BALANCED CATEGORY THEORY

Transcription

1 Theory and Applications o Categories, Vol. 20, No. 6, 2008, pp BALANCED CATEGORY THEORY CLAUDIO PISANI Abstract. Some aspects o basic category theory are developed in a initely complete category C, endowed with two actorization systems which determine the same discrete objects and are linked by a simple reciprocal stability law. Resting on this axiomatization o inal and initial unctors and discrete (op)ibrations, concepts such as components, slices and coslices, colimits and limits, let and right adjunctible maps, dense maps and arrow intervals, can be naturally deined in C, and several classical properties concerning them can be eectively proved. For any object o C, by restricting C/ to the slices or to the coslices o, two dual underlying categories are obtained. These can be enriched over internal sets (discrete objects) o C: internal hom-sets are given by the components o the pullback o the corresponding slice and coslice o. The construction extends to give unctors C Cat, which preserve (or reverse) slices and adjunctible maps and which can be enriched over internal sets too.. Introduction Following [Pisani, 2007c], we urther pursue the goal o an axiomatization o the category o categories based on the comprehensive actorization system. Here, the emphasis is on the simultaneous consideration o both the comprehensive actorization systems (E, M) and (E, M ) on Cat, where E (E ) is the class o inal (initial) unctors and M (M ) is the class o discrete (op)ibrations. Rather than on the duality unctor Cat Cat, we rest on the ollowing reciprocal stability property: the pullback o an initial (inal) unctor along a discrete (op)ibration is itsel initial (inal) (Proposition 2.3). A balanced actorization category is a initely complete category C with two actorization systems on it, which are reciprocally stable in the above sense and generate the same subcategory o internal sets : S := M/ = M / C. It turns out that these simple axioms are remarkably powerul, allowing one to develop within C some important aspects o category theory in an eective and transparent way. In particular, the relection o an object C in S gives the internal set o components π 0 S, while the relections o a point x : in M/ and M / are the Received by the editors and, in revised orm, Transmitted by F. W. Lawvere. Published on Mathematics Subject Classiication: 8A99. Key words and phrases: actorization systems, inal maps and discrete ibrations, initial maps and discrete opibrations, relections, internal sets and components, slices and coslices, colimiting cones, adjunctible maps, dense maps, cylinders and homotopy, arrow intervals, enrichment, complements. c Claudio Pisani, Permission to copy or private use granted. 85

2 86 CLAUDIO PISANI slice /x and the coslice x\, with their maps to. One can deine two unctors C Cat, which on C give the underlying categories (x, y) := C/(/x, /y) and (x, y) := C/(x\, y\). Furthermore, i [x, y] := x\ /y, the internal set (x, y) := π 0 [x, y] enriches both (x, y) and (y, x). Also composition in and can be enriched to a natural symmetrical composition map (x, y) (y, z) (x, z), and it ollows in particular that the underlying categories are dual. In act, the term balanced is intended to relect the act that the objects o C carry two dual underlying categories, coexisting on the same level. It is the choice o one o the two actorization systems which determines one o them as preerred : in act, we will make only terminologically such a choice, by calling inal ( initial ) the maps in E (E ) and discrete ibrations ( opibrations ) those in M (M ). For example, we will see that the underlying unctor C Cat determined by (E, M) preserves discrete ibrations and opibrations, inal and initial points, slices and coslices and let and right adjunctible maps, while the other one reverses them. The structure o balanced actorization category also supports the notion o balanced cylinder and balanced homotopy. In particular, or any x, y : we have a cylinder (see [Lawvere, 994]) (x, y) i e [x, y] c (x, y) with i E, e E and c E E, and pulling back along an element α : (x, y) (corresponding to an arrow o the underlying categories) we get the arrow interval [α]: i e [α] which in Cat is the category o actorizations o α with the two actorizations through identities. Another consequence o the axioms is the ollowing exponential law (Proposition 5.9): i m M/, n M / and the exponential n m exists in C/, then it is in M /. Thus, i m M/ has a complement (see [Pisani, 2007a] and [Pisani, 2007b]): m : S C/ S (π S ) m (where π S : S ), it is valued in M /, and conversely. In the balanced actorization category o posets (see Section 5.), the internal sets are the truth values and the above enrichment gives the standard identiication o posets with 2-valued categories, while the complement operator becomes the usual (classical) one, relating upper-sets and lower-sets. 2. Some categorical notions We here stress some properties o Cat and o the comprehensive actorization systems (E, M) and (E, M ), on which we will model the abstraction o Sections 3 and 4.

3 BALANCED CATEGORY THEORY Slices and intervals. Recall that given two unctors p : P and q : Q, the map (or comma ) construction (p, q) yields as particular cases the slices /x = (id, x) or coslices x\ = (x, id), or any object x :. More generally, p/x = (p, x) and x\q = (x, q) can be obtained as the pullbacks p/x /x x\q x\ () P p Q q In particular, the pullback [x, y] /y (2) x\ is the category o actorizations between x and y, which has as objects consecutive arrows β : x z and γ : z y in, and as arrows the diagonals δ as below: β x z β δ γ z y γ (3) Applying the components unctor π 0 : Cat Set we get π 0 [x, y] = (x, y). Indeed [x, y] is the sum [α], α : x y, where [α] is the arrow interval category o actorizations o α; α = γβ is an initial (terminal) object o [α] i β (γ) is an isomorphism. For any unctor : Y and any x, there is an obvious slice unctor o at x: e,x : /x Y/x (4) 2.2. The comprehensive actorization systems. Recall that a unctor is a discrete ibration i it is orthogonal to the codomain unctor t : 2, and that a unctor p : P is inal i π 0 (x\p) = or any x. For example, an object e : is inal i it is terminal (π 0 (x\e) = π 0 (x, e) = π 0 = ), the identity (or any isomorphism) is inal (π 0 (x\id) = π 0 (x\) =, since x\ has an initial object and so it is connected) and! : is inal i is connected (π 0 (0\!) = π 0 () = ). Final unctors and discrete ibrations orm the let comprehensive actorization system (E, M) on Cat, which is in act the (pre)actorization system generated by t. Since any (E, M)-actorization gives a relection in M/ (see Section 3) we have a let adjoint

4 88 CLAUDIO PISANI i to the ull inclusion o M/ Set op in Cat/. The value o p at x is given by the relection ormula (see [Pisani, 2007a] and [Pisani, 2007b], and the reerences therein): ( p)x = π 0 (x\p) (5) Dually, the domain unctor s : 2 generates the right comprehensive actorization system (E, M ): initial unctors and discrete opibrations (that is the one originally considered in [Street & Walters, 973]). In particular, the relection o an object x : in M/ is the slice projection x : /x (corresponding to the representable preshea (, x)) and the universal property o the relection reduces to the (discrete ibration version o the) Yoneda Lemma. So, a unctor to is isomorphic in Cat/ to the slice projection /x i it is a discrete ibration whose domain has a terminal object over x. Dually, denoting by : Cat/ M / the relection in discrete opibrations, x : x\ is the coslice projection. On the other hand, the relection o in M/ gives the components o, that is, : Cat/ = Cat M/ Set can be identiied with the components unctor π 0. The same o course holds or : Cat M / Set Proposition. The pullback o an initial (inal) unctor along a discrete (op)ibration is itsel initial (inal). Proo. The two statements are clearly dual. Suppose that in the diagram below q is inal, is a discrete opibration and both squares are pullbacks: we want to show that x\p is connected. x\p P Q (6) p q x\ x Since x is a discrete opibration whose domain has an initial object over x, it is isomorphic to the coslice x\y o Y ; thus the vertex o the pullback rectangle is x\p = x\q, which is connected by hypothesis Remark. A unctor : Y is a discrete ibration i the slice unctor (4) is an isomorphism or any x ; or, equivalently, i its object mapping is bijective or any x. Y 3. (E, M) category theory Throughout the section, C will be an (E, M)-category, that is a initely complete category with a actorization system on it. For a brie survey on actorization systems and the

5 BALANCED CATEGORY THEORY 89 associated biibrations on C we reer also to [Pisani, 2007c]. We just recall that an (E, M)- actorization o a map : Y in C e Z gives both a relection o C/Y in M/Y (with e as relection map) and a corelection o \C in \E (with m as corelection map). Conversely, any such (co)relection map gives an (E, M)-actorization. We assume that a canonical actorization o any arrow in C has been ixed, and denote by i the corresponding let adjoint to the ull inclusion i : M/ C/. We now deine and analyze several (E, M)-concepts, which assume their usual meaning when C = Cat with the let comprehensive actorization system (see also [Pisani, 2007c], where the same topics are treated in a slightly dierent way, and other classical theorems are proved). Accordingly, we say that a map e E is inal, while a map m M is a discrete ibration. 3.. Sets and components. I S is a discrete ibration, we say that S is an internal set or also a C-set (or simply a set) and we denote by S := M/ C the relective ull subcategory o C-sets; S is closed with respect to limits in C, and so is itsel initely complete. The relection π 0 := : C S is the components unctor. So a inal map e : S to a set, that is an (E, M)-actorization o the terminal map, gives the set S = π 0 o components o, with the ollowing universal property: Y m e S S m n That is, π 0 is initial among sets with a map rom (and it is also inal among objects with a inal map rom ). In particular, a space is connected (π 0 = ) i is inal, i any map to a set is constant Slices. The other important particular case is the relection x o a point (global element) x :, which is obtained by the (E, M)-actorization o the point itsel: (7) e x /x x x

6 90 CLAUDIO PISANI We say that /x is the slice o at x. The corresponding universal property is the ollowing: e x x a /x u A x m which in Cat becomes the Yoneda Lemma: or any discrete ibration m : A and any object a o A over x, there is a unique actorization a = ue x over through the universal element e x. Note that u is a discrete ibration too, and so displays /x as a slice A/a o A Cones and arrows. Given a map p : P and a point x o, a cone γ : p x is a map over rom p to the slice /x: (8) P γ p /x A cone λ : p x is colimiting i it is universal among cones with domain p: x P λ p γ /x u x y /y (9) That is, a colimiting cone gives a relection o p C/ in the ull subcategory generated by the slices o (the underlying category o ). I γ : p x is inal, it is an absolute colimiting cone. Note that γ is indeed colimiting, since it gives a relection o p in M/. I P =, a cone α : x y is an arrow rom x to y: α /y x Arrows can be composed via the Kleisli construction: i α : x y, β : y z and β is universally deined by the let hand diagram below, then β α is given by the right hand y

7 BALANCED CATEGORY THEORY 9 diagram: β e y /y bβ /z y y z β α α /y bβ /z y x z (0) We so get a category (with the e x as identities) which is clearly isomorphic to the underlying category, and which will be identiied with it whenever opportune. An arrow α : x y is colimiting i it is absolutely so, i it is an isomorphism in. I p : N(p) is the relection o p : P in M/, the relection map e p induces a bijection between cones p x and cones p x, which is easily seen to restrict to a bijection between the colimiting ones: /y y u P e p N(p) λ /x p p x () 3.4. Proposition. I e : T P is a inal map and p : P, then a cone λ : p x is colimiting i λ e : p e x is colimiting. Proo. Since e p e : p e p is a relection map o p e in M/, by the above remark, λ = λ e p is colimiting i λ is colimiting, i λ e p e is colimiting. T e P e p N(p) λ /x p p x (2) 3.5. Slice maps. A map : Y in C acts on cones and arrows via an induced map between slices as ollows (see (4)). By actorizing x as in the let hand diagram below, we get a discrete ibration over Y with a inal point over x. Then there is a (unique) isomorphism A Y/x over Y which respects the selected inal points and, by composition, we get the right square below. Its upper edge e,x, which takes the inal

8 92 CLAUDIO PISANI point o /x to that o Y/x, is the slice map o (at x): e x /x x x e A m x Y Y/x /x x e,x Y/x Y x (3) Note that e,x is uniquely characterized, among (inal) maps /x Y/x over Y, by the equation e,x e x = e x. Indeed, given another such e,x, the dotted arrow induced among the (E, M)-actorizations is the identity, since it takes the universal point e x to itsel: e x Y/x e,x e,x /x Y/x x (4) x x Y The slice map o acts by composition on cones: it takes γ : p x to its image cone γ : p x: P γ γ /x e,x Y/x (5) p x x which has the usual meaning in Cat Proposition. An absolute colimiting cone is preserved by any map. Proo. I in the diagram above γ is inal, so is also γ The underlying unctor. In particular, induces a mapping between arrows x x o and arrows x x o Y, which preserves identities. In act, it is the arrow mapping o a unctor : Y : since in the let hand diagram below e,x is inal, there is a uniquely induced dotted arrow u (over Y ) which makes the upper square commute. So, u e x = u e,x e x = e,x α e x = e,x α = α Y

9 BALANCED CATEGORY THEORY 93 Thus, u = α and the right hand square commutes, showing that composition is preserved too. e,x e /x,x Y/x /x Y/x (6) bα /x e,x u Y/x bα /x e,x cα Y/x Y Furthermore, e g,x e,x = e g,x (since e g,x e,x e x = e g,x e x = e gx ), e g,x /x e,x Y/x e g,x Z/gx (7) x x gx Y g Z so that g = g, and we get the underlying unctor ( ) : C Cat Proposition. The underlying unctor preserves the terminal object, sets, slices and discrete ibrations. Proo. The act that is the terminal category is immediate. For any inal point t : in C, the slice projection /t is an isomorphism, that is a terminal object in C/ and so also in. I : Y is a discrete ibration in C, its slice map at any x is an isomorphism: /x Y/x (8) x x Since e,x induces on elements the object mapping o e,x, is a discrete ibration in Cat by Remark 2.4. The remaining statements now ollow at once. An example in Section 5. shows that connected objects (and so also inal maps) are not preserved in general. Y

10 94 CLAUDIO PISANI 3.9. Adjunctible maps. I the vertex o the pullback square below has a inal point, we say that is adjunctible at y: e /y t q Y/y (9) x p and that the pair x, t : x y is a universal arrow rom to y. In that case, /y is (isomorphic to) a slice /x o. Suppose that : Y is adjunctible at any point o Y and denote by gy := pe the point o corresponding to (a choice o) the inal point e; then we have pullback squares as below. The right hand diagram (whose squares are pullbacks) shows that g : C(, Y ) C(, ) can be extended to a unctor g : Y. (Note that we are not saying that g underlies a map g : Y in C.) /gy q Y/y Y y /gy Y/y (20) gy Y y /gy Y/y 3.0. Proposition. I : Y is an adjunctible map, then there is an adjunction g : Y. Proo. In the let hand diagram below, the slice map e,x is inal, so that any dotted arrow on the let induces a unique dotted arrow on the right, making the upper square (over Y ) commute. Since the lower square is a pullback, the converse also holds true. The diagrams on the right display the naturality o the bijection as composition o commutative squares (see (20) and (6)). Y /x e,x Y/x /x e,x Y/x /x e,x Y/x (2) /gy Y/y /gy Y/y /x e,x Y/x Y /gy Y/y /gy Y/y

11 BALANCED CATEGORY THEORY Corollary. The underlying unctor preserves adjunctible maps Remarks.. Diagram (9) shows that Corollary 3. would be obvious i, along with slices, the underlying unctor would preserve also pullbacks. Up to now, the author does not know i this is true in general. 2. The adjunction o Proposition 3.0 is nothing but the adjunction (where : M/Y M/ is the pullback unctor and gives, when C = Cat, the let Kan extension along ), which or an adjunctible restricts to g (see [Pisani, 2007c]). 3. The diagrams below display the unit and the counit o the adjunction g: /gx q u gx /x e,x Y/x x Y x Y/gy e,gy v /gy q Y/y gy y Y In Cat, u is the unctor α : x x η x α : x gx, and the triangle e,x = q u reduces to the classical expression o the arrow mapping o a let adjoint unctor via transposition and the unit η. Similarly, v is the unctor β ε y β, and the triangle q = v e,gy becomes the classical expression o the transposition bijection o an adjunction via the arrow mapping o the let adjoint unctor and the counit ε Proposition. An adjunctible map : Y preserves colimits: i λ : p x is colimiting, then so is λ : p x. Proo. Suppose that λ : p x is colimiting. We want to show that λ is colimiting as well, that is any cone γ : p y actorizes uniquely through e,x λ : p x. P λ λ γ u e,x /x Y/x u p u l /gy Y/y y gy (22) (23)

12 96 CLAUDIO PISANI Since the rectangle is a pullback, we have a universally induced cone u : p gy and, since λ is colimiting, also a map (arrow) u : /x /gy; the latter corresponds by adjunction (e,x is inal) to a unique u : Y/x Y/y, which is the desired unique actorization o γ through λ. Note that we have ollowed the same steps o the most obvious classical proo, but at this abstract level we need not to veriy or use any naturality condition Dense and ully aithul maps. I in the pullback below e is colimiting, we say that is adequate at y: e /y Y/y (24) while i e is inal we say that is dense. Note that in that case e is an absolute colimiting cone and the diagram above displays at once dense maps as absolutely adequate maps, as well as locally inal maps (see [Adámek et al., 200] or the case C = Cat, where a dierent terminology is used; see also Section 5.6). The density condition can be restated as the act that the counit o the adjunction : M/ M/Y, that is the uniquely induced arrow v on the let, is an isomorphism on the slices o Y : Y y (/y) e v /y q Y/y m p y Y /x q u p /x e,x Y/x x Y x (25) On the other hand, i the unit u o the same adjunction, that is the uniquely induced arrow u on the right, is an isomorphism on the slice /x, we say that is ully aithul at x. Note that i is ully aithul at x, then is adjunctible at x. 4. Balanced actorization categories As shown in the previous section and in [Pisani, 2007c], category theory can in part be developed in any (E, M)-category. However, in order to treat let-sided and rightsided categorical concepts simultaneously and to analyze their interplay, we need to consider two actorization systems, related by appropriate axioms: 4.. Deinition. A balanced actorization category (bc) is a initely complete category C with two actorization systems (E, M) and (E, M ) on it, such that

13 BALANCED CATEGORY THEORY 97. M/ = M / C; 2. the reciprocal stability law holds: the pullback o a map e E (e E ) along a map m M (m M) is itsel in E (E ). While this deinition aims at capturing some relevant eatures o Cat, other interesting instances o bc are presented in Setion 5.. Note that any slice C/ o a bc is itsel a weak bc: the condition M/ = M / may not hold Notations and terminology. As in Section 3, we assume that canonical (E, M) and (E, M )-actorizations have been ixed or the arrows o C, and denote by : C/ M/ and : C/ M / the corresponding relections. For the concepts deined in Section 3, but reerred to (E, M ), we use the standard dual terminology when it is available. Otherwise, we distinguish the (E, M)-concepts rom the (E, M )- ones, by qualiying them with the attributes let and right respectively. So, we say that a map i E is initial, a map n M is a discrete opibration and the coslice o C at x : is given by the (E, M )-actorization o x: i x x\ x We denote by B := M M the class o discrete biibrations. O course, we also have a right underlying unctor ( ) : C Cat, with (x, y) := C/(x\, y\) First properties. We begin by noting that, since S := M/ = M / C (that is, let and right C-sets coincide, and are discrete biibrations over ), then also any (op)ibration over a set is in act a biibration: M/S = M /S = B/S = S/S. Furthermore, a map S to a set is initial i it is inal: E/S = E /S. In that case, S = π 0, where π 0 := = : C S is the components unctor. In particular, E/ = E / C is the ull subcategory o connected objects. Note also that a map S S between sets is inal (or initial) i it is an isomorphism; so, i e : S is a inal (or initial) map rom a set, then S = π Cylinders and intervals. I c : S is a relection in S, that is i c is initial (or equivalently inal), or any element s : S (which is in B), the pullback x [s] (26) c s S

14 98 CLAUDIO PISANI displays [s] as the component [s] o at s; [s] is connected by the reciprocal stability law and is included in as a discrete biibration. (Note that we are not saying that is the sum o its components, although this is the case under appropriate hypothesis.) I urthermore i : S is initial (or inal), then S = S = π0. Thus, i is monomorphic and meets any component [s] in a point, which is an initial point i s o [s]: i s [s] (27) S i S By a balanced cylinder, we mean a cylinder B i C c B e (that is, a map with two sections; see [Lawvere, 994]) where the base inclusions i and e are initial and inal respectively. As a consequence, c is both initial and inal and, i the base is a C-set S, then S = π 0 C and c is a relection in S. Among these discrete cylinders, there are intervals, that is balanced cylinders with a terminal base. Thus, intervals in C are (connected) objects i, e : I with an initial and a inal selected point. In particular, any component o a cylinder with discrete base is an interval: s i s e s [s] (28) s S i e S (Conversely, i E and E are closed with respect to products in C, by multiplying any object B C with an interval, we get a cylinder with base B.) 4.5. Homotopic maps. Two parallel maps, : Y are homotopic i there is a balanced cylinder i C c e and a map ( homotopy ) h : C Y such that = hi and = he (see [Lawvere, 994]) Proposition. The two base inclusions i, e : C o a balanced cylinder are coequalized by any map C S toward a set.

15 BALANCED CATEGORY THEORY 99 Proo. Since the retraction c o the cylinder on its base is inal, the proposition ollows immediately rom the act that, by orthogonality, any map C S to a set actors uniquely through any inal map c : C : C S (29) c 4.7. Corollary. Two homotopic maps, : Y are coequalized by any map Y S toward a set Internal hom-sets. As in Section 2, we denote by [x, y] (or simply [x, y]) the vertex o a (ixed) pullback [x, y] q x,y /y (30) p x,y y x\ x We also denote by c x,y : [x, y] (x, y) the canonical relection map in S. Thus (x, y) (or simply (x, y)) is the C-set π 0 [x, y]. By urther pulling back along the inal (initial) point o the (co)slice, we get: S (3) e e y S i [x, y] /y i x x\ where S and S are sets (since the pullback o a discrete (op)ibration is still a discrete (op)ibration) and i (e) is initial (inal) by the reciprocal stability law. Then b i = c x,y i : S (x, y) and b e = c x,y e : S (x, y) are both isomorphisms and in the above diagram

16 00 CLAUDIO PISANI we can replace S and S by (x, y), i by i b i diagram o pullback squares: (x, y) and e by e b e. So we have the ollowing (32) e x,y e y (x, y) i x,y [x, y] q x,y /y p x,y y i x x\ x and the balanced cylinder: (x, y) i x,y Considering in particular the pullbacks e x,y [x, y] c x,y (x, y) (33) (x, y) qx,y ix,y /y (x, y) px,y ex,y x\ (34) x we see that the elements o α : (x, y) correspond bijectively to let arrows x y and to right arrows y x: C(, (x, y)) = (x, y) = (y, x) (35) Then (x, y) deserves to be called the internal hom-set o arrows rom x to y. (Note that by using [x, y] and (x, y) we have notationally made an arbitrary choice among one o two possible orders, causing an apparent asymmetry in the subsequent theory.) From now on, also an element α : (x, y) will be called an arrow rom x to y. Composing it with q x,y i x,y (p x,y e x,y ) we get the corresponding let (right) arrow (which will be sometimes denoted with the same name). As in (28), to any arrow α : (x, y) there also corresponds an arrow interval [α], the component at α o the balanced cylinder (33): y i e [α] (36) α (x, y) i x,y e x,y [x, y] (x, y)

17 BALANCED CATEGORY THEORY Proposition. Two points x, y : are homotopic i there is an arrow rom x to y. Proo. I α : x y is an arrow o, then [α] gives a homotopy rom x to y. Conversely, given a homotopy and so its image under h is the desired arrow o. i e C h there is a (unique) arrow i e o C, 4.0. Enriched composition. The enriched composition map µ x,y,z : (x, y) (y, z) (x, z) is deined as µ x,y,z := c x,z µ x,y,z, where µ is universally induced by the pullback: (x, y) (y, z) µ l r [x, z] c x,z x\ (x, z) /z (37) in which the maps l and r are the product projections ollowed by p x,y e x,y and q y,z i y,z, respectively. 4.. Proposition. The map µ x,y,z induces on the elements the composition mapping (x, y) (y, z) (x, z). Proo. For any arrow β : (y, z) we have a map [x, β] : [x, y] [x, z]: q x,y [x, y] [x,β] [x, z] p x,y x\ p x,z /y bβ q x,z /z (38) In the diagram below, the let dotted arrow (induced by the universality o the lower pullback square) and the right one ((x, β) := π 0 [x, β] : (x, y) (x, z), induced by the

18 02 CLAUDIO PISANI universality o the relection c x,y : [x, y] (x, y)) are the same, since the horizontal edges o the rectangle are identities: (x, y) i x,y [x, y] c x,y (x, y) (39) (x,β) [x,β] (x,β) (x, z) i x,z [x, z] c x,z (x, z) p x,z So we also have the commutative diagram i x x\ α (x, y) i x,y [x, y] q x,y /y (40) (x,β) [x,β] bβ (x, z) i x,z [x, z] q x,z /z showing that (x, β) acts on an arrow α : (x, y) in the same way as β acts on the corresponding let arrow q x,y i x,y α : /y, that is as β : (x, y) (x, z). Now observe 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 composing [x, β] e x,y : (x, y) [x, z] with the pullback projections p x,z and q x,z we get p x,y e x,y and the constant map through β : /y, so that the map [x, β] e x,y coincides with the composite (x, y) (x, y) Summing up, we have seen that: id β (x, y) (y, z) µ [x, z] (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) id,! So, µ(α, β) = (x, β) α corresponds to µ(α, β), where µ is the composition mapping o, proving the thesis. Dually, or any α : x y there are maps [α, z] : [y, z] [x, z] and (α, z) : (y, z) (x, z) such 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)!, id So, µ(α, β) = (α, z) β corresponds to µ (β, α), where µ is the composition in Corollary. The two underlying categories are related by duality: = op.

19 BALANCED CATEGORY THEORY Internally enriched categories?. Up to now, the author has not been able to prove the associativity o the enriched composition, nor to ind a counter-example. On the other hand, that the identity arrow u x := c x,x u x : (x, x) u i x e x x [x, x] c x,x x\ (x, x) /x (4) satisies the unity laws or the enriched composition, ollows rom the previous section or can be easily checked directly The action o a map on internal hom-sets. Given a map : Y in C, we have an induced map [x,y] : [x, y] Y [x, y]: [x, y] q x,y /y (42) p x,y [x,y] [x, y] q x,y e,y Y/y x\ i,x p x,y x\y Y In the diagram below the let dotted arrow (induced by the universality o the lower pullback square) and the right one ( x,y := π 0 [x,y] : (x, y) Y (x, y), induced by the universality o the relection c x,y : [x, y] (x, y)) are the same, since the horizontal edges

20 04 CLAUDIO PISANI o the rectangle are identities. (x, y) i x,y [x, y] c x,y (x, y) (43) x,y [x,y] x,y (x, y) i x,y [x, y] c x,y (x, y) p x,y i x x\y So we also have the commutative diagram α (x, y) i x,y [x, y] q x,y /y (44) x,y [x,y] e,y (x, y) i x,y [x, y] q x,y Y/y showing that x,y acts on an arrow α : (x, y) in the same way as the slice map o acts on the corresponding let arrow q x,y i x,y α : /y. So, x,y : (x, y) Y (x, y) enriches x,y : (x, y) Y (x, y). We also have the dual commuting diagrams, with e x,y in place o i x,y and so on, showing that x,y also enriches y,x : (y, x) Y (y, x) Corollary. The two underlying unctors are related by duality: ( ) = ( ) op Proposition. The maps x,y are unctorial with respect to enriched composition. Proo. We want to show that the ollowing diagram commutes: (x, y) (y, z) x,y y,z (x, y) (y, z) (45) µ µ (x, z) x,z (x, z)

21 Since the lower rectangle below commutes BALANCED CATEGORY THEORY 05 (x, y) (y, z) x,y y,z (x, y) (y, z) (46) µ µ [x, z] [x,z] [x, z] c c (x, z) x,z (x, z) we have to show that the upper one also commutes, that is that composing with the pullback projections (say, the second one q x,z : [x, z] Y/z) we get the same maps: (x, y) (y, z) π (y, z) (x, y) (y, z) π (y, z) (47) µ q y,z i y,z x,y y,z y,z [x, z] q x,z /z (x, y) (y, z) π (y, z) [x,z] e,z µ q y,z i y,z [x, z] q x,z Y/z [x, z] q x,z Y/z Indeed, this is the case because all the squares commute and the composites on the right are the two paths in the rectangle (44) Enriching discrete ibrations. To any discrete ibration m : A in C there corresponds a preshea m on, as can be seen in several ways:. As we have seen in Proposition 3.8, by applying ( ) : C Cat to m, we obtain a discrete ibration m : A in Cat, corresponding to a preshea m : op Set. 2. Via the inclusion i : M/, any object A M/ is interpreted as a preshea M/(i, A) on. 3. To any object x : o there corresponds the set C/(x, m) o points o A over x, and to any arrow α : x y in there corresponds a mapping mα : C/(y, m) C/(x, m), whose value at a A over y is given by the right

22 06 CLAUDIO PISANI hand diagram below (where â is deined by the let hand one): e y y a /y ba A y m α x (mα)a /y ba A y m (48) Thus, we have mappings (x, y) ym xm, and we now show how they can be enriched as maps in S. (This enriching should not be conused with the m a,b : A(a, b) (ma, mb) o Proposition 4., associated to any map in C.) Following and generalizing the results o Section 4.0, we deine [xm] as the pullback [xm] q x,m A (49) p x,m m x\ x and (xm) := π 0 [xm], with relection map c x,m : [xm] (xm). Then we also have the pullbacks: (xm) i x,m [xm] q x,m A (xm) A (50) p x,m m m i x x\ x x where we can assume that c x,m i x,m is the identity o the set (xm), and which show that the elements o (xm) correspond bijectively to those o xm. Now we deine (see (37)) µ x,y,m : (x, y) (ym) (xm) as c x,m µ : (x, y) (ym) µ l r [xm] c x,m x\ (xm) A (5)

23 BALANCED CATEGORY THEORY 07 where µ is universally induced by the maps l and r, the product projections ollowed by p x,y e x,y and q y,m i y,m, respectively. To prove that this is the desired enrichment, or a ixed point a A over y we consider the map [x, a] : [x, y] [xm]: q x,y [x, y] [x,a] [xm] p x,y x\ p x,m and deine (x, a) := π 0 [x, a] : (x, y) (xm), getting: /y ba q x,m A m (52) (x, y) i x,y [x, y] c x,y (x, y) (x, y) i x,y [x, y] q x,y /y (x,a) [x,a] (x,a) (x,a) [x,a] ba (xm) i x,m [xm] c x,m (xm) (xm) i x,m [xm] q x,m A p x,m i x x\ (53) The right hand diagram above shows that (x, a) acts on an arrow α : (x, y) in the same way as â acts on the corresponding let arrow q x,y i x,y α : /y. Furthermore we 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) id,! Then, µ(α, a) = (x, a) α corresponds to (mα)a, proving the thesis. I n : B is another discrete ibration over and ξ : m n a map over, we have a corresponding morphism o discrete ibrations ξ : m n in Cat/, or equivalently a natural transormation. On the other hand, ξ also induces maps [x, ξ] : [xm] [xn]: q x,m [xm] [x,ξ] [xn] p x,m x\ p x,n A ξ B m q x,n n (54)

24 08 CLAUDIO PISANI and one can check that the maps (x, ξ) := π 0 [x, ξ] : (xm) (xn) deine a morphism o let S-modules on, which enriches the natural transormation ξ. 5. Balanced category theory Balanced category theory is category theory developed in a balanced actorization category C, playing the role o Cat with the comprehensive actorization systems. We here present just a ew o its aspects, others still needing to be analyzed. 5.. The underlying unctors. Summarizing the results o Section 4, we have two underlying unctors: the let one ( ) : C Cat and the right one ( ) : C Cat, where (x, y) = C/(/x, /y) and (x, y) = C/(x\, y\). They are isomorphic up to the duality unctor ( ) op : Cat Cat: Cat ( ) C ( ) By propositions 3.8 and 3., both the underlying unctors preserve the terminal object and sets; the let one preserves inal and initial points, discrete ibrations and opibrations, slices and coslices and let and right adjunctible maps, while the right one reverses them. This apparent asymmetry is only the eect o our naming o the arrows in E, E, M and M. The duality o the underlying unctors (corollaries 4.2 and 4.5) is a consequence o the act that or any object o C we have an S-enriched weak category (that is, the associativity o the composition µ may not hold; see Section 4.3), and or any map : Y an S-enriched unctor x,y : (x, y) Y (x, y) such that the ollowing diagrams in Set commute: Cat ( ) op (55) (z, y) (y, x) C(, (x, y) (y, z)) (x, y) (y, z) (56) µ C(,µ) µ (z, x) C(, (x, z)) (x, z) (y, x) C(, (x, y)) (x, y) (57) y,x C(, x,y) x,y Y (y, x) C(, Y (x, y)) Y (x, y)

25 BALANCED CATEGORY THEORY 09 Then, denoting by S Cat the category o S-enriched weak categories and S-enriched unctors, diagram (55) can be enriched as S Cat ( ) C ( ) op (58) ( ) S Cat where (x, y) = (y, x) = (x, y), while x,y = y,x = x,y : (x, y) (x, y). Furthermore, or any discrete ibration m : A, the corresponding preshea on can also be enriched as a let S-module on the weak S-category, and the unctor M/ Set op can be enriched to a unctor M/ mod(s) (and dually or M / Set ) The tensor unctor. Given p, q C/, their tensor product is the components set ten (p, q) := π 0 (p q) o their product over. We so get a unctor ten : C/ C/ S. In particular, ten(x\, /y) = (x, y) and, i m M/, ten(x\, m) = (xm). I m M/, n M / and C = Cat, ten(m, n) gives the classical tensor product (coend) n m o the corresponding set unctors m : op Set and n : Set The inversion law. I q : Q is any map, and n : D is a discrete opibration over, pulling back the (E, M)-actorization q = q e q along n we get: n e q n q n q n e q D Q N(q) n q q where denotes the product in C/. Since n is a discrete opibration, by the reciprocal stability law n e q : n q n q is a inal map over ; thus its domain and codomain have the same relection in M/: 5.4. Proposition. I n M / and q C/, (n q) = (n q); dually, i m M/, (m q) = (m q) (59)

26 0 CLAUDIO PISANI Similarly, given two maps p : P and q : Q, we have the ollowing symmetrical diagram: p q Q (60) e e q p q i p q N(q) q P i p N(p) Since objects linked by initial or inal maps have the same relection in M/ = S, we get the ollowing inversion law : p ten( p, q) = ten( p, q) = ten(p, q) (6) 5.5. The relection ormula. I p is a point x :, diagram (60) becomes: x\q Q (62) e e q (x q) i [x q] N(q) q i x x\ x and the inversion law becomes the relection ormula : (x q) = ten(x\, q) = π 0 (x\q) = ten(x\, q) S (63) giving the enriched value o the relection q M/ at x. I the unctor M/ S op m ten( \, m) (64) relects isomorphisms, then the ormula ten( \, q) determines q up to isomorphisms. In particular, in this case, q is inal i the inverse image x\q = q (x\) o any slice o is connected: π 0 (x\q) =, x. I C = Cat, we ind again the classical ormula (5): q = π 0 ( \q) which gives the preshea corresponding to the discrete ibration q.

27 BALANCED CATEGORY THEORY 5.6. Dense maps. In Section 3, we have already discussed dense maps, rom a letsided perspective. But density is clearly a balanced concept. Indeed, the ollowing is among the characterizations o dense unctors in Cat given in [Adámek et al., 200]: 5.7. Proposition. I : Y is dense, then or any arrow α o Y the object α//, obtained by pulling back the arrow interval [α] along, is connected. I the unctor (64) relects isomorphisms, the converse also holds. Proo. The pullback square actors through the two pullback rectangles below: α// e [α] (65) /y e m Y/y Y I is dense, the map e is inal by deinition; m is a discrete opibration, because it is the composite o the component inclusion [α] [x, y] and the pullback projection q x,y : [x, y] Y/y, which are both discrete opibrations (see diagrams (30) and (36)). Then, by the reciprocal stability law also e is inal, and the connectedness o [α] implies that o α//. For the reverse implication, note that [α], having an initial point, is a coslice o Y/y: [α] = α\(y/y) (66) Thus, α// is the pullback α\e, and the act that it is connected or any α implies, by the hypothesis, the inality o e Exponentials and complements. Till now, we have not considered any o the exponentiability properties o Cat. In this subsection, we assume that or any C, the discrete ibrations and opibrations over are exponentiable in C/ Proposition. I m M/ and n M / then m n M/ and, dually, n m M /. Proo. I n is a discrete opibration over and q : Q is any object in C/, by Proposition 5.4, n η q : n q n q is a inal map over (recall that e q : q q is the relection map o q C/ in M/, that is the unit η q o the adjunction i : M/ C/). Then it is orthogonal to any discrete ibration over, in particular to m. In turn, this implies that any unit

28 2 CLAUDIO PISANI η q : q q is orthogonal to m n, since in general i L R : A B, an arrow : B B in B is orthogonal to the object RA i L : LB LB is orthogonal to A in A: B h A LB h RA (67) u L u B LB Indeed, in one direction the right hand diagram above gives a unique u, whose transpose u makes the let hand one commute: u = (u L) = (h ) = h. The converse implication is equally straightorward. Then, taking q = m n we see that η m n is a relection map with a retraction, and so m n M/ (see [Pisani, 2007c] or also [Borceux, 994]). Any m M/ has a complement ten(m, ) m : S C/ S (π S ) m (68) where π S : S is the constant biibration over with value S. By Proposition 5.9, m takes values in M /, giving a broad generalization o the act that the (classical) complement o a lower-set o a poset is an upper-set. (These items are treated at length in [Pisani, 2007a] and [Pisani, 2007b]. There, we conversely used the exponential law to prove the inversion law in Cat; however, the present one seems by ar the most natural path to ollow, also in the original case C = Cat.) 5.0. Codiscrete and groupoidal objects. An object C is codiscrete i all its points are both initial and inal (that is, all the (co)slice projections are isomorphisms). An object C is groupoidal i all its (co)slices are codiscrete. (Note that codiscrete objects and sets are groupoidal.) Since these concepts are sel dual and the underlying unctors preserve (or dualize) slices, it is immediate to see that they preserve both codiscrete and groupoidal objects (which in Cat have the usual meaning). 5.. Further examples o balanced actorization categories. I C is an (E, M)-category such that E is M-stable (or, equivalently, satisying the Frobenius law; see [Clementino et al., 996]), then C is a bc with (E, M) = (E, M ). In particular, or any lex category C we have the discrete bc C d and the codiscrete bc C c on C. In C d, E = E = isoc and M = M = arc. Conversely in C c, E = E = arc and M = M = isoc. In C d all object are discrete: S = C, while is the only connected object. Conversely, in C c all objects are connected, while is the only set. So, or any C d ( C c ) id : (! : ) is the relection map o in sets and, or any x :, /x = x\ = x (/x = x\ = id ). Then, or any C d ( C c ), is the (co)discrete category on C(, ).

29 BALANCED CATEGORY THEORY 3 Furthermore, i C d and x, y :, then [x, y] is pointless i x y, while [x, x] =. So the same is true or (x, y). On the other hand, i C c, then [x, y] = and (x, y) =, or any x, y :. Indeed, or any arrow u x,y : x y, the arrow interval [u x,y ] is simply x, y :. On the category Pos o posets, we can consider the bc in which m M (M ) i it is, up to isomorphisms, a lower-set (upper-set) inclusion, and e : P is in E (E ) i it is a coinal (coinitial) mapping in the classical sense: or any x, there is a P such that x ea (ea x). The category o internal sets is S = 2, and the components unctor Pos 2 reduces to the non-empty predicate. For any poset and any point x, the slice /x (x\) is the principal lower-set (upper-set) generated by x. As in Cat, is isomorphic to itsel. This example also shows that the underlying unctor may not preserve inal maps: a poset is internally connected i it is not empty, while = may well be not connected in Cat. Given a map p : P, a colimiting cone λ : p x simply indicates that x is the sup o the set o points pa, a P. The colimit is absolute i such a sup is in act a maximum. A map : Y is adequate i any y Y is the sup o the x which are less than or equal to y. Furthermore, given x, y :, [x, y] has its usual meaning: it is the interval o the points z such that x z y. So, (x, y) is the truth value o the predicate x y, and the enriching o in internal sets gives the usual identiication o posets with categories enriched over 2. By restricting to discrete posets, we get the bc associated to the epi-mono actorization system on Set. Let Gph be the category o relexive graphs, with the actorization systems (E, M) and (E, M ) generated by t : 2 and s : 2 respectively, where 2 is the arrow graph (see [Pisani, 2007a] and [Pisani, 2007b]). In this case is the ree category on, and the underlying unctor is the ree category unctor Gph Cat. Furthermore, i Gph and x, y :, then [x, y] is the graph which has as objects the pairs o consecutive paths α, β, with α : x z and β : z y, while there is an arrow α, β α, β over a : z z i α = aα and β = β a. In Gph, S = Set and the components o [x, y] are the paths x y. Thus, the enriching (x, y) o (x, y) is in act an isomorphism. 6. Conclusions There are at least three well-estabilished abstractions (or generalizations) o category theory: enriched categories, internal categories and 2-category theory. Each o them is best suited to enlighten certain o its aspects and to capture new instances; or example,

30 4 CLAUDIO PISANI monads and adjunctions (via the triangular identities) are surely 2-categorical concepts, while enriched categories subsume many important structures and support quantiications. Here, we have based our abstraction on inal and initial unctors and discrete ibrations and opibrations, whose decisive relations are encoded in the concept o balanced actorization category. In this context, natural transormations and/or exponentials are no more basic notions; rather, we have seen how the universal exactness properties o C and those depending on the two actorization systems, complement each other in an harmonious way (in particular, pullbacks and the relection in discrete (op)ibrations) to give an eective and natural tool or proving categorical acts. Thus we hope to have shown that balanced category theory oers a good perspective on several basic categorical concepts and properties, helping to distinguish the trivial ones (that is, those which depend only on the bc structure o Cat) rom those regarding peculiar aspects o Cat (such as colimits, lextensivity, power objects and the arrow object). The latter have been partly considered in [Pisani, 2007c] and deserve urther study. Summarizing, balanced category theory is simple: a bc is a lex category with two reciprocally stable actorization systems generating the same (internal) sets; expressive: many categorical concepts can be naturally deined in any bc; eective: simple universal properties guide and almost orce the proving o categorical properties relative to these concepts; symmetrical (or balanced ): the category Cat in itsel does not allow one to distinguish an object rom its dual; this is ully relected in balanced category theory; sel-ounded: it is largely enriched over its own internal sets, providing in a sense its own oundation (see or example [Lawvere, 2003] and [Lawvere, 966]). Reerences J. Adámek, R. El Bashir, M. Sobral, J. Velebil (200), On Functors which are Lax Epimorphisms, Theory and Appl. Cat. 8, F. Borceux (994), Handbook o Categorical Algebra (Basic Category Theory), Encyclopedia o Mathematics and its applications, vol. 50, Cambridge University Press. M.M. Clementino, E. Giuli, W. Tholen, (996), Topology in a Category: Compactness, Portugal. Math. 53(4), F.W. Lawvere (966), The Category o Categories as a Foundation or Mathematics, Proceedings o the Conerence on Categorical Algebra, La Jolla, 965, Springer, New York, -20.

31 BALANCED CATEGORY THEORY 5 F.W. Lawvere (994), Unity and Identity o Opposites in Calculus and Physics, Proceedings o the ECCT Tours Conerence. F.W. Lawvere (2003), Foundations and Applications: Axiomatization and Education, Bull. Symb. Logic 9(2), C. Pisani (2007a), Components, Complements and the Relection Formula, Theory and Appl. Cat. 9, C. Pisani (2007b), Components, Complements and Relection Formulas, preprint, ariv:math.ct/ C. Pisani (2007c), Categories o Categories, preprint, ariv:math.ct/ R. Street and R.F.C. Walters (973), The Comprehensive Factorization o a Functor, Bull. Amer. Math. Soc. 79(2), via Gioberti 86, 028 Torino, Italy. pisclau@yahoo.it This article may be accessed at or by anonymous tp at tp://tp.tac.mta.ca/pub/tac/html/volumes/20/6/20-06.{dvi,ps,pd}

32 THEORY AND APPLICATIONS OF CATEGORIES (ISSN 20-56) will disseminate articles that signiicantly advance the study o categorical algebra or methods, or that make signiicant new contributions to mathematical science using categorical methods. The scope o the journal includes: all areas o pure category theory, including higher dimensional categories; applications o category theory to algebra, geometry and topology and other areas o mathematics; applications o category theory to computer science, physics and other mathematical sciences; contributions to scientiic knowledge that make use o categorical methods. Articles appearing in the journal have been careully and critically reereed under the responsibility o members o the Editorial Board. Only papers judged to be both signiicant and excellent are accepted or publication. Full text o the journal is reely available in.dvi, Postscript and PDF rom the journal s server at and by tp. It is archived electronically and in printed paper ormat. Subscription inormation. Individual subscribers receive abstracts o articles by as they are published. To subscribe, send to tac@mta.ca including a ull name and postal address. For institutional subscription, send enquiries to the Managing Editor, Robert Rosebrugh, rrosebrugh@mta.ca. Inormation or authors. The typesetting language o the journal is TE, and L A TE2e strongly encouraged. Articles should be submitted by directly to a Transmitting Editor. Please obtain detailed inormation on submission ormat and style iles at Managing editor. Robert Rosebrugh, Mount Allison University: rrosebrugh@mta.ca TEnical editor. Michael Barr, McGill University: barr@math.mcgill.ca Transmitting editors. Richard Blute, Université d Ottawa: rblute@uottawa.ca Lawrence Breen, Université de Paris 3: breen@math.univ-paris3.r Ronald Brown, University o North Wales: ronnie.probrown (at) btinternet.com Aurelio Carboni, Università dell Insubria: aurelio.carboni@uninsubria.it Valeria de Paiva, erox Palo Alto Research Center: paiva@parc.xerox.com Ezra Getzler, Northwestern University: getzler(at)northwestern(dot)edu Martin Hyland, University o Cambridge: M.Hyland@dpmms.cam.ac.uk P. T. Johnstone, University o Cambridge: ptj@dpmms.cam.ac.uk Anders Kock, University o Aarhus: kock@im.au.dk Stephen Lack, University o Western Sydney: s.lack@uws.edu.au F. William Lawvere, State University o New York at Bualo: wlawvere@acsu.bualo.edu Jean-Louis Loday, Université de Strasbourg: loday@math.u-strasbg.r Ieke Moerdijk, University o Utrecht: moerdijk@math.uu.nl Susan Nieield, Union College: nieiels@union.edu Robert Paré, Dalhousie University: pare@mathstat.dal.ca Jiri Rosicky, Masaryk University: rosicky@math.muni.cz Brooke Shipley, University o Illinois at Chicago: bshipley@math.uic.edu James Stashe, University o North Carolina: jds@math.unc.edu Ross Street, Macquarie University: street@math.mq.edu.au Walter Tholen, York University: tholen@mathstat.yorku.ca Myles Tierney, Rutgers University: tierney@math.rutgers.edu Robert F. C. Walters, University o Insubria: robert.walters@uninsubria.it R. J. Wood, Dalhousie University: rjwood@mathstat.dal.ca