----------------------------------------------------- Preliminaries: Set theory and categories 1. Naive set theory 1.1. Sets. The notion of set formalizes the intuitive idea of `collection of objects'. A set is determined by the edemeats it contains: 'What is an element?' is a forbidden question in naive set theory: the buck must stop somewhere. We can conveniently pretend that a 'universe' of elements is available to us, and we draw from this universe to construct the elements and sets we need, implicitly assuming that all the operations we will explore can be performed within this universe. (This is the tricky point!) In any case, we specify a set by giving a precise recipe determining which elements are in it. 。 multiset：a set in which the elements are allowed to appear `with multiplicity'。 {2, 2} would be distinct from {2} singleton：any set consisting of precisely one element ∃! : there exists a unique... 表达式的顺序很重要(顺序应该是从左向右)。举例对比：(∀a∈Z)(∃b∈Z) b=2a 和(∃b∈Z)(∀a∈Z) b=2a P(S) : power set of S ∪: ∩: \ :the difference. S\T: in S and not in T II: the disjoint union ⨯: the (Cartesian) product; Let S⨯T be the set whose elements are the ordered pairs4 (s, t) of elements of S and T . a #relation# on a set S is simply a subset R of the product S⨯S equivalence class: [a]($_~$) :={b∈S | b~a} the notions of `equivalence relation on S' and `partition of S' are really equivalent. P($_~$) : a set whose elements are the equivalence classes with respect to ~ Definition 1.2. The #quotient of the set S# with respect to the equivalence relation ~ is the set : S/~ := P($_~$) . One way to think about this operation is that the equivalence relation `becomes equality in the quotient(取模？)': that is, two elements of the quotient S/~ are equal if and only if the corresponding elements in S are related by ~. In other words, taking a quotient is a way to turn any equivalence relation into an equality. This observation will be further formalized in `categorical terms' in a short while (§5.3). . the graph of f : Гf := {(a,b)∈A⨯B | b= f(a)} ⊂= A⨯B. . 2. Functions between sets a↦ f(a): The action of a function f :A -> B on an element a∈A B($^A$): a set whose elements are all functions from set A to set B id($_A$):A->A: identity function on A im f: the image of f f|S : the `restriction' of f to the subset S. So, f|S is the composition foi, where i:S->A . Composition is associative： h o (g o f) = (h o g) o f. 2.4. Injections, surjections, bijections. (集合之间的影射关系) injection: one-to-one (w.r.t ↪) surjection: onto (w.r.t ↠) bijection: one-to-one && onto, isomorphism. (w.r.t ≅) (P15)The reason why we focus our attention on injective and surjective maps is that they provide the basic `bricks' out of which any function may be constructed. section: a surjective function will in general have many right-inverses; they are often called sections Proposition 2.1. Assume A!=NULL, and let f : A ->B be a function. Then (1) f has a left-inverse if and only if it is injective. (2) f has a right-inverse if and only if it is surjective. Proposition 2.1 hints that something deep is going on here. The definition of injective and surjective maps given in §2.4 relied crucially on working directly with the elements of our sets; Proposition 2.1 shows that in fact these properties are detected by the way functions are `organized' among sets. Even if we did not know what `elements' means, still we could make sense of the notions of injectivity and surjectivity (and hence of isomorphisms of sets) by exclusively referring to properties of functions. This is a more `mature' point of view and one that will be championed when we talk about categories. To some extent, it should cure the reader from the discomfort of talking about `elements', as we did in our informal introduction to sets, without defining what these mysterious entities are supposed to be. . if f:A->B and T⊆B, If T = {q} consists of a single element of B, f^-1(T) (abbreviated f^-1 (q)) is called the #fiber# of f over q . 2.6. Monomorphisms and epimorphisms. another way to express injectivity and surjectivity: A function f : A->B is a #monornorphism (or manic)# if the following holds: for all sets Z and all functions α', α'': Z->A foα'=foα'' ==> α'=α". . Proposition 2.3. A function is injective if and only if it is a monomorphism. surjective(onto) ~ epimorphism.
2.8. Canonical decomposition. The reason why we focus our attention on in- jective and surjective maps is that they provide the basic `bricks' out of which any function may be constructed. . every function f : A->B determines an equivalence relation ~ on A as follows: for all a', a" ∈A, a' ~a'' <==> f(a')=f(a'').
Theorem 2.7 shows that every function is the composition of a surjection, followed by an isomorphism, followed by an injection. While its proof is trivial, this is a result of some importance, since it is the prototype of a situation that will occur several times in this book. It will resurface every now and then, with names such as `the first isomorphism theorem'. 。 2.9. Clarification. Finally, we can begin to clarify one comment about disjoint unions, products, and quotients, made in §1.4. Our definition of AIIB was the (conventional) union of two disjoint sets A', B' isomorphic to A, B, respectively. It is easy to provide a way to effectively produce such isomorphic copies (as we did in §1.4); but it is in fact a little too easy----many other choices are possible, and one does not look any better than any other. It is in fact more sensible not to make a fixed choice once and for all and simply accept the fact that all of them produce acceptable candidates for AIIB(这就是coset, central). From this egalitarian standpoint, the result of the operation AIIB is not 'well-defined' as a set in the sense specified above. However, it is easy to see (Exercise 2.9) that AIIB is well-defined up to isomorphism: that is, that any two choices for the copies A', B' lead to isomorphic candidates for AIIB. The same considerations apply to products and quotients. 。 The main feature of sets obtained by taking disjoint unions, products, or quotients is not really `what elements they contain' but rather `their relationship with all other sets'. >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> 类比群论里的 First Isomorphism Theorem(<Contemporary Abstract Algebra>P215)：
f is a homomorphism. ψ is an isomorphism. G/Ker(f) ≈ f(G). r: G-->G/Ker f is defined as r(g) = gKer(f) The mapping r is called the natural mapping from G to G/Ker(f) <<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<< Theorem 2.7研究对象是set之间的functions， First Isomorphism Theorem研究对象是group之间的functions. Theorem 2.7是First Isomorphism Theorem的推广。 。 3. Categories Category emphasize less on how you run into a specific set you are looking at and more on how that set may sit in relationship with all other sets. Worse (or better) still, the emphasis is less on studying sets, and functions between sets, than on studying `things, and things that go from things to things' without necessarily being explicit about what these things are . functor: function from category to category . a category consists of a collection of 'objects', and of 'morphisms' between these objects, satisfying a list of natural conditions. . 4. Morphisms groupoid: there are categories in which every morphism is an isomorphism; 5. Universal properties 为什么要发明范畴论： Categories offer a rich unifying language, giving us a bird's eye view of many constructions in algebra (and other fields). In this course, this will be most apparent in the steady appearance of constructions satisfying suitable universal properties. In fact, viewing the construction in terms of its corresponding universal property clarifies why one can only expect it to be defined `up to isomorphism'. 。 5.1. Initial and final objects A category need not have initial or final objects, initial and final objects, when they exist, may or may not be unique . terminal object: initial object or final object . Proposition 5.4. Let C be a category. - If I1,I2 are both initial objects in C, then I1≅I2 - If F1,F2 are both final objects in C, then F1≅F2 Further, these isomozphisms are uniquely determined. . There may be psychological reasons why one initial or final object looks more compelling than others. but this plays no role in how these objects sit in their category. . Example 2.6. If ∼ is an equivalence relation on a set A, there is a (clearly surjective) #canonical projection(denoted by π)# A↠A/∼ obtained by sending every a∈A to its equivalence class [a]($_∼$) . . pair (π, A/~) is an initial object of this category products of sets A,B are final objects in the category C($_{A,B}$) . 假设C(Z),with mophism <=. a⨯b: the categorical product of a and b. The universal property written out above becomes, in this case, for all z∈Z such that z<=a and z<=b, we have z<=a⨯b.事实上， a⨯b是min(a,b), aIIb是max(a,b) ============================================================= º¹²³⁴ⁿ₁₂₃₄·∶αβγδεζηθικλμνξοπρστυφχψω∽≌≅⊥∠⊙⊕∈∩∪∑∫∞≡≠±≈＄㏒㎡㎥㎎㎏㎜ ∈⊂∂Δ◁ ∃∀⊆‾≤≈⨯Г http://www.math.sinica.edu.tw/www/tex/online_latex.jsp
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