On the subdivision of small categories

On the sub division of small categories Ma tias L. del Hoyo Departamen to de Matem´ atica F CEyN, Universidad de Buenos Aires Buenos Aires, Argen tina. Abstract W e present an intrinsic and concrete developmen t of the sub division of small ca t- egories, give some simple examples and derive its fundament al prop erties. As an a p- plication, w e deduce an alternative w ay to compare the homotopy categor ie s of spaces and s mall ca tegories, by us ing pa rtially or dered sets. This y ields a new conceptua l pro of to the well-kno wn fact that these tw o homo top y categories a re equiv a lent . In tro duction W e present the sub division of catego ries from th e homotop y p oint of view, and illustrate this with some simple examples. This sub division is n ot n ew, it h as already app eared in some works by Anderson, Dwyer an d Kan [1, 3]. Here, we d er ive the basic prop erties of the su b division fu nctor C 7→ S d ( C ) from classical resu lts on h omotopy of ca tegories, such as the famous Quillen’s Theorem A. Th is wa y we obtain an intrinsic and geometric-st yle dev elopmen t of the theory . Among the fun damen tal p rop erties of the sub d ivision of categories, we emphasize theo- rems 21 and 32. The fi rst one asserts that an y category b ecomes a p oset after applying the functor S d t wice, and the second relates the classifying spaces of a category a nd its sub d ivision by a homotop y equiv alence. These results suggest that the homotop y type of the classifying space of any small ca tegory can b e mo d elled b y a poset, and therefore that the homotop y categorie s of sm all categories and p osets are equiv alen t. This is pr o ved in theorem 41. Finally , we use some r esu lts of McCord [10 ] to relate the homotop y categories of p osets and t op ological spaces. Com bining these t wo equiv alences w e obtain th e equiv alence of catego ries H o ( T op ) ∼ = H o ( C at ) , whic h migh t b e thought as a c ate goric al description of top olo gic al sp ac es . This and the c ombinatorial description of top olo g ic al sp ac es [4] are related b y Q uillen’s theorem whic h asserts that the nerve functor is an equiv alence at the homotopy lev el [6]. I w ould lik e to thank m y advisor Gabriel Minian for his sev eral suggestions and comments concerning the material of this p ap er. I also would like to thank Man u el Ladra for his kindness durin g my da ys in S an tiago de Comp ostela in which I co rrected this work. 2000 Mathematics Subje ct Cl assific ation: 18G55; 55U35; 18B35. Key wor ds: Sub division; Classifying space; Homotopy category; Posets. 1 On the sub division of small cate gories - M. d el Ho yo 2 1 Preliminaries 1.1 Homotop y categories If M is a category and W is a family of arro ws in M , there exists (eve ntually expanding the base univ erse) a category M [ W − 1 ], called lo c alization of M by W , and a functor p : M → M [ W − 1 ], called lo c alization functor , whic h make s in ve rtible all th e arrows in W and whic h is un iv ersal for this prop ert y . The category M [ W − 1 ] has the same ob jects than M and its arro ws can b e expressed as classes of paths inv olving arro ws of M and formal inv erses of arro ws of W [4]. By usin g this description of the lo calizat ion category , it is easy to pro v e the follo wing resu lt (cf. [13]). 1 Lemma. L et p : M → M [ W − 1 ] b e a c ate goric al lo c alization. Then p induc es a bije ction H om ( F , G ) ∼ − → H om ( F p, Gp ) for every p air of functors F , G : M [ W − 1 ] → N . When M is a category en d o wed with homotopical n otions (e.g. mo del categories, sim- plicial categories, catego ries with cylinders) and when W is th e class of weak equiv alences of M , the lo calization of M b y W is usually called the homotopy c ate gory of M , and is written by H o ( M ). The p aradigmatic example is that of top ological spaces and w eak homotop y equiv alences. W e recall its definition. 2 Definition. The homotop y category H o ( T op ) is the lo calizatio n o f T op , the category of top ological spaces, b y the family of w eak equiv alences. Thus, H o ( T op ) = T op [ W − 1 ] with W = { f : X → Y | f ∗ : π n ( X, x ) → π n ( Y , f ( x )) is an isomorphism ∀ n ∀ x } . 1.2 Homotopical notions in C at The other example we are going to consid er is that of small categories. The category of small categories C at is endow ed with homotopical notions that one can lift from T op by using the cla ssifying space functor B : C at → T op [14]. W e b riefly recall from [12, 14] some definitions and r esu lts concerning this functor. The category ∆ is that whose ob jects are the fi nite ordin als [ q ] = { 0 < 1 < ... < q } and whose arro ws are the ord er p r eserving maps .W e use the follo wing standard n otatio n: for i = 0 , ..., q let s i : [ q + 1] → [ q ] b e the surjection whic h tak es t wice the v alue i , and let d i : [ q − 1] → [ q ] b e the injection whose image do es not con tain the v alue i . If C is a small catego ry , its nerve N C is the simp licial set whose q -simp lices are the c hains X = ( X 0 → X 1 → ... → X q ) of q comp osable arro ws of C . F ormally , a q -simplex X is a functor [ q ] → C w here [ q ] is view ed as a cate gory in t he canonical wa y . F aces and dege neracies of N C are giv en by On the sub division of small cate gories - M. d el Ho yo 3 comp osing ad j acen t arr o w s (or deleting the first or the last arro w) and inserting iden tities, resp ectiv ely . The classifying sp ac e B C of a category C is the geometric realizati on of its nerve, namely B C = | N C | . A fun ctor f : B → C in C at is said to be a we ak e quivalenc e if B f is a homotop y equ iv alence in T op , and a small category C is said to b e c ontr actible if B C is so. There is a homeomorphism B ( C × D ) ≡ B C × B D w hen, for in stance, N C or N D has only finite n on-degenerate simplices. In p articular, denoting I = [1], one has that a functor C × I → D indu ces a con tinuos map B C × B I → B D . Thus, it f ollo ws that a natural transformation f ∼ = g induces a homotop y B f ∼ = B g . S ome simple and usefu l applications of it are the follo wing. 3 Lemma. If a functor admits an adjoint, then it is a we ak e qui v alenc e. Pr o of. It suffices to consider the homotopies arising from the u nit and th e c ounit of the adjunction. 4 Lemma. If a c ate gory has initial or final obje ct, then i t is c ontr actible. Pr o of. In these cases the functor C → ∗ admits an adjoin t, ∗ b eing the one-arro w catego ry . 5 Lemma. L et i : A → B a ful ly faithful inclusion. If ther e is a functor r : B → A and a natur al tr ansformat ion id B ⇒ ir , then i is a we ak e quivalenc e. Pr o of. The natural trans f ormation id B ⇒ ir give s rise to another one i ⇒ ir i , and since i is fully faithful, this is the s ame than a n atural transformation id A ⇒ r i . Th e result no w follo ws f r om the fact that a natural transformation indu ces a homotop y . W e complete this review by reca lling the definition of H o ( C at ). 6 Definition. The homotop y cate gory H o ( C at ) is the lo calizat ion of C at b y the family of w eak equiv alences, that is, H o ( C at ) = C at [ W − 1 ] with W = { f : C → D | B f : B C → B D is a homotop y equiv alence } . 7 Remark. C at admits a d ifferen t homotop y str u cture than the one used here (cf. [11]). The fu nctors w hic h b ecome homotop y equ iv alences after taking the classifying space func- tor are sometimes called top olo gic al we ak e quivalenc es to a v oid confusions. 1.3 Quillen’s Theorem A Quillen’s Th eorem A pro vides a criteria to recognize wh en a fu n ctor is a weak equiv alence. W e fix some notations and recall it from [12, § 1]. If f : C → D is a funtor and if T is an ob ject of D , then the fib er f − 1 T of f o ve r T is the sub category of C wh ose ob jects and arro ws are those which f carr ies in to T and id T , resp ectiv ely . The left fib e r f /T of f ov er T is the category of pairs ( X, u ) w ith X an ob ject of C and u : f X → T , where an arrow b et we en pairs ( X , u ) → ( X ′ , u ′ ) is a map On the sub division of small cate gories - M. d el Ho yo 4 v : X → X ′ in C suc h that u ′ f ( v ) = u . Th e right fib er Y /T is defined dually . By an abuse of notation, w e shall write C T , C /T and T /C for the fib er, left fib er and righ t fib er, resp ectiv ely . 8 Theorem (Qu illen’s Theorem A) . The functor f : C → D is a we ak e quivalenc e if i t satisfies either ( i ) C /T is c ontr actible for every obje ct T of D , or ( ii ) T /C is c ontr actible for every obje ct T of D . Giv en f : C → D , an arro w X u − → Y of C is said to b e c o c artesian if ev ery arro w X v − → Y ′ suc h that f ( u ) = f ( v ) factors as e v ◦ u with f ( e v ) = id f ( Y ) in a uniqu e w a y . Y ′ X ∀ v 6 6 m m m m m m m m m u / / Y ∃ ! e v O O f ( X ) f ( u ) / / f ( Y ) The fun ctor f is a pr e- c ofibr ation if for eac h arro w S φ − → T of D and for eac h X ∈ C S there is a co cartesian arrow X u − → Y o ver φ . The fun ctor f is a c ofibr ation if it is a pre-cofibration and also co cartesian arr o w s are closed under comp ositions. Cartesian arr ows are defined dually , as w ell as pr e- fibr ations and fibr ations . When f : C → D is a p re-cofibration, th e inclusion C T → C /T admits a right adjoin t, called b ase-change , th at push-forward an ob ject ( X , φ ) along a cocartesian arro w X → Y o v er φ . This remark and its dual, com bined with lemma 3, yield the follo wing corollary . 9 Corollary . L et f : C → D b e a functor which is either a pr e- fibr ation or a pr e- c ofibr ation. If C T is c ontr actible for every obje ct T of D , then f is a we ak e q uivalenc e. 2 Sub division of categories 2.1 The construction of S d ( C ) Let C b e a s mall category . With ∆ /C w e mean the left fib er o ver C of the em b ed d ing ∆ → C at . It has the simp lices of N C as ob jects, and giv en X and Y simplices of d imensions q and p , a morphism ( Y , ξ , X ) : X → Y in ∆ /C consists of an order p reserving map ξ : [ q ] → [ p ] s u c h that Y ◦ ξ = X . W e write ξ ∗ instead of ( Y , ξ , X ) when there is n o place to confusion. 10 Remark. Note that if th ere is a map X → Y in ∆ /C , then the sequence X 0 → X 1 → ... → X q is obtained from Y 0 → Y 1 → ... → Y p b y comp osing some arrows and in serting some ident ities ( X is a degeneratio n of a face of Y ). Let X ∈ N C q , and let s : [ q + 1] → [ q ] b e a surjection. If d, d ′ : [ q ] → [ q + 1] are the t w o righ t inv erses of s , then w e say that d ∗ , d ′ ∗ : X → X s are elementary e quivalent , and w e write d ∗ ≈ d ′ ∗ . Note that ≈ is reflexiv e and symmetric. W e define ∼ as th e minor equiv alence relation on the arro ws of ∆ /C whic h is compatible w ith th e composition an d satisfies ξ ∗ ≈ ξ ′ ∗ ⇒ ξ ∼ ξ ′ ∗ . W e say that ξ ∗ and ξ ′ ∗ are e quivalent if ξ ∗ ∼ ξ ′ ∗ . With [∆ /C ] w e denote the quotien t category with the same ob jects than ∆ /C and arro ws the classes under ∼ . On the sub division of small cate gories - M. d el Ho yo 5 11 Definition. T he sub division of C , denoted S d ( C ), is the full su b category o f [∆ /C ] whose ob jects are the n on-degenerate simplices of N C . W e describ e the situation with the follo wing diagram, ∆ /C   S d ( C ) / / [∆ /C ] where S d ( C ) → ∆ /C is just the inclusion and ∆ /C → [∆ /C ] is the functor whic h maps an ob ject to itself and an arr o w ξ ∗ : X → Y to its class [ ξ ∗ ] und er ∼ . 12 Remark. Notice th at this is not the s u b division giv en in [7, I I I-10.1] or [9, IX-5]. Indeed, our construction is equiv alen t to that of [1, § 2], as it ca n b e deduced from lemma 18. Our definition describ es completely the arrows of the s u b division category as homotopy- lik e equiv alences of maps, where a degeneration of a simp lex pla ys the r ole of a cylinder of it. 13 Remark. This sub division give s rise to a functor C at → C at whic h equals the com- p osition c ◦ s d ◦ N , where s d d enotes Kan ’s sub division of simplicial sets [8]. Ho wev er, we b eliev e that the intrinsic definition that we pr esen t here might b e of in terest, as it clarifies some asp ect of sub division of categ ories. 14 Example. If C is the cat egory 0 a / / b / / 1 , then the full sub category of ∆ /C generated by the non-degenerate ob jects is 0 / /   > > > > > > > > a 1 / / @ @         b where a and b denote the non-trivial arro ws of C . This is also the sub division S d ( C ), since no indentifica tion is p ossible. The classifying space B ( S d ( C )) is the 1-sphere S 1 . 15 Example. If C is the t wo-o b ject sim p ly connected group oid 0 ⇄ 1, then N C has t wo non-degenerate simplices on eac h dimension q , say 0101 ... and 1010 ... . If q < p , then there are sev eral arro ws in ∆ /C b et w een a q -simplex and a p -simplex, but is not hard to see that an y tw o of them are equiv alent . Hence, it follo w s that S d ( C ) is the p oset 0 / /   @ @ @ @ @ @ @ @ 01 / / ! ! D D D D D D D D 010 / /   @ @ @ @ @ @ @ @ ... 1 / / ? ? ~ ~ ~ ~ ~ ~ ~ ~ 10 / / = = z z z z z z z z 101 / / ? ? ~ ~ ~ ~ ~ ~ ~ ~ Notice that S d ( C ) is th e colimit of its sub catego ries S d ( C ) 6 n formed by the simplices of dimension 6 n . Since B ( S d ( C ) 6 n ) = S n and since B comm utes w ith directed co limits, it follo ws th at B ( S d ( C )) is homeomorphic to the infin ite dimensional sphere S ∞ . On the sub division of small cate gories - M. d el Ho yo 6 2.2 Some fundamen tal prop erties If X is an ob ject of ∆ /C , w e denote by q X its dimension as a simplex of N C . 16 Definition. A map ξ ∗ : X → Y in ∆ /C is a surje ction if ξ : [ q X ] → [ q Y ] is so. Note that if there is a su rjection X → Y , then X is a degeneration of Y . 17 Lemma. A surje ction ξ ∗ : X → Y in ∆ /C induc es an isomorphism [ ξ ∗ ] : X → Y in [∆ /C ] . Pr o of. It is sufficient to consider the case q X = q Y + 1, for any sur jection can b e expr essed as a composition of some of the s i . T h us, supp ose that ξ = s i : [ q + 1] → [ q ], wh ere q = q Y . Then X = Y s i and the maps ( d i +1 ) ∗ , ( d i +2 ) ∗ : X → X s i +1 are elemen tary equiv alen t. F rom the simplicial ident ities it follo w s that ( s i ) ∗ ( d i +1 ) ∗ = id : Y → Y and that ( d i +1 ) ∗ ( s i ) ∗ = ( s i ) ∗ ( d i +2 ) ∗ ∼ ( s i ) ∗ ( d i +1 ) ∗ = id : X → X. Hence, ( s i ) ∗ : X → Y and ( d i +1 ) ∗ : Y → X are inv erses mo d ulo equiv alences. 18 Lemma. If a functor ∆ /C → D c arries surje ctions into isomorphisms, then it factors as ∆ /C → [∆ /C ] → D in a uniq u e way. Thus, [∆ / C ] is the lo c alization o f ∆ /C by the surje ctions. Pr o of. If it exists, the factoriza tion is un ique b ecause ∆ /C → [∆ /C ] is surjectiv e on ob jects and on arrows. Let f : ∆ /C → D b e a functor whic h carries su rjections in to isomorphisms. If d ∗ ≈ d ′ ∗ : X → X s are elemen tary equiv alen t maps and s ∗ : X s → X is their left in ve rse, th en f ( d ∗ ) = f ( s ∗ ) − 1 = f ( d ′ ∗ ). Thus, the relation ξ ∗ ∼ f ξ ′ ∗ ⇐ ⇒ f ( ξ ∗ ) = f ( ξ ′ ∗ ) is compatible with the comp osition and satisfies ξ ∗ ≈ ξ ′ ∗ ⇒ ξ ∗ ∼ f ξ ′ ∗ . T h erefore, ξ ∗ ∼ ξ ′ ∗ ⇒ ξ ∗ ∼ f ξ ′ ∗ and f factors through [∆ /C ]. Let dim : S d ( C ) → N 0 b e th e fu nctor X 7→ q X whic h assigns to eac h non-degenerate simplex X its dimension. 19 Lemma. If ther e i s a non-identity arr ow X → Y in S d ( C ) , then dim ( X ) < dim ( Y ) . Pr o of. Let [ i ∗ ] : X → Y b e an arro w of S d ( C ), w ith i : [ q X ] → [ q Y ] an order pr eserving map. Then X = Y i : [ q X ] → C and i must b e injectiv e b ecause X is a non-d egenerate simplex of N C . Therefore q X 6 q Y , and q X = q Y if and only if i = id [ q X ] = id [ q Y ] . 20 Corollary . If f : X → Y is an isomorphism in S d ( C ) , then X = Y and f = id X . F ollo win g th e terminology of [5, § 5], w e h av e pro ved that dim : S d ( C ) → N 0 is a line ar extension o f the sub division category S d ( C ), and that the latte r is a dir e ct c ate gory . This is not true for ∆ /C nor [∆ /C ], and here lies one reason for our construction. 21 Theorem. S d 2 ( C ) is a p oset for every c ate gory C . On the sub division of small cate gories - M. d el Ho yo 7 Pr o of. W e must sho w that for ev ery pair X , Y of ob jects of S d 2 ( C ), ( i ) th ere is at m ost one arro w X → Y , and ( ii ) the existence of arro ws X → Y and Y → X implies X = Y . Assertion ( ii ) is an immediate corollary of lemma 19, so let us pro v e ( i ). An ob ject X of S d 2 ( C ) is a non-degenerate simplex of N ( S d ( C )), that is, a c hain of non-trivial comp osable arro ws X = ( X 0 → X 1 → ... → X q X ) of S d ( C ), where X i is a non-degenerate simplex of N C for eac h i . Note that dim ( X i ) < dim ( X i +1 ) b y lemma 19. Fix t wo non-d egenerate simplices X and Y of N ( S d ( C )). W e will s h o w that there exists at most one order map ξ : [ q X ] → [ q Y ] suc h that X = Y ξ , from where ( i ) follo ws. Supp ose that ξ , ξ ′ are su ch that X = Y ξ = Y ξ ′ . As we ha ve p ointe d out, dim ( Y j ) < dim ( Y j +1 ) for all j , so X i = Y ξ ( i ) = Y ξ ′ ( i ) implies that ξ ( i ) = ξ ′ ( i ) and therefore ξ = ξ ′ . 22 Remark. T he same argu m en t of ab o ve prov es that S d ( C ) is a p oset for eve ry dir ect catego ry C in th e sense of [5]. 2.3 F unctorialit y of the subdivision If X is a simplex of N C , w e m igh t think of X as a sequence of comp osable arrows, sa y X = ( f 1 , ..., f q X ). Let p X = # { j | f j 6 = id } b e the n umber of non-iden tit y arrows that app ear in X , and let r ( X ) = ( f i 1 , ..., f i p X ) b e the sequence obtained from X by d eleting the iden tities. Then X is a degeneration of r ( X ) – viewed as simplices of N C – and r ( X ) is a non-degenerate p X -simplex. Moreo v er, X = r ( X ) α X with α X : [ q X ] → [ p X ] the surjectiv e order map d efined b y α X ( i − 1) = α X ( i ) ⇐ ⇒ f i = id . If ξ ∗ : X → Y is an arro w in ∆ /C , w e define r ( ξ ∗ ) as th e arrow r ( X ) → r ( Y ) in S d ( C ) giv en by the comp osition [( α Y ) ∗ ][ ξ ∗ ][( α X ) ∗ ] − 1 in [∆ /C ]. X [ ξ ∗ ] / / [( α X ) ∗ ]   Y [( α Y ) ∗ ]   r ( X ) r ( ξ ∗ ) / / _ _ _ r ( Y ) Note that [( α X ) ∗ ] is in versible b y lemma 17. With these definitions r = r C : ∆ /C → S d ( C ) is a fun ctor which maps surjection in to iden tities, and by lemma 18 it induces a new one [∆ /C ] → S d ( C ), also denoted by r C . Let i C : S d ( C ) → [∆ /C ] b e the ca nonical inclusion. Clearly r C i C = id , and b y lemma 17 w e ha v e that α : id ⇒ i C r C , X 7→ [( α X ) ∗ ] is a natural isomorp h ism. Thus, i C : S d ( C ) → [∆ /C ] is an equiv alence of ca tegories with in v erse r C : [∆ /C ] → S d ( C ). In particular, S d ( C ) is a sk eleton of [∆ /C ] as it follo w s from corollary 20. 23 Lemma. The c onstruction C 7→ S d ( C ) is functorial. Pr o of. A functor f : C → D indu ces a new one f ∗ : ∆ /C → ∆ /D by mapp ing a sim p lex X to f ◦ X . This functor clearly sends surjections into su rjections. Then, it induces a On the sub division of small cate gories - M. d el Ho yo 8 functor [ f ∗ ] : [∆ /C ] → [∆ /D ], whic h d o es n ot necessarily carry S d ( C ) in to S d ( D ). Th us, w e must define S d ( f ) as the comp osition r D [ f ∗ ] i C . S d ( C ) S d ( f )      i C / / [∆ /C ] [ f ∗ ]   S d ( D ) [∆ /D ] r D o o T o pro ve that it is fu nctorial w e m u st v erify that S d ( id ) = id and that S d ( g ) S d ( f ) = S d ( g f ). The first assertion follo ws b ecause S d ( id ) = r C [ id ∗ ] i C = r C i C = id S d ( C ) . Ab out the other, if f : C → D and g : D → E then the natural isomo rp hism α : id ∼ = i D r D induces another one S d ( g f ) = r E [( g f ) ∗ ] i C = r E [ g ∗ ][ f ∗ ] i C ∼ = r E [ g ∗ ] i D r D [ f ∗ ] i C = S d ( g ) S d ( f ) of functors S d ( C ) → S d ( E ). It f ollo ws from corollary 20 that the natural isomorphism S d ( g f ) ∼ = S d ( g ) S d ( f ) must b e the iden tit y , and hence w e hav e pro ve d that S d ( gf ) = S d ( g ) S d ( f ) and that S d is a functor indeed. 24 Remark. It follo ws from prop osition 21 that S d 2 lifts to a functor l : C at → P oS et . P oS et j   C at l : : v v v v v v v v v S d 2 / / C at Here j denotes the canonical inclusion P oS et → C at . Next section we will s h o w that l is a homotop y inv erse for j . 2.4 Relationship b et w een a category and its sub division Recall the fun ctor sup : ∆ /C → C [6, § 3]: Giv en an ob ject X : [ q X ] → C of ∆ /C , sup ( X ) = X q X is the last ob j ect of the sequence X . F or an arro w ξ ∗ : X → Y in ∆ /C , recall that sup ( ξ ∗ ) : X q X → Y q Y is the comp osition of th e arr o w s of Y betw een Y ξ ( q X ) and Y q Y , namely sup ( ξ ∗ ) = Y ( ξ ( q X ) → q Y ). The functor sup maps surjections into iden tities, s in ce a su rjectiv e map [ p ] → [ q ] preserv es final elemen t. I t follo ws from lemma 18 that sup factors through the qu otien t and in d uces a functor [∆ /C ] → C which w ill b e denoted by [ sup ]. ∆ /C   sup " " D D D D D D D D D S d ( C ) i C / / [∆ /C ] [ sup ] / / C 25 Definition. Th e f unctor ε C : S d ( C ) → C is defined as the comp osition [ sup ] ◦ i C of the b ottom of the diagram of ab o v e. On the sub division of small cate gories - M. d el Ho yo 9 26 Lemma. The functor ε C is nat ur al in C . It gives rise to a natur al tr ansformation, denote d ε : S d ⇒ id C at . Pr o of. Giv en a m ap f : C → D in C at , we ha v e the follo wing diagram. S d ( C ) S d ( f )   i C / / [∆ /C ] [ sup ] / / [ f ∗ ]   C f   S d ( D ) i D / / [∆ /D ] [ sup ] / / D Clearly the map sup is natural, and it follo ws from this that [ sup ] is also n atural. Thus, the righ t square of ab o ve is comm u tativ e. T h e left square does not commute, b ut there is a natural isomorphism [ f ∗ ] i C ⇒ i D S d ( f ) whic h consists of a sur j ectiv e map α f X : f X ⇒ i D r D f X for eac h ob ject X of S d ( C ) (see the definition of α in the previous subsection). Finally , as [ sup ] carries surjections in to iden tities, the big square comm utes and the lemma follo ws. Next we sh all p ro v e that the fun ctor ε C : S d ( C ) → C is a w eak homotop y equ iv alence. T o do that, we first study the left fib ers of ε C . Fix some ob ject T of C . Let ( X , f ) b e an ob ject of (∆ /C ) /T , namely the left fib er of sup o v er T . Th us, f : S → T is a m ap in C , and X = ( X 0 → X 1 → ... → X q X − 1 → S ) is an ob ject of ∆ /C wh ose top elemen t is S . W e define r ( X , f ) as the ob ject of ∆ /C obtained b y extending X with f . r ( X, f ) = ( X 0 → X 1 → ... → X q X − 1 → S f − → T ) The assignmen t ( X , f ) 7→ r ( X , f ) is fun ctorial: giv en ( X, f ) ξ ∗ − → ( Y , g ), we define r ( ξ ∗ ) : r ( X, f ) → r ( Y , g ) as th e map of ∆ /C induced b y the order map [ q X + 1] → [ q Y + 1] j 7→ ξ ( j ) (0 6 j 6 q X ) , q X + 1 7→ q Y + 1 . This wa y w e ha ve a fun ctor r : (∆ /C ) /T → (∆ /C ) T in to the fib er, which is some kin d of retraction for the fu lly faithful canonical map i : (∆ /C ) T → (∆ /C ) /T . In d eed, giv en ( X, f ) in (∆ /C ) /T , there is a natural map d ∗ : X → r ( X, f ) in duced b y the in jection d = d q X +1 : [ q X ] → [ q X + 1]. Clea rly , sup ( d ∗ ) = f and d ∗ is also a map in (∆ /C ) /T , hence w e hav e a natural transformation id ⇒ ir : (∆ /C ) /T → (∆ /C ) /T . 27 Lemma. The inclusion (∆ /C ) T → (∆ /C ) /T is a we ak e quivalenc e. Pr o of. F ollo w s fr om lemma 5 and the paragraph of ab ov e. 28 Remark. Note that the map d ∗ : X → r ( X , f ) is not a co cartesian arrow. As an example, consider an arro w f : X → T in C , and let ( X, f ) b e th e corr esp ondent zero- dimension ob ject of (∆ /C ) /T . Then the tw o maps [( d 1 ) ∗ ] , [( d 2 ) ∗ ] : r ( X, f ) = ( X f − → T ) → ( X f − → T id − → T ) are different w ays to factor ( X ) → ( X f − → T id − → T ) through d ∗ . On the sub division of small cate gories - M. d el Ho yo 10 No w w e shall pro ve t hat the functor r : (∆ /C ) /T → (∆ /C ) T giv es rise to a new one [ r ] : [∆ /C ] /T → [∆ /C ] T b et we en the left fib er and the actual fib er of [ sup ]. In order to do that, we ha ve to sh o w that r c arries equiv alent maps int o equiv alen t maps. T o prov e this, we will need a more explicit description of the relation ∼ . 29 Remark. W e say that ξ ∗ ∼ 1 ξ ′ ∗ if there are factorizati ons ξ ∗ = ξ 1 ∗ ξ 2 ∗ ...ξ n ∗ and ξ ′ ∗ = ξ ′ 1 ∗ ξ ′ 2 ∗ ...ξ ′ n ∗ suc h that ξ i ∗ ≈ ξ ′ i ∗ for eac h i . Note th at ∼ 1 is reflexive and symmetric. W e call ∼ 2 to the equiv alence relation generated b y ∼ 1 . Th us, ξ ∗ ∼ 2 ξ ′ ∗ iff there is a sequence ξ ∗ ∼ 1 h 1 ∗ ∼ 1 h 2 ∗ ∼ 1 ... ∼ 1 h N ∗ ∼ 1 ξ ′ ∗ . It is easy to see that ξ ∗ ∼ ξ ′ ∗ ⇐ ⇒ ξ ∗ ∼ 2 ξ ′ ∗ , since ∼ 2 is an equiv alence relation whic h con tains the elemen tary equiv alences and is compatible with the comp osition (this giv es ⇒ ) and ∼ is an equiv alence relation wh ich con tains ∼ 1 (this giv es ⇐ ). 30 Lemma. L et ξ ∗ , ξ ′ ∗ : ( X , f ) → ( Y , g ) b e maps of (∆ /C ) /T such that ξ ∗ ∼ ξ ′ ∗ viewe d as maps of ∆ /C . Then r ( ξ ∗ ) ∼ r ( ξ ′ ∗ ) . Pr o of. First of all, obser ve that if ξ ∗ ≈ ξ ′ ∗ , then r ( ξ ∗ ) ≈ r ( ξ ′ ∗ ). Secondly , if ξ ∗ ∼ 1 ξ ′ ∗ then there are factorizati ons ξ ∗ = ξ 1 ∗ ξ 2 ∗ ...ξ n ∗ and ξ ′ ∗ = ξ ′ 1 ∗ ξ ′ 2 ∗ ...ξ ′ n ∗ suc h that ξ i ∗ ≈ ξ ′ i ∗ for eac h i . A priori these are ju st maps in ∆ /C , but since the target of ξ ∗ and ξ ′ ∗ is an ob ject in (∆ /C ) /T , then w e can think of these maps as arr o w s in the left fib er. By applying the functor r w e obtain factorizations r ( ξ 1 ∗ ) r ( ξ 2 ∗ ) ...r ( ξ n ∗ ) and r ( ξ ′ 1 ∗ ) r ( ξ ′ 2 ∗ ) ...r ( ξ ′ n ∗ ) of r ( ξ ∗ ) and r ( ξ ′ ∗ ) whic h together with previous paragraph imp ly that r ( ξ ∗ ) ∼ 1 r ( ξ ′ ∗ ). Finally , if ξ ∗ ∼ 2 ξ ′ ∗ , then r ( ξ ∗ ) ∼ 2 r ( ξ ′ ∗ ) b y an ind uctiv e argumen t. The lemma follo ws from remark 29. 31 Lemma. The inclusion [∆ /C ] T → [∆ /C ] /T is a we ak e quiv alenc e. Pr o of. F ollo w s fr om lemmas 5 and 30. The follo wing theorem allo w u s to consider S d ( C ) as an algebraic mo del for the homoto py t yp e of B C , lo cally simpler than C . 32 Theorem. The functor ε C : S d ( C ) → C is a we ak e qui v alenc e for every C . Pr o of. The fu nctor ε C factors as [ sup ] ◦ i C . Sin ce i C is an equiv alence of categories, it is a w eak equiv alence (cf. lemma 3) and we just need to prov e that [ sup ] is a weak equiv alence. W e will ap p ly th eorem 8, so w e need to p ro v e that the left fib ers of [ sup ] are cont ractible. By lemma 31 it is sufficien t to p ro v e that the fib er [∆ /C ] T is con tractible for eac h ob ject T of C . Giv en T , we will pro ve that [∆ /C ] T has an initial ob ject and the result will f ollo w from lemma 4. This initial ob ject is T , viewe d as a 0-simplex of N C . If X is any ob ject of [∆ /C ] T , then the ( q X )-th inclusion α : [0] → [ q X ] induces a map [ α ∗ ] : T → X in [∆ /C ] T . On the sub division of small cate gories - M. d el Ho yo 11 If [ β ∗ ] : T → X is any other map in [∆ /C ] T , w e ha v e to pro ve that α ∗ ∼ β ∗ . Consider the order map h : [1] → [ q X ] giv en by h (0 ) = β (0) and h (1) = q X . ( T id − → T ) h ∗ ( ( ( s 0 ) ∗   ( T ) ( d 0 ) ∗ O O ( d 1 ) ∗ O O α ∗ / / β ∗ / / ( X 0 → X 1 → ... → X q X ) h ∗ ( d 0 ) ∗ = α ∗ h ∗ ( d 1 ) ∗ = β ∗ Then α = hd 0 and β = hd 1 , and b ecause X ( β (0) → q X ) = ε C ([ β ∗ ]) = id T it follo w s that X h = T s 0 is a degeneration of T , ( d 0 ) ∗ ≈ ( d 1 ) ∗ and therefore α ∗ ∼ β ∗ . 33 Corollary . The functor S d : C at → C at pr eserves we ak e qui v alenc es. Pr o of. If f : C → D is a w eak equiv alence in C at , it follo ws from theorem 3 2 and th e square S d ( C ) ε C / / S d ( f )   C f   S d ( D ) ε D / / D that S d ( f ) is also a w eak equiv alence. 34 Remark. Given ( X , f ) an ob j ect of (∆ /C ) /T , w e hav e seen in remark 28 that d ∗ : X → r ( X , f ) is n ot a cartesian arro w for sup . Ho w ev er, [ d ∗ ] : X → r ( X, f ) is a co cartesian arro w for [ sup ]. T o s ee that, supp ose that [ ξ ∗ ] : X → Y is an arrow of [∆ /C ] suc h that [ sup ]([ ξ ∗ ]) = f . Th en, ξ ∗ migh t b e consider as an arro w ( X, f ) → ( Y | [ q Y − 1] , Y ([ q Y − 1] → [ q Y ])) in [∆ /C ] /T , and [ ξ ∗ ] factors as [ r ( ξ ∗ )][ d ∗ ] (actually , ξ ∗ = r ( ξ ∗ ) d ∗ ). T o see that this factorizat ion is u nique, sup p ose that another one is giv en, a nd use the fact that r preserves equiv alences. It foll o ws that [∆ /C ] → C is a precofibration, as we ll as S d ( C ) → C . T h us, theorem 32 can b e pro ve d b y using corollary 9. Ho wev er, S d ( C ) → C is not a cofibration in general, sin ce cocartesian arr o w s are not closed un d er composition. This is clear b ecause a co cartesian arro w ov er a non-identit y map m ust increase th e degree in exactly one. 3 Application to homotop y theory 3.1 Homotop y category of P oS et Despite the h omotopy theory of partially ordered sets is largely develo p ed, we could n ot find a definition for the homotop y category H o ( P oS et ). W e construct it here in a suitable form, compatible with the in clusions P oS et → T op and P oS et → T op . W e will use for this purp ose some w ell kn o w n facts ab out A -spaces, p osets and simplicial complexes. Recall that an A -sp ac e , or Alexandr ov sp ac e , is a top ological space in whic h an y arbitrary in tersection of op en subsets is op en. A top ologic al space satisfies the T 0 separabilit y axiom On the sub division of small cate gories - M. d el Ho yo 12 if give n t w o p oin ts on it , there exists an op en su bset that cont ains exactly one of these p oin ts. A T 0 A -space is simply an A -space which satisfies the T 0 axiom. There is a well-kno wn corresp ondence b etw een T 0 A -spaces and preorders. W e r ecall it briefly . If P is a p oset, let a ( P ) b e the top ologic al sp ace with p oin ts the elemen ts of P and with op en basis formed by the subsets { y | y 6 x } , x ∈ P . Clearly , a ( P ) is a T 0 A -space. If X is a T 0 A -space, let s ( X ) b e the p oset with elemen ts the p oin ts of X and with the order x 6 y ⇐ ⇒ y ∈ cl ( x ), where cl ( x ) denotes the closur e of { x } in X . Note that the relation 6 is ant isymmetric b ecause X is T 0 . 35 Lemma. The c onstructions P 7→ a ( P ) and X 7→ s ( X ) ar e fu nctorial, and they define an e q u ivalenc e of c ate gories b etwe en P oS et and the f ul l sub c ate gory of T op whose obje cts ar e the T 0 A -sp ac es. W e recall some constructions from [10]. Giv en X a T 0 A -space, a simplicial complex k ( X ) is constructed with v ertices th e p oin ts of X and simp lices the finite c hains of s ( X ), n amely the sequ en ces of p oints ( x 0 , ..., x q ) satisfying x i +1 ∈ cl ( x i ). The construction X 7→ k ( X ) is fu nctorial. Moreo ver, there is a natural con tinuous map f X : | k ( X ) | → X defined by f X ( u ) = min ( car r ier ( u )), where car r ier ( u ) is the unique op en simplex con taining u . Giv en a simplicial complex K , denote by S ( K ) its set of simplices ord ered b y in clus ion. Define x ( K ) as the T 0 A -space associated to its simplices, namely x ( K ) = aS ( K ). T h e construction K 7→ x ( K ) is fu nctorial, since a a nd S are so. Moreo ver, since k ( x ( K )) is just the barycen tr ic sub division of K , th ere is a natural con tin uous map f K : | K | → x ( K ) defined as the comp osition of the canonical homeomorph ism | K | ∼ − → | k x ( K ) | with the map f x ( K ) . Th e follo wing results are du e to McCord [10]. 36 Prop osition. F or every T 0 A -sp ac e X the map f X : | k ( X ) | → X is a we ak homotopy e quivalenc e. 37 Prop osition. F or every simplicial c omplex K the map f K : | K | → x ( K ) is a we ak homoto py e qu i valenc e. No w w e are in condition to describ e H o ( P oS et ). Recall that j : P oS et → C at is the functor wh ich assigns to eac h p oset P a ca tegory j ( P ) in the usual wa y . T h e functor j admits a left adjoint p : C at → P oS et , which assigns to eac h sm all category C the p oset asso ciated to the pr eorder defined o ve r the ob jects of C b y th e rule X 6 Y ⇐ ⇒ there exists an arrow X → Y . The f unctors a : P oS et → T op and j : P oS et → C at em b ed P oS et as a fu ll reflectiv e sub category of T op and C at . Thus, P oS et inherits tw o definitions for weak equ iv alences b y lifting th ose of T op and C at . Let W a b e the class of maps f : P → Q in P oS et such that a ( f ) : a ( P ) → a ( Q ) is a weak equiv alence in T op , and let W j b e the class of maps f : P → Q in P oS et suc h that j ( f ) : j ( P ) → j ( Q ) is a weak equiv alence in C at or, what is the same, B j ( f ) : B j ( P ) → B j ( Q ) is a weak equiv alence in T op . 38 Prop osition. The classes W a and W j c oincide. On the sub division of small cate gories - M. d el Ho yo 13 Pr o of. F or ea c h p oset P there is a natural homeomorphism B j ( P ) ∼ = | k a ( P ) | b et w een the classifying s pace of j ( P ) and the geometric realiz ation of McCord’s constru ction on a ( P ). Giv en f : P → Q a map in P oS et , consider the follo win g commuta tive diagram. B j ( P ) B j ( f )   ∼ / / | k a ( P ) | f a ( P ) / / | K a ( f ) |   a ( P ) a ( f )   B j ( Q ) ∼ / / | k a ( Q ) | f a ( Q ) / / a ( Q ) Since the maps f a ( P ) and f a ( Q ) are w eak equiv alences in T op (cf. prop osition 36), the con tin uous map B j ( f ) is a w eak equiv alence if and only if a ( f ) is so. 39 Definition. W e sa y that a map f : P → Q in P oS et is a we ak e quivalenc e if f ∈ W a = W j . W e defin e the homotopy c ate gory of P oS et , denoted H o ( P oS et ), as the lo caliza tion of P oS et by the family of w eak equiv alences. 40 Remark. It is clear that pj ( P ) = P . Unfortunately , the comp osition j p does not preserve homotopy typ es – f or instance, a group G is mapp ed by j p in to the one-arro w catego ry . S imilarly , while the comp osition sa is the iden tit y functor ov er P oS et , the other comp osition as fails at the homotopy level – for instance, a Hausdorff space X is mapp ed b y as into a discrete space. Despite last remark, the fu nctors j : P oS et → C at and a : P oS et → T op in d uce equiv a- lences b et we en the h omotop y catego ries. In the next sub section we will construct homo- top y inv erses to the inclusions a and j . 3.2 Categorical description of H o ( T op ) The functors a and j preserve we ak equiv alences. Hence, they in duce fu nctors H o ( a ) and H o ( j ) at the homotop y lev el. C at   P oS et a / /   j o o T op   H o ( C at ) H o ( P oS et ) H o ( a ) / / H o ( j ) o o H o ( T op ) 41 T heorem. The functors H o ( a ) and H o ( j ) ar e e quivalenc es of c ate gories. Henc e, the c ate gories H o ( C at ) and H o ( T op ) ar e e quivalent. This theorem is in timately rela ted with Qu illen’s theorem asserting that N ind uces an equiv alence of categorie s at the homotop y lev el (cf. [6]). One can deriv e one from the other by u sing the w ell known equiv alence H o ( T op ) ∼ = H o ( s S et ). Pr o of. W e prov e firs t that H o ( j ) is an equiv alence of categories. R ecall from remark 24 the definition of l : C at → P oS et . W e hav e seen in corollary 33 that S d p reserv es weak equiv alences. S ince j l = S d 2 , it is clear that l preserves them to o, hence it induces a On the sub division of small cate gories - M. d el Ho yo 14 functor H o ( l ) : H o ( C at ) → H o ( P oS et ). W e assert th at l is a h omotop y inv ers e to j , so w e hav e to prov e that there are natur al isomorp h isms H o ( j l ) = H o ( j ) H o ( l ) ∼ = id H o ( C at ) and H o ( lj ) = H o ( l ) H o ( j ) ∼ = id H o ( P oS e t ) . If w e sho w that there are natural transformations j l ⇒ id C at and lj ⇒ id P oS et whic h assign to any ob ject a w eak equiv alence, th en by comp osing w ith the pro jections we will obtain natural isomorph isms, w h ic h yield another ones H o ( j l ) ∼ = id H o ( C at ) and H o ( lj ) ∼ = id H o ( P oS e t ) b y lemma 1. F or ev ery category C the comp osition ε C ε S d ( C ) : j l ( C ) = S d 2 ( C ) → C is a wea k equ iv- alence by theorem 32, and clearly it is n atural. This giv es the n atural isomorphism H o ( j l ) ∼ = id H o ( C at ) . T h e other n atural isomorph ism can b e obtained as a restriction of this. No w w e prov e that H o ( a ) is an equiv alence of categorie s. W e will construct an inv erse to a by considering for eac h top ologic al space X a simplicial complex K X and a w eak equiv alence | K X | → X , whic h can b e done naturally . W e define a functor b : T op → P oS et b y b ( X ) = S ( K X ) the p oset of simp lices of the asso ciated complex. T o see that b preserves w eak equiv alences it is sufficient to consider the diagram ab ( X ) = aS ( K X )   | K X | f K X o o ∼ / /   X   ab ( Y ) = aS ( K Y ) | K Y | f K Y o o ∼ / / Y where f K X and f K Y are McCord’s wea k equiv alences of prop osition 37. Hence b ind u ces a functor H o ( b ) : H o ( T op ) → H o ( P oS et ). By the same argument used ab o ve, the natural wea k equiv alences | K X | → X and | K X | → ab ( X ) y ield natural isomorph isms at the homotop y lev el, which comp ose to giv e H o ( a ) H o ( b ) = H o ( ab ) ∼ = id H o ( T op ) . The n atur al isomorp h ism H o ( b ) H o ( a ) = H o ( ba ) ∼ = id H o ( P oS e t ) can b e obtained as a restriction of the previous one. 42 Remark. By the w ork of Thomason [15] w e kno w that C at admits a closed mo del structure, w eak equiv alences b eing the ones w e w ork with. By the corrections made by Cisinski [2 ] ov er the pap er of Th omason, we kno w that ev ery cofibran t category un der this structure is a p oset. Th us, the equiv alence H o ( P oS et ) ∼ − → H o ( C at ) can b e dedu ced from the comp osition H o ( C at c ) ∼ − → H o ( P oS et ) ∼ − → H o ( C at ) , where C at c denotes the full su b category of cofibran t ob jects. References [1] D. Anderson. 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