性，以及各种形式的严谨性。

The faintest resonance, though, might be seen in the discussion in Books 13 and 14 of Aristotle's Metaphysics which hits at the perplexity of whether the so-called mathematicals (that ostensibly play their role in the platonic theory of forms) occur uniquely for each mathematical concept, or multiply .
然而，最微弱的共鸣可能在亚里士多德的《形而上学》第十三卷和第十四卷的讨论中显现出来，这讨论触及了所谓的数学（显然在理想形式理论中发挥作用）是否对每个数学概念唯一出现，或是多重的困惑。

To be sure, people have exciting things to talk about, when it comes to the list of topics I have just given. These issues have been the focus of dramatic encounters-famous "conversations," let us call them-that represent turning points in our understanding of what the very mission of mathematics should be. The ancient literature-notably, Plato's comment about how the mathematicians bring their analyses back to the hypotheses that they frame, but no further-already delineates this mission 3 ^3 3. The early modern literature-epitomized by the riveting use that Kant made of his starkly phrased question "how is pure mathematics possible?"-offers a grounding for it. In the century just past, we have seen much drama regarding the grounds for mathematics: the Frege-Russell correspondence, the critique that L.E.J. Brouwer made of the modern dealings with infinity and with Cantor's set theory, Hilbert's response to this critique, leading to his invention of formal systems, and the work of Gödel, itself an extraordinary comment on the relationship between the mission of mathematics and the manner in which it formulates its deductions.

诚然，上述这些主题本身极具探讨价值。这些问题曾是数次激烈交锋的焦点------我们不妨称这些交锋为著名的"对话"------而这些对话，成为了我们对数学使命的理解发生转折的节点。古代文献中，柏拉图就曾指出，数学家的分析最终都会回归到他们设定的假设，却不会再向前推进一步，这一观点已然勾勒出了数学的这一使命。近代早期的文献则为这一使命奠定了基础，其中最具代表性的，是康德以其鲜明的提问"纯粹数学何以可能？"所展开的深刻探讨。在刚刚过去的一个世纪里，数学基础领域上演了诸多精彩的理论交锋：弗雷格与罗素的通信、布劳威尔对现代数学处理无穷和康托尔集合论的批判、希尔伯特对这一批判的回应并由此创立了形式系统，以及哥德尔的研究------其成果本身，就是对数学的使命与其演绎形式之间关系的诠释。

I think you know that students of geometry, calculation, and the like hypothesize the odd and the even, the various figures, the three kinds of angles, and other things akin to these in each of their investigations, as if they know them. They make these their hypotheses and don't think it necessary to give any account of them, either to themselves or others, as if they were clear to everyone. $Plato 1997\] *Republic* Book VI 510c. 想必你知道，几何学、算术等学科的研究者，在每次研究中都会设定奇、偶、各种图形、三种角以及诸如此类的概念作为假设，仿佛这些概念是他们已然熟知的。他们将这些概念当作假设，认为无需向自己或他人解释，仿佛这些概念对所有人而言都是自明的。（柏拉图，1997，《理想国》，第六卷，510c）$

Formal systems remain our lingua franca. The general expectation is that any particular work we happen to do in mathematics should be, or at least should be capable of being, packaged within some formal system or other. When we want to legitimize our modes of operation regarding anything, from real numbers to set theory to cohomology-we are in the habit of invoking axiomatic systems as standardbearer. But when it comes to a crisis of rigorous argument, the open secret is that, for the most part, mathematicians who are not focussed on the architecture of formal systems per se, mathematicians who are consumers rather than providers, somehow achieve a sense of utterly firm conviction in their mathematical doings, without actually going through the exercise of translating their particular argumentation into a brand-name formal system.

形式系统仍是我们的通用语言。人们普遍认为，我们在数学领域所做的任何具体研究，都应被纳入------或至少能够被纳入------某个形式系统之中。当我们想要为自己处理各类问题的方式正名时，无论是研究实数、集合论还是上同调，我们都习惯于将公理系统当作标杆。但当遭遇严谨性论证的危机时，一个公开的秘密是：绝大多数并非专门研究形式系统本身的数学家------即作为形式系统的使用者而非构建者的数学家------无需将自己的具体论证转化为某一知名形式系统的语言，就能对自己的数学研究形成无比坚定的信念。

If we are shaky in our convictions as to the rigor of an argument, an excursion into formal systems is rarely the thing that will shore up faith in the argument. To be sure, it is often very helpful for us to write down our demonstrations very completely using pencil and paper or our all-efficient computers. In any event, no matter how wonderful and clarifying and comforting it may be for mathematician X to know that all of his or her proofs have, so far, found their expression within the framework of Zermelo-Frankel set theory, the chances are that mathematician X, if quizzed on what-exactly-those axioms are, might be at a loss to answer.

若我们对某一论证的严谨性心存疑虑，诉诸形式系统往往无法坚定我们对这一论证的信心。诚然，将证明过程用纸笔或高效的计算机完整地写下来，对我们而言往往大有裨益。但无论如何，即便数学家 X 得知自己迄今为止的所有证明，都能在策梅洛-弗兰克尔集合论的框架下得到表达，这一事实让他感到无比欣慰、豁然开朗，可倘若有人追问他这些公理的具体内容，他很可能也会支支吾吾，答不上来。

Of course, it is important to understand, as fully as we can, what tools we need to assemble in order to justify our arguments. But to appreciate, and discuss, a grand view of the nature of mathematical objects that has taken root in mathematical culture during the past half-century, we must also become conversant with a language that has a thrust somewhat different from the standard fare of foundations. This newer vocabulary has phrases like canonical isomorphism, "unique up to unique isomorphism", functor, equivalence of category and has something to say about every part of mathematics, including the definition of the natural numbers.

当然，尽我们所能地弄清楚为证明论证的合理性所需的工具，具有意义。但要理解并探讨过去半个世纪里在数学文化中扎根的、关于数学对象本质的宏大视角，我们还必须熟悉一套与传统数学基础术语的侧重点有所不同的语言。这套新的术语包含典范同构、"在唯一同构意义下唯一"、函子、范畴等价等概念，并且能应用于数学的各个领域，包括自然数的定义。

2 Defining natural numbers

2 定义自然数

Consider natural numbers; for instance, the number 5. Here are three approaches to the task of defining the number 5.

让我们以自然数为例，比如数字 5。以下是定义数字 5 的三种方法。

We could, in our effort to define the number 5, deposit five gold bars in, say, Gauss's observatory in Göttingen, and if ever anyone wants to determine whether or not their set has cardinality five, they would make a quick trip to Göttingen and try to put the elements of their set in one-one correspondence with the bullion deposited there. Of course, there are many drawbacks to this approach to defining the number "five," the least of which is that it has the smell of contingency. Let us call this kind of approach the bureau of standards attitude towards definition: one chooses a representative exemplar of the mathematical object one wishes to define, and then gives a criterion for any other mathematical object to be viewed as equal to the exemplar. There is, after all, something nice and crisp about having a single concrete exemplar for a mathematical concept.

为定义数字 5，我们可以将 5 根金条存放于某处，比如哥廷根的高斯天文台，倘若有人想判断某个集合的基数是否为 5，只需前往哥廷根，尝试将该集合的元素与存放的金条建立一一对应关系即可。当然，这种定义"5"的方法存在诸多弊端，其中最微不足道的一点，就是它带有明显的偶然性。我们将这种定义方式称为"标准局式定义观"：先选定想要定义的数学对象的一个代表性范例，再给出判定其他数学对象与该范例相等的准则。毕竟，为一个数学概念设定一个具体的范例，这种方式简洁明了，有其可取之处。
The extreme opposite approach to this is Frege's: define a cardinality (for example, five) as an equivalence class in the set of all sets, the equivalence relation A ∼ B A \sim B A∼B being the existence of a one-one correspondence between A and B. The advantage, here, is that it is a criterion utterly devoid of subjectivity: no set is preferred and chosen to govern as benchmark for any other set; no choice (in the realm of sets) is made at all. The disadvantage, after Russell, is well known: the type of universal quantification required in Frege's definition, at least when the equivalence classes involved are considered to be sets, leads to paradox. The Frege-Russell correspondence makes it clear that one cannot, or at least one should not, be too greedy regarding unconditional quantification. To keep clear of immediate paradox, we introduce the word class into our discussion, amend the phrase set of all sets to class of all sets, and hope for the best.

与上述方法截然相反的是弗雷格的做法：将基数（例如 5）定义为所有集合构成的全体在某一等价关系下的等价类，其中等价关系 A ∼ B A\sim B A∼B 定义为集合 A A A 与集合 B B B 之间存在一一对应。这种方法的优势在于，它是一个完全无主观色彩的准则：没有任何一个集合被优先选为其他集合的基准，在集合的范畴内，我们未做任何主观选择。而在罗素发现悖论之后，这种方法的弊端也变得众所周知：弗雷格的定义中所用到的全称量化，至少在将相关等价类视作集合时，会导致悖论的产生。弗雷格与罗素的通信清晰地表明，我们不能------至少不应该------对无条件的量化过于"贪心"。为了避免直接陷入悖论，我们在讨论中引入"类"这一概念，将"所有集合的集合"修正为"所有集合的类"，并姑且以此规避悖论。
A fine compromise between the above two extremes is to do what we all, in fact, do: a strategy that captures the best features of both of the above approaches. What I mean here, by the way, is to indicate what we do, rather than what we say we do when quizzed about our foundations. I allow my notation 1, 2, 3, 4, 5, 6, . . . to play the role of my personal bureau-of-standards within which I happily make my calculations. I think of the set { 1 , 2 , 3 , 4 , 5 } \{1, 2, 3, 4, 5\} {1,2,3,4,5}, for example, as a perfectly workable exemplar for quintuples. Meanwhile you use your notation 1 ′ 1' 1′, 2 ′ 2' 2′, 3 ′ 3' 3′, 4 ′ 4' 4′, 5 ′ 5' 5′, 6 ′ 6' 6′, . . . (or whatever it is) to play a similar role with respect to your work and thoughts, the basic issue being whether there is a faithful translation of structure from the way in which you view natural numbers to the way I do.

介于上述两种极端方法之间的一种折中方案，正是我们实际所采用的方法：这种策略兼具上述两种方法的优点。顺带一提，我这里所指的，是我们实际的做法，而非当被问及数学基础时，我们声称自己所采用的做法。我将记号 1、2、3、4、5、6、... 当作我个人的"标准局"，在这个框架下自如地进行计算。例如，我将集合 { 1 , 2 , 3 , 4 , 5 } \{1,2,3,4,5\} {1,2,3,4,5} 视作表示五元集的一个实用的范例。与此同时，你会使用自己的记号 1 ′ 1' 1′、 2 ′ 2' 2′、 3 ′ 3' 3′、 4 ′ 4' 4′、 5 ′ 5' 5′、 6 ′ 6' 6′、...（或其他任何记号），在你的研究和思考中扮演类似的角色，而问题在于，你对自然数的理解所蕴含的结构，能否忠实地转化为我对自然数的理解所蕴含的结构。

Equivalence (of structure) in the above "compromise" is the primary issue, rather than equality of mathematical objects. Furthermore, it is the structure intrinsic to the whole gamut of natural numbers that plays a role there. For only in terms of this structure (packaged, perhaps, as a version of Peano's axioms) do we have a criterion to determine when your understanding of "natural numbers," and mine, admit "faithful translations" one to another. A consequence of such an approach- which is the standard modus operandi of mathematics ever since Hilbert-is that any single mathematical object, say the number 5, is understood primarily in terms of the structural relationship it bears to the other natural numbers. Mathematical objects are determined by--and understood by-the network of relationships they enjoy with all the other objects of their species.

在上述"折中方案"中，问题是（结构的）等价性，而非数学对象的相等性。此外，自然数系整体所固有的结构，在其中发挥着作用。因为只有基于这一结构（该结构可通过皮亚诺公理的某种形式来刻画），我们才有准则判断你我对"自然数"的理解之间，是否存在相互的"忠实转化"。这种方法是希尔伯特之后数学研究的标准操作模式，其一个推论是：任何单个的数学对象，比如数字 5，其含义主要源于它与其他自然数之间的结构关系。数学对象的定义与理解，皆源于它与同类型的所有其他对象所构成的关系网络。

3 Objects versus structure

3 对象与结构

Mathematics thrives on going to extremes whenever it can. Since the "compromise" we sketched above has "mathematical objects determined by the network of relationships they enjoy with all the other objects of their species," perhaps we can go to extremes within this compromise, by taking the following further step. Subjugate the role of the mathematical object to the role of its network of relationships-or, a further extreme-simply replace the mathematical object by this network.

数学的发展，往往得益于在任何可能的地方走向极致。既然我们上述的"折中方案"提出"数学对象由其与同类型所有其他对象的关系网络所决定"，那么我们可以在这一折中方案的基础上更进一步，走向极致：将数学对象的地位，从属于其关系网络的地位；或者更进一步，直接用这一关系网络取代数学对象本身。

This may seem like an impossible balancing act. But one of the elegant--and surprising-accomplishments of category theory is that it performs this act, and does it with ease.

这看似是一项难以实现的平衡之举，但范畴论的一项精妙而惊人的成就，就是轻松地完成了这一操作。

4 Category Theory as balancing act rather than "Foundations"

4 作为平衡之举的范畴论，而非"数学基础"

There are two great modern formulas-as I'll call them-for packaging entire mathematical theories. There is the concept of formal system, following David Hilbert, as discussed above. There is also the concept of category, the innovation of Samuel Eilenberg and Saunders MacLane. Now, these two formulas have vastly different intents.

现代数学中有两种重要的范式------我姑且如此称呼------可用于整合整个数学理论：一种是前文所讨论的、由大卫·希尔伯特开创的形式系统概念；另一种则是塞缪尔·艾伦伯格与桑德斯·麦克莱恩的创举------范畴的概念。而这两种范式的初衷，有着天壤之别。

A formal system representing a mathematical theory has, within it, all of the mechanics and vocabulary necessary to discuss proofs, and the generation of proofs, in the mathematical theory; indeed, that is mainly what a formal system is all about.

一个表征某一数学理论的形式系统，内部包含了讨论该数学理论中的证明及证明的生成所需的全部机制与术语；事实上，这正是形式系统的内容。

In contrast, a category is quite sparse in its vocabulary: it can say nothing whatsoever about proofs; a category is a mathematical entity that, in the most succinct of languages, captures what a mathematical theory consists: objects of the theory, allowable transformations between these objects, and a composition law telling us how to compose two transformations when the range of the first transformation is the domain of the second.

与之相反，范畴的术语体系极为精简：它对证明问题只字不提。范畴是这样一种数学实体，它以最简洁的语言，捕捉了数学理论的构成：理论的对象、对象之间允许的变换，以及当一个变换的陪域是另一个变换的定义域时，刻画这两个变换如何复合的合成法则。

It stands to reason, then, that the concept of category cannot provide us with anything that goes under the heading of "foundations." Nevertheless, in its effect on our view of mathematical objects it plays a fine balancing role: it extracts-as I hope you will see-the best elements from both a Fregean and a bureau-of-standards attitude towards the formulation of mathematical concepts.

因此，范畴的概念显然无法为我们提供任何可被称作"数学基础"的东西。尽管如此，它在塑造我们对数学对象的认知方面，却起到了平衡作用：正如我希望你能看到的，它从弗雷格式的定义观和标准局式的定义观中，提炼出了各自具有价值的部分，用于刻画数学概念。

5 Example: The category of sets.

5 示例：集合范畴

Even before I describe category more formally, it pays to examine the category of sets as an example. The category of sets, though, is not just "an" example, it is the proto-type example; it is as much an example of a category as Odette is un amour de Swann.

即便在我对范畴给出更正式的定义之前，以集合范畴为例进行分析也是十分有益的。集合范畴并非只是众多范例中的一个，而是原型范例；它作为范畴的范例，就如同奥黛特作为斯万的爱人那般，是极具代表性的。

The enormous complexity to set theory is one of the great facts of life of mathematics. I suppose most people before Cantor, if they ever had a flicker of a thought that sets could occur at all as mathematical objects, would have expected that a rather straightforward theoretical account of the notion would encompass everything that there was to say about those objects. As we all know, nothing of the sort has transpired.

集合论所蕴含的巨大复杂性，是数学领域的一个基本事实。我想，在康托尔之前，即便有人曾隐约想到集合可以作为数学对象存在，也会认为只需通过一个相对简单的理论阐释，就能穷尽关于集合对象的所有内容。但我们都知道，实际情况绝非如此。

The famous attitude of St. Augustine towards the notion of "time," (i.e., "What then is time? If no one asks me, I know what it is. If I wish to explain it to him who asks, I do not know.") mirrors my attitude-and I would suppose, most people's attitude-toward sets. If I retain my naive outlook on sets, all is, or at least seems to be, well; but once I embark on formulating the notion rigorously and specifically, I am either entangled, or else I am forced to make very contingent choices.

圣奥古斯丁对"时间"概念的经典看法------"那么时间究竟是什么？倘若无人问我，我自知其义；倘若我试图向提问者解释，我却一无所知"------恰好反映了我，我想也反映了大多数人，对集合的态度。若我以朴素的视角看待集合，一切都顺理成章，至少看似如此；可一旦我试图对集合的概念进行严谨、具体的刻画，就要么会陷入逻辑的纠缠，要么不得不做出极具偶然性的选择。

Keeping to the bare bones, a set theory will consist of

抛开细枝末节，一个集合论的构成包括：

the repertoire of elements of the theory, and if I wish to refer to one of them I will use a lower case symbol, e.g., a, b, . . .

理论中的元素全体，若需指代其中某一元素，将使用小写符号，例如 a , b , ... a, b, \dots a,b,...。
the repertoire of sets of the theory, and for these I will use upper case symbols, e.g., X , Y , . . . X, Y, ... X,Y,....

理论中的集合全体，若需指代其中某一集合，将使用大写符号，例如 X , Y , ... X, Y, \dots X,Y,...。
the relation of containment telling us when an element x is contained in a set X ( x ∈ X ) X(x \in X) X(x∈X)); each set X is extensionally distinguished by the elements, x ∈ X x \in X x∈X, that are in it.

属于关系，用于刻画元素 x x x 何时包含于集合 X X X（记为 x ∈ X x\in X x∈X）；每个集合 X X X 都由其所含的元素 x ∈ X x\in X x∈X 在外延上确定。
the mappings f : X → Y f: X \to Y f:X→Y between sets of the theory, each mapping f uniquely characterized by stipulating for every x ∈ X x \in X x∈X the (unique) image, f ( x ) ∈ Y f(x) \in Y f(x)∈Y, of that element x.

理论中集合之间的映射 f : X → Y f:X\to Y f:X→Y，每个映射 f f f 都可通过为每个 x ∈ X x\in X x∈X 指定其唯一的像 f ( x ) ∈ Y f(x)\in Y f(x)∈Y 来刻画。
the guarantee that if f : X → Y f: X \to Y f:X→Y and g : Y → Z g: Y \to Z g:Y→Z are mappings in my theory, I can form the composition g ⋅ f : X → Z g \cdot f: X \to Z g⋅f:X→Z by the rule that for any element x ∈ X x \in X x∈X the value ( g ⋅ f ) ( x ) ∈ Z (g \cdot f)(x) \in Z (g⋅f)(x)∈Z is just g ( f ( x ) ) g(f(x)) g(f(x)).

复合性保证：若 f : X → Y f:X\to Y f:X→Y 和 g : Y → Z g:Y\to Z g:Y→Z 是理论中的映射，则可按如下规则构造它们的复合映射 g ⋅ f : X → Z g\cdot f:X\to Z g⋅f:X→Z：对任意 x ∈ X x\in X x∈X，有 ( g ⋅ f ) ( x ) = g ( f ( x ) ) ∈ Z (g\cdot f)(x)=g(f(x))\in Z (g⋅f)(x)=g(f(x))∈Z。

We neither lose nor gain anything by adding the requirement that, for any object X, the identity mapping 1 X : X → X 1_{X}: X \to X 1X:X→X is a bona fide mapping in our set theory, so for convenience let us do that. Also, we see that our composition rule is associative in the standard sense of multiplication, i.e., ( h ⋅ g ) ⋅ f = h ⋅ ( g ⋅ f ) (h \cdot g) \cdot f=h \cdot(g \cdot f) (h⋅g)⋅f=h⋅(g⋅f), when these compositions can be made, and that our identity mappings play the role of "unit."

我们不妨额外要求：对任意集合 X X X，恒等映射 1 X : X → X 1_X:X\to X 1X:X→X 是集合论中的合法映射，这一要求不会让我们的理论增添任何新内容，也不会让我们丢失任何原有内容，因此为了方便，我们做出这一规定。此外，我们容易发现，当复合运算可进行时，上述合成法则满足乘法意义下的结合律，即 ( h ⋅ g ) ⋅ f = h ⋅ ( g ⋅ f ) (h\cdot g)\cdot f=h\cdot(g\cdot f) (h⋅g)⋅f=h⋅(g⋅f)，而恒等映射在复合运算中扮演着"单位元"的角色。

Much has been omitted from this synopsis--all traces of quantifications, for example- and certain things have been hidden. The repeated use of the word repertoire is already a hint that something big is being hidden. It would be downright embarrassing, for example, to have replaced the words "repertoire" in the above description by "set," for besides the blatant circularity, we would worry, with Russell, about what arcane restrictions we would have then to make regarding our universal quantifier, once that is thrown into the picture. "Repertoire" is my personal neologism; the standard word is class and the notion behind it deserves much discussion; we will have some things to say about it in the next section. You may notice that I refrained from using the word "repertoire" when talking about mapings. A subtle issue, but an important one, is that we may boldly require that all the mappings from a given set X to a given set Y form a bona fide set in our theory, and not merely an airy repertoire. This is a source of power, and we adopt it as a requirement. Let us refer, then, to a theory such as we have just sketched as a bare set theory.

这份概述省略了诸多内容------例如所有的量化表述，同时也隐藏了一些问题。我反复使用"全域"一词，已然暗示了其中隐藏着重大问题。例如，若将上述描述中的"全域"替换为"集合"，那将会十分尴尬：因为这不仅存在明显的循环定义问题，还会让我们如同罗素那般，不得不担忧在引入全称量化后，需要为其施加何种晦涩的限制。"全域"是我个人创造的新词，标准的术语是"类"，而这一概念背后的内涵值得深入探讨，我们将在下一节中展开相关讨论。你可能会注意到，我在谈论映射时，刻意避免了使用"全域"一词。一个微妙但重要的点是，我们可以大胆要求：从给定集合 X X X 到给定集合 Y Y Y 的所有映射，在我们的理论中构成一个合法的集合，而非仅仅是一个抽象的"全域"。这一要求是集合论的力量之源，我们将其纳入理论的基本要求中。我们将上述这种极简的集合论称为朴素集合论。

A bare set theory can be stripped down even further by forgetting about the elements and a fortiori the containment relations. What is left?

我们可以对朴素集合论做进一步的精简：完全忽略元素的存在，进而也忽略属于关系。那么剩下的内容是什么？

We still have the objects of the theory, i.e., the repertoire (synonymously: class) of its sets. For any two sets X and Y we have the set of mappings from X to Y; and for any three sets X, Y Z and two mappings f : X → Y f: X \to Y f:X→Y and g : Y → Z g: Y \to Z g:Y→Z we have the mapping that is the composition of the two, g ⋅ f : X → Z g \cdot f: X \to Z g⋅f:X→Z, this composition rule admitting "units" and satisfying the associative law.

我们仍保有理论的对象，即集合的全域（也可称作类）；对任意两个集合 X X X 和 Y Y Y，仍有从 X X X 到 Y Y Y 的映射构成的集合；对任意三个集合 X X X、 Y Y Y、 Z Z Z，以及任意两个映射 f : X → Y f:X\to Y f:X→Y 和 g : Y → Z g:Y\to Z g:Y→Z，仍可构造它们的复合映射 g ⋅ f : X → Z g\cdot f:X\to Z g⋅f:X→Z，且该合成法则仍有"单位元"并满足结合律。

This further-stripped-down bare set theory is our first example of a category: it is the underlying category of the bare set theory.

这种经过进一步精简的朴素集合论，就是我们遇到的第一个范畴示例：它是朴素集合论的底范畴。

The concept of class which will occur in the definition of category, and has already occurred in our proto-example, now deserves some discussion.

"类"这一概念不仅出现在范畴的定义中，也已在我们的原型范例中出现，现在我们有必要对其展开讨论。

6 Class as a library with strict rules for taking out books

6 类：一个有着严格借阅规则的图书馆

I'm certain that there are quite precise formulations of the notion of class, but here is a ridiculously informal user's-eye-view of it. Imagine a library with lots of books, administered by a somewhat stern librarian. You are allowed to take out certain subcollections of books in the library, but not all. You know, for example, that you are forbidden to take out, at one go, all the books of the library. You assume, then, that there are other subcollections of books that would be similarly restricted. But the full bylaws of this library are never to be made completely explicit. This doesn't bother you overly because, after all, you are interested in reading, and not the legalisms of libraries.

我确信"类"的概念有着十分严谨的形式化定义，但在此，我将以一种极为通俗的视角，为读者描绘这一概念。想象有一座藏书丰富的图书馆，由一位略显严厉的图书管理员管理。你可以借阅图书馆中的某些书集，但并非所有书集都能借阅。例如，你明确知道，不允许一次性借阅图书馆里的所有藏书。由此你会推断，还有其他一些书集，也会受到类似的借阅限制。但这座图书馆的完整借阅规则，从未被完全明确地公布。不过这并不会让你过于困扰，因为归根结底，你关注的是阅读本身，而非图书馆的条条框框。

In observing how mathematicians tend to use the notion class, it has occurred to me that this notion seems really never to be put into play without some background version of set theory understood already. In short by a class, we mean a collection of objects, with some restrictions on which subcollections we, as mathematicians, can deem sets and thereby operate on with the resources of our set theory. I'm perfectly confident that this seeming circularity can be--and probably has been--ironed out. But there it is.

通过观察数学家使用"类"这一概念的方式，我发现，这一概念的使用，始终以某个背景集合论的存在为前提。简而言之，类是指一个对象的聚合，而作为数学家，我们只能将其中的某些子聚合视作"集合"，并运用集合论的工具对其进行操作，其余子聚合则受到限制。我完全相信，这种看似循环的定义，是可以被厘清的------而且或许早已被厘清。但目前，这就是我们对"类"的直观理解。

7 Category

7 范畴

A category C \mathcal{C} C is intrinsically a relative notion for it depends upon having a set theory in mind; a bare set theory such as sketched above will do.

范畴 C \mathcal{C} C 本质上是一个相对概念，因为它的定义依赖于一个预设的集合论；上述的朴素集合论就足以满足这一要求。

Fixing on a "bare set theory," a category C \mathcal{C} C (modeled on this bare set theory) is given by the following 4 ^4 4:

选定一个"朴素集合论"，则基于该朴素集合论构造的范畴 C \mathcal{C} C 由如下内容给出 4 ^4 4。

a class of things called the objects of C \mathcal{C} C and denoted O b ( C ) Ob(\mathcal{C}) Ob(C);

一个类，其元素被称为范畴 C \mathcal{C} C 的对象，记为 O b ( C ) Ob(\mathcal{C}) Ob(C)；
given any two objects X, Y of C \mathcal{C} C, a set denoted M o r C ( X , Y ) Mor_{\mathcal{C}}(X, Y) MorC(X,Y), which we think of as the set of transformations from the object X to the object Y; we refer to these transformations as morphisms from X to Y and usually denote such a morphism f as a labelled arrow f : X → Y f: X \to Y f:X→Y;

对任意两个对象 X , Y ∈ O b ( C ) X,Y\in Ob(\mathcal{C}) X,Y∈Ob(C)，对应一个集合 M o r C ( X , Y ) Mor_{\mathcal{C}}(X,Y) MorC(X,Y)，其元素被视作从对象 X X X 到对象 Y Y Y 的变换；我们将这些变换称为从 X X X 到 Y Y Y 的态射，通常将态射 f f f 标记为带箭头的符号 f : X → Y f:X\to Y f:X→Y；
given any three objects X, Y, Z of C \mathcal{C} C and morphisms f : X → Y f: X \to Y f:X→Y and g : Y → Z g: Y \to Z g:Y→Z we are provided with a law that tells us how to "compose" these morphisms to get a morphism g ⋅ f : X → Z g \cdot f: X \to Z g⋅f:X→Z.

选定一个"朴素集合论"后，一个以该朴素集合论为基础的范畴 C \mathcal{C} C 由以下要素构成：

对任意三个对象 X , Y , Z ∈ O b ( C ) X,Y,Z\in Ob(\mathcal{C}) X,Y,Z∈Ob(C)，以及任意两个态射 f : X → Y f:X\to Y f:X→Y 和 g : Y → Z g:Y\to Z g:Y→Z，存在一个合成法则 ，告诉我们如何将这两个态射复合为一个新的态射 g ⋅ f : X → Z g\cdot f:X\to Z g⋅f:X→Z。

Category-theorists will note that I am restricting my attention to what are called locally small
categories.
范畴论研究者会注意到，此处讨论范围限定于所谓局部小范畴。

Intuitively, we are thinking of f and g as "transformations," and composition of them means that we imagine "first" applying f to get us from X to Y and "then" applying g to get us from Y to Z. The rule that associates to such a pair ( f , g ) (f, g) (f,g) the composition, ( f , g ) ↦ g ⋅ f (f, g) \mapsto g \cdot f (f,g)↦g⋅f we think of as a sort of "multiplication law."

直观上，我们将态射 f f f 和 g g g 视作"变换"，而它们的复合，就是先将变换 f f f 作用于对象 X X X 得到对象 Y Y Y，再将变换 g g g 作用于对象 Y Y Y 得到对象 Z Z Z。将态射对 ( f , g ) (f,g) (f,g) 对应到其复合态射 g ⋅ f g\cdot f g⋅f 的规则 ( f , g ) ↦ g ⋅ f (f,g)\mapsto g\cdot f (f,g)↦g⋅f，我们可将其视作一种"乘法法则"。

One also requires that morphisms playing the role of "identity elements" 1 X 1_{X} 1X in M o r C ( X , X ) Mor_{\mathcal{C}}(X, X) MorC(X,X) with respect to this composition law exist; that is, for any morphism f : X → Y f: X \to Y f:X→Y we have f ⋅ 1 X = f f \cdot 1_{X}=f f⋅1X=f; and similarly, for any morphism e : V → X e: V \to X e:V→X we have 1 X ⋅ e = e 1_{X} \cdot e=e 1X⋅e=e. Finally the composition law is assumed to be associative, in the evident sense.

范畴还需满足一个条件：对任意对象 X ∈ O b ( C ) X\in Ob(\mathcal{C}) X∈Ob(C)，集合 M o r C ( X , X ) Mor_{\mathcal{C}}(X,X) MorC(X,X) 中存在关于合成法则的"单位元"态射 1 X 1_X 1X；即对任意态射 f : X → Y f:X\to Y f:X→Y，有 f ⋅ 1 X = f f\cdot 1_X=f f⋅1X=f；同理，对任意态射 e : V → X e:V\to X e:V→X，有 1 X ⋅ e = e 1_X\cdot e=e 1X⋅e=e。最后，合成法则需满足自然的结合律。

As for the word class that enters into the definition, we will, at the very least, want, for any object X in our category, that the singleton set consisting of that object, { X } \{X\} {X}, be viewed as a bona fide set of our set theory.

对于定义中出现的"类"，我们至少要求：对范畴中的任意对象 X X X，由该对象构成的单元素集 { X } \{X\} {X}，是我们预设的集合论中的合法集合。

This concept of category is an omni-purpose affair: we have our categories of sets, where the objects are sets, the morphisms are mappings of sets; we have the category of topological spaces whose objects are the eponymous ones, and whose morphisms are continuous maps. We have the algebraic categories: the category of groups where the morphisms are homomorphisms, the category of rings with unit element where the morphisms are ring homomorphisms (that preserve the unit element), etc. Every branch and sub-branch of mathematics can package their entities in this format. In fact, at this point in its career it is hard to say whether the role of category in the context of mathematical work is more descriptive, or more prescriptive. It frames a possible template for any mathematical theory: the theory should have nouns and verbs, i.e., objects, and morphisms, and there should be an explicit notion of composition related to the morphisms; the theory should, in brief, be packaged by a category. There is hardly any species of mathematical object that doesn't fit into this convenient, and often enlightening, template.

范畴的概念是一个通用工具：我们有集合范畴 ，其对象为集合，态射为集合之间的映射；我们有拓扑空间范畴 ，其对象为拓扑空间，态射为拓扑空间之间的连续映射；我们还有各类代数范畴：群范畴 ，其对象为群，态射为群同态；幺环范畴 ，其对象为含单位元的环，态射为保单位元的环同态，等等。数学的每一个分支，甚至子分支，都能将其研究对象纳入范畴的框架。事实上，在当下的数学研究中，范畴的作用究竟是更多地用于描述数学理论，还是更多地用于规范数学理论的构建，已然难以区分。它为所有数学理论提供了一个通用的范式：一个数学理论应当有"名词"和"动词"，即对象和态射，且态射之间应当有明确的复合概念；简而言之，任何数学理论都可被范畴所整合。几乎所有类型的数学对象，都能适配这一便捷且往往富有启发性的范式。

Template is a feature of categories, for in its daily use, a category avoids any really detailed discussion of its underlying set theory. This clever manner in which category theory engages with set theory shows, in effect, that it has learned the Augustinian lesson. Category theory doesn't legislate which set theory we are to use, nor does it even give ground-rules for what "a" set theory should be. As I have already hinted, one of the beautiful aspects of category theory is that it is up to you, the category-theory-user to supply "a" set theory, a bare category of sets S S S, for example. A category is a B.Y.O.S.T. party, i.e., you bring your own set theory to it. Or, you can adopt an even more curious stance: you can view S S S as something of a free variable, and consequently, end up by making no specific choice!
范式化 是范畴的特征，因为在日常使用中，范畴可以完全避开对其基础集合论的细节讨论。范畴论与集合论的这种巧妙互动方式，实则体现了它对圣奥古斯丁式智慧的借鉴：范畴论既不规定我们必须使用哪一种集合论，也不制定"一个集合论应当满足哪些基本规则"。正如我此前所暗示的，范畴论的精妙之处在于，它将选择集合论的权利交给了使用者------你可以自行选定一个集合论，例如一个朴素的集合范畴 S S S。范畴论就像一场"自带集合论"的聚会，即你需要为其配备自己的集合论。你甚至可以采取一种更独特的立场：将集合范畴 S S S 视作一个自由变量，最终无需做出任何具体的选择！

So, for example, "the" category of rings with unit element is, more exactly, a mold that you can impress on any bare set theory. To be sure, you want your set theory to be sufficiently rich so as to hold this impression: if there were no sets at all in your set theory, you wouldn't get much.

例如，所谓的"幺环范畴"，更准确地说，是一个可以被"烙印"在任意朴素集合论上的模子。当然，你需要你的集合论足够丰富，才能承载这一"烙印"：若你的集合论中根本没有集合，那么这一"模子"也毫无用武之地。

You might wonder why the framers of the notion of category bothered to use two difficult words class and set rather than only one, in their definition. One could, after all, simply require that there be a set of objects of the category, rather than bring in the airy word class. In fact, people do that, at times: it is standard to call a category whose objects form a set a small category, and these small categories do have their uses. Indeed, if you are worried about foundational issues, it is hardly a burden to restrict attention to small categories. But I think the reason that the notion of class is invoked has to do with the ambition we have for categories: categories are meant to offer a fluid vocabulary for whole 'fields of mathematics' like group theory or topology, with a Fregean desire for freedom from the contingency implicit in subjective choices.

你或许会疑惑，范畴概念的创立者为何要在定义中使用"类"和"集合"这两个难以理解的概念，而非只用一个。毕竟，我们本可以直接要求范畴的对象构成一个集合，而非引入抽象的"类"。事实上，人们有时确实会这么做：我们将对象构成集合的范畴称为小范畴，这是一个标准术语，而小范畴也有其适用场景。诚然，若你担忧数学基础问题，将研究范围限制在小范畴内并非难事。但我认为，引入"类"这一概念，源于我们对范畴的期许：范畴旨在为群论、拓扑学等整个数学领域，提供一套灵活的术语体系，同时秉持弗雷格的理念，力求摆脱主观选择中隐含的偶然性。

8 Equality versus isomorphism

8 相等与同构

The major concept that replaces equality in the context of categories is isomorphism. An isomorphism f : A → B f: A \to B f:A→B between two objects A, B of the category C \mathcal{C} C is a morphism in the category C \mathcal{C} C that can be "undone," in the sense that there is another morphism g : B → A g: B \to A g:B→A playing the role of the inverse of f; that is, the composition g f : A → A g f: A \to A gf:A→A is the identity morphism 1 A 1_{A} 1A and the composition f g : B → B f g: B \to B fg:B→B is the identity morphism 1 B 1_{B} 1B. The lesson taught by the categorical viewpoint is that it is usually either quixotic, or irrelevant, to ask if a certain object X in a category C \mathcal{C} C is equal to an object Y. The query that is usually pertinent is to ask for a specific isomorphism from X to Y.

在范畴的框架中，取代"相等"的概念是同构。设 A , B A,B A,B 为范畴 C \mathcal{C} C 中的两个对象，若存在态射 f : A → B ∈ M o r C ( A , B ) f:A\to B\in Mor_{\mathcal{C}}(A,B) f:A→B∈MorC(A,B)，且存在另一个态射 g : B → A ∈ M o r C ( B , A ) g:B\to A\in Mor_{\mathcal{C}}(B,A) g:B→A∈MorC(B,A) 作为 f f f 的逆态射 ，即满足 g ⋅ f = 1 A g\cdot f=1_A g⋅f=1A 且 f ⋅ g = 1 B f\cdot g=1_B f⋅g=1B，则称 f f f 为从 A A A 到 B B B 的同构，记为 f : A ≅ B f:A\cong B f:A≅B。范畴论视角带给我们的启示是：追问范畴 C \mathcal{C} C 中的某个对象 X X X 是否与对象 Y Y Y 相等，通常要么是不切实际的，要么是无关紧要的；而真正有意义的问题，是寻找从 X X X 到 Y Y Y 的具体同构。

Note the insistence, though, on a specific isomorphism; although it may be useful to be merely assured of the existence of isomorphisms between X and Y we are often in a much better position if we can pinpoint a specific isomorphism f : X → Y f: X \to Y f:X→Y characterized by an explicitly formulated property, or list of properties. In some contexts, of course, we simply have to make do without being able to pinpoint a specific isomorphism. If, for example, I manage to construct an algebraic closure of the finite field F 2 F_{2} F2 (i.e., of the field consisting of two elements), and am told that someone halfway around the world has also constructed such an algebraic closure, I know that there exists an isomorphism between the two algebraic closures but--without any further knowledge-I have no way of pinpointing a specific isomorphism. In contrast, despite my ignorance of the manner in which my colleague at the opposite end of the world went about constructing her algebraic closure, I can, with utter confidence, put my finger on a specific isomorphism between the group of automorphisms of my algebraic closure and the group of automorphisms of the other algebraic closure 5 ^5 5. The fact that the algebraic closures are not yoked together by a specified isomorphism is the source of some theoretical complications at times, while the fact that their automorphism groups are seen to be isomorphic via a cleanly specified isomorphism is the source of great theoretical clarity, and some profound number theory.

但需要强调的是，我们始终追求具体的同构 ；尽管仅知晓 X X X 与 Y Y Y 之间存在同构有时已颇具价值，但如果能找到一个由明确性质（或一系列性质）刻画的具体同构 f : X → Y f: X \to Y f:X→Y，我们往往能获得更深入的结论。当然，在某些情境下，我们确实无法确定具体的同构，只能退而求其次。例如，若我构造出了有限域 F 2 \mathbb{F}_2 F2（即由两个元素构成的域）的一个代数闭包，同时得知地球另一端的某人也构造出了这样一个代数闭包，那么我能确定这两个代数闭包之间存在同构，但在没有更多信息的情况下，我无法找到这个具体的同构。与之相反，即便我完全不了解这位远方同事构造代数闭包的具体方法，也能无比确定地找到一个具体的同构，将我的代数闭包的自同构群，与她的代数闭包的自同构群联系起来。这两个代数闭包之间缺乏明确的同构，有时会带来一些理论上的复杂问题；而它们的自同构群之间存在清晰的显式同构，却能为我们带来极大的理论清晰性，甚至催生出一些深奥的数论结论。

for these automorphism groups are both topologically generated by the field automorphism
consisting of squaring every element.
这是因为这两个自同构群在拓扑意义下，均由对域中每个元素取平方所给出的域自同构生成。

A uniquely specified isomorphism from some object X to an object Y characterized by a list of explicitly formulated properties-this list being sometimes, the truth be told, only implicitly understood-is usually dubbed a "canonical isomorphism." The "canonicality" here depends, of course, on the list. It is this brand of equivalence, then, that in category theory replaces equality: we wish to determine objects, as people say, "up to canonical isomorphism."

若从对象 X X X 到对象 Y Y Y 的同构是唯一确定 的，且由一系列明确的性质所刻画（诚然，这些性质有时只是被隐含地理解），那么这个同构通常被称为典范同构 。当然，这里的"典范性"由这些刻画性质所决定。而范畴论中用以取代"相等"的，正是这种等价关系：如人们所言，我们只需将对象在典范同构的意义下确定下来即可。

9 An example of categorical vocabulary: Initial Objects

9 范畴术语的示例：始对象

We also have at our immediate disposal, a broad range of concepts that can be defined purely in terms of the structure that we have already elucidated. For example, if we are given a category C \mathcal{C} C , an initial object Z Z Z of C \mathcal{C} C is an object Z Z Z that has the property that given any object X X X of C \mathcal{C} C there is a unique morphism of the category i X : Z → X i_{X}: Z \to X iX:Z→X from Z Z Z to X X X; that is, the set M o r C ( Z , X ) Mor_{\mathcal{C}}(Z, X) MorC(Z,X) consists of the single element { i X } \{i_{X}\} {iX} .

我们还能直接基于前文阐释的范畴结构，定义一系列相关概念。例如，给定范畴 C \mathcal{C} C，若其对象 Z Z Z 满足：对范畴 C \mathcal{C} C 中的任意对象 X X X，都存在唯一的态射 i X : Z → X ∈ M o r C ( Z , X ) i_X: Z \to X \in Mor_{\mathcal{C}}(Z,X) iX:Z→X∈MorC(Z,X)，即集合 M o r C ( Z , X ) Mor_{\mathcal{C}}(Z,X) MorC(Z,X) 是仅含一个元素 i X i_X iX 的单元素集，则称 Z Z Z 为范畴 C \mathcal{C} C 的始对象。

Suppose that a category C \mathcal{C} C has an initial object Z Z Z . There may, very well, be quite a number of objects vying for the role of initial object of this category C \mathcal{C} C . But given another contender, call it Z ′ Z' Z′ , there is a unique morphism i Z ′ : Z → Z ′ i_{Z'}: Z \to Z' iZ′:Z→Z′ since Z Z Z is an initial object, and a unique morphism i Z ′ : Z ′ → Z i_{Z}': Z' \to Z iZ′:Z′→Z since Z ′ Z' Z′ is. Also, again since Z Z Z is an initial object, there can only be one morphism from Z Z Z to Z Z Z , and the identity morphism 1 Z : Z → Z 1_{Z}: Z \to Z 1Z:Z→Z fills this role just fine, so we must have that i Z ′ ⋅ i Z ′ = 1 Z i_{Z}' \cdot i_{Z'}=1_{Z} iZ′⋅iZ′=1Z and, for similar reasons, i Z ′ ⋅ i Z ′ = 1 Z ′ i_{Z'} \cdot i_{Z}'=1_{Z'} iZ′⋅iZ′=1Z′ . In summary, i Z ′ i_{Z'} iZ′ and i Z ′ i_{Z}' iZ′ are (inverse) isomorphisms, and provide us with canonical, in fact the only, isomorphisms between Z Z Z and Z ′ Z' Z′ . One way of citing this is to say, as people do, that the initial object of a category- if it exists-is unique up to unique isomorphism. To be sure it is not unique as "object," but rather, as "something else." It is this difference, what the "something else" consists in, that we are exploring.

假设范畴 C \mathcal{C} C 有一个始对象 Z Z Z，该范畴中很可能存在多个对象都符合始对象的特征。若取另一个候选始对象 Z ′ Z' Z′，由于 Z Z Z 是始对象，存在唯一的态射 i Z ′ : Z → Z ′ i_{Z'}: Z \to Z' iZ′:Z→Z′；而由于 Z ′ Z' Z′ 也是始对象，存在唯一的态射 i Z ′ : Z ′ → Z i_Z': Z' \to Z iZ′:Z′→Z。此外，仍因 Z Z Z 是始对象，从 Z Z Z 到自身的态射唯一，而恒等态射 1 Z : Z → Z 1_Z: Z \to Z 1Z:Z→Z 恰好满足这一要求，因此必有 i Z ′ ⋅ i Z ′ = 1 Z i_Z' \cdot i_{Z'} = 1_Z iZ′⋅iZ′=1Z；同理可得 i Z ′ ⋅ i Z ′ = 1 Z ′ i_{Z'} \cdot i_Z' = 1_{Z'} iZ′⋅iZ′=1Z′。综上， i Z ′ i_{Z'} iZ′ 与 i Z ′ i_Z' iZ′ 互为逆同构，它们构成了 Z Z Z 与 Z ′ Z' Z′ 之间典范的、且唯一的 同构。对此，人们通常的表述是：一个范畴的始对象若存在，则在唯一同构的意义下唯一。诚然，始对象并非作为"对象本身"是唯一的，而是作为某种"更本质的东西"是唯一的。我们所要探究的，正是这种差异，以及这一"更本质的东西"的内涵。

10 Defining the natural numbers as an "initial object."

10 将自然数定义为"始对象"

For this discussion, let us start by considering "the" initial object in the category of rings with unit. As we shall see, such an initial object does exist, given that the underlying bare set theory is not ridiculously impoverished. Such an initial object is "unique up to unique isomorphism," as all initial objects are. What is it?

在本节的讨论中，我们先考虑幺环范畴中的始对象。我们将看到，只要基础的朴素集合论并非极度贫乏，这个始对象就必然存在；且与所有始对象一样，该始对象"在唯一同构的意义下唯一"。那么这个始对象究竟是什么？

Well, by the definition of initial object, it must be a ring Z \mathbb{Z} Z (with a unit element) that admits a unique ring homomorphism (preserving unit elements) to any ring with unit. Since the ring of ordinary integers Z \mathbb{Z} Z has precisely this property (there being one and only one ring homomorphism from Z \mathbb{Z} Z to any ring with unit, the one that sends 1 ∈ Z 1 \in \mathbb{Z} 1∈Z to the unit of the range ring) "the" initial object in the category of rings with unit is nothing more nor less than Z \mathbb{Z} Z but, of course, only "up to unique isomorphism."

根据始对象的定义，这个始对象必然是一个幺环 Z \mathbb{Z} Z，且从该幺环到任意幺环，都存在唯一的保单位元环同态。而普通的整数环 Z \mathbb{Z} Z 恰好满足这一性质：从 Z \mathbb{Z} Z 到任意幺环，都存在唯一的环同态，即将 1 ∈ Z 1 \in \mathbb{Z} 1∈Z 映射为目标幺环的单位元的同态。因此，幺环范畴的始对象，正是整数环 Z \mathbb{Z} Z ------当然，是在"唯一同构的意义下"。

The previous paragraph situated Z \mathbb{Z} Z among its fellow rings with unit element. Let us fashion a similar discussion for the Natural Numbers, highlighting the type of structure that Peano focussed on, when formulating his famous axioms.

上文将整数环 Z \mathbb{Z} Z 置于幺环范畴中进行了分析，接下来我们对自然数做类似的讨论，重点关注皮亚诺在构建其著名的皮亚诺公理时，所聚焦的自然数的结构特征。

For this, I want to define a category denoted P \mathcal{P} P that I will call the Peano category.

为此，我将定义一个范畴 P \mathcal{P} P，并称其为皮亚诺范畴。

The objects, O b ( P ) Ob(\mathcal{P}) Ob(P) , of the Peano category consists of triples ( X , x , s ) (X, x, s) (X,x,s) where X X X is a set, x ∈ X x \in X x∈X is an element (call it a base point), and s : X → X s: X \to X s:X→X is a mapping of X X X to itself ( a "self-map" which we might call the successor map).

皮亚诺范畴的对象 O b ( P ) Ob(\mathcal{P}) Ob(P) 是三元组 ( X , x , s ) (X, x, s) (X,x,s)，其中： X X X 是一个集合； x ∈ X x \in X x∈X 是集合中的一个固定元素，称为基点； s : X → X s: X \to X s:X→X 是集合 X X X 到自身的映射，称为后继映射。

Given two objects X = ( X , x , s ) \mathcal{X}=(X, x, s) X=(X,x,s) and Y = ( Y , y , t ) \mathcal{Y}=(Y, y, t) Y=(Y,y,t) of P \mathcal{P} P , a morphism F : X → Y F:\mathcal{X} \to \mathcal{Y} F:X→Y in the category P \mathcal{P} P is a mapping of sets f : X → Y f: X \to Y f:X→Y with the property that:

设 X = ( X , x , s ) \mathcal{X}=(X, x, s) X=(X,x,s) 和 Y = ( Y , y , t ) \mathcal{Y}=(Y, y, t) Y=(Y,y,t) 是皮亚诺范畴 P \mathcal{P} P 中的两个对象，范畴 P \mathcal{P} P 中从 X \mathcal{X} X 到 Y \mathcal{Y} Y 的态射 F : X → Y F: \mathcal{X} \to \mathcal{Y} F:X→Y 是一个集合映射 f : X → Y f: X \to Y f:X→Y，且满足以下两个条件：

f f f preserves base points; i.e., f ( x ) = y f(x)=y f(x)=y , and
保基点 ： f ( x ) = y f(x) = y f(x)=y；
f f f respects the self-maps s s s and t t t , in the sense that f ⋅ s = t ⋅ f f \cdot s=t \cdot f f⋅s=t⋅f , i.e., we have for all elements z ∈ X z \in X z∈X , f ( s ( z ) ) = t ( f ( z ) ) f(s(z))=t(f(z)) f(s(z))=t(f(z)) .
We will denote by M o r P ( X , Y ) Mor_{\mathcal{P}}(\mathcal{X}, \mathcal{Y}) MorP(X,Y) the set of morphisms of the Peano category from X \mathcal{X} X to Y \mathcal{Y} Y , i.e., the set of such F F F 's.
保后继映射 ： f ∘ s = t ∘ f f \circ s = t \circ f f∘s=t∘f，即对任意 z ∈ X z \in X z∈X，都有 f ( s ( z ) ) = t ( f ( z ) ) f(s(z)) = t(f(z)) f(s(z))=t(f(z))。
我们将皮亚诺范畴中从 X \mathcal{X} X 到 Y \mathcal{Y} Y 的所有态射构成的集合记为 M o r P ( X , Y ) Mor_{\mathcal{P}}(\mathcal{X},\mathcal{Y}) MorP(X,Y)，即所有满足上述条件的映射 F F F 的集合。

For any choice of bare set theory, we have thereby formed the category which we will call P \mathcal{P} P . If our bare set theory, on which the category is modeled, is at all decent-e.g., is one of the standard set theories containing the set of natural numbers N = { 1 , 2 , 3 , . . . } \mathbb{N}=\{1,2,3, ...\} N={1,2,3,...} , then N \mathbb{N} N may be viewed as an object of P \mathcal{P} P , its base point being given by 1 ∈ N 1 \in \mathbb{N} 1∈N , and the self-map s : N → N s: \mathbb{N} \to \mathbb{N} s:N→N being given by the rule that sends a natural number to its successor, i.e., n ↦ n + 1 n \mapsto n+1 n↦n+1 .

无论选择何种朴素集合论，我们都能按上述方式构造皮亚诺范畴 P \mathcal{P} P。若作为范畴基础的朴素集合论足够完善------例如，是包含自然数集 N = { 1 , 2 , 3 , ... } \mathbb{N} = \{1,2,3,\dots\} N={1,2,3,...} 的标准集合论之一------那么自然数集 N \mathbb{N} N 可被视作皮亚诺范畴 P \mathcal{P} P 中的一个对象，其基点为 1 ∈ N 1 \in \mathbb{N} 1∈N，后继映射 s : N → N s: \mathbb{N} \to \mathbb{N} s:N→N 为自然数的后继运算，即 n ↦ n + 1 n \mapsto n+1 n↦n+1。

Given any object X = ( X , x , s ) \mathcal{X}=(X, x, s) X=(X,x,s) in O b ( P ) Ob(\mathcal{P}) Ob(P) there is one and only one morphism from N \mathbb{N} N to X \mathcal{X} X in the category P \mathcal{P} P ; it is given by the mapping of sets sending 1 ∈ N 1 \in \mathbb{N} 1∈N to the base point x ∈ X x \in X x∈X (for, indeed, any morphism in P \mathcal{P} P is required to send base point to base point) and the mapping is then "forced," from then on, by the formula f ( n + 1 ) = s f ( n ) f(n+1)=s f(n) f(n+1)=sf(n) .

对皮亚诺范畴 O b ( P ) Ob(\mathcal{P}) Ob(P) 中的任意对象 X = ( X , x , s ) \mathcal{X}=(X, x, s) X=(X,x,s)，从 N \mathbb{N} N 到 X \mathcal{X} X 的态射唯一存在 ：该态射是这样的集合映射------将 1 ∈ N 1 \in \mathbb{N} 1∈N 映射到基点 x ∈ X x \in X x∈X（皮亚诺范畴的态射必然保基点），而后由公式 f ( n + 1 ) = s ( f ( n ) ) f(n+1) = s(f(n)) f(n+1)=s(f(n)) 唯一确定后续所有元素的映射关系。

In summary, there is a unique morphism in P \mathcal{P} P from the natural numbers to any object in the category. That is, the natural numbers, N \mathbb{N} N , is an initial object of P \mathcal{P} P .

综上，皮亚诺范畴 P \mathcal{P} P 中，从自然数集 N \mathbb{N} N 到该范畴的任意对象，都存在唯一的态射。也就是说，自然数集 N \mathbb{N} N 是皮亚诺范畴 P \mathcal{P} P 的始对象。

Moreover, as any initial object in any category is uniquely characterized, up to unique isomorphism, by its role as initial object, the natural numbers when viewed as initial object of P \mathcal{P} P is similarly pinned down.

此外，正如任意范畴中的始对象都能通过其始对象的角色，在唯一同构的意义下被唯一刻画，将自然数集视作皮亚诺范畴的始对象时，它也能以同样的方式被精准界定。

This strategy of defining the Natural Numbers as "an" initial object in a category of (what amounts to) discrete dynamical systems, as we have just done, is revealing, I think; it isolates, as Peano himself had done, the fundamental role of mere succession in the formulation of the natural numbers. It also follows the third of the three formats we listed for defining natural numbers; it is, in a sense that deserves to be understood, a compromise strategy between a bureau-of-standards kind of definition, and a Fregean universal quantification approach. Notice, though, its elegant shifting sands. At the very least, we have a definition that depends upon a selection of a set theory, as well as an agreement to deal with the object Z \mathbb{Z} Z pinned down "up to unique isomorphism." We have even further to go, but first let us discuss how our definition differs in approach from the standard way of expressing Peano's axioms.

我认为，我们将自然数定义为（本质上的）离散动力系统范畴中的一个始对象，这一策略极具启发性：它如同皮亚诺本人所做的那样，将后继运算 从自然数的定义中提炼了出来，凸显了其基础性作用。同时，这一定义也契合我们此前列出的第三种自然数定义范式；在某种值得深入理解的意义上，它是标准局式定义 与弗雷格全称量化定义 之间的折中策略。但需要注意的是，这一定义并非绝对固定，而是有着精妙的灵活性：它至少依赖于集合论的选择，且要求我们在"唯一同构的意义下"研究对象 Z \mathbb{Z} Z。我们的探讨尚未结束，不过在此之前，先分析这一定义与皮亚诺公理的标准表述在方法上的差异。

11 Light, shadow, dark

11 明、影、暗

In elementary mathematics classes, we usually describe Peano's axioms that characterize the natural numbers roughly as follows.

在初等数学课程中，我们通常将刻画自然数的皮亚诺公理大致描述如下。

The natural numbers N \mathbb{N} N is a set with a chosen element 1 ∈ N 1 \in \mathbb{N} 1∈N and an injective ("successor") function s : N → N s: \mathbb{N} \to \mathbb{N} s:N→N such that 1 ∉ s ( N ) 1 \notin s(\mathbb{N}) 1∈/s(N) and such that mathematical induction holds, in the sense that if P ( n ) P(n) P(n) is any proposition which may be formulated for all n ∈ N n \in \mathbb{N} n∈N , and for which P ( 1 ) P(1) P(1) is true, and which has the further property that whenever P ( n ) P(n) P(n) is true then P ( s ( n ) ) P(s(n)) P(s(n)) is true, then P ( n ) P(n) P(n) is true for all n ∈ N n \in \mathbb{N} n∈N .

自然数集 N \mathbb{N} N 是一个集合，满足：存在选定元素 1 ∈ N 1 \in \mathbb{N} 1∈N，存在单射的后继函数 s : N → N s: \mathbb{N} \to \mathbb{N} s:N→N，且 1 ∉ s ( N ) 1 \notin s(\mathbb{N}) 1∈/s(N)；同时满足数学归纳法 ：若命题 P ( n ) P(n) P(n) 对所有 n ∈ N n \in \mathbb{N} n∈N 有定义，且 P ( 1 ) P(1) P(1) 为真，并且若 P ( n ) P(n) P(n) 为真则必有 P ( s ( n ) ) P(s(n)) P(s(n)) 为真，那么对所有 n ∈ N n \in \mathbb{N} n∈N， P ( n ) P(n) P(n) 均为真。

This, of course, has shock-value: it recruits the entire apparatus of propositional verification to its particular end. The fact that, especially when taken broadly, mathematical induction has extraordinary consequences, is amply illustrated by the ingenious work of Gentzen. To formalize things, we tame these axioms by explicitly providing a setting in which the words proposition and true make sense.

当然，这一公理体系有着令人震撼的特点：它将整个命题逻辑的验证体系，服务于自然数的定义。根岑的精妙研究充分表明，数学归纳法的内涵极为丰富，尤其是在广义上，能推导出诸多深刻的结论。为了将皮亚诺公理化形式化，我们需要明确设定一个框架，让"命题"和"真"这些概念有明确的含义。

The easiest way of comparing the Peano axioms with the Peano category as modes of defining natural Numbers, is to ask what each of these formats:

要比较皮亚诺公理与皮亚诺范畴这两种自然数定义方式，最简便的方法是分析这两种范式分别：

shines a spotlight on?
凸显了什么？
keeps in the shadows?
淡化了什么？
keeps in the dark?
完全回避了什么？

Both ways of pinpointing natural numbers are fastidiously explicit about the fact that a certain discrete dynamical system is involved: each shine their spotlight on the essence of iteration, the successor function. The Peano axioms do this by focussing in somewhat more detail on the elementary properties of this successor function, requiring as those axioms do, that 1 not be in the image of s, and that s be injective. The Peano category approach does this by simply considering the entire species of discrete dynamical systems with chosen base point.

两种定义方式都明确指出，自然数是一类离散动力系统 ：二者均凸显了迭代与后继函数。皮亚诺公理通过细致刻画后继函数的初等性质来实现这一点，例如要求 1 不在后继函数的像中、后继函数是单射；而皮亚诺范畴的方式，则是直接将所有带选定基点的离散动力系统作为研究对象。

Both modes of definition need a way of insisting on a certain "minimality" for the structure of natural numbers that they are developing. The Peano axioms formulate this "minimality" by dependence upon the domino effect of truth in a mathematically inductive context. The Peano category approach formulates "minimality" by considering the position of the natural numbers as a discrete dynamical system, among all discrete dynamical systems.

两种定义方式都需要刻画自然数结构的极小性 。皮亚诺公理通过数学归纳法中"真值的多米诺效应"，来定义这种极小性；而皮亚诺范畴的方式，则是将自然数视作离散动力系统范畴中的始对象，通过其在所有离散动力系统中的位置，来刻画这种极小性。

The Peano axiom approach calls up the full propositional apparatus of mathematics. But the details of the apparatus are kept in the shadows: you are required to "bring your own" propositional vocabulary if you wish to even begin to flesh out those axioms. The Peano category approach keeps all this in the dark: no mention whatsoever is made of propositional language.

皮亚诺公理的方式，调用了整个数学的命题逻辑体系，但却淡化了该体系的细节：若要深入阐释皮亚诺公理，人们需要自行设定命题逻辑的相关术语。而皮亚诺范畴的方式，则完全回避了命题逻辑：在其定义中，未提及任何命题语言相关的内容。

The Peano axiom approach requires -at least explicitly-hardly any investment in some specific brand of set theory. At most one set is on the scene, the set of natural numbers itself. In contrast, the Peano category approach forces you to "bring your own set theory" to make sense of it.

皮亚诺公理的方式，至少在显式层面 ，几乎不依赖于任何特定的集合论：整个公理体系中，最多只涉及一个集合，即自然数集本身。与之相反，皮亚诺范畴的方式则要求人们自行配备集合论，否则无法理解其定义。

When we gauge the differences in various mathematical viewpoints, it is a good thing to contrast them not only by what equipment these viewpoints ultimately invoke to establish their stance, for ultimately they may very well require exactly the same things, but also to pay attention to the order in which each piece of equipment is introduced and to the level of explicitness required for it to play its role.

在比较不同的数学观点时，我们不仅要对比这些观点最终为确立自身立场所调用的工具（因为它们最终所需的工具可能完全相同），更要关注这些工具被引入的顺序 ，以及工具发挥作用所需的显式化程度，这一点至关重要。

12 Representing one theory in another

12 一个理论在另一个理论中的表示

If categories package entire mathematical theories, it is natural to imagine that we might find the shadow of one mathematical theory (as packaged by a category C \mathcal{C} C ) in another mathematical theory (as packaged by a category D \mathcal{D} D ). We might do this by establishing a "mapping" of the entire category C \mathcal{C} C to the category D \mathcal{D} D . Such a "mapping" should, of course, send basic features (i.e., objects, morphisms) of C \mathcal{C} C to corresponding features of the category D \mathcal{D} D , and moreover, it must relate the composition law of morphisms in C \mathcal{C} C to the corresponding law for morphisms of D \mathcal{D} D ; we call such a "mapping" a functor from C \mathcal{C} C to D \mathcal{D} D .

既然范畴能整合整个数学理论，那么我们自然可以设想：将一个由范畴 C \mathcal{C} C 整合的数学理论，在另一个由范畴 D \mathcal{D} D 整合的数学理论中找到对应影子 。要实现这一点，我们需要建立从整个范畴 C \mathcal{C} C 到范畴 D \mathcal{D} D 的一个"映射"，该映射需满足：将 C \mathcal{C} C 的基本要素（对象、态射）映射到 D \mathcal{D} D 的对应要素，且保持态射的合成法则------我们将这种"映射"称为从 C \mathcal{C} C 到 D \mathcal{D} D 的函子。

To give a functor F F F from C \mathcal{C} C to D \mathcal{D} D , then, we must stipulate how we associate to any object X X X of C \mathcal{C} C a well-defined object F ( X ) F(X) F(X) of D \mathcal{D} D , and to any morphism between objects f : X → Y f: X \to Y f:X→Y of C \mathcal{C} C a well-defined morphism F ( f ) : F ( X ) → F ( Y ) F(f): F(X) \to F(Y) F(f):F(X)→F(Y) between corresponding objects of D \mathcal{D} D ; and, as mentioned, this relationship between objects and morphisms in C \mathcal{C} C to objects and morphisms in D \mathcal{D} D must respect identity morphisms, and the composition laws of these categories 7 ^7 7. Let us denote such a functor F F F from C \mathcal{C} C to D \mathcal{D} D by a broken arrow F : C ⇢ D F: \mathcal{C}\dashrightarrow \mathcal{D} F:C⇢D .

要定义从范畴 C \mathcal{C} C 到范畴 D \mathcal{D} D 的函子 F F F，需明确：

对 C \mathcal{C} C 中的任意对象 X X X，指定 D \mathcal{D} D 中的一个明确定义的对象 F ( X ) F(X) F(X) 与之对应；
对 C \mathcal{C} C 中的任意态射 f : X → Y f: X \to Y f:X→Y，指定 D \mathcal{D} D 中的一个明确定义的态射 F ( f ) : F ( X ) → F ( Y ) F(f): F(X) \to F(Y) F(f):F(X)→F(Y) 与之对应；
且上述对应关系需保恒等态射 和保态射合成法则 。我们将从 C \mathcal{C} C 到 D \mathcal{D} D 的函子 F F F 记为虚线箭头： F : C ⇢ D F: \mathcal{C} \dashrightarrow \mathcal{D} F:C⇢D。

By definition, then, a functor F F F from C \mathcal{C} C to D \mathcal{D} D associates to each object U U U of C \mathcal{C} C an object of D \mathcal{D} D, call it F ( U ) F(U) F(U); and to each morphism h : U → V h: U \to V h:U→V of C \mathcal{C} C, a morphism of D \mathcal{D} D, call it F ( h ) : F ( U ) → F ( V ) F(h): F(U) \to F(V) F(h):F(U)→F(V).
因此，根据定义，从范畴 C \mathcal{C} C 到范畴 D \mathcal{D} D 的函子 F F F 满足：对 C \mathcal{C} C 中的任意对象 U U U，指定 D \mathcal{D} D 中的一个对象与之对应，记为 F ( U ) F(U) F(U)；对 C \mathcal{C} C 中的任意态射 h : U → V h: U \to V h:U→V，指定 D \mathcal{D} D 中的一个态射与之对应，记为 F ( h ) : F ( U ) → F ( V ) F(h): F(U) \to F(V) F(h):F(U)→F(V)。

If the morphism 1 U : U → U 1_U: U \to U 1U:U→U is the identity, then we require that F ( 1 U ) = 1 F ( U ) F(1_U) = 1_{F(U)} F(1U)=1F(U), i.e., that it be the identity morphism of the object F ( U ) F(U) F(U) in D \mathcal{D} D. When we say that the functor F F F respects composition laws we mean that if g : V → W g: V \to W g:V→W is a morphism of C \mathcal{C} C (so that we can form the composition g ⋅ h : U → W g \cdot h: U \to W g⋅h:U→W in the category C \mathcal{C} C) we have the law
若态射 1 U : U → U 1_U: U \to U 1U:U→U 是恒等态射，则要求 F ( 1 U ) = 1 F ( U ) F(1_U) = 1_{F(U)} F(1U)=1F(U)，即 F ( 1 U ) F(1_U) F(1U) 是 D \mathcal{D} D 中对象 F ( U ) F(U) F(U) 的恒等态射。当我们称函子 F F F 保合成法则 时，意指：若 g : V → W g: V \to W g:V→W 是 C \mathcal{C} C 中的态射（从而可在范畴 C \mathcal{C} C 中构造合成态射 g ⋅ h : U → W g \cdot h: U \to W g⋅h:U→W），则有如下法则：
F ( g ⋅ h ) = F ( g ) ⋅ F ( h ) : F ( U ) → F ( W ) . F(g \cdot h) = F(g) \cdot F(h): F(U) \to F(W). F(g⋅h)=F(g)⋅F(h):F(U)→F(W).

In this way, we have a vocabulary for establishing bridges between whole disciplines of mathematics; we have a way of representing grand aspects of, say, topology in algebra (or conversely) by establishing functors from the category of topological spaces to the category of groups (or conversely): construct the pertinent functors from the one category to the other!

函子为我们提供了一套术语，让我们能在不同的数学分支之间搭建桥梁：例如，要将拓扑学的内容表示在代数学中（或反之），只需建立从拓扑空间范畴到群范畴的函子（或反之）------构造出这个函子即可！

The easiest thing to do, at least in mathematics, is to forget, and the forgetting process offers us some elementary functors, such as the functor from topological spaces to sets that passes from a topological space to its underlying set, thereby forgetting its topology. Of course one should also pay one's respects to the simplest of functors, the identity functor C ⇢ 1 C C \mathcal{C}\dashrightarrow^{\mathfrak{1}_{\mathcal{C}}}\mathcal{C} C⇢1CC , which when presented with any object U U U of C \mathcal{C} C it gives it back to you intact, as it does each morphism.

至少在数学中，最容易做到的事就是遗忘，而遗忘的过程能为我们带来一些初等函子。例如，从拓扑空间范畴到集合范畴的遗忘函子 ：将拓扑空间映射到其底集合，从而"遗忘"其拓扑结构。当然，我们也不能忽略最简单的函子------恒等函子 C ⇢ 1 C C \mathcal{C} \dashrightarrow^{\mathfrak{1}_{\mathcal{C}}} \mathcal{C} C⇢1CC：将范畴 C \mathcal{C} C 中的任意对象和态射，都映射为其自身。

The more profound bridges between fields of mathematics are achieved by more interesting constructions. But there is a ubiquitous type of functor, as easy to construct as one can imagine, and yet extraordinarily revealing. Given any object X X X in any category C \mathcal{C} C we will construct (in section 14) an important functor (we will denote it F X F_{X} FX ) from C \mathcal{C} C to S \mathcal{S} S , the category of sets upon which C \mathcal{C} C was modeled. This functor F X F_{X} FX will be enough to "reconstruct" X X X , but-as you might guess-only "up to canonical isomorphism."

数学不同分支之间更深刻的桥梁，由更精妙的构造所搭建。但存在一种极为普遍的函子，其构造简单易懂，却能揭示极为深刻的数学内涵。给定任意范畴 C \mathcal{C} C 中的任意对象 X X X，我们将在第 14 节中构造一个重要的函子 F X F_X FX：从范畴 C \mathcal{C} C 到其基础集合范畴 S \mathcal{S} S。这个函子 F X F_X FX 足以让我们"重构"对象 X X X ------当然，如你所想，只是在"典范同构的意义下"。

But before we do this, we need to say what we mean by a morphism from one functor to another.

但在此之前，我们需要先定义函子之间的态射。

13 Mapping one functor to another

13 函子间的态射

If we are given two functors, F , G : C ⇢ D F, G: \mathcal{C}\dashrightarrow \mathcal{D} F,G:C⇢D , by a morphism of functors μ : F → G \mu: F \to G μ:F→G we mean that we are given, for each object U U U of C \mathcal{C} C , a morphism μ U : F ( U ) → G ( U ) \mu_{U}: F(U) \to G(U) μU:F(U)→G(U) in the category D \mathcal{D} D which respects the structures involved 8 ^8 8.

设 F , G : C ⇢ D F, G: \mathcal{C} \dashrightarrow \mathcal{D} F,G:C⇢D 是从范畴 C \mathcal{C} C 到范畴 D \mathcal{D} D 的两个函子，函子间的态射 μ : F → G \mu: F \to G μ:F→G 指的是：对 C \mathcal{C} C 中的任意对象 U U U，指定 D \mathcal{D} D 中的一个态射 μ U : F ( U ) → G ( U ) \mu_U: F(U) \to G(U) μU:F(U)→G(U)，且该指定满足结构相容性。

in the sense that for every pair of objects U , V U, V U,V of C \mathcal{C} C, and morphism h : U → V h: U \to V h:U→V in Mor C ( U , V ) \text{Mor}{\mathcal{C}}(U, V) MorC(U,V) we have the equality
其含义为：对范畴 C \mathcal{C} C 中任意一对对象 U , V U, V U,V，以及 Mor C ( U , V ) \text{Mor}{\mathcal{C}}(U, V) MorC(U,V) 中的任意态射 h : U → V h: U \to V h:U→V，均有如下等式：
G ( h ) ⋅ μ U = μ V ⋅ F ( h ) , G(h) \cdot \mu_U = \mu_V \cdot F(h), G(h)⋅μU=μV⋅F(h),
both left and right hand side of this equation being morphisms F ( U ) → G ( V ) F(U) \to G(V) F(U)→G(V) in the category D \mathcal{D} D.
该等式的左右两边均为范畴 D \mathcal{D} D 中从 F ( U ) F(U) F(U) 到 G ( V ) G(V) G(V) 的态射。

To offer a humble example, for any functor F : C ⇢ D F: \mathcal{C}\dashrightarrow \mathcal{D} F:C⇢D , we have the identity morphism of the functor F F F to itself, F → 1 F F F \stackrel{\mathfrak{1}{F}}{\to} F F→1FF which associates, to an object U U U of C \mathcal{C} C , the identity morphism 1 F ( U ) : F ( U ) → F ( U ) 1{F(U)}: F(U) \to F(U) 1F(U):F(U)→F(U) (this being the identity morphism of the object F ( U ) F(U) F(U) in the category D \mathcal{D} D ). You might think that this example is not very enlightening, but it already holds its surprises; in any event, in the next section we shall visit an important large repertoire of morphisms of functors.

举一个简单的例子：对任意函子 F : C ⇢ D F: \mathcal{C} \dashrightarrow \mathcal{D} F:C⇢D，存在从函子 F F F 到自身的恒等态射 F → 1 F F F \stackrel{\mathfrak{1}{F}}{\to} F F→1FF，即对 C \mathcal{C} C 中的任意对象 U U U，指定 D \mathcal{D} D 中的恒等态射 1 F ( U ) : F ( U ) → F ( U ) 1{F(U)}: F(U) \to F(U) 1F(U):F(U)→F(U)（ 1 F ( U ) 1_{F(U)} 1F(U) 是 D \mathcal{D} D 中对象 F ( U ) F(U) F(U) 的恒等态射）。你可能认为这个例子并无太多启发性，但它实则蕴含着一些值得探究的细节；无论如何，我们将在下一节中介绍一类重要且丰富的函子态射。

Once we have settled on the definition of morphism of functors, our way is clear to define isomorphism of functors for the definition of this notion follows the natural format for the definition of isomorphism related to absolutely any species of mathematical object. Namely, an isomorphism of the functor F : C ⇢ D F: \mathcal{C}\dashrightarrow \mathcal{D} F:C⇢D and the functor G : C ⇢ D G: \mathcal{C}\dashrightarrow \mathcal{D} G:C⇢D is a morphism of functors μ : F → G \mu: F \to G μ:F→G for which there is a morphism of functors going the other way, ν : G → F \nu: G \to F ν:G→F such that ν ⋅ μ : F → F \nu \cdot \mu: F \to F ν⋅μ:F→F and μ ⋅ ν : G → G \mu \cdot \nu: G \to G μ⋅ν:G→G are equal to the respective identity morphisms (of functors).

一旦定义了函子间的态射，我们就能自然地定义函子的同构 ------其定义范式与所有数学对象的同构定义一致：设 F , G : C ⇢ D F, G: \mathcal{C} \dashrightarrow \mathcal{D} F,G:C⇢D 是两个函子，若存在函子态射 μ : F → G \mu: F \to G μ:F→G 和 ν : G → F \nu: G \to F ν:G→F，使得 ν ∘ μ = 1 F \nu \circ \mu = \mathfrak{1}_F ν∘μ=1F 且 μ ∘ ν = 1 G \mu \circ \nu = \mathfrak{1}_G μ∘ν=1G（ 1 F , 1 G \mathfrak{1}_F, \mathfrak{1}_G 1F,1G 分别为 F , G F, G F,G 的恒等态射），则称函子 F F F 与 G G G 同构。

To understand the notion of isomorphism of functors I find it particularly illuminating to consider, for the various categories of interest, what the automorphisms of the identity functor consist of. Note, to take a random example, that if V \mathcal{V} V is the category of vector spaces over a field k k k , then multiplication by any nonzero scalar (i.e., element of k ∗ k^{*} k∗ ) is an automorphism of the identity functor. That is, let 1 V : V ⇢ V 1_{\mathcal{V}}: \mathcal{V}\dashrightarrow \mathcal{V} 1V:V⇢V denote the identity functor; for any fixed nonzero scalar λ ∈ k ∗ \lambda \in k^{*} λ∈k∗ we can form (for all vector spaces U U U over k k k ) the morphism in V \mathcal{V} V λ U : U → U \lambda_{U}: U \to U λU:U→U defined by x ↦ λ ⋅ x x \mapsto \lambda \cdot x x↦λ⋅x , and this data can be thought of as giving an isomorphism of functors λ : 1 V ≅ 1 V \lambda: 1_{\mathcal{V}} \cong 1_{\mathcal{V}} λ:1V≅1V .

要理解函子同构的概念，我发现一个极具启发性的方法：对各类我们感兴趣的范畴，分析其恒等函子的自同构 。举一个随机的例子：设 V \mathcal{V} V 是域 k k k 上的向量空间范畴，那么非零标量乘法 就是该范畴恒等函子的自同构。即，设 1 V : V ⇢ V 1_{\mathcal{V}}: \mathcal{V} \dashrightarrow \mathcal{V} 1V:V⇢V 为 V \mathcal{V} V 的恒等函子，对任意固定的非零标量 λ ∈ k ∗ \lambda \in k^* λ∈k∗，对 k k k 上的任意向量空间 U U U，定义 V \mathcal{V} V 中的态射 λ U : U → U \lambda_U: U \to U λU:U→U 为 x ↦ λ ⋅ x x \mapsto \lambda \cdot x x↦λ⋅x，则这一系列态射构成了恒等函子的自同构： λ : 1 V ≅ 1 V \lambda: 1_{\mathcal{V}} \cong 1_{\mathcal{V}} λ:1V≅1V。

14 An object "as" a functor from the theory-in-which-it-lives to set theory

14 作为"其所属理论到集合论的函子"的对象

Given an object X X X of a category C \mathcal{C} C , we shall define a specific functor (that we will denote F X F_{X} FX ) that encodes the essence of the object X X X . The functor F X F_{X} FX will, in fact, determine X X X up to canonical isomorphism.

给定范畴 C \mathcal{C} C 中的一个对象 X X X，我们将定义一个具体的函子 F X F_X FX，它能编码对象 X X X 的全部特征；事实上，这个函子 F X F_X FX 能在典范同构的意义下 唯一确定对象 X X X。

This functor F X F_{X} FX maps the category C \mathcal{C} C to the category S \mathcal{S} S of sets (the same category of sets on which C \mathcal{C} C is "modeled," as we've described in section 7 above).

函子 F X F_X FX 是从范畴 C \mathcal{C} C 到集合范畴 S \mathcal{S} S 的映射（ S \mathcal{S} S 是我们在第 7 节中定义的、范畴 C \mathcal{C} C 所基于的集合范畴）。

Here is how it is defined. The functor F X F_{X} FX assigns to any object Y Y Y of C \mathcal{C} C the set of morphisms from X X X to Y Y Y ; that is, F X ( Y ) : = M o r C ( X , Y ) F_{X}(Y):=Mor_{\mathcal{C}}(X, Y) FX(Y):=MorC(X,Y) .

其定义如下：对范畴 C \mathcal{C} C 中的任意对象 Y Y Y，函子 F X F_X FX 将其映射为从 X X X 到 Y Y Y 的所有态射构成的集合，即 F X ( Y ) : = M o r C ( X , Y ) F_{X}(Y) := Mor_{\mathcal{C}}(X, Y) FX(Y):=MorC(X,Y)。

Now, M o r C ( X , Y ) Mor_{\mathcal{C}}(X, Y) MorC(X,Y) is indeed a set, i.e., an object of S \mathcal{S} S , so we have described a mapping from objects of C \mathcal{C} C to sets, Y ↦ F X ( Y ) = M o r C ( X , Y ) Y \mapsto F_{X}(Y)=Mor_{\mathcal{C}}(X, Y) Y↦FX(Y)=MorC(X,Y) .

显然， M o r C ( X , Y ) Mor_{\mathcal{C}}(X, Y) MorC(X,Y) 是一个集合，即集合范畴 S \mathcal{S} S 中的对象，因此我们已定义了从 C \mathcal{C} C 的对象到 S \mathcal{S} S 的对象的映射： Y ↦ F X ( Y ) = M o r C ( X , Y ) Y \mapsto F_{X}(Y) = Mor_{\mathcal{C}}(X, Y) Y↦FX(Y)=MorC(X,Y)。

Moreover, to every morphism g : Y → Z g: Y \to Z g:Y→Z of C \mathcal{C} C , our functor F X F_{X} FX assigns the mapping of sets F X ( g ) : F X ( Y ) = M o r C ( X , Y ) → F X ( Z ) = M o r C ( X , Z ) F_{X}(g): F_{X}(Y)=Mor_{\mathcal{C}}(X, Y) \to F_{X}(Z)=Mor_{\mathcal{C}}(X, Z) FX(g):FX(Y)=MorC(X,Y)→FX(Z)=MorC(X,Z) given simply by composition with g g g ; i.e., if f ∈ F X ( Y ) f \in F_{X}(Y) f∈FX(Y) , the mapping F X ( g ) F_{X}(g) FX(g) sends f f f to g ⋅ f ∈ F X ( Z ) g \cdot f \in F_{X}(Z) g⋅f∈FX(Z) : f ↦ g ⋅ f f \mapsto g \cdot f f↦g⋅f .

此外，对范畴 C \mathcal{C} C 中的任意态射 g : Y → Z g: Y \to Z g:Y→Z，函子 F X F_X FX 将其映射为集合之间的映射 F X ( g ) : F X ( Y ) → F X ( Z ) F_X(g): F_X(Y) \to F_X(Z) FX(g):FX(Y)→FX(Z)，该映射的定义为与 g g g 的复合 ：即对任意 f ∈ F X ( Y ) = M o r C ( X , Y ) f \in F_X(Y) = Mor_{\mathcal{C}}(X,Y) f∈FX(Y)=MorC(X,Y)，有 F X ( g ) ( f ) = g ∘ f ∈ F X ( Z ) = M o r C ( X , Z ) F_X(g)(f) = g \circ f \in F_X(Z) = Mor_{\mathcal{C}}(X,Z) FX(g)(f)=g∘f∈FX(Z)=MorC(X,Z)，简记为 f ↦ g ⋅ f f \mapsto g \cdot f f↦g⋅f。

In this way, every object X X X of any category C \mathcal{C} C gives us a functor, F X F_{X} FX from C \mathcal{C} C to S \mathcal{S} S .

通过这种方式，任意范畴 C \mathcal{C} C 中的任意对象 X X X，都能诱导出一个从 C \mathcal{C} C 到集合范畴 S \mathcal{S} S 的函子 F X F_X FX。

Also any morphism h : X ′ → X h: X' \to X h:X′→X in C \mathcal{C} C gives rise to a morphism of functors η : F X → F X ′ \eta: F_{X} \to F_{X'} η:FX→FX′ by this simple formula: for an element Y Y Y of C \mathcal{C} C , the morphism h h h gives us a mapping of sets η Y : F X ( Y ) → F X ′ ( Y ) \eta_{Y}: F_{X}(Y) \to F_{X'}(Y) ηY:FX(Y)→FX′(Y) by sending any f : X → Y f: X \to Y f:X→Y in F X ( Y ) F_{X}(Y) FX(Y) to f ⋅ h : X ′ → Y f \cdot h: X' \to Y f⋅h:X′→Y in F X ′ ( Y ) F_{X'}(Y) FX′(Y) . The rule associating to an object Y Y Y the mapping of sets η Y \eta_{Y} ηY produces our morphism of functors η : F X → F X ′ \eta: F_{X} \to F_{X'} η:FX→FX′ .

同时，范畴 C \mathcal{C} C 中的任意态射 h : X ′ → X h: X' \to X h:X′→X，也能诱导出函子间的态射 η : F X → F X ′ \eta: F_X \to F_{X'} η:FX→FX′，其定义十分简洁：对 C \mathcal{C} C 中的任意对象 Y Y Y，由态射 h h h 定义集合映射 η Y : F X ( Y ) → F X ′ ( Y ) \eta_Y: F_X(Y) \to F_{X'}(Y) ηY:FX(Y)→FX′(Y)，即将任意 f : X → Y ∈ F X ( Y ) f: X \to Y \in F_X(Y) f:X→Y∈FX(Y) 映射为 f ∘ h : X ′ → Y ∈ F X ′ ( Y ) f \circ h: X' \to Y \in F_{X'}(Y) f∘h:X′→Y∈FX′(Y)；将对象 Y Y Y 对应到集合映射 η Y \eta_Y ηY 的这一规则，就构成了函子态射 η : F X → F X ′ \eta: F_{X} \to F_{X'} η:FX→FX′。

The fundamental, but miraculously easy to establish, fact is that the object X X X is entirely retrievable (however, only up to canonical isomorphism, of course) from knowledge of this functor F X F_{X} FX . This fact, a consequence of a result known as Yoneda's Lemma, can be expressed this way:

一个极为基础但证明却异常简洁的事实是：函子 F X F_X FX 能唯一地还原出对象 X X X（当然，仅在典范同构的意义下）。这一事实是米田引理的直接推论，可表述为：

Theorem : Let X , X ′ X, X' X,X′ be objects in a category C \mathcal{C} C . Suppose we are given an isomorphism of their associated functors η : F X ≅ F X ′ \eta: F_{X} \cong F_{X'} η:FX≅FX′ . Then there is a unique isomorphism of the objects themselves, h : X ′ ≅ X h: X' \cong X h:X′≅X that gives rise-as in the process described above-to this isomorphism of functors.
定理：设 X , X ′ X, X' X,X′ 为范畴 C \mathcal{C} C 中的两个对象，若其诱导的函子 F X F_X FX 与 F X ′ F_{X'} FX′ 同构，即存在 η : F X ≅ F X ′ \eta: F_{X} \cong F_{X'} η:FX≅FX′，则存在唯一的对象同构 h : X ′ ≅ X h: X' \cong X h:X′≅X，且该同构恰好是诱导上述函子同构的态射。

The beauty of this result is that it has the following decidedly structuralist, or Wittgensteinian language-game, interpretation: an object X X X of a category C \mathcal{C} C is determined (always only up to canonical isomorphism, the recurrent theme of this article!) by the network of relationships that the object X X X has with all the other objects in C \mathcal{C} C .

这一结果的精妙之处，在于它蕴含着鲜明的结构主义 （或维特根斯坦"语言游戏"）解读：范畴 C \mathcal{C} C 中的对象 X X X，由其与 C \mathcal{C} C 中所有其他对象的关系网络唯一确定（当然，始终是在典范同构的意义下------这是本文反复强调的主题）。

Yoneda's lemma, in its fuller expression, tells us that the set of morphisms (of the category C \mathcal{C} C ) from an object X X X to an object Y Y Y is naturally in one-one correspondence with the set of morphisms of the functor F Y F_{Y} FY to the functor F X F_{X} FX .

完整的米田引理还告诉我们：范畴 C \mathcal{C} C 中从对象 X X X 到对象 Y Y Y 的态射集 M o r C ( X , Y ) Mor_{\mathcal{C}}(X,Y) MorC(X,Y)，与函子 F Y F_Y FY 到函子 F X F_X FX 的态射集之间，存在自然的一一对应。

In brief, we have (or rather, Yoneda has) reconstructed the category C \mathcal{C} C , objects and morphisms alike, purely in terms of functors to sets; i.e., in terms of networks of relationships that deal with the entire category at once 9 ^9 9.

简而言之，我们（更准确地说，是米田）仅通过到集合范畴的函子 ，就重构了整个范畴 C \mathcal{C} C ------包括其所有对象和态射；也就是说，我们通过整个范畴的全局关系网络，完成了范畴的重构。

The connection between Yoneda's lemma and structuralist and/or Wittgensteinian attitudes towards meaning was discussed in Michael Harris's review of Mathematics and the Roots of Post-modern Theory [Harris 2003]
米田引理与结构主义和/或维特根斯坦主义意义观之间的关联，在迈克尔·哈里斯对《数学与后现代理论的根源》一书的书评中有所讨论 [Harris 2003]。

With all this, Yoneda's Lemma is one of the many examples of a mathematical result that is both extraordinarily consequential, and also extraordinarily easy to prove 10 ^{10} 10.

米田引理是众多"结论极为深刻、证明却极为简洁"的数学结论的典型代表。

A full proof, for example, is given neatly and immediately via a single diagram in the wikipedia entry"oneda's_lemma" . For an accessible introductory reference to the ideas of category theory, see the article by Daniel K. Biss [Biss 2003]. For a more technical, but still relatively gentle, account of category theory, see Saunders MacLane's Categories for the working mathematician [Mac Lane 1971].
完整的证明，例如，在维基百科条目"米田引理"中，通过一个简洁的图示即刻给出。关于范畴论思想的入门介绍，可参考丹尼尔·K·比斯的文章[Biss 2003]。欲了解更为技术化但仍相对易懂的范畴论阐述，请参阅桑德斯·麦克莱恩的《数学家的范畴论》[Mac Lane 1971]。

15 Representable functors

15 可表函子

The following definition (especially as it pervades the mathematical work of Alexander Grothendieck) marked the beginning of a significantly new viewpoint in our subject.

下述定义（尤其是在亚历山大·格罗滕迪克的数学研究中被广泛应用后），标志着数学领域迎来了一个全新的视角。

A functor F : C ⇢ S F: \mathcal{C}\dashrightarrow \mathcal{S} F:C⇢S , from a category C \mathcal{C} C to the category of sets S \mathcal{S} S on which it is modeled, is said to be represented by an object X X X of C \mathcal{C} C if an isomorphism of functors F ≅ F X F \cong F_{X} F≅FX is given. The functor F F F is said, simply, to be representable if it can be represented by some object X X X .

设函子 F : C ⇢ S F: \mathcal{C} \dashrightarrow \mathcal{S} F:C⇢S 从范畴 C \mathcal{C} C 到其基础集合范畴 S \mathcal{S} S，若存在 C \mathcal{C} C 中的对象 X X X，使得函子 F F F 与 F X F_X FX 同构（ F ≅ F X F \cong F_{X} F≅FX），则称函子 F F F 由对象 X X X 表示 ；若存在这样的对象 X X X，则称函子 F F F 为可表函子。

If you consult the theorem quoted at the end of the last section you see that Yoneda's lemma, then, guarantees that if a functor F F F is representable, then F F F determines the object X X X that represents it up to unique isomorphism.

结合上一节末尾的定理可知，米田引理保证了：若函子 F F F 是可表函子，则它能在唯一同构的意义下 ，唯一确定表示它的对象 X X X。

One of the noteworthy lessons coming from subjects such as algebraic geometry is that often, when it is important for a theory to make a construction of a particular object that performs an important function, we have a ready description of the functor F F F that it would represent, if it exists. Often, indeed, the basic utility of the object X X X that represents this functor F F F comes exactly from that: that X X X represents the functor! 11 ^{11} 11 Although a specific construction of X X X may tell us more about the particularities of X X X , there is no guarantee that all the added information a construction provides-or any of it-furthers our insight beyond guaranteeing representability of F F F .

从代数几何等学科中，我们能得到一个重要的启示：当一个数学理论需要构造一个具有特定功能的对象时，我们往往能先直接描述出这个对象若存在则必然表示的函子 F F F；事实上，表示函子 F F F 的对象 X X X，其价值恰恰在于它是函子 F F F 的表示对象 。尽管对 X X X 的具体构造能让我们了解其更多细节，但这些细节并非总能帮助我们深化对问题的理解------除了验证函子 F F F 的可表性之外。

Students of algebra encounter this very early in their studies: the tensor product is (happily) nowadays first taught in terms of its functorial characterization, with its construction only coming afterwards; this is also the case for fiber products , for localization in commutative algebra; indeed this is the pattern of exposition for lots of notions in elementary mathematics, as it is for many of the grand constructions in modern algebraic geometry.
代数学的学习者在入门阶段就会遇到这一模式：如今，张量积的教学往往会先介绍其函子刻画，之后才给出具体构造；纤维积、交换代数中的局部化等概念也遵循这一模式。事实上，初等数学中的诸多概念，以及现代代数几何中的许多重大构造，都采用这种先刻画、后构造的讲解范式。

Some of the important turning points in the history of mathematics can be thought of as moments when we achieve a fuller understanding of what it means for one "thing" to represent another "thing." The issue of representation is already implicit in the act of counting, as when we say that these two mathematical units "represent" those two cows. Leibniz dreamed of a scheme for a universal language that would reduce ideas "to a kind of alphabet of human thought" and the ciphers in his universal language would be manipulable representations of ideas.

数学史上的一些重要转折点，本质上都是我们对"一个事物表示另一个事物的内涵"有了更深刻理解的时刻。"表示"的概念早在计数行为中就已隐含：例如，我们说两个数学符号"表示"两头牛。莱布尼茨曾构想过一套"通用语言"，能将人类的思想还原为"一种思想字母表"，而这套语言中的符号，就是对思想的可操作表示。

Kant reserved the term representation (Vorstellung) for quite a different role. Here is the astonishing way in which this concept makes its first appearance in the Critique of Pure Reason 12 ^{12} 12:

康德为"表象（Vorstellung）"这一术语赋予了截然不同的含义。在《纯粹理性批判》中，这一概念的首次出现极具震撼力：

There are only two possible ways in which synthetic representations and their objects ... can meet one another. Either the object (Gegenstand) alone must make the representation possible, or the representation alone must make the object possible.

综合表象与其对象的联结，仅有两种可能的方式：要么对象独自决定表象的可能性，要么表象独自决定对象的可能性。

$Kant 1961\], p. 125. \[Kant 1961\]，第 125 页。$

It is this either-or, this dance between object and representation, that animates lots of what follows in Kant's Critique of Pure Reason . With meanings quite remote from Kant's, the same two terms, object and representation, each provide grounding for the other, in our present discussion.

正是这种二选一的关系，这种对象与表象之间的互动，构成了康德《纯粹理性批判》后续内容的脉络。在我们的讨论中，"对象"与"表示"这两个术语的含义虽与康德的定义相去甚远，但也呈现出相互支撑、彼此奠基的关系。

Students of algebra encounter this very early in their studies: the tensor product is (happily) nowadays first taught in terms of its functorial characterization, with its construction only coming afterwards; this is also the case for fiber products, for localization in commutative algebra; indeed this is the pattern of exposition for lots of notions in elementary mathematics, as it is for many of the grand constructions in modern algebraic geometry.

代数学的学习者在入门阶段就能接触到这一思想：如今，张量积的教学通常先介绍其函子刻画，再给出具体构造；纤维积、交换代数中的局部化也是如此。事实上，初等数学中的诸多概念，以及现代代数几何中的许多重大构造，都遵循这一讲解范式。

Nowadays, whole subjects of mathematics are seen as represented in other subjects, the "represented" subject thereby becoming a powerful tool for the study of the "representing" subject, and vice versa.

如今，整个数学分支都可被视作在另一个分支中的表示，而被表示的分支也因此成为研究表示分支的有力工具，反之亦然。

It sometimes happens that the introduction of a term in a mathematical discussion is the signal that an important shift of viewpoint is taking place, or is about to take place. An emphasis on "representability" of functors in a branch of mathematics suggests an ever so slight, but ever so important, shift. The lights are dimmed on mathematical objects and beamed rather on the corresponding functors; that is, on the networks of relationships entailed by the objects. The functor has center stage, the object that it represents appears almost as an afterthought. The lights are dimmed on equality of mathematical objects as well, and focussed, rather, on canonical isomorphisms, and equivalence.

在数学研究中，一个术语的引入往往标志着一个重要的视角转变------或即将发生的转变。在一个数学分支中强调函子的"可表性"，就意味着一种微妙却至关重要的视角转变：人们不再聚焦于数学对象本身，而是将目光投向其对应的函子，即对象所蕴含的关系网络 ；函子成为研究的中心，而表示函子的对象则几乎成为附属品。同时，人们也不再关注数学对象的相等性 ，而是将焦点放在典范同构 与等价性上。

16 The Natural Numbers as functor

16 作为函子的自然数

Allow me to define, for any category, a particularly humble functor. If C \mathcal{C} C is a category with underlying set theory S \mathcal{S} S define a functor I : C ⇢ S I: \mathcal{C}\dashrightarrow \mathcal{S} I:C⇢S as follows:

让我们为任意范畴定义一个极为简单的函子。设范畴 C \mathcal{C} C 以集合论 S \mathcal{S} S 为基础，定义函子 I : C ⇢ S I: \mathcal{C}\dashrightarrow \mathcal{S} I:C⇢S 如下：

If X X X is an object of C \mathcal{C} C, let I ( X ) : = { X } I(X):= \{X\} I(X):={X}; that is, the set I ( X ) I(X) I(X) is the singleton consisting in the set containing only one element: the object X X X. If f : X → Y f: X \to Y f:X→Y is any morphism in C \mathcal{C} C, I ( f ) : { X } → { Y } I(f):\{X\} \to \{Y\} I(f):{X}→{Y} is the unique mapping of singleton sets. We may think of our functor I I I as a singleton functor : it is a functor from C \mathcal{C} C to the category of sets that assigns to each object of the category C \mathcal{C} C a singleton set. Any two "singleton functors" are (uniquely) isomorphic as functors. Is our functor I I I representable?

对 C \mathcal{C} C 中的任意对象 X X X，令 I ( X ) : = { X } I(X):=\{X\} I(X):={X}，即集合 I ( X ) I(X) I(X) 是仅以对象 X X X 为唯一元素的单元素集 ；对 C \mathcal{C} C 中的任意态射 f : X → Y f:X\to Y f:X→Y，令 I ( f ) : { X } → { Y } I(f):\{X\}\to\{Y\} I(f):{X}→{Y} 为单元素集之间的唯一映射。我们将这个函子 I I I 称为单点函子 ：它是从范畴 C \mathcal{C} C 到集合范畴的函子，将 C \mathcal{C} C 的每个对象都映射为一个单元素集。任意两个单点函子作为函子都是（唯一）同构的。那么，我们的单点函子 I I I 是可表函子吗？

The answer here is clean. The functor I I I is representable if and only if our category C \mathcal{C} C has an initial object. For if Z Z Z is an initial object, then F Z F_{Z} FZ, by the very meaning of initial object, is a singleton functor ( there is a unique morphism from Z Z Z to any object X X X of the category). Therefore F Z F_{Z} FZ is isomorphic as a functor to I I I. Conversely, any object that represents I I I has the feature that it needs for us to deem it an initial object of C \mathcal{C} C.

答案十分明确：单点函子 I I I 是可表函子，当且仅当范畴 C \mathcal{C} C 存在始对象 。原因如下：若 Z Z Z 是 C \mathcal{C} C 的始对象，根据始对象的定义，函子 F Z F_Z FZ 是一个单点函子（从 Z Z Z 到 C \mathcal{C} C 的任意对象都存在唯一态射），因此 F Z F_Z FZ 与 I I I 作为函子同构；反之，任何表示单点函子 I I I 的对象，都满足始对象的定义，即该对象是 C \mathcal{C} C 的始对象。

This viewpoint gives us a way of pinning down the natural numbers from a different angle, which at first glance may seem quite strange.

这一视角为我们提供了一种全新的、乍看之下略显奇特的自然数定义方式。

The natural numbers is defined uniquely, up to unique isomorphism, as an object of the Peano category P \mathcal{P} P that represents the singleton functor I I I.
自然数在唯一同构的意义下被唯一定义为：皮亚诺范畴 P \mathcal{P} P 中表示单点函子 I I I 的对象。

There is an aspect to this definition that Frege might have liked: nothing "bureau-of-standards-like," nothing that smacks of a subjective choice of some particular exemplar, has entered this description. But where, in this definition, are the tangible, familiar, natural numbers? You may well ask this question; for-despite the crispness of the above definition-the concept embodied by the good old symbols 1, 2, 3, . . . appears to have holographically smeared itself over the panoply of little "discrete dynamical systems" given by the objects of P \mathcal{P} P. And the category P \mathcal{P} P itself, remember, is but a template, dependent upon an underlying set theory. But we have even further to go.

这一定义有一个弗雷格可能会认可的特点：其中没有任何"标准局式"的设定，也没有任何主观选择特定范例的痕迹。但你可能会问，在这个定义中，那些具体、熟悉的自然数在哪里？尽管上述定义十分简洁，但 1、2、3......这些我们熟知的符号所代表的概念，似乎被"全息式"地融入了皮亚诺范畴 P \mathcal{P} P 的所有对象------即一众"离散动力系统"之中。而且别忘了，皮亚诺范畴 P \mathcal{P} P 本身也只是一个范式，其定义依赖于基础的集合论。而我们的探讨还能更进一步。

17 Equivalence of Categories

17 范畴的等价

If the grand lesson is that equivalence has some claim to priority over equality in the mathematical theories packaged by categories, why are categories themselves untouched by this insight? The answer is that they are not. With this brief Q & A, to say nothing of the title of this section, you will not be surprised to find that what is next on the agenda is

如果范畴论带给我们的启示是：在由范畴整合的数学理论中，等价性 优先于相等性，那么为何范畴本身不遵循这一原则？答案是，范畴同样遵循。通过这一简单的问答，再加上本节的标题，你不会对接下来的内容感到意外：

Definition

定义

A functor F : C ⇢ D F: \mathcal{C}\dashrightarrow \mathcal{D} F:C⇢D from the category C \mathcal{C} C to D \mathcal{D} D is called an equivalence of categories if there is a functor going the other way, G : D ⇢ C G: \mathcal{D}\dashrightarrow \mathcal{C} G:D⇢C such that G ∘ F G \circ F G∘F is isomorphic to the identity functor from C \mathcal{C} C to C \mathcal{C} C, and F ∘ G F \circ G F∘G is isomorphic to the identity functor from D \mathcal{D} D to D \mathcal{D} D. And we are specifically interested in the nature of many of our categories, only up to equivalence . So with this elementary vocabulary, entire theories are allowed to shift-up to equivalence.

设函子 F : C ⇢ D F: \mathcal{C}\dashrightarrow \mathcal{D} F:C⇢D 是从范畴 C \mathcal{C} C 到 D \mathcal{D} D 的映射，若存在反向的函子 G : D ⇢ C G: \mathcal{D}\dashrightarrow \mathcal{C} G:D⇢C，使得复合函子 G ∘ F G \circ F G∘F 与 C \mathcal{C} C 的恒等函子同构，且 F ∘ G F \circ G F∘G 与 D \mathcal{D} D 的恒等函子同构，则称函子 F F F 为范畴的等价 ，并称范畴 C \mathcal{C} C 与 D \mathcal{D} D 等价。我们研究多数范畴的性质时，只需考虑其在等价意义下 的表现。因此，借助这一基础术语，我们可以认为：整个数学理论都能在等价的意义下进行转换。

18 Object and problem

18 对象与问题

Following Kronecker, we sometimes allow ourselves to think, say, of 2 \sqrt{2} 2 as nothing more than a cipher that obeys the standard rules of arithmetic and about which all we know is that its square is 2. This characterization, to be sure, doesn't pin it down, here, for having named our cipher − 2 -\sqrt{2} −2 has precisely the same description. Nevertheless, there is no contradiction: 2 \sqrt{2} 2 we have given birth to a specific creature of mathematics, and − 2 -\sqrt{2} −2 is just another creature with (evidently!) a different name. It is a clarifying move (in fact, the essence of algebra) to usher into the mathematical arena, and to name, certain mathematical objects that are unspecified beyond the sole fact that they are a solution to a certain explicit problem; in this case: a solution to the polynomial equation X 2 = 2 X^{2}=2 X2=2.

遵循克罗内克的思想，我们有时会将数学对象简化为符号来理解：例如，将 2 \sqrt{2} 2 仅视作一个遵循算术基本法则的符号，我们对它的唯一认知就是其平方等于 2。诚然，这一描述无法唯一确定该对象------因为 − 2 -\sqrt{2} −2 也满足完全相同的描述。但这并不存在矛盾：我们将 2 \sqrt{2} 2 定义为一个特定的数学对象，而 − 2 -\sqrt{2} −2 只是另一个有着不同名称的数学对象而已。将这类除了"是某一明确问题的解"之外无其他限定的数学对象引入数学领域并为之命名，是一种极具启发性的做法；在上述例子中， 2 \sqrt{2} 2 就是多项式方程 X 2 = 2 X^2=2 X2=2 的一个解。

When we do such a thing, what is sharply delineated is the problem , the object being a tag for (a solution to) the problem.

当我们以这种方式定义数学对象时，被清晰界定的是问题本身 ，而对象只是该问题（的一个解）的标记。

In the same spirit, any functor, explicitly given, from a category C \mathcal{C} C to the category of sets S \mathcal{S} S that the category is modeled on, F : C ⇢ S F: \mathcal{C}\dashrightarrow \mathcal{S} F:C⇢S may be construed as formulating an explicit problem:

以同样的思路，任意一个显式定义的、从范畴 C \mathcal{C} C 到其基础集合范畴 S \mathcal{S} S 的函子 F : C ⇢ S F: \mathcal{C}\dashrightarrow \mathcal{S} F:C⇢S，都可以被诠释为一个明确的问题：

PROBLEM

问题

Find "an" object X X X of the category C \mathcal{C} C together with an isomorphism of functors ι : F X ≅ F \iota: F_{X} \cong F ι:FX≅F.

找到范畴 C \mathcal{C} C 中的一个对象 X X X，使得存在函子同构 ι : F X ≅ F \iota: F_{X} \cong F ι:FX≅F。

In a word, solve the above problem for the unknown X X X. To be sure, if we find two solutions, ι : F X ≅ F \iota: F_{X} \cong F ι:FX≅F and ι ′ : F X ′ ≅ F \iota': F_{X'} \cong F ι′:FX′≅F, then ι − 1 ∘ ι ′ : F X ′ ≅ F X \iota^{-1} \circ \iota': F_{X'} \cong F_{X} ι−1∘ι′:FX′≅FX is an isomorphism of representable functors and so, by Yoneda's Lemma, is induced from a unique isomorphism X ′ ≅ X X' \cong X X′≅X --- i.e., the solution is unique, up to unique isomorphism.

简言之，求解上述问题中的未知对象 X X X。显然，若我们找到两个解 ι : F X ≅ F \iota: F_{X} \cong F ι:FX≅F 和 ι ′ : F X ′ ≅ F \iota': F_{X'} \cong F ι′:FX′≅F，则复合映射 ι − 1 ∘ ι ′ : F X ′ ≅ F X \iota^{-1} \circ \iota': F_{X'} \cong F_{X} ι−1∘ι′:FX′≅FX 是可表函子之间的同构；根据米田引理，该函子同构由唯一的对象同构 X ′ ≅ X X' \cong X X′≅X 诱导而来------即该问题的解在唯一同构的意义下是唯一的。

The moral here, is that it is the problem that is explicit, while the object (that represents the solution of the problem) follows the theme of this essay: it is unique up to unique isomorphism.

由此我们可以得出结论：在范畴论的框架中，问题是显式的 ，而作为问题解的表示对象则遵循本文的主题------在唯一同构的意义下唯一。

19 Object and equality

19 对象与相等性

The habitual format for discussions regarding the grounding of mathematics shines a bright light on modes of formulating assertions, organizing and justifying proofs of those assertions, and on setting up the substrate for it all-which is invariably a specific set theory. In doing this a battery of choices will be made. These choices smack of contingency, of viewing the clarity of mathematics through some subjective lens or other.

传统的数学基础研究，往往将焦点放在以下方面：命题的表述方式、证明的组织与验证方法，以及为这一切搭建基础框架------而这一框架几乎总是某一特定的集合论。在构建这一框架的过程中，人们会做出一系列选择，而这些选择带有明显的偶然性，仿佛是通过某种主观视角来审视数学的严谨性。

I imagine that all of us want to ignore-when possible-the contingent, and seize the essential, aspect of any idea. If we are of the make-up of Frege, who relentlessly strove to rid mathematical foundations of subjectivism (Frege excoriated the writings of Husserl-incorrectly, in my opinion-for ushering psychologism into mathematics), we look to universal quantification as a possible method of effacing the contingent- drowning it, one might say, in the sea of all contingencies. But this doesn't work.

我想，我们所有人都希望在可能的情况下，忽略思想中的偶然因素，抓住其本质。倘若我们和弗雷格一样，执着于将主观主义从数学基础中剔除（弗雷格曾严厉批判胡塞尔的著作，认为其将心理主义引入了数学------我认为这一批判并不恰当），我们会将全称量化视作消除偶然性的一种方法------或者说，将偶然性淹没在所有可能的偶然情形之中。但这种方法并不奏效。

A stark alternative-the viewpoint of categories- is precisely to dim the lights where standard mathematical foundations shines them brightest. Instead of focussing on the question of modes of justification, and instead of making any explicit choice of set theory, the genius of categories is to provide a vocabulary that keeps these issues at bay. It is a vocabulary that can say nothing whatsoever about proofs, and that works with any-even the barest--choice of a set theoretic language, and that captures the essential template nature of the mathematical concepts it studies, showing these concepts to be-indeed-separable from modes of justification, and from the substrate of ever-problematic set theory. Separable but not forever separated, effecting the kind of aphairesis that Aristotle might have wanted, for, as we have said, you must bring your own set theory, and your own mode of proof, to this party.

而一种截然相反的选择------范畴论的视角------则恰好是在传统数学基础研究的焦点处"调暗灯光"。范畴论并未聚焦于证明的验证方式，也未明确选择某一特定的集合论；其精妙之处在于，提供了一套能将这些问题悬置起来的术语体系。这套体系对证明问题只字不提，可适配任意一种集合论语言（即便是最朴素的），并能捕捉到所研究数学概念的范式特征，证明这些概念确实能与证明方式、与始终存在争议的集合论基础相分离。当然，这种分离并非永久的，而是实现了亚里士多德所追求的那种抽象分离------正如我们此前所说，范畴论就像一场聚会，你需要自带集合论和证明方式才能参与其中。

With the other lights low, the mathematical concepts shine out in this new beam, as pinned down by the web of relations they have with all the other objects of their species. What has receded are set theoretic language and logical apparatus. What is now fully incorporated, center stage under bright lights, is the curious class of objects of the category, a template for the various manners in which a mathematical object of interest might be presented to us. The basic touchstone is that, in appropriate deference to the manifold ways an object can be presented to us, objects need only be given up to unique isomorphism , this being an enlightened view of what it means for one thing to be equal to some other thing.

当其他"灯光"调暗后，数学概念在这一新的视角下变得清晰可见------它们由其与同类型所有其他对象的关系网络 所界定。逐渐隐去的是集合论的语言和逻辑体系，而被推到聚光灯下的，是范畴中那类独特的对象------它为我们感兴趣的数学对象的各种呈现方式，提供了一个统一的范式。范畴论的基本准则是：充分尊重对象呈现方式的多样性，只需将对象在唯一同构的意义下确定下来------这就是对"一物何时等同于另一物"这一问题的深刻解答。

References

参考文献

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