Balanced Category Theory II

BALANCED CA TEGOR Y THEOR Y II CLA UDIO PISANI A BSTRACT . In the ﬁrst pa rt, we further a dv ance the study of category theory in a strong balanced factorization category C [Pisani, 200 8], a ﬁnitely co mplete categor y en- dow ed with tw o recipro ca lly stable fa c torization systems such that M / 1 = M ′ / 1. In particular some asp ects related to “internal” (co)limits and to Cauc hy completeness are considered. In the second pa r t, we maintain that also some asp ects of to po logy can be eﬀectively synthesized in a (weak) balance d facto r ization categ o ry T , whose ob jects s hould be considered a s p os sibly “ inﬁnitesimal” and suitably “regular ” top ologica l spaces. While in C the classes M and M ′ play the role of discrete ﬁbrations and opﬁbrations, in T they play the ro le of lo cal homeomor phisms and pe rfect maps , s o that M / 1 and M ′ / 1 are the sub catego ries of discr e te and compact spaces r esp ectively . One so gets a dir ect abstract link b etw een the sub jects, with mutual b eneﬁts. F or example, the slice pro jection X/x → X and the coslice pro jection x \ X → X , obtained as the second factors of x : 1 → X acco rding to ( E , M ) and ( E ′ , M ′ ) in C , c o rresp ond in T to the “inﬁnitesimal” neighbor ho o d o f x ∈ X and to the closur e of x . F urthermore, the o pen- closed complementation (genera liz e d to r ecipro cal stability) b ecomes the key to ol to internally tr e a t, in a coherent way , so me categorica l conc e pts (such a s (co)limits of presheaves) which a re clas s ically r e la ted by duality . Conten ts 1 Int ro ductio n 2 2 Bicartesia n arrows of bimo dules 4 3 F actorizatio n systems 6 4 Balanced fa ctorization categor ie s 10 5 Cat as a strong balanced factoriza tion categ o ry 12 6 Slices and colimits in a bfc 14 7 Int erna l asp ects o f bala nc e d ca teg ory theo ry 16 8 The tenso r functor and the internal ho m 20 9 Retracts of slices 24 2000 Mathematics Sub ject Clas siﬁcation: 18A05, 18A22 , 18 A30, 18A32, 18B30 , 18D99 , 54 B30, 54C10. Key words and phrases: factorization systems, r ecipro cal stability , discrete ﬁbrations and opﬁbra - tions, ﬁnal and initial ma ps, bimo dules, bicar tesian arrows, retra cts, slices and coslice s , int erna l sets, comp onents, internal colimits and limits, tensor pr o duct, top ological spaces, lo cal homeomor phisms and per fect maps, discr ete and compact spaces, connected and lo cally connected spaces, inﬁnitesimal neigh b or- ho o d, conv erge nce , ﬁnite coverings, simply connected spaces, op en-clo s ed complementation, exp onentials. c  Claudio Pisa ni, 20 09. P ermiss ion to copy for priv ate use gr anted. 1 2 10 Conclusion of the ﬁrst part 28 11 Univ ersa l prop erties in to po logy 29 12 T o po logical spa c es a nd discr ete ﬁbrations 32 13 Balanced top olo gy 35 14 Conclusion of the second part 38 1. In tro du ct ion In [Pisani, 2008] w e a rgued that a go o d deal of basic category theory can b e car r ied out in an y strong “ba la nced fa ctorization cat ego ry” (bfc). Recall that a ﬁnitely complete category C is a bfc if it is endow ed with t w o factorization systems ( E , M ) a nd ( E ′ , M ′ ) whic h are reciprocally stable: the pullbac k of a map in E (resp. E ′ ) along a map in M ′ (resp. M ) is itself in E (resp. E ′ ). W e sa y tha t C is a “strong” bfc if, furthermore, M / 1 = M ′ / 1 (the category S of “in ternal sets”). W e refer to “w eak” bfc’s when we wish to emphasize t ha t this condition is not required to hold. The motiv ating example of a strong bfc is Cat , with the comprehensiv e factorizatio n systems: M and M ′ are the class es of discrete ﬁbrations and opﬁbrations, while E and E ′ are the classes of ﬁnal and initial functors, so that M / 1 = M ′ / 1 ≃ Set (while E / 1 = E ′ / 1 are the connected categories). In the ﬁrst part of the presen t pap er, we review and further dev elop some asp ects of balanced category theory . In particular, we consider the bifunctors ¯ ⊗ X : C / X × C / X → S and their restrictions ⊗ X : M ′ / X × M / X → S , where n ⊗ X m := π 0 ( n × X m ) is the in ternal set of comp o nen ts of (the total of ) the pro duct ov er X (and reduces in Cat to the tensor pro duct of the corresp onding set-functors). No w, while the biﬁbrations asso ciated to the factorization systems of the bfc C a re summarized, in t erms of indexed categories, by the adjunctions f ! ⊣ f ∗ : C / Y → C / X ∃ f ⊣ f ∗ : M / Y → M / X ; ∃ ′ f ⊣ f ∗ : M ′ / Y → M ′ / X for an y f : X → Y in C , and the reﬂections ↓ X ⊣ i X : M / X → C / X ; ↑ X ⊣ i ′ X : M ′ / X → C / X for an y X ∈ C (and in particular π 0 := ↓ 1 = ↑ 1 ⊣ i : S → C ), the recipro cal stabilit y axiom allo ws us to obtain also the follo wing “coadjunction” la ws: f ∗ n ⊗ X m ∼ = n ⊗ Y ∃ f m ; n ⊗ X f ∗ m ∼ = ∃ ′ f n ⊗ Y m n ¯ ⊗ X q ∼ = n ⊗ X ↓ X q ; p ¯ ⊗ X m ∼ = ↑ X p ⊗ X m natural in m ∈ M / X (or M / Y ), n ∈ M ′ / Y (or M ′ / X ) and p, q ∈ C / X . With this to olkit, we are in a p osition to stragh tforwardly pro v e familiar prop erties of colimits of “internal-set-v alued” maps m ∈ M / X or n ∈ M ′ / X , and a lso that, f o r an y 3 x : 1 → X in C , there is a bicartesian arr ow ↑ X x → ↓ X x of the bimo dule ten X : ( M ′ / X ) op → M / X , obtained by comp osing ⊗ X with the points functor S (1 , − ) : S → Set . Th us the sub categories X o f “ slices (pro jections)” ↓ X x : X/x → X in M / X and X ′ of “coslices (pro jections)” ↑ X x : x \ X → X in M ′ / X ar e dual. F urthermore, under a “Nullstellensatz” h yp othesis, w e pro v e that the bicartesian arrow s of ten X corresp ond to the retracts of slices in M / X ( o r coslices in M ′ / X ), so oﬀering an alternativ e p erspectiv e on Cauch y completion also in the classical case C = Cat . It is also shown ho w these retracts ma y arise as reﬂections of ﬁg ures P → X whose shap e P is a n “atom” (suc h as the monoid with an idempotent non-iden tit y arrow f or C = Cat ) . In the second part, most of whic h can b e read indip enden tly from the ﬁrst one, we sk etc h how some relev ant asp ects of top ology can b e dev elop ed in a bfc to o. While p erfect maps are kno wn to f orm the second factor of a factorization system on the category T op of top ological spaces , w e in tend to sho w that, b y replacing T op with a suitable category T , it is reasonable to assume that the same is true for lo cal ho meomorphisms and that recipro cal stabilit y holds therein. The existence of a reﬂection π 0 : T → M / 1 in “sets” suggests that the spaces X ∈ T are “lo cally connected”, and in fact t he neigh b orho ods X / x are connected that is, the map ! X/x : X/x → 1 is in E . Some homotopical prop erties o f spaces can b e studied through“ﬁnite cov erings” that is, maps in B = M ∩ M ′ ; for instance, a space is “simply connected” if ! ∗ X : B / 1 → B /X is a n equiv alence. By the recipro cal stability law , spaces in T are also lo cally simply connected, so that ﬁnite co v erings are in fact lo cally trivial (Corollary 13.8). Th us w e maintain that (w eak) bfc’s form a common k ernel shared b y category the- ory and top olog y , and that b o t h the sub jects are enlighted by this p oin t of view. F or example, the recipro cal stability law allows us, on the top ological side, to extend (via exp o nen tiation) the classical complemen tarit y b et w een op en and closed parts to lo cal homeomorphisms and p erfect maps in T , with eviden t conceptual adv a ntages; on the other side, it pro vides a sort of in ternal dua lity for categor ical concepts (as sk etc hed ab ov e) whic h often turns out to be more eﬀectiv e than an “o b vious” dualit y functor. 1.1. Outline. After three preliminary “technic al” sections on bimo dules, factoriza- tion systems and balanced factorizatio n categories, and after recalling some concepts of balanced category theory , w e emphasize in sections 7 , 8 and 9 the cen tral ro le of the recip- ro cal stabilit y la w in treating “ in ternal asp ects” of (balanced) category theory . Namely , w e study (co)limits of in ternal preshea v es in M / X or M ′ / X , and the ro le of t he retracts of the represen table ones (that is, (co)slice pro jections). In the last three sections we sk etc h the idea o f balanced top olog y; in par ticular, w e presen t some “ evidences ” of the fact that the recipro cal stability la w should hold in a n appropria t e “top olog ical” category T , in whic h lo cal homeomorphisms and p erfect maps are assumed as the basic concepts. 4 2. Bicartesi a n arrows of bimo dules In this section w e collect some basic facts a b out bimo dules that will b e used in the sequel; while most of them are w ell kno wn, others (Prop osition 2.2) are new to our kno wledge. W e assume that the reader is familiar with the deﬁnition o f ﬁbration. Recall that a bimo dule t : X → Y can b e seen as a bifunctor t : X op × Y → Set or as a functor t : T → 2, where 2 is t he arrow category < : 0 → 1. W e pa ss from one represen tation to the other, dep ending o n the con v enience. A bimo dule is “represen table o n the righ t” if it is a preﬁbration (or, equiv alently , a ﬁbration): for any y ∈ Y , t ( − , y ) : X op → Set is represen t a ble: t ( − , y ) ∼ = X ( − , y ). Dually , a bimo dule is “represen table on the left” if it is an op(pre)ﬁbration. It is a biﬁbration iﬀ it is birepresen table, that is corresp o nds (up to choice ) to an adjunction X ⇀ Y . Giv en a bimo dule t : T → 2, if g a = bf is a square in T as b elo w, w e write g ( a, b ) f . y g   x f   a o o y ′ x ′ b o o 1 0 o o If a is op cartesian, then the relation ( a, b ) is a function X ( x, x ′ ) → Y ( y , y ′ ) (whic h in the case of represen table bimo dules, if b is op cartesian to o , b ecomes the hom-set mapping of a corresp onding functor X → Y ). In or der to graphically emphasize this, when a is op cartesian we write h a, b ) in place of ( a, b ) , and similarly ( a, b i if b is cartesian. If b oth conditions hold, w e hav e a bijection h a, b i , whic h in the case of represen table bimodules b ecomes the hom- set bij ection of a corresp onding adjunction X ⇀ Y (note tha t , in that case, the na t ur a lit y of the bijection is giv en simply b y comp osition-juxtap osition of squares). On the other hand, if a = b we write ( a ) in place of ( a, a ). So g ( a ) f means that f and g are endomorphisms and g a = af ; in particular, for iden tities, y ( a ) x simply means that a : x → y . F or represen table bimo dules y h a ) x sa ys that y is the image of x according to a corresponding f unctor , with a as the univ ersal elemen t. W e will b e here mainly concerned with the ternary relation g h a i f (or in particular, for iden tities-ob jects, y h a i x ), sa ying that a is bicartesian (or “biuniv ersal”) and f and g are related (as ab ov e) by it. In that case, w e sa y that sa y that f and g are “conjug ate” b y a . Often we are in terested to existen tially quantify this relation ov er some of the three v ar ia bles; for example, w e write y h−i x if x a nd y are conjuga te b y some arro w, or we sa y that x is (or has a) conjugate if this holds for some y ∈ Y . 2.1. Remark. [Fixed categories] It is easily seen by the ab o v e remarks that, for any bimo dule t , there is an equiv alence b etw een the full subcategories X t and Y t of conjugate 5 ob jects in X and Y resp ectiv ely; if t he bimo dule is birepresen table, w e get t he classical fact that an a djunction restricts to an equiv alence among the full sub categories of ob jects with isomorphic units or counits r esp ectiv ely . (Indeed, the units and counits are conjug ate to isomorphisms in X or Y , and the latter a re the bicartesian arrows ov er 0 or 1.) Similarly , if in the ab o ve situation all cartesian arr ows are also o p cartesian t hen Y t = Y and so t he righ t adjoint is fully faithful. No w we pro v e that b y splitting conjug a te idemp oten ts, one gets conjugate o b jects; for clarit y of not a tions, we no w consider a bimo dule t : X → X ′ and use primes to denote ob jects or arrows in X ′ . 2.2. Theorem. If e ′ h−i e ar e c onjugate idemp otents w hich split thr ough y ′ and y r esp e ctively, then y ′ h−i y . Mor e pr e cise l y, if e ′ h u i e , e ′ = i ′ r ′ and e = ir , then y ′ h r ′ ui i y . Let us sho w that y ′ h r ′ ui ) y that is, t ha t r ′ ui is op cartesian; that it is cartesian as w ell is pro v ed dually . x ′ r ′   x r   u o o y ′   i ′ E E y r ′ ui o o t } } z z z z z z z z z z z z z z z i E E z ′ W e need the follo wing 2.3. Lemma. The r etr action − ◦ r ′ , − ◦ i ′ : X ′ ( x ′ , z ′ ) → X ′ ( y ′ , z ′ ) r escrits to a r etr action b etwe en the set F of arr o ws f ′ : x ′ → z ′ such that f ′ u = tr and the set G of arr ows g ′ : x ′ → z ′ such that g ′ ( r ′ ui ) = t . The theorem follows from the lemma: since F is a terminal set by the h ypothesis, the same holds for its retract G , sho wing that r ′ ui is cartesian. Proo f. If f ′ ∈ F then f ′ i ′ ∈ G : f ′ i ′ ( r ′ ui ) = f ′ e ′ ui = f ′ uei = f ′ ui = tr i = t If g ′ ∈ G then g ′ r ′ ∈ F : g ′ r ′ u = g ′ r ′ e ′ u = g ′ r ′ ue = g ′ ( r ′ ui ) r = tr 2.4. Coro llar y. Given a bim o dule t : X → Y , if X and Y ar e Cauchy c omplete the same holds for the ﬁxe d c ate gories X t and Y t . In Section 8, w e will treat bimodules X op → Y , that is bifunctor s X × Y → Set . W e lea v e to the reader the simple task of rephrasing t he ab ov e results to ﬁt this situation. 6 3. F actorizat i on systems W e assume that the reader is f amiliar with the basic facts ab out orthogo nalit y a nd factor- ization systems. W e b egin b y presen ting some pro p erties that will b e useful in the sequel and conclude b y recalling the biﬁbration asso ciated to a factorization system on a ﬁnitely complete category . 3.1. Proposition. If L ⊣ R : C → C ′ is an adjunction, Lf ⊥ g iﬀ f ⊥ Rg . Proo f. By duality , it is suﬃcien t to pro v e o ne direction; suppose tha t Lf ⊥ g and tha t the righ t hand square b elow comm utes. LA ′ h ∗ / / Lf   A g   LB ′ k ∗ / / u > > B A ′ h / / f   RA Rg   B ′ k / / u ∗ = = RB Then, b y the naturality of the tra nsp o se bijections, the left square commutes as well, giving a unique diagonal u ; it s transp ose u ∗ , again by naturality , is easily c hec k ed to be the desired unique diago nal. 3.2. Proposition. L e t ( E , M ) b e a factorization system on a c ate gory C . T h e fol lowing ar e e quivalent for a map e : P → X : 1. ther e ex i s ts n : X → Y in M such that any squar e n ◦ e = m ◦ l , with m ∈ M , has a unique diagonal; 2. for any triangle e = m ◦ l , with m ∈ M , ther e is a unique se ction of m extendin g l ; 3. e ∈ E . Proo f. The ab o v e conditions sa y , resp ective ly , that the squares b elow (with m ∈ M and the map n ∈ M in the ﬁrst one b eing ﬁxed) ha v e a unique diagonal: P l / / e   A m   X n / / u ? ? Y P l / / e   A m   X u > > X P l / / e   A m   X h / / u ? ? Y 7 (1) ⇒ (2). T o ﬁnd the unique section u , w e comp o se out with n ﬁnding an unique u suc h that u ◦ e = l and n ◦ m ◦ u = n : P e / / l & & e @ @ @ @ @ @ @ @ @ @ @ n ◦ e $ $ X id   u / / N m ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ n ◦ m z z X n   Y It remains to sho w that m ◦ u = id X ; this follo ws from t he unicit y o f the diagonal m ◦ s in the square b elo w, since m ◦ u ◦ e = m ◦ l = e : P e / / e   X n   X n / / mu > > Y (2) ⇒ (3). Consider the factorizatio n e = me ′ and the uniquely induced diagonal on the left: P e ′ / / e   A m   X u > > X P e ′ / / e ′   A m   A m / / um ? ? X The square on the right sho ws that also um = id, so that m is an isomorphism. T rivially (3) ⇒ (1), and the pro of is complete. 3.3. Coro llar y. An ( E , M ) -factorization of a ma p f : P → Y in C P e / / f @ @ @ @ @ @ @ @ @ @ @ X m   Y gives b oth a r eﬂe ction of f ∈ C / Y in M / Y (with e as r eﬂe ction map ) and a c or eﬂe ction of f ∈ P \C in P \E (with m as c or eﬂe c tion map). Conversely, a ny such (c o)r eﬂe ction map gives an ( E , M ) -factorization. Proo f. One direction is straigh tforward. F or the con v erse, note tha t t o sa y that e : ( P , ne ) → ( X , n ) is a reﬂection of ne in M / Y is exactly condition (1) of Prop osition 3.2. The rest f ollo ws b y dualit y . 8 3.4. Proposition. L et M b e a pul lb ack-stable class of map s in a ﬁnitely c omplete c ate gory C . T h e fol lowing ar e e quivalent for a map e : P → X : 1. for any triangle e = m ◦ l , with m ∈ M , ther e is a unique se ction of m extendin g l ; 2. e ⊥ m , for any m ∈ M ; 3. the pul lb ack functor e ∗ : C / X → C / P gives a bi j e ction C / X (1 X , m ) ∼ = C / P (1 P , e ∗ m ) , for an y m ∈ M / X , b etwe en se ctions of m and s e ctions of e ∗ m . Proo f. (1) ⇐ ⇒ (2). One implication is trivial. F o r the o t her one, recall the adjunction h ! ⊣ h ∗ : C / Y → C / X and, giv en g : B → Y , denote the corresp onding map to the terminal in C / Y b y ˆ g : ( B , g ) → ( Y , id Y ). Squares in C with edges f , g and h and their diagonals corresp ond to squares in C / Y with edges h ! ˆ f , ˆ g and ˆ h and their diagonals: A l / / f   B g   X h / / u ? ? Y ( A, hf ) g ! ˆ l / / h ! ˆ f   ( B , g ) ˆ g   ( X , h ) ˆ h / / u : : ( Y , id Y ) so that f ⊥ g in C iﬀ h ! ˆ f ⊥ ˆ g in C / Y for any h : X → Y . By Prop osition 3.1, h ! ˆ f ⊥ ˆ g in C / Y iﬀ ˆ f ⊥ h ∗ ˆ g (= d h ∗ g ) in C / X , that is iﬀ an y square as the righ t hand b elow has a unique diagonal: ( A, f ) / / ˆ f   ( h ∗ B , h ∗ g ) h ∗ ˆ g   ( X , id X ) u 9 9 ( X , id X ) A / / f   h ∗ B h ∗ g   X u = = X Since M is pullback -stable, by the hy p othesis the last condition holds for f = e and g = m (for an y m ∈ M ) so that e ⊥ m , for an y m ∈ M . (1) ⇐ ⇒ (3). Again b y the adjunction e ! ⊣ e ∗ : C / X → C /P , condition (3) sa ys that there is a bijection C / X ( 1 X , m ) ∼ = C /P ( e, m ), whic h is easily seen to corresp ond t o the one of condition (1). 3.5. Coro llar y. If C is ﬁnitely c omplete, Pr op osition 3.2 holds true for pr efactor- ization systems as wel l. Proo f. The implication (2) ⇒ (3) follo ws from Prop osition 3.4 abov e. 3.6. Coro llar y. L e t M b e a pul lb a ck-stable class of maps in a ﬁnitely c omplete c ate gory C and f : X → Y a map in C . If f ∗ : M / Y → M / X is ful ly faithful, then f ⊥ m , for any m ∈ M . 9 3.7. Coro llar y. L et ( E , M ) b e factorization system on a ﬁ nitely c omplete c ate go ry. A map f : X → Y is in E iﬀ the f unc tor f ∗ : M / Y → M / X giv es a bije ction b etwe en the h om-sets M / Y (1 Y , m ) ∼ = M / X ( 1 X , f ∗ m ) , for any m ∈ M / X . In p articular, if f ∗ : M / Y → M / X is ful ly faithful, then f ∈ E . 3.8. The bifibra tion associa ted to ( E , M ) . Let ( E , M ) a factorization system on a ﬁnitely complete categor y . By restricting the co domain ﬁbration to the arrow s in M w e get a subﬁbration M → → C whic h is a biﬁbratio n: the cartesian arrow s ar e the pullbac k squares again a nd the op cartesian ar r ows a re the squares with the top ro w in E : A ′ m ′   + + + + + + + + + + + + + + + + + + + + ) ) T T T T T T T T T T T T T T T T T T T T T A / / m   B n   X f / / Y B ′ n ′                       A e / / m   5 5 j j j j j j j j j j j j j j j j j j j j B n   > > X f / / Y (Note that the co domain biﬁbratio n C → → C itself can b e thought of as associated to the factorization system (Iso C , Ar C ).) F rom the indexed p o in t of view, w e thus ha v e the family M / X , X ∈ C , of full sub cat- egories i X : M / X ֒ → C / X , and adjunctions ∃ f ⊣ f ∗ : M / Y → M / X for an y arro w f : X → Y in C . (No confusion should arise from using the same sym b ol for b oth t he pullback functor f ∗ : C / Y → C / X and its “ r estriction” M / Y → M / X .) By Remark 2.1, f ∗ : M / Y → M / X is fully fa ithful iﬀ an y cartesian arro w o v er f is op cartesian as well that is, iﬀ pulling back f along maps in M one gets maps in E . Con v ersely , ∃ f : M / X → M / Y is fully fa it hful iﬀ squares with the top row in E are pullbac ks. W e also recall that for p : P → X in C / X , ↓ X p := ∃ p 1 P is a reﬂection of p in M / X : M / X ( ∃ p 1 P , m ) ∼ = M / P ( 1 P , p ∗ m ) ∼ = C / P (1 P , p ∗ m ) ∼ = C / X ( p ! 1 P , m ) = C / X ( p, m ) W e th us ha v e the adjunction ↓ X ⊣ i X : M / X → C / X in whic h the reﬂection map (unit) p → ↓ X p is giv en b y the following op car t esian arrow with domain 1 P : P e p / / A ↓ X p   P p / / X 10 So, it pro jects in C to the ﬁrst factor e p of a n ( E , M )-fa cto r ization of p , while ↓ X p is its second factor (see also Corollary 3.3). Note also tha t the biﬁbration M / X restricts to a “slices” sub opﬁbratio n, for med b y those ob jects in M → whic h admit an op cartesian p oint. (In Cat , thes e are the slice pro jections, so that we obtain the opﬁbrat io n corresp onding to the “identit y” indexed category; see Section 5.) 4. Balanced fa ct oriza tion catego r ies 4.1. Definition. A balanced factorization category (bfc) is a ﬁnitely c omplete c ate gory C with two factorization s ystems ( E , M ) and ( E ′ , M ′ ) satisfying the recipro cal stabilit y la w (rsl): t he pullbac k of a map in E (resp. E ′ ) a long a map in M ′ (resp. M ) is itself in E (resp. E ′ ). (In [Pisani, 2008], these w ere called “w eak” bfc). If furthermore M / 1 = M ′ / 1, we sa y that C is a strong bfc. 4.2. Remark. An y slice C / X of a bfc is itself a bfc, with the classes M X , M ′ X , E X and E ′ X of the maps in C / X mapp ed to M , M ′ , E and E ′ b y the pro jection C / X → X ; it is strong iﬀ X is a “group oidal ob ject” that is, if M / X = M ′ / X . T ypical istances are, for a category X , the slice C at / X (see Section 5 ) and, for a p o set X , the p oset P X of the parts of X with the lo w er-sets (resp. upp er-sets) inclusions as M (resp. M ′ ). Bot h of them are strong if X is a group oid. If ( E , M ) is a factorizatio n system on a ﬁnitely complete C satisfying the F rob enius la w that is, maps in E are pullbac k-stable along maps in M , then w e obtain a “ symmetrical” bfc by p osing E ′ = E and M ′ = M ; all its ob jects are group oidal and all its slices are symmetrical again. An exemple of symmetrical bfc is the category of group oids, with M the class of cov ering maps. (Other istances of bfc’s are presen ted in [Pisani, 2008].) W e now draw some consequences of the ab ov e axioms whic h will b e used in the sequel. Throughout this sec tion, w e assum e that C is a (w eak) bfc. 4.3. Proposition. Pul ling b ack an ( E , M ) -factorization f = m ◦ e along a map n ∈ M ′ in C o ne ge ts an ( E , M ) -factorization n ∗ f = ( m ∗ n ) ∗ e ◦ n ∗ m . Proo f. Consider the pullback squares b elo w. Since n ∗ m ∈ M and m ∗ n ∈ M ′ , the result follo ws b y applying the rsl to the left one: A e ′ / /   B m ∗ n   n ∗ m / / C n   X e / / Y m / / Z 11 4.4. Proposition. If K ∈ M ′ / 1 and e ∈ E then the map e × K is also in E . Proo f. Considering the pullbac k squares b elow, the pro jection p is in M ′ and so b y the rsl e × K ∈ E : X × K e × K / /   Y × K p   / / K   X e / / Y / / 1 4.5. Remark. Of course, any prop ert y in a bfc (suc h as the ab o v e o nes) has a “ dua l” prop ert y , obtained b y exc hanging M with M ′ and E with E ′ . 4.6. Proposition. [The exponential law ] If m ∈ M / X , n ∈ M ′ / X and m n exists in C / X , then it is in M / X ; dual ly, n m ∈ M ′ / X . Proo f. By Remark 4.2, w e can assume X = 1: if S ∈ M / 1, K ∈ M ′ / 1 and the expo- nen tial S K exists in C , then it is in M / 1 t ha t is, e ⊥ S K for any e ∈ E ; by Prop osition 3.1, this amoun t to e × K ⊥ S for an y e ∈ E , whic h follo ws from Prop o sition 4.4. 4.7. Proposition. Supp ose that, in the cub e b elo w, the b ottom, the left and the right fac es ar e pul lb ack s. I f e ∈ E and n ∈ M ′ then e ′ ∈ E . C   e ′ / / ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ D   ~ ~ } } } } } } } } A   e / / B   Z / / ~ ~ ~ ~ ~ ~ ~ ~ ~ W n ~ ~ } } } } } } } } X f / / Y Proo f. Apply the rsl to the top face, whic h is a pullbac k as w ell. 4.8. Remark. In [La wv ere, 1970] it was remarked that the Bec k and F ro b enius conditions do not hold in the eed Set X op , X ∈ Cat (see Section 5 b elo w); the ab o v e prop osition sa ys that the Bec k condition do es hold when restricted to pullback squares, in the base category C , whose righ t edge is in M ′ (and con v ersely for Set X ). Th us, w e can say that the biﬁbrations associated to a bfc (see Section 3) satisfy the “mixed” Bec k la w. 12 5. Cat a s a stro ng bala nced fa ct oriza tion catego r y Balanced category theory is an abstraction of categor y theory based on an axiomatization of Cat . It mainly aims to oﬀer a simple but remark ably p o w erful conceptual frame in whic h sev eral categorical concepts and prop erties b ecome quite transparent. How ev er, it also shows that category t heory can b e dev elop ed, for instance, relativ ely to a group oid X that is, in Cat / X , where the category S of internal sets (see Section 7 ) is the b o olean top os Set X op ∼ = Set X of t he cov erings of X , or in the category P os of p osets ( see [Pisani, 2008]) where S = 2. The a bstractio n is ba sed on the fact that Cat is a strong bfc with the classes M of discrete ﬁbrations and E of ﬁnal functors o n one side and the classes M ′ of discrete opﬁbrations and E ′ of initial functors on t he other side. Recall that p : P → X is ﬁnal (resp. initial) iﬀ π 0 ( x \ p ) = 1 (resp. π 0 ( p/x ) = 1) for any x ∈ X . Among ﬁnal (resp. initial) f unctors there are the right (left) adjoin t ones, since in this case x \ p (resp. p/x ) has an initial (resp. terminal) ob ject. 5.1. Remark. W e note that ﬁnal (resp. initial) functors arise as those whic h are “ A - asph ´ erique” in the sense of [Maltsiniotis, 2005], where A is the “structures d’asph´ e ricit´ e ` a gauche ” (resp. “droite”) giv en b y the connected categories; this fact (whic h, somewhat surprisingly , is not men tioned there) yields sev era l prop erties of ﬁnal a nd initial functors. The indexed category M / X ≃ Set X op , X ∈ Cat , was axiomatized (among other things) in t he late sixties b y Lawv ere as an instance of elemen tary existen tial do ctrine (eed) satisfying the “comprehension sc heme”. So, for example, left Ka n extensions app ear as existen t ia l quan tiﬁcations left adjoint to substitutions: ∃ f ⊣ f ∗ : Set Y op → Set X op . That the biﬁbratio n corresp onding t o this eed is asso ciated t o a factorization system w as sho wn in [Street & W alters, 1973]: 5.2. The compre hensive f actoriza t ion systems. One easily v eriﬁes that ( E , M ) and ( E ′ , M ′ ) are the prefactorization sys tems generated, resp ectiv ely , by the co domain and the doma in functors t, s : 1 → 2 of the arro w. After Section 3, to see that these are in fact fa cto r izat io n systems it is enough to c hec k that M / X is r eﬂectiv e in Cat / X , whic h follo ws b y a simple generalization of the Y o neda lemma. One also easily c hec ks that the recipro cal stabilit y la w holds (see [Pisani, 2008]): 5.3. Proposition. Cat , with the c om pr e hensive factorization s ystems, is a str ong b alanc e d factorization c ate gory in wh ich M / 1 = M ′ / 1 ≃ Set . 5.4. Remark. By Remark 5.1, initial (ﬁnal) functors are in fact stable with respect to pullback s along an y (op)ﬁbration (not only t he discrete ones); indeed, the latter are smo oth functors for any a sphericit y structure [Maltsiniotis, 2005]. Th us, one of the f ea- tures that distinguishes C at among o ther (strong) bfc’s is the fact tha t ﬁnal or initial maps are stable with resp ect to pullbac ks along a n y pro jection: if e : X → Y is in E then also e × K : X × K → Y × K is in E , for any K ∈ Cat (and not o nly f or K ∈ M ′ / 1, as in Prop osition 4.4). 13 The fo llo wing pro p osition is an example of an eﬀectiv e use of the recipro cal stabil- it y la w; it giv es c haracterizations of absolutely dense (or “connected”) functors, follow- ing [Ad´ amek et al., 2001]: 5.5. Proposition. L e t f : X → Y b e a functor and let [ α ] → Y b e the i n terval c ate gory of fa ctorizations of the arr ow α in Y with its pr oje ction. The fol lowing ar e e quivalent: 1. f ∗ [ α ] is c onne cte d, for any α i n Y ; 2. f is lo c al ly ﬁnal: in the pul lb ack squar e b el o w e i s ﬁnal, for any y ∈ Y ; f /y e / /   Y /y   X f / / Y (1) 3. f is lo c al ly initial: y \ f → y \ Y is in E ′ for an y y ∈ Y ; 4. f ∗ : M / Y → M / X is ful l and faithful; 5. f ∗ : M ′ / Y → M ′ / X is ful l and faithful. Proo f. First note that [ α ] → Y is the comp osite of a coslice and a slice pro jection: [ α ] ∼ = α \ ( Y /y ) / / Y /y / / Y Th us, in the pullbac k diag ram b elo w n ∈ M ′ and, if e ∈ E a lso e ′ ∈ E b y the recipro cal stabilit y la w: f ∗ [ α ] e ′ / /   " " E E E E E E E E [ α ]   n ! ! B B B B B B B B f /y e / / { { w w w w w w w w w Y /y } } z z z z z z z z X f / / Y Since π 0 [ α ] = 1, also π 0 f ∗ [ α ] (obtained by fa ctorizing f ∗ [ α ] → 1 = f ∗ [ α ] → [ α ] → 1 according to ( E , M )) is 1. Con v ersely , t o sa y that f ∗ [ α ] is connected for an y α is to say that e ∈ E , by deﬁnition. Since condition (1) is self-dual, the equiv alence of the ﬁrst three conditions is pro v ed. Recalling the adjunction ∃ f ⊣ f ∗ : M / Y → M / X of Section 3 (where ∃ f m is ob- tained b y factorizing f m according to ( E , M ), generalizing π 0 : C at → Set ≃ M / 1) , Diagram (1) show s that lo cal ﬁnalit y of f is equiv alen t to the fact that the counit ∃ f f ∗ Y /y → Y /y is an isomorphism, for an y y ∈ Y . By t he prop erties of the “Y oneda” inclusion of slices of Y into M / Y , it is also equiv a lent to the fact that the an y counit ∃ f f ∗ m → m is an isomorphism, that is f ∗ : M / Y → M / X is full and faithful. 14 5.6. Remark. After r eading Section 6 b elow (see in particular Prop osition 6.5), it will b e clear that D iagram (1) can b e in terpreted as exhibiting y as an absolute colimit of f /y → Y , th us explaining the term “absolute densit y”; see also [Pisani, 2008], w ere it is sho wn that part of the ab ov e prop osition holds true in an y strong bfc. 5.7. Remark. If f : X → Y is also full and faithful, then f ∗ : M / Y → M / X is an equiv alence. Indeed, in this case the adjoint bimo dules correspo nding to f a r e an equiv alence in the bicategory Bim of bimo dules, whic h induces an equiv alence b et w een Bim (1 , X ) ≃ Set X and B im (1 , Y ) ≃ Set Y . Alternativ ely , recall that the left Kan extension along a fully f a ithful functor is indeed an extension; thus a functor f : X → Y is fully f a ithful iﬀ the unit m → f ∗ ∃ f m is a n isomorphism, for an y m ∈ M / X , iﬀ ∃ f : M / X → M / Y is f ully faithful. 5.8. Coro llar y. Any absolutely dense map is b oth initial and ﬁnal. Proo f. This follo ws from Corollary 3 .7 . Alternativ ely , note that if f is lo cally ﬁnal, since π 0 ( Y /y ) = 1 also π 0 ( f /y ) = 1 (see Diagram (1)) , tha t is f is initial. F or example, the insertion of a category in its g roup oidal reﬂection is b oth initial and ﬁnal. 6. Slices an d co limi t s in a bfc Throughout this section, w e assume that C is a bfc. W e adopt the general p olicy of denoting t he v a r ious concepts in C as t he corresp onding ones in Cat . Thus , fo r instance, the maps in M and E a re called discrete ﬁbratio ns and ﬁnal maps resp ectiv ely , and so on. As in in ternal category theory , there ar e t w o a sp ects o f balanced category theory . On the one ha nd, the ob jects and arrows of C are (generalized) categories and functor, and we can consider concepts suc h as limits or colimits o f maps f : X → Y and adjunctible maps. As show n in [Pisani, 200 8] and as w e partly recall b elow, familiar prop erties (suc h as t he preserv ation of limits by adjunctible maps) can b e prov ed therein in a more transparent w a y . F or these a sp ects, the recipro cal stability la w pla y no real role, so that w e could in fact conside r this a s ( E , M )-categor y theory (see [Pisani, 2007b]). On the other hand (a nd more in terestingly) there are “internal” asp ects, in whic h ob jects in M / X o r M ′ / X are considered as (con trav a r ian t or co v arian t) internal-set-v alued functors. In the next section, we sho w ho w the rsl is what make s the in ternal theory to w ork. (Some “ in ternal” asp ects, how ev er, suc h as the Y oneda Lemma b elow , dep end o nly on the factorizat io n syste ms axioms.) 15 6.1. Slices. By f actorizing an “ob ject” (p oin t) x : 1 → X according to ( E , M ) and ( E ′ , M ′ ), w e obtain the slice and the coslice pro jection resp ectiv ely of X at x : 1 e x / / x A A A A A A A A A A A A X/x ↓ x   X 1 e ′ x / / x B B B B B B B B B B B B x \ X ↑ x   X So, as remarke d in Section 3, w e ha v e ↓ X x = ∃ x 1 1 . One of the consequen t unive rsal prop erties is usually known (in Cat ) as the Y oneda Lemma: 1 e x / / x A A A A A A A A A A A A a & & X/x   u / / A m } } | | | | | | | | | | | | X On the other hand, the slice pro jection X/x → X is also the “biggest” (that is, ﬁnal) ob ject ov er X with a ﬁnal point ov er x (see Corollary 3.3). 6.2. Cones and colimits. Giv en a map p : P → X a nd a p oin t x of X , a cone γ : p → x (resp. γ : x → p ) is a map in C / X from p to the slice pro jection ↓ X x (resp. coslice pro jection ↑ X x ): P γ / / p ! ! B B B B B B B B B B B B X/x   X P γ / / p ! ! B B B B B B B B B B B B x \ X   X A cone λ : p → x (resp. λ : x → p ) is colimiting ( r esp. limiting ) if it is univers al among cones with domain p : P λ / / p ! ! B B B B B B B B B B B B γ ( ( X/x   u / / X/y | | y y y y y y y y y y y y X That is, a colimiting cone giv es a reﬂection of p ∈ C / X in the full sub catego ry X generated b y the slice pro jections of X . The follow ing prop erty is often tak en a s a deﬁnition of ﬁnal functors in Cat . (The con v erse holds in any C with “p o w er o b jects”; see [Pisani, 2007b].) 6.3. Proposition. Pr e c omp o sing with ma ps in E do es not aﬀe ct c olimits. 16 Proo f. If e : Q → P is in E , then factorizing p : P → X and pe : Q → X w e get isomorphic f actors in M ; th us, p and pe ∈ C / X ha v e the same reﬂection in M / X and so also in X (if they exist). The ab ov e result can obviously b e “ dualized” for limits; more inte restingly , w e will sho w in Section 7 ho w the recipro cal stabilit y la w allows us to internalize it (see Prop o - sition 7.4). 6.4. Remark. With resp ect to the classical treatment o f ( co) limits, the pr esen t approac h has sev eral adv an tages also in the case C = Cat : considering the colimit f unctor on X as a (partia l) reﬂection Cat / X → X makes the pro ofs of the fo llowing prop erties quite straigh tforw ard (see also [P ar ´ e, 1973]). 1. The colimit x of 1 X , if it exists, is terminal in X (since the reﬂection λ : 1 X → X / x is then an iso); by Prop osition 6.3, the same is true f or any ﬁnal functor e : P → X . 2. The colimit of the empt y functor 0 → X is an initial o b ject; if p : P → X and q : Q → X ha v e colimits x p and x q , the colimit of [ p, q ] : P + Q → X is x p + x q . 3. If P is connected (so that P → 1 is in E ) and p : P → X is constan t through x : 1 → X , then b y Prop o sition 6 .3 x is the colimit of p ; similarly , if p is lo cally constan t (that is, factors through π 0 P ) then it s colimit is the copro duct of the corresp onding family . Giv en a cone p → x o v er X a nd a map f : X → Y , w e get a cone f p → f x b y comp osing with an o p cartesian arro w o v er f (whose co do ma in is a slice pro jection again since it has, by comp osition, an op cartesian p oint as w ell): P γ / / p ! ! B B B B B B B B B B B B X/x   e / / Y /f x   X f / / Y Th us, w e sa y tha t f preserves colimits if it ta kes colimit cones γ : p → x to colimiting cones eγ : f p → f x . 6.5. Proposition. [Absolute colimits; see also [P ar ´ e, 1973]] I f a c one γ : p → x is in E , then it is c olim iting and is pr eserve d by any map. 7. In ternal a sp ects of balanced categ ory theor y Throughout this section, w e assume that C is a str ong bfc. F ollowing Section 3, the bfc C gives rise to t w o subﬁbrations of the co domain biﬁbration whic h are themselv es biﬁbrations. F rom an indexed (or eed) p oint of view w e th us ha v e adjunctions f ! ⊣ f ∗ : C / Y → C / X 17 ∃ f ⊣ f ∗ : M / Y → M / X ; ∃ ′ f ⊣ f ∗ : M ′ / Y → M ′ / X for an y f : X → Y in C . (No confusion should arise from using the same sym b ol f ∗ for three diﬀeren t functors, since all of them are obtained by pulling ba c k.) As clearly explained in [Lawv ere, 1992], w e th us hav e v arying “ quantities” with b oth extensiv e and in tensiv e asp ects. Within the “gros” categories C / X , there are the “p etit” ones of left and righ t “discrete quan tities”: i X : M / X ֒ → C / X and i ′ X : M ′ / X ֒ → C / X . (F or the “top ological” (w eak) bfc T o f Section 13, it w ould be of course more appropriate to sp eak of “discrete” and “compact” quan tities or spaces o v er X .) 7.1. Extensive aspects of discrete quantities. Since w e a r e now w orking in a strong bfc, the constan t left and righ t discrete quan tities coincide: S := M / 1 = M ′ / 1; w e refer to them as in ternal sets . Thus w e ha v e a comp onents functor π 0 ⊣ i : S → C where π 0 X is the total (in the sense of ∃ or ∃ ′ ) of the bidiscrete quantit y 1 X , whic h can b e obtained b y factorizing ! X : X → 1 according to ( E , M ) o r ( E ′ , M ′ ): π 0 X := ∃ ! X 1 X = ∃ ′ ! X 1 X More generally , w e ha v e left adjoin ts π X 0 ⊣ (! X ) ∗ ◦ i whic h can b e obtained as π X 0 := π 0 ◦ (! X ) ! : C / X → S that is, if p : P → X , π X 0 p = π 0 P : P e / / p   π 0 P   X ! X / / 1 Note that the total ∃ ! X m of a left discrete quantit y m : A → X o v er X can b e obtained as π 0 A = π X 0 m (more precis ely , π X 0 i X m , where i X : M / X ֒ → C / X ). Similarly , ∃ ′ ! X n = π X 0 n . 7.2. Internal-set -v alued maps . P assing no w to in tensiv e (that is, con tra v arian t) asp ects, f o r an y p oin t x : 1 → X we get the (in ternal) set x ∗ m by ev aluating a left or righ t discrete quan tity m ov er X . F urthermore, internal sets are included as constan tly v ar ying quan tities ov er X b y ! ∗ X : S → M / X (and, of course, ev aluating ! ∗ X S at any x returns S itself ). Th us a discrete ﬁbratio n (or o pﬁbration) m ∈ M / X in C can b e considered as an “in ternal-set-v alued” map. In this p ersp ectiv e the functor f ∗ : M / Y → M / X can b e seen 18 as precomp o sition of in ternal-set-v alued ma ps o v er Y with f : X → Y , as is eviden t from the pullbac k squares b elo w: x ∗ f ∗ m = ( f x ) ∗ m / /   A f ∗ m   / / B m   1 x / / X f / / Y In Cat , ev aluation of m at x giv es of course the v alue at x of the presheaf asso ciated to m . On the o ther hand, by the adjunction ∃ f ⊣ f ∗ : M / Y → M / X , the op cartesian arro w (in M → → C ) A e / / m   B ∃ f m   X f / / Y corresp onds to the left Kan extension along f . In particular, the tota l of m : A → X in M / X , t he in ternal set π 0 A , corresp onds to the (internal) colimit of m . (Classically , one sa ys that the colimit of a presheaf is giv en b y the comp onen ts o f its catego ry of elemen ts). Of course, similar considerations hold for discrete opﬁbrations. Th us, a s noted ab o v e, the functor π X 0 : C / X → S restricts to giv e, for discrete ﬁbrations or opﬁbrations, the internal- colimit functors M / X → S and M ′ / X → S : M / X i X   ∃ ! X ' ' N N N N N N N N N N N N N N N N N N C / X π X 0 / / S M ′ / X i ′ X O O ∃ ′ ! X 7 7 p p p p p p p p p p p p p p p p p p 7.3. The role of the re ciprocal st ability la w. Consider the pullbac k square b elo w with n ∈ M ′ , A f ′ / / f ∗ n   B n   X f / / Y 19 Note that f ′ , as a map ov er Y , is the counit f ′ = ε n : f ! f ∗ n → n of the a djunction f ! ⊣ f ∗ : C / Y → C / X . Th us b y applying π Y 0 : C / Y → S we get a natural transformation π Y 0 ε : π Y 0 f ! f ∗ = π X 0 f ∗ → π Y 0 : M ′ / Y → S (which corresp onds “externally” to the canonical X/x q u     Q f / / q ' ' P P P P P P P P P P P P P P P P P P P P P P 0 0 P p ! ! C C C C C C C C C C C C / / X/x p   X where x p and x q are colimits of p and q = pf resp ectiv ely). Now, if f ∈ E then also f ′ ∈ E b y the recipro cal stabilit y law , and since π 0 : C → S takes maps in E to isomorphisms, π Y 0 ε is in fact a nat ur a l isomorphism: 7.4. Proposition. Pr e c omp osition with a ﬁnal map p r eserves internal c olimits o f discr ete opﬁbr ations; that is, for any e : X → Y in E , ther e ar e isomorphisms π X 0 e ∗ n ∼ = π Y 0 n natur al in n ∈ M ′ / Y . In part icular, the (in ternal) v alue of n ∈ M ′ / X at a “ ﬁnal p oin t” e : 1 → X in E g ives the (in ternal) colimit of n . No w w e apply a similar pro cedure to obtain other “coherence” results, supp ort ed by the recipro cal stability law, that will b e used in the next section. Considering the diagrams of Section 4: A e ′ / /   B m ∗ n   n ∗ m / / C n   X e / / p 7 7 Y m / / Z C   e ′ / / ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ D   ~ ~ } } } } } } } } A m   e / / B ∃ f m   Z n ∗ f / / f ∗ n ~ ~ ~ ~ ~ ~ ~ W n ~ ~ } } } } } } } } X f / / Y (2) w e see t ha t the ﬁnal maps e ′ are r esp ectiv ely the components ε n,p : n ! n ∗ p → n ! n ∗ ↓ p ; ε n,m : f ! ( f ∗ n ) ! ( f ∗ n ) ∗ m → n ! n ∗ ∃ f m of natural transformations b et w een functors C / Z × M ′ / Z → C / Z ; M / X × M ′ / Y → M / Y 20 By applying π Z 0 and π Y 0 w e get natural transformations π Z 0 ε n,p : π Z 0 ( n ! n ∗ p ) = π Z 0 ( n × Z p ) → π Z 0 ( n ! n ∗ ↓ p ) = π Z 0 ( n × Z ↓ p ) π Y 0 ε n,m : π Y 0 ( f ! ( f ∗ n ) ! ( f ∗ n ) ∗ m ) = π X 0 ( f ∗ n × X m ) → π Y 0 ( n ! n ∗ ∃ f m ) = π Y 0 ( n × Y ∃ f m ) where w e hav e used the fa ct that, if p : P → X a nd f : X → Y , π X 0 p = π Y 0 f ! p and, if also q : Q → X , then π X 0 p ! p ∗ q = π X 0 q ! q ∗ p = π X 0 ( p × X q ): R q ∗ p / / p ∗ q   p × X q ? ? ? ? ? ? ?   ? ? ? ? ? ? ? Q q   P p / / X Since the e ′ are in E and π 0 tak es ﬁnal maps to isomorphisms, we get: 7.5. Proposition. F or any X ∈ C , ther e ar e isomorphisms π X 0 ( n × X p ) → π X 0 ( n × X ↓ p ) natur al in n ∈ M ′ / X and p ∈ C / X . F or any f : X → Y ther e ar e isomo rphisms π X 0 ( f ∗ n × X m ) → π Y 0 ( n × Y ∃ f m ) natur al in m ∈ M / X and n ∈ M ′ / Y . 8. The tensor funct o r a nd the internal hom Throughout the section, C is assumed to b e a str ong bfc. F or an y X ∈ C w e deﬁne the functor ¯ ⊗ X := π X 0 ◦ × X : C / X × C / X → S Th us, if p : P → X and q : Q → X , we hav e p ¯ ⊗ X q = π X 0 ( p × X q ) = π 0 ( P × X Q ) By restricting ¯ ⊗ X to M ′ / X × M / X , w e obtain the tensor functor ⊗ X := ¯ ⊗ X ◦ ( i ′ X × i X ) : M ′ / X × M / X → S By prop ositions 7.4 and 7.5, it immediately follo ws: 21 8.1. Proposition. F or any X ∈ C , ther e ar e isomorphisms n ¯ ⊗ X p ∼ = n ⊗ X ↓ p (3) natur al in n ∈ M ′ / X and p ∈ C / X . F or any f : X → Y ther e ar e isomo rphisms f ∗ n ⊗ X m ∼ = n ⊗ Y ∃ f m (4) natur al in m ∈ M / X and n ∈ M ′ / Y . F urthermor e, if e : X → Y is a ﬁnal map, ther e ar e isomorphisms e ∗ n ⊗ X 1 X ∼ = n ⊗ Y 1 Y (5) natur al in n ∈ M ′ / Y . 8.2. Remark. In fact, b oth equations (3) and (5) fo llow from Equation (4). Indeed, since the F rob enius law f ! ( p × X f ∗ q ) ∼ = f ! p × Y q clearly holds for the f actorization system (Iso C , Ar C ) (that is for the codo ma in ﬁbration), w e get isomorphisms f ! p ¯ ⊗ Y q ∼ = p ¯ ⊗ X f ∗ q natural in p ∈ C / X , q ∈ C / Y , for an y f : X → Y . So, if p : P → X and n ∈ M ′ / X , n ¯ ⊗ X p ∼ = n ¯ ⊗ X p ! 1 P ∼ = p ∗ n ¯ ⊗ P 1 P ∼ = p ∗ n ⊗ P 1 P ∼ = n ⊗ X ∃ p 1 P ∼ = n ⊗ X ↓ p If e : X → Y is in E , then ↓ Y e = 1 Y so that e ∗ n ⊗ X 1 X ∼ = n ⊗ Y ∃ e 1 X ∼ = n ⊗ Y ↓ Y e ∼ = n ⊗ Y 1 Y 8.3. The coadjunction la ws and t he tens or-hom duality. W e hav e so obtained some “coadjunction” laws which, remark ably , ar e the exact counterparts of the gen uine adjunction la ws constituting the log ic of the biﬁbrations originated by ( E , M ) and ( E ′ , M ′ ). Let us emphasize this sort of dualit y: M / X ( m, f ∗ m ′ ) M / Y ( ∃ f m, m ′ ) M ′ / X ( n, f ∗ n ′ ) M ′ / Y ( ∃ f n, n ′ ) ; f ∗ n ⊗ X m n ⊗ Y ∃ f m n ⊗ X f ∗ m ∃ ′ f n ⊗ Y m (6) C / X ( p, m ) M / X ( ↓ X p, m ) C / X ( q , n ) M ′ / X ( ↑ X q , n ) ; p ¯ ⊗ X m ↑ X p ⊗ X m n ¯ ⊗ X q n ⊗ X ↓ X q (7) F urthermore, from the (6 ) ab o v e and “surjectivit y” of ﬁnal maps ( ∃ e 1 X = ↓ e = 1 Y ), it follo ws that, if e ∈ E a nd i ∈ E ′ , M / X ( 1 X , e ∗ m ) M / Y (1 Y , m ) M ′ / X (1 X , i ∗ n ) M ′ / Y (1 Y , n ) ; 1 X ⊗ X i ∗ m 1 Y ⊗ Y m e ∗ n ⊗ X 1 X n ⊗ Y 1 Y (8) 22 (Note that, by Corollary 3.7, the con v erse holds for the left ones.) No w, as explained b efore, the in ternal set 1 X ⊗ X m = π X 0 m can be seen as the colimit of the internal-set- v alued map m ∈ M / X ; “dually”, the (external) set of sec tions M / X (1 X , m ) can b e seen as the limit of m . W e will discuss in Section 8.5 b elo w under whic h h yp othesis this limit can be in ternalized to o. Of course , in Cat b oth giv e the usual limit and colimit of (the presheaf corresp o nding to) m . More generally , p ¯ ⊗ X m can b e seen a s an internal-set-v alued w a y to “test” the quan tity m , “dual” to the standard set-v alued testing by ﬁgures of shap e p ∈ C / X . As discussed at lengh t in [Pisani, 2 007a] and [Pisani, 2 0 05], the tensor functor ¯ ⊗ can b e see n as a sort of “meets” predicate, so that p ¯ ⊗ X m giv es (the in ternal set of ) the w ay s in whic h p meets m (so as C / X ( p, m ) giv es the w a ys in whic h p is contained in, or b elongs to, m ). In the strong bfc P os mentioned at the b eginning of Section 5 (that is the example at the end of [Pisani, 2008]), one has S = 2 = { true ⊢ false } and ⊗ X : M ′ / X × M / X → 2 is indeed the t w o-v alued meets predicate for upp er and lo w er subsets of the p oset X : n ⊗ X m = true iﬀ n ∩ m is non-v oid. 8.4. Remarks. 1. In this p ersp ectiv e, the in ternal colimit functor 1 X ⊗ X − = π X 0 : M / X → S b ecomes a “non-v oid” predicate (and similarly M / X (1 X , − ) : M / X → Set is a “whole” predicate). Preserv ation of in ternal colimits (of discrete opﬁbrat ions) b y precomp osition with ﬁnal maps (equations (8) a b o v e) then b ecomes the fact that the “surjectivit y” of e : X → Y ( ∃ e 1 X = 1 Y ) imply that t aking inv ers e imag es preserv es (and reﬂects) non- v oidness. Similarly , preserv ation of limits (of discrete ﬁbrations) sa ys that taking inv erse images (preserv es and) reﬂects wholeness. 2. Conv ers ely , the “ meets” and the “b elongs to” predicates can b e reduced to the “non-v oid” (colimit) and the “whole” (limit) predicates b y the (co)adjunction laws : p ¯ ⊗ X m ∼ = 1 P ⊗ P p ∗ m ; C / X ( p, m ) ∼ = M / P ( 1 P , p ∗ m ) 3. If x : 1 → X is a p oin t, x ¯ ⊗ X m ∼ = x ∗ m is the internal v alue of m at x as discussed ab ov e, while C / X ( x, m ) is the set of p o in ts of the to tal of m whic h are (in the ﬁb er) o v er x . 4. F rom equations (6) and (7) w e get the classical form ulas for the left Kan extension ∃ f m of m ∈ M / X along f : X → Y : y ∗ ∃ f m ∼ = y ¯ ⊗ Y ∃ f m ∼ = ↑ Y y ⊗ Y ∃ f m ∼ = f ∗ ↑ Y y ⊗ X m The last t erm b eing π Y 0 ( f ∗ ↑ Y y × X m ) = π X 0 ( m ∗ ( f ∗ ↑ Y y )), in Cat we get the coend or the colimit formula resp ectiv ely (see the second of diagrams (2)). 23 8.5. Internal limits and internal hom. W e hav e argued ab o v e that there is a sort of dua lity b etw ee n the hom and the tensor functors; on the other hand there is a great diﬀerence: while the latter is v alued in S , the fo rmer is v alued in Set ; t o b etter compare them we need either to unenric h ⊗ to Set or to enric h hom to S . The ﬁrst option will b e follo w ed in Section 9.2, where we consider the “ten” bimo dule, obtained b y comp osing ⊗ with the p oints functor; w e no w brieﬂy consider the other one. So as w e obta ined in ternal colimits of discrete (op)ﬁbrations by restricting the left adjoin ts π X 0 ⊣ (! X ) ∗ ◦ i : S → C / X , and then used these to deﬁne the (more general) in ternal tensor functor, w e no w need t o assume the righ t adjoin ts | − | X : C / X → S to the “constan t” inclusions, whic h restricted to (o p)ﬁbrations g iv e the internal limit (or “in ternal sections”) functors. (These functors in fact exis t in Cat where, since S = Set and ! ∗ X S = S · 1 X , they are the sections functors | p | X = C / X (1 X , p ).) Assuming furthermore that, as in Cat , discrete ﬁbrations m ∈ M / X and opﬁbratio ns n ∈ M ′ / X are exp onentiable in C / X (so tha t also n × X m , m p , etc. are exponentiable), it is natural to deﬁne the “ internal hom” (partial) functors (o ve r X ) as the exp onential fo llow ed b y in ternal sections: hom X ( p, q ) = | q p | X ∈ S While hom X ma y b e not deﬁned on the whole C / X × C / X , it is of course deﬁned when the second comp o nen t is exp onen tiable. W e denote b y hom M / X : M / X × M / X → S t he restriction of hom X . (Not e that m m ′ , with m, m ′ ∈ M / X , ma y b e not in M / X .) 8.6. Proposition. If the str ong bfc C admits internal limits | − | X : C / X → S , we have the “internal adjunctions” hom X ( p, m ) ∼ = hom M / X ( ↓ X p, m ) ; hom M / X ( m, f ∗ l ) ∼ = hom M / Y ( ∃ f m, l ) natur al in p ∈ C / X , m ∈ M / X and l ∈ M / Y ; the s a me holds of c ourse for discr ete opﬁbr ations. Proo f. S ( S, | m p | X ) ∼ = C / X (! ∗ X S, m p ) ∼ = C / X (! ∗ X S × p, m ) ∼ = M / X ( ↓ (! ∗ X S × X p ) , m ) ∼ = ∼ = M / X ( ! ∗ X S × X ↓ p, m ) ∼ = C / X (! ∗ X S, m ↓ p ) ∼ = S ( S, | m ↓ p | X ) where, in the passage fr om the ﬁrst to the second row , w e ha v e applied Prop osition 4.3 to the ( E , M )-factorization of p , since the constan t ! ∗ X S is a discrete biﬁbration. The second deduction is similar: S ( S, | ( f ∗ l ) m | X ) ∼ = C / X (! ∗ X S, ( f ∗ l ) m ) ∼ = M / X ( ! ∗ X S × m, f ∗ l ) ∼ = ∼ = M / Y ( ∃ f (! ∗ X S × X m ) , l ) ∼ = M / Y ( ∃ f ( f ∗ ! ∗ Y S × X m ) , l ) ∼ = ∼ = M / Y ((! ∗ Y S × Y ∃ f m ) , l ) ∼ = C / Y (! ∗ Y S, l ∃ f m ) ∼ = C / Y ( S, | l ∃ f m | Y ) here, in the passage from the second to t he third ro w, w e hav e applied the mixed Beck la w (see Prop osition 4.7 and the second of dia g rams (2)). 24 8.7. Coro llar y. F inal maps pr eserve internal limits of discr ete ﬁbr ations. Proo f. hom M / X (1 X , e ∗ m ) ∼ = hom M / Y ( ∃ e 1 X , m ) ∼ = hom M / Y (1 Y , m ) Th us, as for in ternal colimits, in order to coheren tly internalize limits a nd hom (“nat- ural tr ansformations”) of in ternal-set-v a lued maps w e need the rsl in an essen tial w a y . W e conclude this section by comparing the hom M / X and hom M ′ / X (and hom X ) with ⊗ X (and ¯ ⊗ X ), obtained as the horizon tal comp ositions in the diagrams b elo w: M / X i X   colim X ' ' N N N N N N N N N N N N N N N N N N h 1 X , M / X i z z M ′ / X × M / X i ′ X × i X / / C / X × C / X × X / / C / X π X 0 / / S M ′ / X i ′ X O O colim ′ X 7 7 p p p p p p p p p p p p p p p p p p h M ′ / X, 1 X i d d M / X i X   lim X & & L L L L L L L L L L L L L L L L L L L h 1 X , M / X i u u M / X × M / X i X × i X , , C / X × C / X exp X / / C / X |−| X / / S M ′ / X × M ′ / X i ′ X × i ′ X 2 2 M ′ / X i ′ X O O lim ′ X 8 8 r r r r r r r r r r r r r r r r r r r h 1 X , M ′ / X i i i 9. Retracts of sl ices 9.1. Components and the Nullste llensa tz hypothesis. As p o in ted out b y La wv ere in sev eral pap ers, for categories of cohesion C , whose ob jects are to b e though t of as spaces of some kind, t here is a basic ch ain of adjoints p ! ⊣ p ∗ ⊣ p ∗ : C → S ( with suitable prop erties), con trasting it with a category S of (relativ ely) discrete spaces. In that situatio n, he refers to the Nullstellensatz condition as the requiremen t that (assuming p ∗ fully faithful) the natura l map p ∗ X → p ! X , f rom the p oin ts functor to t he comp onen ts (or “pieces”) functor, is a n epimorphism. In our setting, we ha v e π 0 ⊣ i : S → C , but w e do not a ssume in general a further right adjoin t. Notwiths tanding, w e will use a weak form of the Nullstellensatz: if w e denote b y | | − | | := C (1 , − ) : C → Set 25 the (external) po ints functor, a nd b y [ − ] X : X → π 0 X the unit of the (in ternal) compo- nen ts reﬂection, w e require that the mapping | | [ − ] X | | : | | X | | → | | π 0 X | | is surjectiv e, for an y X ∈ C . Note that for an y eleme nt s ∈ | | π 0 X | | (that is, s : 1 → π 0 X ) of the set of compo nen ts of X , w e ha v e a “comp onent” [ s ] ֒ → X , that is the sub o b ject give n b y the f ollo wing pullbac k: [ s ]   / / X   1 s / / π 0 X (9) Note also that a ﬁgure p : P → X b elongs to (t ha t is, factors thro ug h) a comp o nen t, iﬀ the comp osite [ p ] : P → π 0 X is constan t. In particular, any ﬁgure with a connected shap e that is, with ! P ∈ E (for instance a p oint), b elongs to a comp onen t. Thus, the Nullstellensatz condition | | π 0 X | | = { [ x ] | x : 1 → X } ma y b e rephrased b y sa ying that eac h comp onent has a p o int (which b elongs to it). F urthermore, for a map f : X → Y , t he correspo nding mapping | | π 0 X | | → | | π 0 Y | | acts as [ x ] 7→ [ f x ]. 9.2. The bimodule ten. It is w ell known that the rectracts of slices (represen t able preshea v es) in M / X ha v e an imp ortant role in Cat ; for instance, they generate the Cauc h y completion of X and can b e c haracterized in sev eral w a ys. In order to dev elop a similar analysis in C , w e need to consider t he “unenric hmen t” mentioned in Section 8.5, by taking the p o in ts of ⊗ ; namely w e deﬁne the bimo dules ten X : ( M ′ / X ) op → M / X by ten X ( n, m ) := | | n ⊗ X m | | 9.3. Proposition. F or any x : 1 → X ther e is a bic artesian arr ow ↑ X x → ↓ X x for ten X . Proo f. Rec alling the notatio ns of Section 2 , w e b egin by show ing that ↑ X x ( −i ↓ X x , that is that ten X ( ↑ X x, − ) : M / X → Set is represen ted b y ↓ X x : | | ↑ X x ⊗ m | | ∼ = | | ∃ ′ x 1 1 ⊗ m | | ∼ = | | 1 1 ⊗ x ∗ m | | ∼ = | | x ∗ m | | ∼ = ∼ = S (1 , x ∗ m ) ∼ = M / X ( ∃ x 1 1 , m ) ∼ = M / X ( ↓ X x, m ) 26 No w, it is easy to see tha t t he cartesian arrow (univ ersal eleme nt) is giv en by the comp o- nen t [ h e ′ x , e x i ] of the elemen t h e ′ x , e x i : 1 → x \ X × X X/x (with e x ∈ E and e ′ x ∈ E ′ ): 1 % % J J J J J J J J J J J J J J J e x # # e ′ x & & x \ X × X X/x   / / X/x ↓ x   x \ X ↑ x / / X Th us, b y symmetry , the cartesian arro w is also op cartesian, and the pro of is complete. 9.4. Coro llar y. T he ful l sub c ate gories X ֒ → M / X and X ′ ֒ → M ′ / X , gener ate d by the slic es and the c oslic es pr oje ctions r esp e ctively, ar e d ual. W e are no w in a p osition to pro v e: 9.5. Proposition. Under the Nul lstel lensatz hyp othesis, al l the (op) c artesian arr ows of ten X ar e in fact bic artesian, and the c o n jugate ob j e cts in M / X ( M ′ / X ) ar e the r etr acts of (c o)slic es pr oje c tion s . So the ﬁxe d c ate gories for ten X ar e the Ca uchy c ompletions of X ֒ → M / X and X ′ ֒ → M ′ / X . Proo f. Observ e that, b y the h yp othesis, for n : D → X ∈ M ′ / X and m : A → X ∈ M / X , ten( n, m ) = { d ⊗ a | d : 1 → D , a : 1 → A, nd = ma } where w e p ose, as usual for Cat , d ⊗ a := [ h d, a i ]. Th us, if n h s ) m is cartesian, w e ha v e that s = v ⊗ w ∈ ten( n, m ) is ov er a n x = nv = mw ∈ X . Let i : m → ↓ x be the unique map in M / X suc h that ten X ( n, i ) : v ⊗ w 7→ v ⊗ e x , and r : ↓ x → m b e the unique map in M / X suc h that r : e x 7→ w (where ↓ x ◦ e x : 1 → X is an ( E , M )-factorizat io n of x ). Then, ten X ( n, r i ) : v ⊗ w 7→ v ⊗ w and the cartesianess of v ⊗ w implies r i = id m that is, m is a retract of ↓ x . By Prop osition 9.3 ab ov e, ↑ x h−i ↓ x and since M ′ / X is ﬁnitely complete, the idem- p oten t e ′ , conjugate to e = ir , splits as e ′ = i ′ r ′ . Th us, b y Prop osition 2.2, m ha s a conjugate in M ′ / X and the result fo llows. 9.6. A toms. Intuitiv ely , an ob ject P ∈ C is an atom if it is so small that any non- v oid op en or closed part ov er it is the whole P , and y et so big that t he whole P is itself non-v oid (see also [Pisani, 2 0 07a] and [Pisani, 200 5]). No w (see Remark 8.4 ) | | π P 0 m | | = | | 1 P ⊗ P m | | = ten P (1 P , m ) can b e seen a s the (external) truth v alue of the “non- voidness ” of m ∈ M / P , while hom P (1 P , m ) is the (external) truth v alue of its “wholeness” ( where for simplicit y w e denote by hom P the hom-functor on 27 C / P or also its restriction to M / P or M ′ / P ). Thus w e formalize the ab o v e idea b y the conditions ten P (1 P , m ) ∼ = hom P (1 P , m ) ; ten P ( n, 1 P ) ∼ = hom P (1 P , n ) (for m ∈ M / P and n ∈ M ′ / P ) whic h express the fact t ha t the (external) limit and colimit functors, for discrete ﬁbratio ns and opﬁbrations, are isomorphic. In fact, the t w o conditions are equiv alen t b ecause, for this particular case, the r esults of Prop osition 9.5 can b e summarize d in the follow ing corollary-deﬁnition: 9.7. Proposition. Under the Nul lstel lensatz hyp othesis, the fol lowing ar e e quivelent for an obje c t P of C : 1. P is an atom ; 2. 1 P h−i 1 P , for ten P ; 3. ten P (1 P , m ) ∼ = hom P (1 P , m ) , natur al ly in m ∈ M / P ; 4. ten P ( n, 1 P ) ∼ = hom P (1 P , n ) , natur al ly in n ∈ M ′ / P . A ty pical case o f an ato m is an ob ject X ∈ C with a “zero” p oint x : 1 → X in E ∩ E ′ , since in that case x \ X ∼ = X/x ∼ = 1 X . In particular, the terminal ob ject itself 1 ∈ C is an atom. F or C = Cat , the ab ov e conditions are related again to the Nullstellensatz condition, no w referred to the (colimit and limit) adjunctions p ! ⊣ p ∗ ⊣ p ∗ : Set X op ≃ M / X → Set (see [La Palme et al., 2004]). Indeed, X is an atom iﬀ p ! ∼ = p ∗ or, equiv a len tly , if the same holds for p ! , p ∗ : Set X → Set . (Note that atoms in Cat are connected since π 0 X = | | π 0 X | | = ten X (1 X , 1 X ) = hom X (1 X , 1 X ) = 1 so that the corresp onding p ∗ is fully faithful.) The most relev ant instance of a (non p oin t-like) atom in Cat is the monoid e with an unique idemp oten t non- iden tit y arro w e . Indeed, the p oints-sec tions-limit functor a nd the comp onen ts-colimit functor Set e → Set are isomorphic ( to the ﬁxed p oints of the endomapping asso ciated to e ). 9.8. Remark. Of course, in presence of internal limits | − | X : C / X → S , as discusse d in Section 8.5, one may deﬁne “in ternal atoms” that is, ob jects P ∈ C suc h tha t π P 0 m ∼ = | m | P , naturally in m ∈ M / P o r m ∈ M ′ / P . 9.9. Proposition. T he r eﬂe ction ↓ X x (r esp. ↑ X x ) of ﬁgur es x : P → X with atomic shap es ar e in the ﬁxe d c ate gory of the c onjugate obje cts in M / X (r es p. M ′ / X ). 28 Proo f. If P ∈ C is an atom and x : P → X , the adjunction and (unenric hed) coadjunction la ws giv e rise to isomorphisms ten X ( ↑ X x, m ) ∼ = ten X ( ∃ ′ x 1 P , m ) ∼ = ten P (1 P , x ∗ m ) ∼ = ∼ = hom P (1 P , x ∗ m ) ∼ = hom X ( ∃ x 1 P , m ) ∼ = hom X ( ↓ X x, m ) natural in m ∈ M / X (and similarly for n ∈ M ′ / X ). Thus the result follo ws fr o m Prop o- sition 9.5. In Cat one so gets in fact al l conjug a te preshea v es: indeed, a retract m of the r ep- resen table ↓ x can b e obtained as the reﬂection ↓ e of t he atomic ﬁgure e : e → X whic h represen ts the corresp onding idemp oten t ( m ∼ = ↓ e since b o t h of them split the same idem- p oten t in M / X ; see also [Pisani, 2007]). Another prop ert y whic h c haracterizes conjugate preshea v es in Cat is the co contin uit y of the functor Set X op → Set represen ted by it: Set X op ( ↓ e, − ) ∼ = hom X ( ↓ e, − ) ∼ = hom X ( e, − ) ∼ = ten X ( e, − ) ∼ = ten X ( ↑ e, − ) whic h giv es, f or m ∈ Set X op , the elemen ts of mx ﬁxed b y me : mx → mx . F or a general bfc C , we hav e a par tial in ternal v ersion of that fact, as a consequence of the follo wing: 9.10. Proposition. [Complemen ts] If the c onstant biﬁbr ations ! ∗ X S , with S ∈ S , ar e exp onentiable in C / X , then the functor n ⊗ X − : M / X → S has a right adjoint (! ∗ X − ) n . Proo f. First no te that, by the exp o nen tial law (Prop osition 4.6), (! ∗ X − ) n is indeed v alued in M / X . Then w e ha v e: S ( n ⊗ X m, S ) ∼ = S ( π X 0 ( n × X m ) , S ) ∼ = C / X ( n × X m, ! ∗ X S ) ∼ = M / X ( m, (! ∗ X S ) n ) As discussed at lengh t in [Pisani, 2007 a], [Pisani, 2007] and [Pisani, 2005] (see also Section 11 b elo w), this right adjoint w ell deserv es to b e called the “complemen t” of n . No w, if ↓ e is a conjugate ob ject and the p oints functor | | − | | : S → Set preserv es itself colimits, then the same holds for hom X ( ↓ e, − ) ∼ = ten X ( ↑ e, − ) ∼ = | | ↑ e ⊗ X −| | 10. Conclusion o f the ﬁr st p a rt W e hop e to ha v e show n t hat the adjunction and coadjunction la ws asso ciated to a strong balanced factorization category are a p o w erful to ol to syn t hetically treat some basic as- p ects of category t heory . In this “categorical logic of categor ies”, a straightforw ard common generalization of the “meets” predicate a nd of the (in ternal) colimit functor, namely t he tensor functor, is related to the hom functor (generalizing the “b elongs to ” predicate and the sections or limit functor) b y a useful sort of duality , whic h is disciplined by the r ecipro cal stabilit y la w. F urther suggestions can b e drawn b y comparing it with the w eak er logic associated to “top ological” w eak bfc’s, whic h is brieﬂy illustrated in the seq uel. 29 11. Univ ersal prop er t ies in to p ology It is commonly ack now ledged that t he main reason of the eﬀectiv eness of category theory is its role as a language apt to deﬁne a nd elab o r ate the univers al prop erties which p erv ade mathematics. F or instance, the univ ersal deﬁnition of pro duct giv es, in Set and Set op , the ob jectiﬁed vers ion of pro duct and sum of natural n um b ers, and the righ t adjoint to X × − giv es exp onen tials (and in general implies t he distributiv e la w). Shifting fr o m Set to P X (the slice Set /X restricted to monomorphisms), one similarly gets the b o olean algebra of the part s of X ∈ Set (with implication as exponential). Our presen t aim is to sk etc h ho w some of the univ ers al prop erties that p erv ade top o logy can b e used to organize and guide o ur top ological thinking. 11.1. Or thogonality in topology. Let us consider the concepts of connectednes s and densit y . In the category T op of top ological spaces, a space X is connected iﬀ any map to a discrete space is constan t, that is if the map X → 1 is or t hogonal to S → 1, fo r an y discrete S : X / /   S   1 / / ? ? 1 Shifting fr o m T op to P X (the slice T op /X restricted to monomorphisms), and replacing discrete spaces with closed parts, w e get density : a part P of X is dense iﬀ any map (that is, inclusion in) to a closed part D is constant (that is, it facto r s through the terminal part X ∈ P X ): P / /   D   X / / > > X In T op , lo cal homeomorphisms (resp. p erfect maps) to a space X can b e seen b o th as v ar ia ble discrete (resp. compact) spaces ov er X and as generalized (non monomorphic) op en (resp. closed) parts of X . Th us, one is led to consider lo cal homeomorphisms and p erfect maps as the basic conce pts, and t o in v estigate whic h are the g eneral coun terparts of the ab o v e orthogonalit y conditions. 11.2. F actoriza tion systems. It is kno wn t hat p erfect (that is, prop er and separated) maps are the second factor of a factorization sys tem ( E ′ , M ′ ) on T op ( which generalizes the Stone-Cec h compactiﬁcation; see e.g. [Clemen tino et al., 1996]). Since K = M ′ / 1 is the subcategory of compact (separated) spaces, E ′ / 1 includes the co discrete spaces. On the other hand, lo cal homeomorphisms are not the second factor of a factorization system ( E , M ) on T op : assuming that they are so corresp onds in tuitiv ely to a ssume b oth 30 some lo cal connectedness prop ert y on spaces, and the existence inﬁnitesimal neighbouring spaces. Indeed, ( E , M )-factorization gives reﬂections C / X → M / X and in particular π 0 : C → S , where S = M / 1 is the category of “inte rnal sets” or “discrete spaces”. In T op , only the (we akly) lo cally connected spaces (whic h are the sum of their comp onents) ha v e suc h a reﬂection. By considering instead the “ o pp osite” case of monomorphic ﬁgures P ֒ → X , one should obtain the smallest open part including the ﬁgure, that is the (“inﬁnitesimal”) neigh b orho o d of P ֒ → b P ֒ → X of P in X : P / /   O   b P / / ? ? X The same diag ram sho ws that, if O → X is any ma p in M and the ﬁgure p : P → X lifts to q : P → O , then its “neigh b orho o d” b p : b P → X lifts uniquely to give a neigh b orho od b q : b P → O of q , whic h v ery strongly resem bles a deﬁnition of lo cal ho meomorphism! 11.3. The reciprocal st ability la w. There are sev eral evidences of the fact that the recipro cal stabilit y laws should hold in an appropriate category T o f “t o p ological spaces”: 1. On one side, the (an tip erfect, p erfect) factorization in T op is indeed pullbac k-stable along lo cal homeomorphisms. This generalizes the fact that if P ֒ → X is dense and O ֒ → X is op en, then P ∩ O ֒ → O is dense. (The related fact that op en maps reﬂect densit y is take n as a basis for a deﬁnition of o p en maps in [Clemen tino et al., 2004].) 2. F or the other stabilit y la w, w e presen t three particular cases. If X ∈ T is a T 1 space (that is, its p oin ts x : 1 → X are in M ′ ), the pullbac k squares b elow sho w that t he (discrete) ﬁb er b P x of the etale reﬂection of a map p : P → X is giv en b y the compo nents of the ﬁb er space P x : P x e ′ / /   b P x m ′ / / n   1 x   P e / / b P m / / X Indeed, since x ∈ M ′ and m ∈ M , also n ∈ M ′ and m ′ ∈ M . Then, by the recipro cal stability , e ∈ E implies e ′ ∈ E . Th us the top row giv es the discrete reﬂection o f P x , that is b P x = π 0 P x . In fact, giving the quotien t top ology to the set of ﬁb ers components, one gets t he etale reﬂection for some classes of maps in T op (see e.g. [Johnstone, 198 2]). 31 3. If K ∈ T is compact (that is, K → 1 is in M ′ ), the following pullback diagram similarly sho ws that if b P is a neigh b orho o d of P ֒ → X then K × b P is a neighborho o d of K × P ֒ → K × X : K × P e ′ / /   K × b P m ′ / / n ′′   K × X / / n ′   K n   P e / / b P m / / X / / 1 In classical terms (in T op ), op en sets K × O form a basis for the op en sets in K × X con taining K × P , when O runs through o p en sets in X containing P (whic h is of course not t r ue if K ֒ → X is, for instance, a straig ht line in the plane). 4. Similarly , if D a closed part of X ∈ T ( t hat is, the monomorphism D ֒ → X is in M ′ ), we ha v e D ∩ P e ′ / /   D ∩ b P m ′ / / n ′   D n   P e / / b P m / / X In classical terms, the o p en sets D ∩ O form a basis for the op en sets in D con taining D ∩ P , when O runs through op en sets in X containing P (whic h is of course not true in general fo r a non-closed part D ). 5. A consequ ence of the recipro cal stabilit y la w is the exp onen tial la w (Prop osition 4 .6 ): if m ∈ M / X , n ∈ M ′ / X and the exp onen tial m n exists in T / X , then it is in M / X (and con v ersely). In particular, if K is compact and S is discrete, K S is compact and S K is discrete. The ﬁrst one is a consequence, in T op , of T yc honoﬀ theorem, or else it f ollo ws fro m the ﬁrst p oin t ab o v e. F or the second one, note that a compact lo cally connected space has a ﬁnite num ber of comp onents. Thus , the compact-op en top ology sho ws that S K is discrete : namely S K = S π 0 K . 6. Aga in b y the exp o nen tial law, an y “ﬁnite co v ering” b of X ∈ T (that is, b ∈ ( M ∩ M ′ ) /X ; see Section 13.5) yields a “ b -complemen tation” ¬ b := b − : M / X → M ′ / X (and conv erse ly); if T has an initial ob ject and 0 → 1 is a “ﬁnite set” in ( M ∩ M ′ ) / 1, w e get a complemen tation ¬ ! ∗ X 0 , whic h generalizes the classical one b etw een op en and closed parts in T op . Th us the latter is only the trace left on monomorphic parts of a less p erfect but m uc h more pregnan t “duality ” b et w een p erfect maps and lo cal homemorphisms, whic h is f ully ex pressed by the recipro cal stabilit y la w. 32 11.4. Rela ted work. “Categorical” or “univ ersal” top ology has a long history and has dev elop ed in man y diﬀerent threads (whic h in part reﬂect t he v ariety of the concepts that can b e considered as basic in top ology itself ). The presen t w ork b elongs to the one that lo ok for a pr o p er categorical foundation of top ology via a suitable axiomatization of “top olog ical” categories that is, categories T whose ob jects can be eﬀectiv ely consid ered as top olo gical spaces of kind T (in the same sense, sa y , tha t the ob jects o f a top os T can b e considered as sets of kind T ). In this direction (but not concerned sp eciﬁcally with classical top ology) w e hav e al- ready men tioned the fundamen tal w ork of Bill La wv ere who dev elops in sev eral pa p ers an analysis o f the ob jects of a category C b y con trasting them with “discrete” ob jects; furthermore the latter can often b e deﬁned inside C b y means of a sp ecial ob ject (for instance, the arrow category in Cat , or a “tiny” T ∈ T suc h that X T is the tangen t bundle of X ). Here, a similar role may b e play ed by a “F rec het space” (see Section 12.3), whic h giv es “discrete” (or “ etale”) and “compact” (or “p erfect”) ob jects at an y slice T / X . An yw a y , we do not assume expo nentiabilit y a nd the exis tence of “interior” righ t adjo in ts T / X → M / X as basic; rather, these pr o p erties can b e considered as further p ossible axioms (see Section 13). On the other hand in [Tholen 1999] a nd [Clemen tino et al., 2004], it is presen ted a n abstraction of T op based on closed maps and it is dev elop ed a great amoun t of classical top ology therein. In spite o f the strictly related basic concepts, ho w ev er, that approac h diﬀers f rom our s in sev eral resp ects. F or instance, w e simultane o usly consider p erfect maps and lo cal homeomorphisms (rather than seeing them as tw o separated instances of the same abstraction) and w e use facto r izat io n systems to condense their basic prop erties and recipro cal relat io nships (rather than to handle images of “sub ob jects”, whic h are no t particularly relev ant t o us). Sev eral factorization systems on T op hav e b een considered in the literature and man y of t hem ha v e b een studie d in [Johnstone, 19 8 2], in the context o f top oses a s generalized spaces. Among these, there see ms not to b e (ev en in the generalized con text) a natural pair of reciprocally stable fa cto r izat io n systems, so that the question of a concrete mo del for “balanced to p ology” remains op en. W e men tion also the recen t w ork [Anel, 2009], concerned with the construction of a G rothendiec k topo logy asso ciated to a factorization system, esp ecially in the con text of algebraic g eometry; there, the etale-prop er (or p erfect) “dualit y” seems to emerge again in guises related to t he prese nt w ork. 12. T op o logi cal spaces and di screte ﬁbrati ons The analogy betw ee n lo cal ho meomorphisms and discrete ﬁbrations a nd betw een p erfect maps and discrete opﬁbrations is one of the main motiv es of our common abtraction of T op a nd Cat as w eak bfc’s. W e here review tw o “ explainations” of this ana lo gy . 12.1. Comp actness and discretene ss in slices of T op and of Cat . F ollow- ing [Bourbaki, 1961], a space X ∈ T op is compact if it is 33 1. quasi-compact, tha t is all the pro jections p : T × X → T are closed, 2. and separated, that is the diagonal map ∆ : X → X × X is closed. This deﬁnition can b e extended to any ﬁnitely complete category C with a functor ( − ) ∗ : C → T op : an ob ject X ∈ C is compact if it is 1. quasi-compact, tha t is all the maps p ∗ : ( T × X ) ∗ → T ∗ are closed, 2. and separated, that is the map ∆ ∗ : X ∗ → ( X × X ) ∗ is closed. With the pro jection ( − ) ∗ : T op / X → T op , the compact (resp. quasi-compact) ob jects of T op / X are the p erfect (resp. prop er) maps to X (see [Bourbaki, 1961]). Replacing “closed” with “op en” in the ab o v e deﬁnitions, w e similarly get discrete spaces in T op , and lo cal homeomorphisms (resp. op en maps) in T op / X . Considering the functor ( − ) ∗ : Cat → T op that sends a category X to the (Alexan- droﬀ ) space X ∗ ∈ T op of its thin reﬂection, it is easy to see that all categories are quasi-compact, while the separated, and hence also the compact ones, coincide with the discrete ones. Comp osing with Cat / X → Cat , we get a functor ( − ) ∗ : Cat / X → T op g iv- ing, as compact ob jects, the discrete opﬁbrations o v er X . D ually , local homeomorphisms in Cat / X are the discrete ﬁbrations ov er X . Of course, b y redeﬁning closed parts a s monomorphic p erfect maps, one gets the “up w ard-closed” full subcategor ies that is, the closed parts of X ∈ Cat are t hose of X ∗ , but considered a s full subcategory inclusions (and similarly for open parts). 12.2. Discre te (op)fibra tions via or thogonality. W e ha v e just seen a deﬁni- tion of discre te opﬁbrations o v er X ∈ Cat as compact ob jects in Cat / X . But they can b e deﬁned more na t ura lly as those functors whic h are ortho gonal to the domain s : 1 → 2 of the ar r ow catego r y; n : D → X is in M ′ / X iﬀ an y squ are 1 a / / s   D n   2 l / / l ′ ? ? X has a unique diagonal. That is, giv en an ob ject a ∈ D , an y a rro w l in X with do main na has a unique lifting (along n ) to an arrow l ′ with domain a . (Quasi-compact o b jects are those for whic h the lifting l ′ exists but not nece ssarly unique, as can b e che c k ed b y using T = 2 a s test ob j ect.) Dually , discrete ﬁbrations are those functors whic h are orthogonal to the co domain functor t : 1 → 2 . 34 12.3. Perfect maps and local homeomorphisms via convergence. P erfect maps n : D → X in T op can b e deﬁned by a similar “con v ergence lifting” prop ert y (see [Bourbaki, 1961]); if ν is an ultraﬁlter in D suc h that nν con v erges t o x ∈ X , then ν con v erges to a unique a ov er x . Brieﬂy , p erfect maps a re “ultr a ﬁlter opﬁbrations”. No w, one would exp ect that (in view of the ab ov e “dual” c haracterizations) lo cal homeomorphisms m : O → X in T op can b e dually deﬁned as “ ultr aﬁlter ﬁbra tions”: if a ∈ O and ξ conv erges to ma ∈ X , then there is a unique ν in O o v er ξ con v erging to a . In f act, in [Clemen tino et al., 2005] it is sho wn that lo cal homeomorphisms are the pul lb ack stable ultraﬁlter ﬁbrations. W e can mak e the link explicit b y a ssuming that in our “top ological” bfc T the fac- torization systems are generated by a “F reche t” ob j ect (instead of the bip ointed a rro w ob ject o f Cat ): 1 e / / F F ′ i o o (with e ∈ E and i ∈ E ′ ). F ′ should b e though t of as a “free sequence”, which is included in F as a “conv ergen t sequence”. In that case, lo cal homeomorphisms a nd p erfect maps ar e the F rec het discrete ﬁbrat io ns and opﬁbrations, that is maps in M and M ′ are deﬁned b y the follo wing unique liftings prop erties: 1 / / e   O   F / / ? ? X F ′ / /   D   F / / > > X (F or monomorphic maps, these giv e t he classical conv ergence characterization of op en and closed parts: O ֒ → X is op en when any “conv ergen t sequence” in X , con v erging to a p oin t in O , is itself (deﬁnitiv ely) in O ; D ֒ → X is closed when for an y “sequence” in D , con v erging to a p oin t x ∈ X , one has x ∈ D .) 12.4. Remark. Con v ergence is o ne of the basic ideas of top ology . Its f ormalization in T op through ultraﬁlters has b een prov ed fruitful in sev eral resp ects, giving often more in tuitiv e counterparts of deﬁnitions and prop erties. Beside the ab ov e men tioned c har- acterization of p erfect maps and lo cal homeomorphisms (and so also of compact spaces, closed and op en parts, etc.), ultraﬁlters can also b e used to deﬁne top ological spaces themselv es and to c haracterize the exp onen tiable o nes (see for instance [Pisani, 1999] and [Clemen tino et al., 2 003]). On the other hand, t he use o f ultr aﬁlters in to p ology has some draw back s; apart from the lack of constructivit y , their practical use is often rather akw ard (as in the pro of, in [Clemen tino et al., 2003], of the exponentiabilit y of p erfect maps). F urthermore, sometimes the results are not exactly ho w one could reasonably exp ect. F or instance, the f a ct that an ultraﬁlter ﬁbrations may not b e a lo cal homeo- morphism, with the accompan ying coun ter-example, app ears rather as a ﬂa w of classical top ological spaces and of the ultra ﬁlter analysis o f con v ergence, allo wing suc h “patholog- 35 ical” space s. In our con text, it seems to b e p ossible a more direct and intuitiv e approac h to inﬁnitesim al asp ects a nd to their analysis via con v ergence. 13. Balanced t op ol ogy Balanced top ology is based on the assumption that T is a (w eak) bfc, whose ob jects are to b e thought of as some kind of top ological spaces, p ossibly inﬁnitesimal and suitably regular. W e here brieﬂy sk etc h some prop erties that follow fro m this assumption, and hin t at some other p ossible axioms that ma y render T a b etter appro ximation of the idea of a top ological category . 13.1. Terminology and not a tion. W e refer to the ob jects o f T as (top ological) spaces , to the maps in M as lo cal homeomorphisms (or also “discrete” or “ etale” maps) and to maps in M ′ as p erfect maps . Maps in B = M ∩ M ′ are the ﬁnite co v erings . Maps in E (r esp. E ′ ) will b e called ﬁnal (resp. initial ) maps (altough t other names ha v e b een used for the latter in T op ). The ob jects (maps) in S := M / 1 are the discrete spaces or (internal) sets . The ob jects (maps) in K := M ′ / 1 are the compact spaces . The o b jects (maps) in S 0 := S ∩ K = B / 1 (the ﬁnite cov erings of 1) are the ﬁnite sets . (Note that, for T = Cat , ﬁnite sets ma y b e not... ﬁnite.) The ob jects (maps) in E / 1 are the connected spaces . Letting P X b e the slice T / X restricted to monomorphisms, Ø X := P X ∩ M / X ar e the op en parts of X , and D X := P X ∩ M ′ / X are the closed parts of X . The parts in D X ∩ Ø X = P X ∩ B / X ar e clopen . The reﬂection π 0 : T → S is the comp onents functor, and π 0 X is the set of comp onen ts of X . A space X is ﬁnite if its set of comp onen ts is ﬁnite. A space X is separated if the diag o nal ∆ : X → X × X is in M ′ . A space is T 1 if its p oin ts are closed. A space X is groupoidal if M / X = M ′ / X = B / X . If P ֒ → X , its ( E , M )-factorization P → b P → X is the neigh b orho o d o f P in X . If it is monomorphic as w ell, it is b oth the smallest op en part con taining P and the big gest part of X con taining P as a ﬁnal part (see Corollary 3.3). The prop osition b elo w simply express es prop erties of factorizat io n syste ms rephrase d in the a b o v e language: 13.2. Proposition. • Perfe ct maps and lo c al home o morphisms o v er a sp ac e a r e close d with r esp e ct to al l the limits which exist in T ; in p articular ﬁnite limits of c omp ac t (r esp. discr ete) sp ac es ar e themse lves c omp act (r esp. discr ete). • I f T is (ﬁnitely) c o c omplete, so ar e p erf e ct m aps and lo c al home omorphism s ov e r a sp ac e ( i n p articular, K and S ). • Perfe ct maps and lo c al home omorphisms a r pul lb ack stable. The pul lb a c k of a p erfe ct map (r esp. lo c al home omorphism ) alo ng a ma p with a c omp a c t (r esp. discr ete) domain, has itself a c o m p act (r esp. discr ete) domain. (Brieﬂy, a p erfe ct map has c omp act ﬁ b ers over c omp act p arts.) 36 • Any c omp act sp ac e is sep ar ate d and T 1 . Any discr ete sp ac e has an op en diagonal and op en p oints. • Th e e qualizer of two p ar al lel maps to a sep ar ate d (r esp. discr ete) sp ac e is close d (r esp. op en). • Any map b etwe en c omp act sp ac es is p erfe ct. Any ma p b etwe en discr ete sp ac es is a lo c al ho me omorphism. • A sp ac e is c o n ne cte d iﬀ any map to a discr ete sp ac e is c onstant. • F or any ﬁgur e P → X with a c onne cte d shap e, its neighb orho o d b P → X h a s a c onne cte d shap e as wel l; in p articular, any sp ac e is lo c a l ly c onne cte d. The f ollo wing are some “top ologically reasonable” consequences of t he recipro cal sta- bilit y la ws: 13.3. Proposition. • Pul ling b ack neighb orho o ds a l o ng pr op er maps , one gets neighb orho o ds again; in p articular, interse cting with close d p arts or multiplying by c omp act sp ac es pr eserves neighb orho o ds. • Th e exp on ential law ho lds for exp on e ntiable sp ac es ( se e Pr op osition 4.6). • Th e ﬁb er of a ﬁna l map over a close d c onne cte d p a rt, is c onne cte d (e.g., over p oints, for T 1 sp ac es, or over closur es o f p oints if the discr ete ar e sep ar ate d). • Th e c omp onents of a ﬁnite sp ac e ar e c onne cte d and cl o p en. Proo f. Most of these hav e b een a lready discussed at the b eginning of this second part; for the last one, r ecall Dia gram (9) and that the p oin ts of a ﬁnite (internal) set are clop en. 13.4. Fur ther topological axioms. The f ollo wing prop erties hold in T op , and so are p o ssible axioms for T : • T is extens iv e and 1 ∈ T is (externally) connected. • 1 ∈ T is group oidal and t w o-v alued: 1 and 0 a re the o nly (cl)op en part of it. • There is a “Sierpinski” space , whic h classiﬁes op en parts. • There are “in terior” coreﬂections C / X → M / X , for an y X ∈ T . • D iscrete spaces ar e separated and T 1 . • Perfec t maps and lo cal homeomorphisms are exponen tiable. 37 13.5. Some homotopical proper ties. Since a p erfect lo cal homeomorphism b et w een lo cally connected top olog ical spaces is a ﬁnite co v ering (see [Bourbaki, 1961]), it is natural to deﬁne the class of ﬁnite co v erings in T as B = M ∩ M ′ ; t hen B / X should reﬂect the π 1 -homotopy t yp e of X ∈ T . W e sa y that maps f : X → Y and g : Y → X are a π 1 -equiv alence if t hey induce an equiv alence b et w een B / X and B / Y . In particular, a space I is “simply connected” if it π 1 -equiv alent to 1 ∈ T , that is, if the ﬁnite in ternal sets inclusion S 0 → B / X is an equiv alence. In Cat , w e ha v e B / X ≃ Set X ′ , where X ′ is the group oidal reﬂection of X . Thu s, for example, an adjunction f ⊣ g : X → Y is a π 1 -equiv alence in Cat , since it giv es an equiv alence f ′ ⊣ g ′ : X ′ → Y ′ . In par ticular, a categor y with a terminal (o r initial) ob ject is simply connected. Another instance of simply connected category is any connected p oset. The follow ing is a consequenc e of Prop o sition 3.6: 13.6. Coro llar y. The π 1 -e quivalenc es have the uniq ue lifting pr op e rty with r esp e ct to ﬁnite c o verings. In a top o lo gical bfc T , a map i : A ֒ → X in E can b e see n as the inclusion of A in one of its p ossible neigh b orho ods in an “ampler” space (e.g., X itself ). Th us, the f o llo wing result ma y b e rephrased b y sa ying that an (inﬁnitesimal) neigh b o r ho o d of a space A whic h retracts on A has the same π 1 -homotopy t yp e of A itself. 13.7. Proposition. A r etr action r, i : A → X with i ∈ E is a π 1 -e quivalenc e. Proo f. Let b : B → X b e any ﬁnite cov ering o f X . In t he diagr a m b elow, the left hand square is a pullback and the rig ht hand one is obtained b y factorizing t he map r b : B → A according to ( E , M ): i ∗ B e ′ / / i ∗ b   B b   e / / ∃ r B b ′   A i / / X r / / A By the rsl, e ′ is in E , and so also e ◦ e ′ is in E ; since b ′ ◦ e ◦ e ′ = i ∗ b and i ∗ b, b ′ ∈ M , the map e ◦ e ′ is also in M , and so it is an iso. Th us the adjunction ∃ r ⊣ r ∗ : M / A → M / X restricts to an adjunction ∃ r ⊣ r ∗ : B / A → B / X . Since r ∈ E , again b y t he rsl the counit ∃ r r ∗ b ′ → b ′ is an iso for a ny b ′ ∈ B / A . It remains t o sho w that the unit b → r ∗ ∃ r b is an iso as w ell. 38 Pulling bac k b ′ along r i = id A w e get another isomor phism e ′′′ ◦ e ′′ : i ∗ r ∗ ∃ r B → ∃ r B : i ∗ r ∗ ∃ r B e ′′ / /                           r ∗ ∃ r B e ′′′   5 5 5 5 5 5 5 5 5 5 5 5 b ′′                           i ∗ B e ′ / / i ∗ b   s C C B b   e / / u E E ∃ r B b ′   A i / / X r / / A The mediating iso s is easily seen to b e a map o v er A suc h that u ◦ e ′ = e ′′ ◦ s , where u is univ ersally induced to the pullback r ∗ ∃ r B . Th us the latt er is b oth in M and in E that is, it is an isomorphism. In particular, any ﬁnite co v ering b of the neighbouring space X/x of a p oin t x : 1 → X is “constan t” that is, b = ! ∗ X/x S , f or a ﬁnite set S ∈ S 0 : 13.8. Coro llar y. Any sp a c e X ∈ T is lo c al ly simply c onne cte d and any ﬁnite c overing b ∈ B / X is “lo c al ly trivial”: pul lin g b ack b along a neighb orho o d X/x → X one gets a c onstant c ove ring. 14. Conclusion o f the second pa rt W e hav e sho wn tha t assuming that T is a (F rec het generated) bfc allo ws o ne to capture sev eral relev a n t features of top ology . Although this “vers ion” of top ology ma y app ear o v er- simpliﬁed, it has the adv an tage to oﬀer a direct and intuitiv e approach b oth to “lo cal” ( or “inﬁnitely close”) asp ects of spaces, and also to some “global” (o r homotopical) pro p erties. In fact, an y space X ∈ T has a “left topo logy” M / X of “op en” ﬁgures and a “rig h t top ology” M ′ / X of “closed” ﬁgures inte racting b y the recipro cal stability la w (wich gen- eralizes the complemen tation law in classical top ology). 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