A Peter May Picture Book, Part 1

Transcription

1 A Peter May Picture Book, Part 1 Steve Balady Auust 17, 2007 This is the beinnin o a larer project, a notebook o sorts intended to clariy, elucidate, and/or illustrate the principal ideas in A Concise Course in Alebraic Topoloy. Many o the worthwhile exercises suested in the text will be done in relatively complete detail, one-line proos expanded, constructions illustrated, and deinitions iven as assertions that a certain diaram commutes (which, when I try to explain to my riends, I am walked away rom) iven explanations anyway. The bulk o this paper ocuses on the cateorical intuition that the book assumes, with the concepts illustrated in the cateory o topoloical spaces. Followin that is a careul illustration o a proo that well illustrates the utility o such intuition, which I irst presented as part o a directed readin proram durin the previous academic year. This paper concludes with a list o currently intended illustrations or chapters 5-10, which will be added to a workin version o this document as they are done. p. 16, on limits and colimits in Top and Grp At this point, anyone who hasn t seen cateory theory beore will probably be utterly despondent. Some pictures will help. We ll do the product and coproduct, initial and terminal object, equalizer and coequalizer, pullback and pushout in the cateories o topoloical spaces and o roups these are the most important examples, and in practice the only ones that come up all that oten. Let s start with the product in Top. Remember, in this cateory the objects are topoloical spaces, and the arrows are continuous maps between spaces. Eectively, with the product we start with a collection o unrelated topoloical spaces. We want the limit o this collection, and assert that it is their Cartesian product endowed with the product topoloy, alon with the canonical projection maps onto each coordinate. 1

2 The ollowin is a diaram indicatin this or two spaces X and Y ; see also Fiure 1. X p X Z X p Y X Y Z Y Y Z To say that this diaram commutes is to say that the Cartesian product endowed with the product topoloy has two special properties: irst, it is itsel endowed with projection maps onto each coordinate; second, that any other space with such projections has a unique projection onto the Cartesian product. We shall check this assertion in eneral. Let I be an indexin set; we assert that the product o X i, i I is Π i I X i endowed with the product topoloy, alon with its projection maps p j : Π i I X i X j or j I. On the set-theoretic level, this is relatively clear: iven any other space Q with maps Q i : Q X i, we et a actorization throuh the Cartesian product. Speciically, deine the map : Q Π i I X i by q (Q i (q)) i I ; uniqueness up to bijection is orced by deinition. We need only impose a topoloy on the Cartesian product as a set; since the product topoloy is homeomorphic to the weak topoloy enerated by the projection maps, and the weak topoloy is the weakest topoloy relative to which the projection maps are continuous, since Q i are actually continuous maps, we are done. Dually, we assert that the coproduct o topoloical spaces is the disjoint union endowed with the topoloy enerated by its disjoint subspaces, alon with the standard inclusion maps. The diaram X Z X i X i Y X Y Z says that each coordinate embeds in the disjoint union, and also that the disjoint union embeds uniquely in any other space endowed with embeddin maps. See also Fiure 2. A terminal object is the speciic case o the product on the empty set; its dual, the coproduct on the empty set, is called an initial object. It should be clear now that or topoloical spaces, the terminal object is the one-point space with its only topoloy and the initial object is the empty set with its only topoloy. Z Y Y 2

3 Now we restrict ourselves to a pair o spaces X and Y, but we also have a pair o continuous maps, between them. We call the limit o this diaram the equalizer o and. As in the case o the product, its name is suestive in Top, it is the subspace o X on which and are equal. In the ollowin, we set L = {x X (x) = (x)} and l to be the standard inclusion map. l L X m M Y Dually, the coequalizer o X and Y with maps and will be a quotient space; speciically, the quotient by the smallest equivalence relation or which, or all x X, (x) (x). In the ollowin, we call it C, with q the quotient map. X Y q C d D The pullback is the limit o A C B. For topoloical spaces, it is the subspace o A B iven by {(x, y) x X, y Y, (x) = (y)} with its standard maps (projection to each actor ollowed by inclusion), and is usually written as A C B. D A C B A B C 3

4 The dual to the pullback is the pushout; it is the colimit o A C B. For a topoloical space, it is a quotient o A B, and as with the coequalizer it is the quotient induced by the weakest equivalence relation such that (x) (y). It is usually written as A C B. An illustration provides ar more intuition than paes o ormality; thus, see Fiure 3. C A B A C B In eneral, the intuition used or calculatin the limitin objects in Top carry throuh to Grp with minor modiications. We assert the ollowin or roups: the product is aain the Cartesian product with its natural roup structure, the coproduct is the ree product amalamated over the identity, an equalizer is the larest subroup contained in {a (a) = (a)}, a coequalizer is the quotient by the larest normal subroup contained in {a (a) = (a)}, and similarly or the pullback and pushout. We prove only the ollowin (which will be useul or the theorem on pae 36, which we later illustrate): i G is a roup and N is normal in G, the pushout o G i N 0 in the cateory o roups is isomorphic to G/N. D N i G k 0 G/N r j H We must show that any object makin the diaram commute actors uniquely throuh G/N and its associated maps. Assume that H, j : 0 H, r : G H does so, and consider n N; we trace all o its possible paths. Followin i irst, we see that n n, where the latter is in G. On the other hand, or jk to be a roup homomorphism, we must have that jk(n) = 0. Since the diaram commutes, jk(n) = ri(n) = 0, which immediately implies that ker(r) N. Identiyin H by the irst isomorphism theorem as G/ker(r), the above ives us a unique actorization throuh G/N, which veriies the universal property o the pushout. 4

5 p. 17, completeness is equivalent to the existence o products and equalizers I m relatively convinced that in this case, the proo is a worthwhile exercise should be interpreted to mean the proo is an awul diaram-chasin mess. The idea, however, is well worth seein. We present the basic idea o the standard proo, and reer the reader to Borceux or the orey details; eectively, we eneralize the idea above that a pullback is a subset o the cartesian product (a product and an equalizer). Thinkin dually, a pushout is a quotient o a disjoint union (a coproduct and a coequalizer), and by reversin all the arrows, the ollowin will also prove that cocompleteness is equivalent to the existence o coproducts and coequalizers. Let Cbe a cateory. I Cis complete, it contains in particular all products and equalizers, so that s ine. So assume that Chas all o its products and equalizers, let Dbe a small cateory (that is, a set o objects equipped with a collection o arrows between objects), and let F be a unctor rom Dto C. This is just a way o oranizin inormation. For example, the equalizer o, : A B written in this way would be the limit o F x, F y : F 1 F 2 where Dis the cateory with objects 1 and 2 and arrows x, y : 1 2, F 1 = A, F 2 = B, F x =, F y =. What we want is the equalizer o a pair o maps α, β. L l F D F (t()) D D D Here s() is the source (domain, corane) o, t() is its taret (rane, codomain), α((x D ) D D ) = (x t() ) D, β((x D ) D D ) = (F (x s() )) D, and L, l is the equalizer o α, β. We assert that L is the limit o F. In our example above, α(x A, x B ) = (x B, x B ), β(x A, x B ) = ((x A ), (x A )), and equalizin α, β is the assertion that (x A ) = (x A ); that is, L is the equalizer o,. From here the proo is airly straiht-orward: show that L, l ives a cone on F, and then that any other cone actors uniquely throuh it. I can oer no more intuition than that ained by starin at the diarams which prove this, so I leave the remainder to Borceux. p. 36, an illustration o the proo that every roup is π 1 o some space See Fiures 4 and 5. p. 39, on the adjunction o product and Hom in CGHTop See Fiure 6. 5

6 To be continued... p. 41, some intuition or the HEP p. 42, a mappin cylinder p. 42, a coibration is an inclusion with closed imae p. 43, illustratin the retraction which a coibration eects p. 47, on duality, intuition or the CHP p. 47, a mappin path space p. 49, a iber bundle is a ibration, with picture p. 55, S m S n = S m+n, with picture p. 56, [Σ 2 X, Y ] is an Abelian roup p. 58, an illustration that Ci Ci/CA is a homotopy equivalence p. 58, an illustration that up to homotopy equivalence, each pair o maps in our coiber sequence is the composite o a map and the inclusion o its taret in its coiber p. 59, a homotopy iber p. 61, illustration o unit and counit between loops and suspension p. 63, illustration o the maps in the lon exact sequence o pairs p. 64, a ew calculations actually calculated p. 65, the Hop map is a iber bundle p. 66, a word and picture on chane o basepoint p. 67, homotopy equivalence induces weak equivalence p. 68, the point o the technical lemma, with picture, explanation and condolances p. 73, how the lemma implies the HELP p. 74, the Whitehead theorem, and what it is not Reerences J. Peter May, A Concise Course in Alebraic Topoloy. University o Chicao Press, Chicao, may/ F. Borceux, Handbook o cateorical alebra 1. Cambride University Press, Cambride, Basic cateory theory. 6