Math Homotopy Theory Additional notes

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1 Math Homotopy Theory Addtonal notes Martn Frankland February 4, 2013 The category Top s not Cartesan closed. problem. In these notes, we explan how to remedy that 1 Compactly generated spaces Ths secton and the next are essentally taken from [3, 1,2]. 1.1 Basc defntons and propertes Defnton 1.1. Let X be a topologcal space. A subset A X s called k-closed n X f for any compact Hausdorff space K and contnuous map u: K X, the premage u 1 (A) K s closed n K. The collecton of k-closed subsets of X forms a topology, whch contans the orgnal topology of X (.e. closed subsets are always k-closed). Notaton 1.2. Let kx denote the space whose underlyng set s that of X, but equpped wth the topology of k-closed subsets of X. Because the k-topology contans the orgnal topology on X, the dentty functon d: kx X s contnuous. Defnton 1.3. A space X s compactly generated (CG), sometmes called a k-space, f kx X s a homeomorphsm. In other words, every k-closed subset of X s closed n X. Example 1.4. Every locally compact space s CG. Example 1.5. Every frst-countable space s CG. More generally, every sequental space s CG. Example 1.6. Every CW-complex s CG. Notaton 1.7. Let CG denote the full subcategory of Top consstng of compactly generated spaces. Notaton 1.8. The constructon of kx defnes a functor k : Top CG, called the k-fcaton functor. Proposton 1.9. Let X be a CG space and Y an arbtrary space. Then a functon f : X Y s contnuous f and only f for every compact Hausdorff space K and contnuous map u: K X, the composte fu : K Y s contnuous. 1

2 Proposton For any space X, we have k 2 X = kx, so that kx s always compactly generated. Proposton Let X be a CG space and Y an arbtrary space. Then a functon f : X Y s contnuous f and only t s contnuous when vewed as a functon f : X ky. Proposton 1.11 can be reformulated n the followng more suggestve way, as a unversal property. For any CG space X and contnuous map f : X Y, there exsts a unque contnuous map f : X ky satsfyng f = d f,.e. makng the dagram f ky d Y X f commute. Note that f has the same underlyng functon as f. Ths exhbts ky Y as the closest approxmaton of Y by a CG space. Corollary The k-fcaton functor k : Top CG s rght adjont to the ncluson ι: CG Top. In other words, CG s a coreflectve subcategory of Top. The dentty functon ιkx X s the count of the adjuncton, whereas the unt W kιw s the dentty map for any CG space W. Proposton The category CG s complete. Lmts n CG are obtaned by applyng k to the lmt n Top. 2. The category CG s cocomplete. Colmts n CG are computed n Top. Proof. Let I be a small category and F : I CG an I-dagram. Let us wrte X := F () and, by abuse of notaton, lm X := lm I F. (1) Vewng the CG spaces X as spaces ιx, we can compute the lmt of the dagram ιf snce Top s complete (c.f. Homework 4 Problem 2). Applyng k yelds the CG space k(lm ιx ). For any CG space W, we have a natural somorphsm Hom CG (W, k(lm ιx )) = Hom Top (ιw, lm ιx ) = lm Hom Top (ιw, ιx ) = lm Hom CG (W, X ) where the last equalty comes from the fact that CG s a full subcategory of Top. Ths proves k(lm ιx ) = lm X. (2) We can compute the colmt X = colm ιx of the dagram ιf snce Top s cocomplete (c.f. Homework 4 Problem 2 and Remark afterwards). Snce X s a quotent of a coproduct of CG spaces ιx, X s also CG, by [3, Prop. 2.1, Prop. 2.2]. Moreover t s the desred colmt 2

3 n CG. For any CG space Y, we have a natural somorphsm Hom CG (X, Y ) = Hom Top (ιx, ιy ) = Hom Top (ι colm ιx, ιy ) = Hom Top (colm ιx, ιy ) = lm Hom Top (ιx, ιy ) whch proves X = colm X. = lm Hom CG (X, Y ) In partcular, products n CG may not agree wth the usual product n Top. Notaton For CG spaces X and Y, wrte X 0 Y = ιx ιy for ther usual product n Top, and wrte X Y = k(x 0 Y ) for ther product n CG. 1.2 Mappng spaces Defnton Let X and Y be CG spaces. For any compact Hausdorff space K, contnuous map u: K X, and open subset U Y, consder the set W (u, K, U) := {f : X Y contnuous fu(k) U}. Denote by C 0 (X, Y ) the set of contnuous maps from X to Y, equpped wth the topology generated by all such subsets W (u, K, U). Ths topology s called the compact-open topology. Note that C 0 (X, Y ) need not be CG. Wrte Map(X, Y ) := kc 0 (X, Y ). Theorem For any CG spaces X, Y, and Z, the natural map s a homeomorphsm. ϕ: Map(X Y, X) Map(X, Map(Y, Z)) (1) The fact that ϕ s bjectve tells us that CG s Cartesan closed, n the unenrched sense. The theorem s even better: CG s Cartesan closed, n the enrched sense. Note that CG s enrched n tself, gven that the composton map s contnuous. Map(X, Y ) Map(Y, Z) Map(X, Z) Remark The exponental object Map(X, Y ) s often denoted Y X. The somorphsm (1), whch can be wrtten as Z X Y = (Z Y ) X s often called the exponental law. 3

4 2 Weakly Hausdorff spaces The category CG would be good enough to work wth, but we can also mpose a separaton axom to our spaces. Defnton 2.1. A topologcal space X s weakly Hausdorff (WH) f for every compact Hausdorff space K and every contnuous map u: K X, the mage u(k) X s closed n X. Remark 2.2. Hausdorff spaces are weakly Hausdorff, snce u(k) s compact and thus closed n X f X s Hausdorff. Ths justfes the termnology. Moreover, weakly Hausdorff spaces are T 1, snce the sngle pont space s compact Hausdorff. Thus we have mplcatons Hausdorff weakly Hausdorff T 1. Example 2.3. Every CW-complex s Hausdorff, hence n partcular WH. Proposton 2.4. If X s a WH space, then any larger topology on X s stll WH. In partcular, kx s stll WH. Proof. Let X be the set X equpped wth a topology contanng the orgnal topology,.e. the dentty functon d: X X s contnuous. For any compact Hausdorff space K and contnuous map u: K X, the composte du: K X s contnuous and so ts mage du(k) X s closed n X. Thus u(k) = d 1 du(k) s closed n X. Proposton 2.5. Any subspace of a WH space s WH. Proof. Let X be a WH space and : A X the ncluson of a subspace. For any compact Hausdorff space K and contnuous map u: K A, the composte u: K X s contnuous and so ts mage u(k) X s closed n X, and thus n A as well. Notaton 2.6. Let CGWH denote the full subcategory of CG consstng of compactly generated weakly Hausdorff spaces. Defnton 2.7. For any CG space X, let hx be the quotent of X by the smallest closed equvalence relaton on X (see [3, Prop. 2.22]). Then hx s stll CG snce t s a quotent of a CG space [3, Prop. 2.1], and t s WH snce we quotented out a closed equvalence relaton on X [3, Cor. 2.21]. Ths defnes a functor h: CG CGWH called weak Hausdorfffcaton By constructon, the quotent map q : X hx satsfes the followng unversal property. For any CGWH space Y and contnuous map f : X Y, there exsts a unque contnuous map f : hx Y satsfyng f = fq,.e. makng the dagram X q hx f f Y commute. Ths exhbts X hx as the closest approxmaton of X by a CGWH space. 4

5 Corollary 2.8. The functor h: CG CGWH s left adjont to the ncluson functor ι: CG CGWH. In other words, CGWH s a reflectve subcategory of CG. The quotent map q : X hx s the unt of the adjuncton, whereas the count hιw W s the dentty map for any CGWH space W. Proposton 2.9. n CG. 1. The category CGWH s complete. Lmts n CGWH are computed 2. The category CGWH s cocomplete. Colmts n CGWH are obtaned by applyng h to the colmt n CG. Proof. Let I be a small category and F : I CGWH an I-dagram. (1) The lmt X = lm ιx computed n CG, whch exsts snce CG s complete (by 1.13), s stll WH. Indeed, an arbtrary product n CG of CGWH spaces s stll WH [3, Cor. 2.16], and so s an equalzer n CG of two maps (by 2.5 and 2.4). Therefore X s also the lmt n CGWH, by the same argument as 1.13 (2). (2) We have h(colm ιx ) = colm X n CGWH by the same argument as 1.13 (1). To summarze the stuaton, there are two adjont pars as follows: ι CG Top k h ι CGWH Proposton If X s a CG space and Y s a CGWH space, then Map(X, Y ) s CGWH. Consequently, the category CGWH s enrched n tself. Note that t s also Cartesan closed (n the enrched sense). Indeed, for any X, Y, and Z n CGWH, the natural map s a homeomorphsm. ϕ: Map(X Y, X) Map(X, Map(Y, Z)) 5

6 3 A convenent category of spaces In ths secton, we explan n what sense t s preferable to work wth the category CGWH nstead of Top. We follow the treatment n [1], tself nspred by [2]. Defnton 3.1. A convenent category of topologcal spaces s a full replete (meanng closed under somorphsms of objects) subcategory C of Top satsfyng the followng condtons. 1. All CW-complexes are objects of C. 2. C s complete and cocomplete. 3. C s Cartesan closed. Note that CG and CGWH are replete full subcategores of Top, snce both condtons of beng CG or WH are nvarant under homeomorphsm. Let us summarze the dscusson as follows. Proposton 3.2. The categores CG and CGWH are convenent. In fact, there are other desrable propertes for a convenent category of spaces. For nstance, one would lke that closed subspaces of objects n C also be n C. Both CG and CGWH satsfy ths addtonal condton. Proposton 3.3. Consder nclusons of spaces A B X. If A s k-closed n B and B s k-closed n X, then A s k-closed n X. Proof. Let K be a compact Hausdorff space and u: K X a contnuous map. Then the premage u 1 (B) s closed n K, hence compact (and also Hausdorff). Consder the restrcton u u 1 (B) : u 1 (B) B. Snce A s k-closed n B, the premage u 1 u 1 (B) (A) = u 1 (A) s closed n u 1 (B) and thus n K as well. Corollary 3.4. A closed subspace of a CG space s also CG. Proof. Let X be a CG space and B X a closed subspace. Let A B be a k-closed subset of B. Then A s k-closed n X (snce B s closed n X), hence closed n X (snce X s CG). Therefore A s closed n B. Corollary 3.5. A closed subspace of a CGWH space s also CGWH. 6

7 4 Some applcatons Here s a toy example llustratng the use of the exponental law. Proposton 4.1. For any spaces X and Y, the natural map [X, Y ] bjecton. Proof. Snce the map Map(X I, Y ) Map(I, Map(X, Y )) = π 0 Map(X, Y ) s a s a bjecton, two maps f, g : X Y are homotopc f and only f they are connected by a (contnuous) path n Map(X, Y ). Here s another beneft of workng n the category CG. Proposton 4.2. If X and Y are CW-complexes, then X Y nherts a CW-structure, where a p-cell of X and a q-cell of Y produce a (p + q)-cell of X Y. Here we do mean the product X Y s CG, not n Top. A pror, the product X 0 Y n Top could fal to be a CW-complex. See Hatcher (A.6) for detals. References [1] Ronald Brown, nlab: Convenent category of topologcal spaces (2012), avalable at org/nlab/show/convenent+category+of+topologcal+spaces. [2] N. E. Steenrod, A convenent category of topologcal spaces, Mchgan Math. J. 14 (1967), [3] Nel Strckland, The category of CGWH spaces (2009), avalable at shef.ac.uk/courses/homotopy/cgwh.pdf. 7

For instance, ; the five basic number-sets are increasingly more n A B & B A A = B (1)

For instance, ; the five basic number-sets are increasingly more n A B & B A A = B (1) Secton 1.2 Subsets and the Boolean operatons on sets If every element of the set A s an element of the set B, we say that A s a subset of B, or that A s contaned n B, or that B contans A, and we wrte A