Balanced category theory

BALANCED CA TEGOR Y THEOR Y CLA UDIO PISANI A BSTRACT . Some asp ects of basic category theory are dev elop ed in a finitely complete category C , endow ed with tw o factor ization systems which determine the same discrete ob jects and are linked by a simple re cipro cal stability law. Resting on this axioma tization o f final and initial functors and discrete (op)fibrations, concepts suc h as comp onents, slices and coslices, colimits and limit s, le ft and right adjunctible maps, dense maps and arrow int erv als, can be natura lly defined in C , and s ev era l classical proper ties concerning them can b e effectively proved. F or an y o b ject X of C , by re s tricting C / X to the slices or to the coslices of X , tw o dual “underlying ca tegories” ar e obtained. These ca n b e enriched ov er internal s ets (discrete ob jects) of C : internal hom-sets ar e g iv en b y the co mponents o f the pullback of the cor resp onding slice and coslice of X . The co nstruction extends to give functor s C → Cat , whic h prese r ve (or reverse) slices and adjunctible maps and which can be enriched over internal sets to o. Conten ts 1 Int ro duction 1 2 Some categ orical notions 3 3 ( E , M )-categor ies and ( E , M ) categor y theor y 5 4 Balanced factor ization categor ies 13 5 Balanced categ ory theory 25 6 Conclusions 30 1. In tro duction F ollowing [Pisani, 2 0 07c], w e further pursue the goal of an axiomatization of the cat ego ry of categories based on the comprehe nsiv e factorization system (cfs f or short). Here, the emphasis is on the sim ultaneous consideration of b o th the cfs’s ( E , M ) and ( E ′ , M ′ ) on Cat , where E ( E ′ ) is the class of final (initial) functors and M ( M ′ ) is the class of discrete (op)fibrations. R ather than on the dualit y functor Cat → Cat , w e rest on the following 2000 Ma thematics Sub ject Class ific a tion: 18 A99. Key w ords a nd phras e s: factorizatio n sys tems , final maps and discr ete fibrations, initial maps and discrete o pfibrations, reflections, internal sets and comp onents, slices and coslices , colimiting cones, ad- junctible maps, dens e maps, cylinders and homotopy , arr ow in terv als, enr ichment , complements. c  Cla udio Pisa ni, 2008 . Permission to copy for pr iv ate use granted. 1 2 “recipro cal stabilit y” property: the pul l b ack of an initial ( final) functor along a discr ete (op)fibr ation is itself initial (fin a l) (Prop osition 2.3). A “balanced factorization category” is a finitely complete category C with t w o factor- ization systems on it, whic h are recipro cally stable in the ab o v e sense and generate the same sub category of “in ternal sets”: S := M / 1 = M ′ / 1 ֒ → C . It turns out that these simple axioms are remark ably p ow erful, allo wing to dev elop within C some imp ortant asp ects of categor y theory in an effectiv e and transparen t w a y . In particular, the reflection of an ob ject X ∈ C in S giv es the internal set of comp onents π 0 X ∈ S , while the reflections of a “p o in t” x : 1 → X in M / X and M ′ / X are the slice X/x and the coslice x \ X , with their maps to X . One can define tw o functors C → Cat , whic h on X ∈ C giv e the “underlying” categories X ( x, y ) := C / X ( X/x, X/ y ) and X ′ ( x, y ) := C / X ( x \ X, y \ X ). F urthermore, if X [ x, y ] := x \ X × X X/y , the internal set X ( x, y ) := π 0 X [ x, y ] enric hes b oth X ( x, y ) and X ′ ( y , x ). Also comp osition in X and X ′ can b e enric hed to a natural “symmetrical” comp osition ma p X ( x, y ) × X ( y , z ) → X ( x, z ), and it follows in particular that the underlying categories are dual. In fact, the term “balanced” is in tended to reflect the fact that the ob jects of C carry t w o dual underlying categories, coexisting on the same lev el. It is the c hoice of one of the tw o factorization systems whic h determine s one of them as “preferred”: in fact, w e will mak e only terminologically suc h a c hoice, b y calling “final” (“initial”) the maps in E ( E ′ ) and “discrete fibrations” (“opfibrations”) those in M ( M ′ ). F or example, w e will see that t he underlying functor C → Cat determined b y ( E , M ) preserv es discrete fibrations and opfibrations, final and initial p oints , slices and coslices and left and right adjunctible maps, while the other one rev erses them. The structure of ba la nced factorization categor y also supp orts the notion of “balanced” cylinder and “balanced” homot op y . In particular, for any x, y : 1 → X w e ha ve a cylinder (see [Law v ere, 1 994]) X ( x, y ) i / / e / / X [ x, y ] c / / X ( x, y ) with i ∈ E ′ , e ∈ E and c ∈ E ∩ E ′ , and pulling back along an elemen t of α : 1 → X ( x, y ) (corresp onding to an arrow x → y as defined ab o v e) w e get the “arr o w in terv al” [ α ]: 1 i / / e / / [ α ] whic h in Cat is the category of factorizations of α with the tw o fa cto r izat io ns through iden tities. Another consequence of the axioms is t he following exp onen tial la w (Prop osition 5.9): if m ∈ M / X , n ∈ M ′ / X a nd the expo nen tial n m exists in C / X , then it is in M ′ / X . As a consequenc e, if m ∈ M / X has a “complemen t” (see [Pisani, 2007a] and [Pisani, 2007b]): ¬ m : S → C / X S 7→ ( S × X ) m then it is v alued in M ′ / X (and conv erse ly). In the balanced fa cto r izat io n category of p osets (see Section 5.11), the in ternal sets ar e the truth v alues and the ab ov e enric hmen t g ives 3 the standard iden tification of p osets with 2-categories, while the complemen t op erator b ecomes the usual (classical) one, relating upp er-sets and low er-sets . 2. Some categor ical noti ons W e here stress some prop erties of Cat and of t he comprehensiv e factorization systems ( E , M ) and ( E ′ , M ′ ), on whic h we will mo del the abstraction o f Sections 3 and 4. 2.1. Slices and inte r v als. Recall that give n tw o functors p : P → X and q : Q → X , the map (or “comma”) construction ( p, q ) yields a s pa rticular cases t he slices X/x = (id , x ) or coslices x \ X = ( x, id) , for an y ob ject x : 1 → X . More generally , p/x = ( p, x ) and x \ q = ( x, q ) can b e obt a ined as the pullbac ks p/x / /   X/x   P p / / X x \ q / /   x \ X   Q q / / X (1) In particular, the pullback [ x, y ] / /   X/y   x \ X / / X (2) is the category of factorizations b etw een x and y , whic h has as o b jects consecutiv e arrows β : x → z and γ : z → y in X , a nd as arrows the diagonals δ a s b elow: x β ′ / / β   z ′ γ ′   z γ / / δ ? ?              y (3) Applying the comp o nents functor π 0 : Cat → Set w e get π 0 [ x, y ] = X ( x, y ). Indeed [ x, y ] is the su m P [ α ] , α : x → y , where [ α ] is t he “interv al category” of facto r izations of α ; α = γ β is an initial (terminal) ob ject of [ α ] iff β ( γ ) is an isomorphism. F or any functor f : X → Y and an y x ∈ X , there is an ob vious “slice functor” o f f at x : e f ,x : X/ x → Y /f x (4) 4 2.2. The comprehensive f a ctoriza tion systems. Recall that a f unctor is a discrete fibration if it is orthogonal to the co domain functor t : 1 → 2, or, equ iv alen tly , if the ob ject mapping of the slice functor e f ,x is bijectiv e, for an y x ∈ X . A functor p : P → X is final if π 0 ( x \ p ) = 1 fo r an y x ∈ X . F or example, an ob ject e : 1 → X is final iff it is terminal ( π 0 ( x \ e ) = π 0 X ( x, e ) = π 0 1 = 1), the iden tit y (or an y isomorphism) X → X is final ( π 0 ( x \ id) = π 0 ( x \ X ) = 1, since x \ X has an initial ob ject and so it is connected) and ! : X → 1 is fina l iff X is connected ( π 0 (0 \ !) = π 0 ( X ) = 1). Final functors and discrete fibrations form the “left” comprehensiv e factorization sys- tem ( E , M ) on Cat , whic h is in fact the (pre)factorizatio n sys tem generated by t . Since an y ( E , M )-factorization giv es a reflection in M / X (see Section 3) w e ha v e a left adjoint ↓ X ⊣ i X to the full inclusion of M / X ≃ S et X op in Cat / X . The “reflection form ula” (see [Pisani, 20 0 7a] and [Pisani, 2007b], and the references therein) ( ↓ p ) x ∼ = π 0 ( x \ p ) (5) giv es its v alue at x . Dually , the domain functor s : 1 → 2 generates the “right” cfs ( E ′ , M ′ ): initial functors and discrete opfibrations (this is the cfs origina lly considered in [Street & W alters, 1973]). In particular, the reflection of an ob ject x : 1 → X in M / X is the slice pro jection ↓ x : X/x → X (cor r esponding to the represen table presh eaf X ( − , x )) and the univ ersal prop ert y of the reflection r educes to the (discrete fibration ve rsion of the) Y oneda Lemma. So, a functor to X is isomorphic in Cat / X to the slice pro jection X/x → X iff it is a discrete fibration whose domain has a terminal ob ject o v er x . Dually , if ↑ X ⊣ j X : M ′ / X → Cat is the reflection in discrete opfibrations, ↑ x : x \ X → X in M ′ / X is the coslice pro j ection. On the other hand, t he reflection of X → 1 in M / 1 give s the comp onen ts of X , that is, ↓ 1 : Cat / 1 ∼ = Cat → M / 1 ≃ Set can b e iden tified with the comp onen ts functor π 0 . The same of course holds for ↑ 1 : Cat → M ′ / 1 ≃ Set . 2.3. Proposition. The pul lb ack of an initial (final) functor along a discr ete (op)fibr ation is itself initial (fin a l). Proo f. The t w o statemen ts are clearly dual. If in the diagram b elo w q is final, f is a discrete opfibratio n, b oth squares are pullbac ks and x = ↓ x ◦ i is a ( E ′ , M ′ )-factorization of x : 1 → X , x \ p / /   P / / p   Q q   1 i / / x \ X ↑ x / / X f / / Y (6) w e w ant to show that x \ p is connected. Since f ◦ ↑ x is a discrete opfibration whose domain has an initial ob ject, it is a coslice of Y : x \ X ∼ = f x \ Y ( f x = ( f ◦ ↓ x ) ◦ i is a ( E ′ , M ′ )-factorization); then the v ertex o f the pullback rectangle is x \ p ∼ = f x \ q , whic h is connected by hypothesis. 5 3. ( E , M )-categories a nd ( E , M ) categor y theor y Throughout the section, C will b e an ( E , M )-category , that is a finitely complete catego ry with a factorizatio n system on it. 3.1. F a ctoriza tion systems and (co)reflections. F or a brief surv ey on factor- ization systems and the asso ciated bifibrations on C w e refer also to [Pisani, 2007c]. W e just recall that an ( E , M )-factorization of a map f : X → Y in C X e / / f @ @ @ @ @ @ @ @ @ @ @ Z m   Y giv es b oth a reflection of f ∈ C / Y in M / Y (with e as reflection map) and a coreflection o f f ∈ X \C in X \E (with m as coreflection map). Con v ersely , an y suc h (co)reflection map giv es an ( E , M )-factorizatio n. This can b e restated in terms of a rro w in terv als: say t hat f = hg is a left (r ig h t) p oint of [ f ] if g ( h ) is in E ( M ). Then by orthogonalit y there is a unique map in [ f ] from an y left point to an y righ t one, and an ( E , M )-factorization of f is a “middle p oint” of the in terv a l [ f ]: it is final among left p oin ts and initial among right ones. The ab ov e po in t of view ma y b e reinforced b y g eometrical in tuition (as discussed at length in [Pisani, 2007c]). A map f : X → Y is a figure in Y of shap e X (or a cofigure o f X in Y ). A left p oin t of [ f ] factorizes the figure thro ug h a n “infinitesimal mo dification” of X , while a rig ht p oin t factorizes it through a “lo cal aspect” of Y . The middle po in t of the in terv al [ f ] X e f / / N ( f ) m f / / Y factorizes the figure through its “neigh b o rho o d” N ( f ). If monic maps are concerned, a left p oint is an “infinitesimal enlargemen t” o f X with resp ect to the embedding f in Y , while a right p oint is a n “op en” part o f Y whic h contains f . Then the (infinitesimal) neigh b orho o d N ( f ) is b oth the smallest op en part of Y containing f and the bigg est infinitesimal enlargemen t of X with resp ect to f . In this ve in, maps m : X → Y in M are c haracterized as “lo cal homeomorphisms” (if t : T → X is a fig ure of X , then m ◦ m t : N ( t ) → Y giv es the neigh b orho o d N ( mt ) → Y ); on the other hand, maps e : X → Y in E preserv e “ global asp ects”: if h : Y → Z is a cofigure of Y and e h is the corresp onding maximal infinitesim al enlargemen t of Y in Z , then e h ◦ e is a maximal “infinitesimal enlargemen t” X → N ( he ) of X in Z . Ho w ev er, our intuition of “infinitesimal mo dification” should be broad enough to in- clude, sa y , the dense inclusion of a par t of a top ological space in ano t her one, a nd also the reflection map of a space in its set of comp onen ts X → π 0 X . F or example, if C = Cat , comp onen ts are one of the “g lo bal asp ects” preserv ed (in the ab ov e sense) b y final func- tors. 6 3.2. ( E , M ) ca teg or y theor y. W e no w define and analyze sev eral ( E , M )-concepts, whic h a ssume their usual meaning when C = Cat with the left comprehensiv e factorization system (see also [Pisani, 20 07c], where the same topics are treated in a sligh tly differen t w a y , and other classical theorems are pro v ed). W e sa y tha t a map e ∈ E is final , while a map m ∈ M is a discrete fibration (df ). W e assume that a “canonical” factorization of an y arro w in C has b een fixed, a nd denote b y ↓ X ⊣ i X the corresponding left adjoint to the full inclusion i X : M / X ֒ → C / X . 3.3. Sets and components. If S → 1 is a discrete fibratio n, w e sa y that S is an in ternal set or also a C -set (or simply a set) a nd w e denote by S := M / 1 ֒ → C the reflectiv e full sub category of C -sets; S is closed with resp ect t o limits in C , a nd so is itself finitely complete. The reflection π 0 := ↓ 1 : C → S is the comp onen ts functor. So a final map e : X → S to a set, that is a factorization of the terminal map X → 1 , giv es the set S ∼ = π 0 X of comp onen ts of X , with the follo wing univ ersal prop ert y: X e / /   @ @ @ @ @ @ @ @ @ @ @ & & S m   / / S ′ n   ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1 (7) That is, π 0 X is initial among sets with a map from X and, as remark ed in Section 3.1, it is also final a mong ob jects with a final map from X . In pa r t icular, a space X is connected ( π 0 X ∼ = 1) iff X → 1 is final, iff an y map to a set is constan t. 3.4. Slices. The other imp ortant particular case is the r eflection ↓ x of a “ p oin t” (global elemen t) x : 1 → X , whic h is obtained b y factorizing the p oin t itself: 1 e x / / x A A A A A A A A A A A A X/x ↓ x   X W e say t hat X/x = N ( x ) is the slice o f X a t x (or the “neighborho o d” of x , as in Section 3.1 and in [Pisani, 2007c]). The corresp onding univers al prop ert y is t he following: 1 e x / / x A A A A A A A A A A A A a & & X/x ↓ x   u / / A m } } | | | | | | | | | | | | X (8) whic h in Cat b ecomes the Y oneda Lemma: giv en a discrete fibratio n m : A → X and an ob ject a o f A ov er X , there is a unique factorizatio n a = ue x o v er X t hr o ugh the “univ ersal elemen t” e x . Note tha t u is a discrete fibration to o, and so displa ys X/x as a slice A/a o f A . 7 3.5. Cones and arro ws. Giv en a map p : P → X and a p oin t x o f X , a cone γ : p → x is a map ov er X f r o m p to the slice X/x : P γ / / p ! ! B B B B B B B B B B B B X/x ↓ x   X A cone λ : p → x is colimiting if it is univ ersal among cones with domain p : P λ / / p ! ! B B B B B B B B B B B B γ ( ( X/x ↓ x   u / / X/y ↓ y | | y y y y y y y y y y y y X (9) That is, a colimiting cone giv es a reflection of p in the full subcategory X ֒ → M / X generated b y the slices o f X ( the underlying category of X ) . If γ : p → x is final, it is an absolute colimiting cone. Note t ha t γ is indeed colimiting, since it giv es a reflection of p in M / X . If P = 1, a cone α : x → y is an arro w from x to y : 1 α / / x A A A A A A A A A A A A X/y ↓ y   X Arro ws can b e comp osed via the Kleisli construction: if α : x → y , β : y → z and b β is univ ersally defined by the left hand dia g ram b elo w, then β ◦ α is giv en by the righ t hand diagram: 1 e y / / y A A A A A A A A A A A A β ( ( X/y ↓ y   b β / / X/z ↓ z | | y y y y y y y y y y y y X 1 α / / x A A A A A A A A A A A A β ◦ α ( ( X/y ↓ y   b β / / X/z ↓ z | | y y y y y y y y y y y y X (10) W e so get a category whic h is clearly isomorphic to the underlying category X , and whic h w e will b e identified with it whenev er opp o rtune. Note that the identities e x are fixed b y the choice of a “canonical” ( E , M )-facto r ization of p oints. An arro w α : x → y is colimiting iff it is absolutely so, iff it is an isomorphism in X . 8 If ↓ p : N ( p ) → X is the reflection of p : P → X in M / X , the reflection map e p induces a bijection b etw een cones p → x and cones ↓ p → x , whic h is easily seen to restrict to a bijection b et w een the colimiting ones: X/y ↓ y   P e p / / p ! ! C C C C C C C C C C C C 6 6 m m m m m m m m m m m m m m m m m m m m m m m N ( p ) < < y y y y y y y y y y y y ↓ p   λ / / X/x u O O ↓ x | | x x x x x x x x x x x x X (11) 3.6. Proposition. If e : T → P is a final map and p : P → X , then a c one λ : p → x is c olimiting, iff λ ◦ e : p ◦ e → x is c olimiting. Proo f. Since e p ◦ e : p ◦ e → ↓ p is a reflection map of p ◦ e in M / X , b y the a b o v e remark, λ = λ ′ ◦ e p is colimiting iff λ ′ is colimiting, iff λ ′ ◦ e p ◦ e is colimiting. T e / / P e p / / p ! ! C C C C C C C C C C C C N ( p ) ↓ p   λ ′ / / X/x ↓ x | | x x x x x x x x x x x x X (12) 3.7. Arro w maps . A map f : X → Y in C acts on cones and arro ws via an induced map b et w een slices as follo ws (see (4)) . By factorizing f ◦ ↓ x as in t he left hand diagram b elo w, w e get a df o v er Y with a final p o int ov er f x . Then t here is a (unique) isomorphism A → Y /f x o v er Y whic h respects the selected final points. By comp osition, w e get the righ t square b elo w. Its upp er edge e f ,x , whic h tak es the final p oin t of X/x to that of Y /f x , is the arrow map (or slice map ) of f (at x ): 1 e x / / x   ; ; ; ; ; ; ; ; ; ; ; ; ; ; X/x e / / ↓ x   A m   ∼ / / Y /f x ↓ f x                 X f / / Y X/x e f ,x / / ↓ x   Y /f x ↓ f x   X f / / Y (13) Note that e f ,x is uniquely characterized, among final maps X/x → Y /f x , by the equation e f ,x ◦ e x = e f x . Indeed, giv en a no ther such e ′ f ,x , the dotted arrow induced among the 9 ( E , M )-factorizations is the iden tit y , since it take s the “univ ersal p oint” e f x to itself: Y /f x ↓ f x                              1 e x / / X/x e f ,x / / e ′ f ,x 6 6 l l l l l l l l l l l l l l l l l l l l l l l l l l l l l ↓ x   Y /f x ↓ f x   < < X f / / Y (14) The arrow map of f acts by comp osition o n cones: it take s γ : p → x t o its “ image cone” f γ : f p → f x : P γ / / p B B B B B B B B B B B B B B B f γ ) ) X/x e f ,x / / ↓ x   Y /f x ↓ f x   X f / / Y (15) whic h has the usual meaning in Cat . 3.8. Proposition. A n absolute c olimiting c one is p r ese rv e d by any m ap. Proo f. If in the diagram ab ov e γ is final, so is also f γ . 3.9. The underl ying functor. In particular, f induces a mapping b et w een arrows x → x ′ of X and arrows f x → f x ′ of Y , whic h preserv es iden tities. In fact, it is the a rro w mapping of a functor f : X → Y : since in the left hand diagram b elow e f ,x is final, there is a uniquely induced dotted arrow u (ov er Y ) whic h mak es the upp er square commute. So, u ◦ e f x = u ◦ e f ,x ◦ e x = e f ,x ′ ◦ b α ◦ e x = e f ,x ′ ◦ α = f α Th us, u = c f α and the right hand square comm utes, showin g that comp osition is pr eserv ed to o. X/x   b α   e f ,x / / Y /f x u     X/x ′   e f ,x ′ / / Y /f x ′   X f / / Y X/x b α   e f ,x / / Y /f x c f α   X/x ′ e f ,x ′ / / Y /f x ′ (16) 10 F urthermore, e g ,f x ◦ e f ,x = e g f ,x (since e g ,f x ◦ e f ,x ◦ e x = e g ,f x ◦ e f x = e g f x ), X/x e f ,x / / e gf ,x * * ↓ x   Y /f x e g,f x / / ↓ f x   Z /g f x ↓ g f x   X f / / Y g / / Z (17) so that g ◦ f = g ◦ f , and w e get the underlying functor ( − ) : C → Cat . 3.10. Proposition. The underlying functor pr eserves the terminal obje ct, sets, slic es and discr ete fibr ations. Proo f. The fact that 1 is the terminal category is immediate. F or any final p oin t t : 1 → X in C , the slice pro j ection X/ t → X is an isomorphism, tha t is a terminal ob ject in C / X and so also in X . If f : X → Y is a df in C , its arr ow map at a n y x is an isomorphism: X/x ∼ / / ↓ x   Y /f x ↓ f x   X f / / Y (18) Th us, any arrow y → f x in Y has a unique lifing along f to an arrow x ′ → x in X , tha t is f is a discrete fibrat io n in Cat . The remaining statemen ts no w follo w at o nce. An example in Section 5.11 sho ws that connected ob jects ( a nd so also final maps) are not preserv ed in general. 3.11. Adjunctible m aps. If the vertex of the pullbac k square b elow has a final p oin t, w e sa y that f is adjunctible at y : 1 e / / x @ @ @ @ @ @ @ @ @ @ @ @ t ' ' f /y p   q / / Y /y ↓ y   X f / / Y (19) and that the pa ir h x, t : f x → y i is a univ ersal arro w from f to y . In that case, f /y is (isomorphic to) a slice X/x of X . Supp ose that f : X → Y is adjunctible at an y p oin t of Y and denote b y g y := pe the p oin t of X corresponding to (a c hoice of ) the final p o in t e , as in the left hand diagram 11 b elo w. The righ t hand diagram (whose squares are pullbac ks) sho ws that g : C (1 , Y ) → C (1 , X ) can b e extended to a functor g : Y → X . (Note that we are not sa ying t ha t there is a map g : Y → X in C whic h giv es g as its underlying map.) 1 e gy / / g y ! ! C C C C C C C C C C C C X/g y ↓ g y   q / / Y /y ↓ y   X f / / Y X/g y     / / Y /y     X/g y ′   / / Y /y ′   X f / / Y (20) 3.12. Proposition. If f : X → Y is an adjunctible map , then ther e is an adjunction f ⊣ g : Y → X . Proo f. Supp ose that in the left hand diagram b elo w the low er square is a pullbac k and that e is the arrow map of f . Since e is final, a ny dotted arrow o n the left induces a unique dotted arrow on the rig h t whic h make s the upp er square (o v er Y ) commute. Since the lo w er square is a pullbac k, the con v erse also holds true. The bijection is natural, as sho wn b y the dia g rams o n t he rig ht. X/x     e / / Y /f x     X/g y   / / Y /y   X f / / Y X/x     e / / Y /f x     X/g y   / / Y /y   X/g y ′ f / / Y /y ′ X/x ′     e ′ / / Y /f x ′     X/x   e / / Y /f x   X/g y f / / Y /y (21) 3.13. Coro llar y. The underlying functor pr eserves adjunctible maps. 3.14. Remarks. 1. Dia g ram (19) sho ws that Corollary 3.13 w ould b e ob vious if, along with slices, the underlying functor w ould preserv e also pullbac ks. Up to no w, the author do es not kno w if this is true in general. 2. The adjunction of Propo sition 3.12 is nothing but the adjunction ∃ f ⊣ f ∗ (where ∃ f : M / X → M / Y giv es , when C = Cat , the left K a n extension a long f ), whic h for an adj unctible f restricts to f ⊣ g (see [Pisani, 2007c]). 12 3. The diagrams b elo w displa y the unit and the counit of adjunction f ⊣ g : X/g f x q ) ) R R R R R R R R R R R R R R R R R R R R R R ↓ g f x   4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 X/x u c c e f ,x / / ↓ x   Y /f x ↓ f x   X f / / Y Y /f g y v | | ↓ f g y                         X/g y q / / ↓ g y   e f ,gy 5 5 l l l l l l l l l l l l l l l l l l l l l l Y /y ↓ y   X f / / Y (22) In Cat , u is the functor α : x ′ → x 7→ η x ◦ α : x ′ → g f x , and the triangle e x,y = q ◦ u reduces to the classical expression of the arro w mapping of a left adjoint functor via transp osition and the unit η . Similarly , v is the functor β 7→ ε y ◦ β , and the triangle q = v ◦ e f ,g y b ecomes the classical expression of the transp osition bijection of an adjunction via the arrow mapping of the left adjoint functor and the counit ε . 3.15. Proposition. A n adjunc tible map f : X → Y pr eserves c olimits: if λ : p → x is c olimiting, then so is f λ : f p → f x . Proo f. Supp ose tha t λ : p → x is colimiting. W e wan t to sho w that f λ is colimiting as w ell, that is an y cone γ : f p → y fa cto r izes uniquely through e f ,x ◦ λ : f p → f x . P f λ $ $ J J J J J J J J u   p   λ z z u u u u u u u γ   X/x u ′ # # e f ,x / /   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Y /f x   / / / / / / / / / / / / / / / / / / u ′′ # # X/g y l / /   Y /y   X f / / y (23) Since the r ectangle is a pullbac k, w e hav e a univers ally induced cone u : p → g y , and since λ is colimiting also u ′ : X/ x → X/g y , whic h by adjunction ( e f ,x is final) corresp onds to a unique u ′′ : Y /f x → Y /y , whic h is the desired unique factorization of γ through f λ . Note that w e ha v e follo w ed the same steps of the most obvious classical pro of, but at this abstract lev el w e need not to v erify or use an y nat ur a lit y condition. 13 3.16. Dense and full y f aithful map s. If in the pullback b elow e is colimiting w e sa y that f is adequate at y : f /y   e / / Y /y ↓ y   X f / / Y (24) while if e is final w e say that f dense . No te that in that case e is an absolute colimiting cone and the diagra m ab ov e display s at once dense maps as “ absolutely adequate” ma ps, as we ll as “lo cally final” maps (see [Ad´ amek et al., 2001] for the case C = C at , whe re a differen t terminology is used; see also Section 5.6). The densit y condition can b e restated as the fact that the counit of the adjunction f ∗ ⊣ ∃ f : M / X → M / Y , that is the uniquely induced arrow v on the left, is an isomorphism on the slices of Y : ∃ f ( f /y ) v { { m                         f /y q / / p   e 5 5 l l l l l l l l l l l l l l l l l l l l l l Y /y ↓ y   X f / / Y f /f x q ) ) R R R R R R R R R R R R R R R R R R R R R R p   3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 X/x u b b e f ,x / / ↓ x   Y /f x ↓ f x   X f / / Y (25) On the other hand, if the unit u of the same adj unction, that is the uniquely induced arro w u on the rig h t, is an isomorphism on the slice X/x , we say that f is fully faithfu l at x . Note that if f is fully faithful at x , then f is adjunctible at f x . 4. Balanced factori zatio n cat egories As shown in the previous section and in [Pis ani, 2 0 07c], category theory can in part b e dev elop ed in any ( E , M )-category . Ho w ev er, in order to treat “ left- sided” and “righ t- sided” categorical concepts sim ultaneously and to ana lyze their in terpla y , w e need to consider two factorizat io n sys tems, satisfying appropriate axioms: 4.1. Definition. A balanced factor ization c at egor y (bfc) is a finitely c omp lete c ate gory C with two factorization systems ( E , M ) and ( E ′ , M ′ ) on it, such that M / 1 = M ′ / 1 ֒ → C and satisfying the recipro cal stabilit y la w (rsl) : the pul lb ack of a map e ∈ E ( e ′ ∈ E ′ ) along a map m ′ ∈ M ′ ( m ∈ M ) is itself in E ( E ′ ). While this definition aims at capturing some r elev an t features of Cat , other in terses- ting instances of bfc are presen t ed in Setion 5.11. Note that any slice C / X o f a bfc is itself a “weak ” bfc: t he requiremen t M / 1 = M ′ / 1 ma y not hold. 14 4.2. Not a tions and te rminology. As in Section 3, w e assu me that “canonical” ( E , M ) and ( E ′ , M ′ )-factorizations ha ve b een fixed for the arrows of C , and denote by ↓ X : C / X → M / X and ↑ X : C / X → M ′ / X the corresp onding reflections. F or the concepts defined in Section 3, but referred to ( E ′ , M ′ ), w e use the standard “dual” terminolog y when it is a v ailable. Otherwise , we distinguish the ( E , M )-concepts from the ( E ′ , M ′ )- ones, by qualifying them with the attributes “left” and “right” resp ectiv ely . So, we sa y that a map i ∈ E ′ is initial and a map n ∈ M ′ is a discret e opfibration (dof ). The coslice of X ∈ C at x : 1 → X is: 1 i x / / x B B B B B B B B B B B B x \ X ↑ x   X W e denote by B := M ∩ M ′ the class of “bifibrations”. Of course, w e also ha v e a “righ t” underlying functor ( − ) ′ : C → Cat , with X ′ ( x, y ) := C / X ( x \ X , y \ X ). 4.3. First proper t ies. W e b egin b y noting t ha t, since S := M / 1 = M ′ / 1 ֒ → C (that is, “left” and “righ t” C -sets coincide, and are bifibrations o v er 1), then also an y (op)fibration o v er a set is in fact a bifibration: M /S = M ′ /S = B /S = S /S . F urthermore, a map X → S to a set is initial iff it is final: E /S = E ′ /S . In that case , S ∼ = π 0 X , where π 0 := ↑ 1 = ↓ 1 : C → S is the comp onen t functor. In particular, E / 1 = E ′ / 1 ֒ → C is the full sub category of connected o b jects. Note also tha t a map S → S ′ b et w een sets is final (o r initial) iff it is an isomorphism; so, if e : S → X is a final (or initial) map fro m a set, then S ∼ = π 0 X . 4.4. Cylinders and inter v als. If c : X → S is a r eflection in S , that is if c is initia l (or equiv alen tly final), for an y elemen t s : 1 → S (whic h is in B ), the pullbac k [ s ] / /   X   1 s / / S (26) displa ys [ s ] as the “ comp onen t” [ s ] ֒ → X of X at s ; [ s ] is connected b y the rsl and is included in X as a discrete bifibratio n. (Note that we are not sayin g that X is the sum of its comp onen ts, although this is t he case under appropriate hy p othesis.) If furthermore i : S ′ → X is initia l (or final), then S ′ ∼ = S ∼ = π 0 X . Th us, i is monomorphic and meets 15 an y comp onen t [ s ] in a p oint, whic h is an initial p o in t i s of [ s ]: 1 i s / /   [ s ] / /   1 s   S ′ i / / X / / S (27) By a balanced cylinder , w e mean a cylinder B i / / e / / X c / / B (that is, a map with t w o sections; see [La wv ere, 1994]) where the base inclusions i and e are initial and final resp ectiv ely . As a consequence, c is b oth initial and final, and if the base is a C -set S , then S ∼ = π 0 X and c is a reflection in S . Among these “discrete cylinders”, t here are “interv als”, that is ba la nced cylinders with a t erminal base. Th us, in terv a ls in C are ( connected) ob jects i, e : 1 → I with an initial and a final selected p oint. In particular, an y comp o nen t of a cylinder with discrete base is an in terv al: 1 i s / / e s / /   [ s ] / /   1 s   S i / / e / / X / / S (28) (Con v ersely , if E and E ′ are closed with res p ect to pro ducts in C → , b y mu ltiplying any ob ject B ∈ C with an in terv al, w e get a cylinder with base B .) 4.5. Homotopic maps. Tw o par a llel maps f , f ′ : X → Y are homotopic if there is a bala nced cylinder X i / / e / / C c / / X and a map ( ˇ homotop y”) h : C → Y suc h that f = hi and f ′ = he (see [La wv ere, 1994]). 4.6. Proposition. The two b ase inclusions i, e : X → C of a b alanc e d cylinder ar e c o e qualize d by any map C → S towar d a set. Proo f. Since the retraction c of the cylinder o n its base is final, the prop osition follo ws immediately from t he fact that, by orthogo nalit y , any map C → S to a set factors uniquely through an y fina l map c : C → X : C / / c   S   X / / ? ? 1 (29) 16 4.7. Coro llar y. Two homotopic maps f , f ′ : X → Y ar e c o e qualize d b y any map Y → S towar d a set. 4.8. Internal hom-sets. As in Section 2, w e denote b y X [ x, y ] (or simply [ x, y ]) the v ertex o f a (fixed) pullbac k [ x, y ] q x,y / / p x,y   X/y ↓ y   x \ X ↑ x / / X (30) W e also denote by c x,y : X [ x, y ] → X ( x, y ) the “canonical” reflection map in S . Th us X ( x, y ) (or simply ( x, y )) is the C - set π 0 X [ x, y ]. By further pulling back along the final (initial) p oin t of the (co)slice, w e get: S ′ / / e   1 e y   S i / /   [ x, y ] / /   X/y   1 i x / / x \ X / / X (31) where S and S ′ are se ts (since the pullbac k of a discrete ( o p)fibration is still a discrete (op)fibration) and i ( e ) is initial (final) b y the rsl. Then b i = c x,y ◦ i : S → ( x, y ) and b e = c x,y ◦ e : S ′ → ( x, y ) are b oth isomorphisms a nd in t he ab ov e diagram w e can replace S and S ′ b y ( x, y ), i b y i ◦ b − 1 i and e b y e ◦ b − 1 e . So w e hav e the follow ing dia g ram of pullbac k squares: ( x, y ) / / e x,y   1 e y   ( x, y ) i x,y / /   [ x, y ] q x,y / / p x,y   X/y ↓ y   1 i x / / x \ X ↑ x / / X (32) 17 and the ba la nced cylinder: ( x, y ) i x,y / / e x,y / / [ x, y ] c x,y / / ( x, y ) (33) Considering in particular the pullback s ( x, y ) q x,y ◦ i x,y / /   X/y   1 x / / X ( x, y ) p x,y ◦ e x,y / /   x \ X   1 y / / X (34) w e see t ha t the elemen ts of α : 1 → X ( x, y ) corresp ond bijectiv ely to “left” arrows x → y and to “rig h t” a rro ws y → x : C (1 , X ( x, y )) ∼ = X ( x, y ) ∼ = X ′ ( y , x ) (35) Then X ( x, y ) deserv es to b e called the “internal hom-set of arrows from x to y ”. (Note that b y using [ x, y ] and ( x, y ) w e hav e notationally made an arbitrary choice among one of t w o p o ssible orders, causing an app erent asymmetry in the subsequen t theory .) F rom no w on, also an elemen t α : 1 → X ( x, y ) will b e called a n “arrow f r om x to y ”. Comp osing it with q x,y ◦ i x,y ( p x,y ◦ e x,y ) w e get the corresponding left (righ t) arrow (whic h will b e sometimes denoted with the same name). As in (28), to an y ar r ow α : 1 → X ( x, y ) there also cor r esp onds an “ar r ow in terv al” [ α ], the comp o nent at α of the balanced cylinde r ( 3 3): 1 i / / e / / α   [ α ] / /   1   ( x, y ) i x,y / / e x,y / / [ x, y ] / / ( x, y ) (36) 4.9. Proposition. Two p oints x, y : 1 → X ar e homotopic iff ther e is an arr ow fr o m x to y . Proo f. If α : x → y is an arrow of X , then [ α ] → X gives a homotop y fr om x to y . Con v ersely , giv en a homotop y 1 i / / e / / C h / / X there is a (unique) arro w i → e of C , and so its image under h is the desired arro w of X . 18 4.10. Enriched composition. The “enriche d comp osition” map µ x,y ,z : X ( x, y ) × X ( y , z ) → X ( x, z ) is defined as µ x,y ,z := c x,z ◦ µ ′ x,y ,z , where µ ′ is univ ersally induced b y the pullbac k: ( x, y ) × ( y , z ) l                     r   < < < < < < < < < < < < < < < < < < µ ′   [ x, z ] x x q q q q q q q q q q q & & M M M M M M M M M M M c x,z   x \ X ' ' N N N N N N N N N N N N ( x, z ) X/z x x p p p p p p p p p p p p X (37) in whic h the maps l and r are the pro duct pro jections f o llo w ed by p x,y ◦ e x,y and q y , z ◦ i y , z , resp ectiv ely . 4.11. Proposition. The map µ x,y ,z induc es on the elements the c omp osition mapping X ( x, y ) × X ( y , z ) → X ( x, z ) in X . Proo f. F or an y arrow β : 1 → X ( y , z ) w e ha v e a map [ x, β ] : [ x, y ] → [ x, z ]: [ x, y ] [ x,β ] # # q x,y / / p x,y   . . . . . . . . . . . . . . . . . . . . . . X/y   . . . . . . . . . . . . . . . . . . . . . . b β " " E E E E E E E E [ x, z ] q x,z / / p x,z   X/z   x \ X / / X (38) In the diagra m b elow , the left dotted arrow (induced b y the univ ersalit y of the low er pullbac k square) and the right one (( x, β ) := π 0 [ x, β ] : ( x, y ) → ( x, z ), induced b y the univ ersalit y of the reflection c x,y : [ x, y ] → ( x, y )) are t he same, since the ho rizon tal edges 19 of the rectangle are iden tities: ( x, y )   i x,y / / ( x,β )   [ x, y ] c x,y / / [ x,β ]   ( x, y ) ( x,β )   ( x, z ) i x,z / /   [ x, z ] c x,z / / p x,z   ( x, z ) 1 i x / / x \ X (39) So we also hav e the comm utative diagra m 1 α / / ( x, y ) i x,y / / ( x,β )   [ x, y ] q x,y / / [ x,β ]   X/y b β   ( x, z ) i x,z / / [ x, z ] q x,z / / X/z (40) sho wing that ( x, β ) acts on an a r r o w α : 1 → X ( x, y ) in the same wa y as b β acts on the corresp onding left a rro w q x,y ◦ i x,y ◦ α : 1 → X/ y , t ha t is as − ◦ β : X ( x, y ) → X ( x, z ). No w observ e that since ( x, β ) = c x,z ◦ [ x, β ] ◦ i x,y , and since i x,y and e x,y are trivially homotopic, also ( x, β ) = c x,z ◦ [ x, β ] ◦ e x,y . But comp osing [ x, β ] ◦ e x,y : ( x, y ) → [ x, z ] with the pullbac k pro jections p x,z and q x,z w e get p x,y ◦ e x,y and t he constant map thro ugh β : 1 → X/y , so that the map [ x, β ] ◦ e x,y coincides with the comp osite ( x, y ) ∼ / / ( x, y ) × 1 id × β / / ( x, y ) × ( y , z ) µ ′ / / [ x, z ] Summing up, w e hav e seen that: ( x, β ) = π 0 [ x, β ] = c x,z ◦ [ x, β ] ◦ i x,y = c x,z ◦ [ x, β ] ◦ e x,y = µ ◦ (id × β ) ◦ h id , ! i So, µ ( α , β ) = ( x, β ) ◦ α corresp onds to µ ( α, β ), where µ is the comp osition in X , pro ving the thesis. “Dually”, for an y α : x → y there are maps [ α , z ] : [ y , z ] → [ x, z ] and ( α, z ) : ( y , z ) → ( x, z ) suc h that ( α, z ) = π 0 [ α, z ] = c x,z ◦ [ α, z ] ◦ i y , z = c x,z ◦ [ α, z ] ◦ e y , z = µ ◦ ( α × id ) ◦ h ! , id i So, µ ( α, β ) = ( α , z ) ◦ β corresp onds to µ ′ ( β , α ), where µ ′ is the compo sition in X ′ . 4.12. Coro llar y. The two underlying c ate gories ar e r elate d by d uali ty: X ′ ∼ = X op . 20 4.13. Internall y e nriched ca tegories? Up to no w, the author has not b een able to prov e the asso ciativit y of the enrich ed comp o sition, nor to find a coun ter-example. On the other ha nd, that the iden tit y arrow u x := c x,x ◦ u ′ x : 1 → ( x, x ) 1 i x                     e x   9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 u ′ x   [ x, x ] y y s s s s s s s s s s % % K K K K K K K K K K c x,x   x \ X & & L L L L L L L L L L L ( x, x ) X/x y y r r r r r r r r r r r X (41) satisfies the unit y laws for the enric hed compo sition, f ollo ws from the previous section or can b e easily c hec k ed directly . 4.14. The action of a map on internal hom-se ts. Giv en a map f : X → Y in C , w e hav e an induced map f [ x,y ] : X [ x, y ] → Y [ f x, f y ]: [ x, y ] f [ x,y ] $ $ q x,y / / p x,y   X/y   e f ,y # # F F F F F F F F [ f x, f y ] q f x,f y / / p f x,f y   Y /f y   x \ X i f ,x $ $ I I I I I I I I I / / X f # # F F F F F F F F F F f x \ Y / / Y (42) In the diagram b elo w the left dotted a rro w (induced b y the univ ers ality of the lo w er pullbac k square) and the righ t one ( f x,y := π 0 f [ x,y ] : X ( x, y ) → Y ( f x, f y ), induced by the univ ersalit y of the reflection c x,y : [ x, y ] → ( x, y )) are t he same, since the ho rizon tal edges 21 of the rectangle are iden tities. ( x, y )   i x,y / / f x,y   [ x, y ] c x,y / / f [ x,y ]   ( x, y ) f x,y   ( f x, f y ) i f x,f y / /   [ f x, f y ] c f x,f y / / p f x,f y   ( f x, f y ) 1 i f x / / f x \ Y (43) So we also hav e the comm utative diagra m 1 α / / ( x, y ) i x,y / / f x,y   [ x, y ] q x,y / / f [ x,y ]   X/y e f ,y   ( f x, f y ) i f x,f y / / [ f x, f y ] q f x,f y / / Y /f y (44) sho wing tha t f x,y acts on an arrow α : 1 → X ( x, y ) in the same wa y as the arrow map o f f acts on the corresp onding left a rro w q x,y ◦ i x,y ◦ α : 1 → X/y . So, f x,y : X ( x, y ) → Y ( f x, f y ) enric hes f x,y : X ( x, y ) → Y ( f x, f y ). W e also ha v e the “dual” comm uting diagrams, with e x,y in place of i x,y and so on, sho wing that f x,y also enric hes f ′ y , x : X ′ ( y , x ) → Y ′ ( f y , f x ). 4.15. Coro llar y. The two underlying functors ar e r elate d by duality: ( − ) ′ ∼ = ( − ) op . 4.16. Proposition. The maps f x,y ar e “functorial” with r esp e ct to enriche d c omp o - sition. Proo f. W e w an t to sho w that the f ollo wing diagram comm utes: ( x, y ) × ( y , z ) f x,y × f y ,z / / µ   ( f x, f y ) × ( f y , f z ) µ   ( x, z ) f x,z / / ( f x, f z ) (45) 22 Since the low er rectangle b elo w commutes ( x, y ) × ( y , z ) f x,y × f y ,z / / µ ′   ( f x, f y ) × ( f y , f z ) µ ′   [ x, z ] f [ x,z ] / / c   [ f x, f z ] c   ( x, z ) f x,z / / ( f x, f z ) (46) w e ha v e to sho w that the upp er one also commutes , that is t ha t comp osing with the pullbac k pro jections (sa y , the second o ne q f x, f z : [ f x, f z ] → Y /f z ) w e get t he same maps: ( x, y ) × ( y , z ) π / / µ ′   ( y , z ) q y ,z ◦ i y ,z   [ x, z ] q x,z / / f [ x,z ]   X/z e f ,z   [ f x, f z ] q f x,f z / / Y /f z ( x, y ) × ( y , z ) π / / f x,y × f y ,z   ( y , z ) f y ,z   ( f x, f y ) × ( f y , f z ) π / / µ ′   ( f y , f z ) q f y ,f z ◦ i f y ,f z   [ f x, f z ] q f x,f z / / Y /f z (47) Indeed, this is the case b ecause all the squares commute and the comp o sites on the righ t are the tw o pa t hs of the rectangle (44). 4.17. Enriching discrete fibra tions. T o an y discrete fibration m : A → X in C there corresponds a presheaf m on X , as can b e seen in sev eral wa ys: 1. As w e ha v e seen in Prop osition 3.10), by applying ( − ) : C → Cat to m , w e obtain a discrete fibration m : A → X in Cat , corresponding to a presheaf m : X op → Set . 2. Via the inclusion i : X ֒ → M / X , any ob ject A ∈ M / X is “in terpreted” as a presheaf M / X ( i − , A ) on X . 3. T o an y ob ject x : 1 → X of X there cor r esp onds t he set C / X ( x, m ) of p oin ts of A o v er x , and to any arro w α : x → y in X there corresp onds a mapping mα : C / X ( y , m ) → C / X ( x, m ), whose v alue a t a ∈ A ov er y is give n b y t he right 23 hand diagram b elow (where b a is defined b y the left hand one): 1 e y / / y A A A A A A A A A A A A a & & X/y ↓ y   b a / / A m ~ ~ | | | | | | | | | | | | X 1 α / / x A A A A A A A A A A A A ( mα ) a & & X/y ↓ y   b a / / A m ~ ~ | | | | | | | | | | | | X (48) Th us, w e ha v e mappings X ( x, y ) × y m → xm , a nd w e now sho w how they can b e enric hed as maps in S . (This enric hing should not b e confused with the m a,b : A ( a, b ) → X ( ma, mb ) of Prop osition 4.1 1, asso ciated to an y map in C .) F ollowing and generalizing the results of Section 4.10, we define [ xm ] a s the pullbac k [ xm ] q x,m / / p x,m   A m   x \ X ↑ x / / X (49) and ( xm ) := π 0 [ xm ], with reflection ma p c x,m : [ xm ] → ( xm ). Then w e also hav e the pullbac ks: ( xm )   i x,m / / [ xm ] q x,m / / p x,m   A m   1 i x / / x \ X ↑ x / / X ( xm )   / / A m   1 x / / X (50) where we can assume that c x,m ◦ i x,m is the identit y of the set ( xm ), a nd whic h sho w tha t the elemen t s of ( xm ) corresp ond bij ectiv ely to tho se of x m . No w w e define (see ( 37)) µ x,y ,m : X ( x, y ) × ( y m ) → ( xm ) as c x,m ◦ µ ′ : ( x, y ) × ( y m ) l                     r   9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 µ ′   [ xm ] x x q q q q q q q q q q q % % K K K K K K K K K K K c x,m   x \ X ' ' N N N N N N N N N N N N ( xm ) A x x r r r r r r r r r r r r r X (51) 24 where µ ′ is univ ersally induced b y the maps l and r , the pro duct pro jections follow ed b y p x,y ◦ e x,y and q y , m ◦ i y , m , respectiv ely . T o pro v e that this is the desired enric hmen t, for a fixed p oin t a ∈ A o v er y w e consider the map [ x, a ] : [ x, y ] → [ xm ]: [ x, y ] [ x,a ] # # q x,y / / p x,y   . . . . . . . . . . . . . . . . . . . . . . X/y   , , , , , , , , , , , , , , , , , , , , , , b a B B B B B B B B [ xm ] q x,m / / p x,m   A m   x \ X / / X (52) and define ( x, a ) := π 0 [ x, a ] : ( x, y ) → ( xm ), getting: ( x, y )   i x,y / / ( x,a )   [ x, y ] c x,y / / [ x,a ]   ( x, y ) ( x,a )   ( xm ) i x,m / /   [ xm ] c x,m / / p x,m   ( xm ) 1 i x / / x \ X ( x, y ) i x,y / / ( x,a )   [ x, y ] q x,y / / [ x,a ]   X/y b a   ( xm ) i x,m / / [ xm ] q x,m / / A (53) The righ t hand diag r am ab o v e sho ws that ( x, a ) acts on a n arrow α : 1 → X ( x, y ) in the same w a y a s b a acts on the corresp onding left arrow q x,y ◦ i x,y ◦ α : 1 → X/y . F urthermore w e obtain that ( x, a ) = π 0 [ x, a ] = c x,m ◦ [ x, a ] ◦ i x,y = c x,m ◦ [ x, a ] ◦ e x,y = µ ◦ (id × a ) ◦ h id , ! i Then, µ ( α, a ) = ( x, a ) ◦ α corresp onds to ( mα ) a , pro ving the thesis. If n : B → X is ano ther discrete fibrat ion o v er X and ξ : m → n a map ov er X , we ha v e a corresp onding morphism of discrete fibrations ξ : m → n in Cat / X , o r equiv alently a nat ural transformation. O n the other hand, ξ also induces maps [ x, ξ ] : [ xm ] → [ xn ]: [ xm ] [ x,ξ ] " " q x,m / / p x,m   . . . . . . . . . . . . . . . . . . . . . . A m   * * * * * * * * * * * * * * * * * * * * * * ξ   = = = = = = = = [ xn ] q x,n / / p x,n   B n   x \ X / / X (54) 25 and one can c hec k that the maps ( x, ξ ) := π 0 [ x, ξ ] : ( xm ) → ( xn ) define a morphism of left S -mo dules on X , which enrich es the natural transformation ξ . 5. Balanced categor y theo ry Balanced category theory is categor y theory dev elop ed in a balanced factorizat io n category C , play ing the role of Cat with the comprehensiv e factorization systems. W e here presen t just a few asp ects of it; others still need to b e analyzed. 5.1. The underl ying functors. Summarizing the results of Section 4, we hav e t w o “underlying” functors: the “left” one ( − ) : C → Cat and the “rig h t” one ( − ) ′ : C → Cat , where X ( x, y ) = C / X ( X/x, X/y ) and X ′ ( x, y ) = C / X ( x \ X , y \ X ). They are isomorphic up to the dualit y functor ( − ) op : Cat → Cat : Cat ( − ) op   C ( − ) 5 5 j j j j j j j j j j j j j j j j ( − ) ′ ) ) T T T T T T T T T T T T T T T T Cat (55) By prop ositions 3.10 and 3 .13, b oth t he underlying f unctors preserv e t he terminal ob ject and sets; the left one preserv es final a nd initia l p oin ts, discrete fibratio ns and opfibra t io ns, slices and coslices and left and right a djunctible maps, while the rig h t one rev erse s them. This appar ent asymmetry is only the effect of o ur naming of t he ar r ows in E , E ′ , M and M ′ . The duality of the underlying functors (corollaries 4.1 2 and 4 .15) is a consequence of the fact that for any ob ject X of C we ha v e an S -enric hed w eak category (that is, the asso ciativity of the comp osition µ may not hold; see Section 4.13), and fo r an y map f : X → Y an S -enric hed functor f x,y : X ( x, y ) → Y ( f x, f y ) suc h that the fo llo wing diagrams in Set comm ute: X ′ ( z , y ) × X ′ ( y , x ) µ ′   C (1 , X ( x, y ) × X ( y , z )) ∼ / / ∼ o o C (1 ,µ )   X ( x, y ) × X ( y , z ) µ   X ′ ( z , x ) C (1 , X ( x, z )) ∼ / / ∼ o o X ( x, z ) (56) X ′ ( y , x ) f ′ y ,x   C (1 , X ( x, y )) ∼ / / ∼ o o C (1 ,f x,y )   X ( x, y ) f x,y   Y ′ ( f y , f x ) C (1 , Y ( f x, f y )) ∼ / / ∼ o o Y ( x, y ) (57) 26 Then, denoting by S Cat ∗ the category of S -enric hed w eak categories and S - enric hed functors, diagram (55) can b e enric hed as S Cat ∗ ( − ) op   C ( − ) 4 4 i i i i i i i i i i i i i i i i ( − ) ′ * * U U U U U U U U U U U U U U U U S Cat ∗ (58) where X ( x, y ) = X ′ ( y , x ) = X ( x, y ), while f x,y = f ′ y , x = f x,y : X ( x, y ) → X ( f x, f y ). F urthermore, f o r an y df m : A → X , the corresp onding presheaf on X can also b e enric hed a s a left S -mo dule on the w eak S -category X , and the functor M / X → Set X op can b e enriche d to a functor M / X → X mo d ( S ) (and dually for M ′ / X → Set X ). 5.2. The tensor functor. G iven p, q ∈ C / X , their tensor pro duct is the com- p onen ts set ten X ( p, q ) := π 0 ( p × X q ) of their pro duct o v er X . W e so get a functor ten X : C / X × C / X → S . In particular, ten( x \ X , X/y ) = X ( x, y ) and, if m ∈ M / X , ten( x \ X, m ) = ( xm ). If m ∈ M / X , n ∈ M ′ / X and C = C at , ten( m, n ) g iv es the classical tensor pro duct (co end) n ⊗ m of the corresp onding set functors m : X op → Set and n : X → Set . 5.3. The inversion la w. If q : Q → X is an y map, and n : D → X is a dof o v er X , pulling bac k the ( E , M )-factorization q = ↓ q ◦ e q along n we get: n × q n × e q / /   n ′ F F F F F F F F F F F " " F F F F n × ↓ q | | x x x x x x x x x x x x x x x x x " " F F F F F F F F F F F F F F F F D n # # F F F F F F F F F F F F F F F F F Q q   e q / / N ( q ) ↓ q | | x x x x x x x x x x x x x x x x X (59) where × denotes the pro duct in C / X . Since n ′ is a dof, b y the rsl n × e q : n × q → n × ↓ q is a final map o v er X ; th us its domain and co domain ha v e the same r eflection in M / X : 5.4. Proposition. If n ∈ M ′ / X an d q ∈ C / X , ↓ X ( n × q ) ∼ = ↓ X ( n × ↓ X q ) ; dual ly, if m ∈ M / X , ↑ X ( m × q ) ∼ = ↑ X ( m × ↑ X q ) 27 Similarly , given tw o maps p : P → X and q : Q → X , w e ha v e the followin g symmet- rical diagram: ↑ p × q / / e   Q e q   p × ↓ q i / /   ↑ p × ↓ q / /   N ( q ) ↓ q   P i p / / N ( p ) ↑ p / / X (60) Since ob jects link ed b y initia l or final maps ha v e the same reflection in M / 1 = S , we get the follo wing “in v ersion la w”: ten( ↑ p, q ) ∼ = ten( ↑ p, ↓ q ) ∼ = ten( p, ↓ q ) (61) 5.5. The re flection formula. If p is a po in t x : 1 → X , diagram (60) becomes: x \ q / / e   Q e q   ( x ↓ q ) i / /   [ x ↓ q ] / /   N ( q ) ↓ q   1 i x / / x \ X ↑ x / / X (62) and the inv ersion law b ecomes the “reflection form ula”: ( x ↓ q ) = ten ( x \ X, ↓ q ) ∼ = π 0 ( x \ q ) = ten( x \ X, q ) ∈ S (63) giving the enriche d v alue of the reflection ↓ q ∈ M / X at x . If the functor M / X → S X op m 7→ ten( −\ X , m ) (64) reflects isomorphisms, t hen the formula ten ( −\ X, q ) determines ↓ q up to isomorphisms . In particular, in this case, q is final iff the in v erse image x \ q = q ∗ ( x \ X ) of any slice of X is connected : π 0 ( x \ q ) = 1 , x ∈ X . If C = Cat , w e find again the “classical” formula (5): ↓ q ∼ = π 0 ( −\ q ) whic h giv es the presheaf corresponding to the discrete fibration ↓ q . 28 5.6. Dense maps. In Section 3, we hav e already discussed dense maps, from a “left- sided” p ersp ectiv e. But densit y is clearly a “balanced” concept. Indeed, the follo wing is among the characterizations of dense functors in Cat given in [Ad´ amek et a l., 2001]: 5.7. Proposition. If f : X → Y is dense, then for any arr o w α of Y the obje ct α/ /f , obtaine d by pul ling b ack the arr ow interval [ α ] al o ng f , is c onne cte d . If the functor (64) r efle cts i s omorphisms, the r everse implic ation also holds. Proo f. The pullbac k square fa cto r s through the t w o pullbac k rectangles b elow : α/ /f e ′ / /   " " E E E E E E E E [ α ]   m ′ ! ! B B B B B B B B f /y e / / { { x x x x x x x x x Y /y } } z z z z z z z z X f / / Y (65) If f is dense, the map e is final by definition; m ′ is a discrete opfibration, b ecause it is the comp o site of the comp onent inclusion [ α ] → [ x, y ] and the pullbac k pro jection q x,y : [ x, y ] → Y /y , whic h are b oth discrete o pfibrations (see diag r ams (3 0 ) and ( 3 6)). Then, b y the rsl also e ′ is final, and the connectedne ss o f [ α ] implies that of α / /f . F or the second par t , note that [ α ], ha ving an initial p o in t, is a coslice of Y /y : [ α ] ∼ = α \ ( Y /y ) (66) Th us, α/ /f is the pullbac k α \ e ′ , and the fact that it is connected fo r an y α implies, b y the h ypo thesis, the finalit y of e ′ . 5.8. Exponentials and complements. Till now, we ha v e not considered an y of the exp onen tiability prop erties of Cat . In this subsection, w e assume that fo r any X ∈ C , the discrete fibratio ns and opfibrations ov er X are exp onen tiable in C / X . 5.9. Proposition. If m ∈ M / X and n ∈ M ′ / X then m n ∈ M / X and, dual ly, n m ∈ M ′ / X . Proo f. If n is a dof ov er X and q : Q → X is a n y o b ject in C / X , b y Prop osition 5.4, n × η q : n × q → n × ↓ q is a final map o v er X (recall that e q : q → ↓ q is the reflecton map of q ∈ C / X in M / X , that is the unit η q of the adjunction ↓ X ⊣ i X : M / X ֒ → C / X ). Then it is ortho g onal to an y df o v er X , in par t icular to m . In turn, this implies tha t any unit η q : q → ↓ q is orthogonal to m n , since in general if L ⊣ R : A → B , an arrow f : B → B ′ in B is or t ho gonal to the 29 ob ject R A iff Lf : LB → LB ′ is orthogonal to A in A : B h / / f   A   B ′ / / u ∗ > > 1 LB h ∗ / / Lf   RA   LB ′ / / u = = 1 (67) Indeed, in one direction the rig h t hand diagram ab o v e g iv es a unique u , whose tra nspo se u ∗ mak es the left hand one comm ute: u ∗ ◦ f = ( u ◦ Lf ) ∗ = ( h ∗ ) ∗ = h . The conv ers e implication is equally straigh tforw ard. Then, taking q = m n w e see that η m n is a reflection map with a retractio n, and so m n ∈ M / X (see [Pisani, 2007c] or also [Borceux, 1994]). An y m ∈ M / X has a “complemen t” ten( m, − ) ⊣ ¬ m : S → C / X S 7→ ( S × X ) m (68) where S × X is the constan t bifibration o v er X with v alue S . By Prop osition 5 .9, ¬ m tak es v alues in M ′ / X , giving a broad g eneralization of the fact that the ( classical) com- plemen t of a lo w er-set of a p oset X is an upp er-set. (These items are trated at length in [Pisani, 200 7a] and [Pisani, 2007b]. There, w e con v ersely used the expo nen tial law to pro v e the inv ersion law in Cat ; ho w ev er, the presen t one seems by far the most natural path to fo llo w, also in the original case C = Cat .) 5.10. Codiscrete and grupoid al objects. An ob ject X ∈ C is co discrete if all its p oin ts a re b oth initial and final (that is, all the (co)slice pro jections are isomorphisms). An ob ject X ∈ C is grup oidal if a ll its (co)slices a re co discrete. (Note that co discrete ob jects and sets are grup oida l.) Since these concepts are auto dual and the underlying functors preserv e (or dualize) slices, it is immediate to see that they preserv e b oth co discrete and grup oidal ob j ects (whic h in Cat ha v e the usual meaning). 5.11. Fur ther examples of balanced f actoriza tion ca tegories. • If C is a n ( E , M )-category suc h that E is M -stable (or, equiv alen tly , satisfying the F rob enius la w; see [Clemen tino et al., 199 6]), then C is a bfc with ( E , M ) = ( E ′ , M ′ ). • In particular, for any lex categor y C we hav e the “discrete” bfc C d and the “co dis- crete” bfc C c on C . In C d , E = E ′ = iso C and M = M ′ = ar C . Con v ersely in C c , E = E ′ = ar C and M = M ′ = iso C . In C d all ob ject are discrete: S = C , while 1 is the only connected ob ject. Conv erse ly , in C c all o b jects are connected, while 1 is the only set. So, for an y X ∈ C d ( X ∈ C c ) id : X → X (! : X → 1) is t he r eflection map of X in sets and, for an y x : 1 → X , X/ x = x \ X = x ( X/x = x \ X = id X ). Then, for a ny X ∈ C d ( X ∈ C c ), X is the (co)discrete category on C (1 , X ). 30 F urthermore, if X ∈ C d and x, y : 1 → X , then X [ x, y ] is p oin tless if x 6 = y , while X [ x, x ] = 1 . So the same is true for X ( x, y ). On the other hand, if X ∈ C c , then X [ x, y ] = X and X ( x, y ) = 1, for any x, y : 1 → X . Indeed, f or an y arrow u x,y : x → y , the arrow interv al [ u x,y ] is simply x, y : 1 → X . • On the category Pos of p o sets, w e can consider the bfc in whic h m ∈ M ( M ′ ) iff it is, up to isomorphisms, a low er- set (upp er-set) inclusion, and e : P → X is in E ( E ′ ) iff it is a cofinal (coinitial) mapping in the classical sense: for any x ∈ X , there is a ∈ P such that x ≤ ea ( ea ≤ x ). The category o f internal sets is S = 2, a nd the comp onen t functor Pos → 2 reduces to t he “non-empt y” predicate. F or any p oset X and a ny p oint x ∈ X , the slice X/x ( x \ X ) is the principal low er-set (upp er-set) generated b y x . As in Cat , X is isomorphic to X itself. This example also sho ws that the underlying functor may not preserv e final maps: a p oset X is “in ternally” connected iff it is not empt y , while X ∼ = X may w ell b e not connected in Cat . Giv en a map p : P → X , a colimiting cone λ : p → x simply indicates that x is the sup of t he set of p oin ts pa, a ∈ P . The colimit is absolute iff suc h a sup is in f a ct a maxim um. A map f : X → Y is adequate iff an y y ∈ Y is t he sup of the f x whic h are less than or equal to y . F urthermore, giv en x, y : 1 → X , [ x, y ] ⊆ X has it s usual meaning: it is the in terv a l of the p oin ts z ∈ X suc h that x ≤ z ≤ y . So, X ( x, y ) is the truth v alue of the predicate x ≤ y , and the enric hing of X in in ternal sets giv es the usual iden tification of p osets with catego ries enric hed ov er 2 . By restricting to discrete p osets, we get the bfc asso ciated to t he epi-mono factor- ization system on Set . • Let Gph b e the category of reflexiv e graphs, with the factorization systems ( E , M ) and ( E ′ , M ′ ) generated b y t : 1 → 2 and s : 1 → 2 resp ectiv ely , where 2 is the arrow graph ( see [Pisani, 2007a] and [Pisani, 2 007b]). In this case X is the free category on X , and the underlying functor is the free catego r y functor Gph → Cat . F urthermore, if X ∈ Gph a nd x, y : 1 → X , then X [ x, y ] is the graph whic h ha s as ob jects the pairs of consecutiv e pa ths h α , β i , with α : x → z and β : z → y , while there is an arro w h α , β i → h α ′ , β ′ i o v er a : z → z ′ iff α ′ = aα and β = β ′ a . In Gph , S = Set and the comp onen ts of X [ x, y ] ar e the paths x → y . Th us, the enric hing X ( x, y ) of X ( x, y ) is in fact an isomorphism. 6. Conclusi ons There are at least three w ell-estabilished abstractions (or generalizations) of category theory: enric hed categories, in ternal categories and 2-cat ego ry theory . Eac h of them is b est suited to enligh ten certain of its a sp ects and to capture new instances; for example, 31 monads and adjunctions (via the triangular iden tities) are surely 2 -categorical concepts, while enric hed categories subsume man y imp ortan t structures and supp o rt quan tifications. Here, w e ha v e based our abstraction on final and initial functors and discrete fibra- tions and opfibrations, whose decisiv e relatio ns are enco ded in the concept of balanced factorization category; in this con text, natural transformations and/or exponentials are no more ba sic notions. Ra ther, w e ha v e seen ho w the univ ers al exactness prop erties of C and those dep ending on the tw o factorization systems, complemen t eac h other in a n harmonious w a y (in particular, pullbac ks and t he reflection in discrete (op)fibrations) to giv e an effectiv e and natural to ol for prov ing categorical facts. While not so ric h of significant instances differen t from Cat as other t heories, w e hop e to ha v e sho wn that balanced categor y theory offers a go o d persp ectiv e o n sev eral basic catego rical concepts and prop erties, helping t o distinguish the “ t rivial” ones (that is, those whic h dep end only on the bfc structure of Cat ) from those regarding p eculiar asp ects of Cat (suc h as colimits, lextensivit y , p ow er o b jects and the arro w ob ject). The latter ha v e b een partly considered in [Pisani, 2007c] and deserv e f urther study . Summarizing, balanced category theory is • simple : a bfc is a lex category with tw o recipro cally stable factor izat io n systems generating the same (inte rnal) sets; • expr essive : many categor ical concepts can b e naturally defined in an y bfc; • effe ctive : simple univ ersal prop erties guide and almost “fo r ce” the proving of cate- gorical prop erties relativ e to these concepts; • symmetric al (or “balanced”): the category Cat in itself do es no t allo w to distinguish an ob ject fr o m its dual; this is fully reflected in balanced category theory; • self-fo und e d : it is largely enriche d on its ow n in ternal sets, pro viding in a sense its o wn foundation (see fo r example [La wv ere, 20 03] and [La wv ere, 19 66]). References J. Ad´ amek, R. El Bashir, M. Sobral, J. V elebil (20 0 1), On F unctors whic h are Lax Epi- morphisms, The ory and Appl. Cat. 8 , 5 0 9-521. F. Borceux (1994 ), Handb o ok of C ate gori c al Algebr a 1 (Basic Cate gory The ory) , Encyclo- p edia of Mathematics and its applications, v ol. 50, Cam bridge Univ ersit y Press. M.M. Clemen tino, E. Giuli, W. Tholen, (1996), T op ology in a Category: Compactness, Portugal. Math. 53 (4), 397–433. F.W. Lawv ere (196 6 ), The Cate gory of Cate gories as a F oundation for Mathema tics , Pro ceedings of the Conference on Categorical Algebra, La Jolla, 196 5, Springer, New Y ork, 1-20. 32 F.W. Law v ere (1994), Unity and Iden tity of O pp o sites in Calculus and Physics , Pro ceed- ings of the ECCT T ours Conference. F.W. L a wv ere (2003), F o undations and Applications: Axiomatization a nd Education, Bul l. Symb. L o gic 9 (2), 213-224. C. Pisani (2007a), Compo nen ts, Complemen ts and the Reflection F or m ula, The ory and Appl. Ca t. 19 , 19-40. C. Pisani (2007b), Comp onen ts, Complemen t s and Reflection F ormulas, preprint, math.- CT/0701457. C. Pisani ( 2007c), Catego ries of Categories, preprin t, math.CT/0709.0837 R. Street and R.F.C. W alters (19 73), The Comprehensiv e F actorization of a F unctor, Bul l. A mer. Math. So c. 79 ( 2 ), 936-941. via Giob erti 86, 10128 T ori n o, Italy. Email: piscla u@yahoo.it