NATURAL WEAK FACTORIZATION SYSTEMS

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1 NATURAL WEAK FACTORIZATION SYSTEMS MARCO GRANDIS AND WALTER THOLEN Abstract. In order to acilitate a natural choice or morphisms created by the (let or right) liting property as used in the deinition o weak actorization systems, the notion o natural weak actorization system in the category K is introduced, as a pair (comonad, monad) over K 2. The link with existing notions in terms o morphism classes is given via the respective Eilenberg Moore categories. Dedicated to Jiří Rosický at the occasion o his sixtieth birthday 1. Introduction Weak actorization systems (L, R) play a key role or Quillen model categories, deined in terms o (coibrations, trivial ibrations) and (trivial coibrations, ibrations). While the two players L and R have good stability properties under some colimits and limits, respectively, unlike the counterparts appearing in the orthogonal actorization systems, they ail to be closed under the ormation o all colimits and limits, coequalizers and equalizers among them. The general reason or that, o course, lies in the act that morphisms provided by the (right or let) liting property are not chosen naturally, even in the presence o a unctorial realization or the system. While the notion o lax actorization algebra presented in [RT1] leads to a natural extension o the presentation o orthogonal actorization systems as Eilenberg Moore algebras or the squaring monad on CAT (see [C], [KT], [RW]) it does not give a remedy or the deect just described. This paper, thereore, takes a new look at what unctorial weak actorization systems ought to be and, ater a careul recollection o the existing notions, proposes to deine such systems in the category K by a pair (comonad, monad) in K 2, under suitable conditions. A irst step in the passage towards a pair o morphism classes in K is made by considering the respective Eilenberg Moore categories. Their (co)algebras are morphisms in K that come with a (co)algebraic structure, and it is that structure that contains all inormation to construct litings naturally. O course, as in all Eilenberg Moore categories, all (co)limits Date: October 6, Partially supported by MIUR Research Projects. Partial inancial assistance by NSERC is grateully acknowledged. 1

2 2 MARCO GRANDIS AND WALTER THOLEN are now created over K 2. I both the comonad and monad are idempotent, the structures become properties, and the approach takes us back to the traditional presentation o orthogonal actorization systems in terms o two subclasses o mork. Those results are presented in Sections 2 and 3, and examples ollow in Section 4. These encompass the examples treated in [RT1]. Furthermore, all coibrantly generated weak actorization systems in locally initelypresentable categories are natural, but we must leave a presentation o the rather lengthy and cumbersome proo to a later paper. 2. Natural weak actorization systems 2.1. Morphisms (u, v): g in the category K 2 are commutative squares u v o morphisms in the category K. The two projections give the (vertical) domain and the codomain unctors dom, cod: K 2 K, and there is a natural transormation κ: dom cod with κ =. According to [RT1], a unctorial realization o a weak actorization system o K is given by a unctor F : K 2 K and natural transormations λ: dom F, ρ: F cod such that commutes and, or all obk 2, F λ ρ κ dom cod λ L F := {g s : λ g = s g, ρ g s = 1}, ρ R F := {g p : ρ g = g p, p λ g = 1}. Now, (L F, R F ) is indeed a weak actorization system (ws) o K in the sense o [AHRT], and any ws (L, R) that admits a unctorial realization (F, λ, ρ) (so that all properties above are satisied, with L F, R F traded or L, R) necessarily satisies L = L F, R = R F (see Theorem 2.4 o [RT1]). These data provide or every morphism a commutative diagram g ρ L F L F λ A λ L λ R ρ F R B ρ RF where we have written L or λ (considered as an object o K 2 ), and R or ρ, and where ρ L, λ R have splittings s, p with (1) λ L = s λ, ρ L s = 1, ρ R = ρ p, p λ R = 1.

3 NATURAL WEAK FACTORIZATION SYSTEMS 3 Unortunately, these splittings (that are used or constructing litings ) need to be chosen each time and add a non constructive aspect to the notion o unctorial realization o ws The natural transormation λ in 2.1 is equivalently described by a unctor L: K 2 K 2 with (2) doml = dom, codl = F, κl = λ. Now ρ may be described by a natural transormation Φ: L 1 K 2 (3) domφ = 1 dom, codφ = ρ; explicitly, Φ is the commutative square A 1 A A with λ F ρ B Likewise, when we present ρ as a unctor R: K 2 K 2 with (4) domr = F, codr = cod, κr = ρ, then λ may be presented as a natural transormation Λ: 1 K 2 R with (5) domλ = λ, codλ = 1 cod ; hence, Λ is the commutative square A λ F B 1 B B Now, let us suppose that there is a natural choice or the splittings s, p satisying (1). Hence, we suppose that there are natural transormations σ : F F L, π : F R F with (6) λl = σ λ, ρl σ = 1 F, ρr = p π, π λr = 1 F. Equivalently, σ and π can be described by natural transormations Σ: L LL and Π: RR R with domσ = 1 (7) dom, codσ = σ, ΦL Σ = 1 L, codπ = 1 cod, domπ = π, Π ΛR = 1 R. Explicitly, Σ and Π are respectively the commutative diagrams ρ A 1 A A F R π F r λ λ L ρ R ρ F σ F L B 1 B B

4 4 MARCO GRANDIS AND WALTER THOLEN It seems natural to assume that (L, Φ, Σ) and (R, Λ, Π) actually orm a comonad and a monad on K 2, respectively, so that in addition to (7) one has (8) (9) LΦ Σ = 1 L, ΣL Σ = LΣ Σ, Π ΛR = 1 R, Π ΠR = Π RΠ. Alternatively, in addition to (6) one requires F (1 a, ρ ) σ = 1 F, σ L σ = F (1 A, σ ) σ, π F (λ, 1 B ) = 1 F, π π R = π F (σ F, 1 B ). This leads to the Deinition 2.4 below, or which the ollowing setting is considered Let CAT//K be the 2 category whose objects are unctors with values in K, whose arrows F : U V are commutative triangles A U F V K o unctors, and whose 2-cells are natural transormations α: F G with V α = 1 U. A monad (T, η, µ) on U in CAT//K is simply a monad (T, η, µ) on A in CAT with UT = U, Uη = 1 U, Uµ = 1 U. A comonad on U in CAT//K is described analogously Deinition. A natural weak actorization system (natural ws) in a category K is a pair (L, R) such that (1) L = (L, Φ, Σ) is a comonad on dom in CAT//K, (2) R = (R, Λ, Π) is a monad on cod in CAT//K, (3) codl = domr, codφ = κr, domλ = κl. From our discussion in 2.1, 2.2 one sees immediately: 2.5. Proposition. (i) Let us be given a unctor F : K 2 K and natural transormations λ: dom F, ρ: F cod, σ : F F L, π : F R F, (where L, R: K 2 K 2 respectively represent λ, ρ as in 2.2). I ρ λ = κ: dom cod and (6) holds, then (F, λ, ρ) is a unctorial ws (with a natural choice o splittings). (ii) There is a bijection between natural ws (L, R) = (L, Φ, Σ; R, Λ, Π) on K, as in 2.4, and systems (F, λ, ρ, σ, π) as in (i) which satisy ρ λ = κ and the conditions (6), (9). Given a natural ws (L, R), one deines the system: F := codl = domr, λ := domλ, ρ := codφ, σ := codσ, π := domπ. B

5 NATURAL WEAK FACTORIZATION SYSTEMS 5 Conversely, given such a system, one deines the associated natural ws as in For a natural ws (L, R), let L L and R R denote the Eilenberg Moore categories o L and R, respectively. Hence, an object in L L is a pair (, (i, s): L) such that 1 Φ L (i,s) (i,s) L L(i,s) Σ LL commutes in K 2. Since necessarily i = 1, with 2.5 we can simply write obl L = {( : A B, s: B F ) λ = s, ρ s = 1 B, σ s = F (1 A, s) s}; a morphism (u, v): (, s) (g, t) in L L is a morphism (u, v): g in K 2 which satisies t v = F (u, v) s. Hence, objects in L L are, in the setting o 2.1, simply morphisms o L F (see 2.1) that come with a given splitting s which, in addition, must be compatible with the (co)multiplicative structure o L; morphisms o L L must respect the given splittings. Similarly one obtains obr R = {( : A B, p: A F ) ρ = p, pλ = 1 A, pπ = p F (p, 1 B )}, with morphisms (u, v): (, p) (g, q) in R R satisying u p = q F (u, v) Corollary. Let (L, R) be a natural ws o K. Then, in the notation o 2.1 and 2.2, every morphism : A B actors as A λ F ρ B with (λ, σ ) L L and (ρ, π ) R R. Furthermore, or all (, s) L L and (g, q) R R and (u, v): g in K 2, there is a naturally chosen diagonal morphism w as in A u C w g B v D namely: w = q F (u, v) s. I K has colimits (resp. limits) o a given type, then L L (resp. R R ) also has them, ormed as in K 2. Proo. The orgetul unctors L L K 2 and R R K 2 create colimits and limits, respectively.

6 6 MARCO GRANDIS AND WALTER THOLEN 3. Orthogonal actorization systems 3.1. Recall that an orthogonal actorization system in a category K may be given by a pair (L, R) o classes o morphisms o K, both closed under composition with isomorphisms, such that K = R L, and or all L, g R and (u, v): g in K 2 there is a unique morphism w with w = u and gw = v. Equivalently, it may be described by a unctor F : K 2 K with F E = 1 K and F (1, ) L 1 F := {g F (g, 1) iso}, F (, 1) R 1 F := {g F (1, g) iso} (see Theorem A o [KT]); here E : K K 2 is the ull embedding with A 1 A, and the morphisms (1, ), (, 1) stem rom the generic actorization (1 A,) (,1 B ) 1 A 1 B E=(,) in K 2, or every : A B in K. Note that such a unctor F gives rise to a natural ws, with λ = F (1, ), ρ = F (, 1) and σ, π both isomorphisms. Since orthogonal actorization systems are weak actorization systems ([AHS], [AHRT]), one has L 1 F = L F and R 1 F = R F Theorem. Orthogonal actorization systems o K are equivalently described as those natural ws (L, R) or which L and R are idempotent. In this case the Eilenberg Moore categories L L and R R are equivalent to L F and R F, considered as ull corelective and relective subcategories o K 2, with F as in 2.5. Proo. It is clear that an orthogonal actorization system gives rise to a natural ws (L, R) with L, R idempotent, see 3.1. Conversely, having such a natural ws, in the notation on 2.5 one has σ, π iso. In order to be able to apply Theorem A o [KT], we just have to show F E = 1 K ; in act, it is suicient to show F E = 1 K (see 2.2 o [KT]). Hence, we must show that or every object A in K, the morphisms l = λ 1A and r = ρ 1A are isos (see 2.6 o [RT1]). To this end, one considers the morphism (l, 1 A ): 1 A r in K 2 and notices that F (l, 1 A ) is iso since π 1A is iso (by (9) o 2.2). Now we actor (l, 1 A ) as 1 A (1 A,l) l (l,r) r. By idempotency o R and L, we may assume ρ l = 1 B and λ r = 1 B (with B = F 1 A ), so that an application o F to the actorization o (l, 1 A ) leads

7 NATURAL WEAK FACTORIZATION SYSTEMS 7 to the ollowing commutative diagram A 1 A A l B l B F (1 A,l) l 1 B F (l,r) B 1 B Now one has: r A 2 l B 1 B l r = F (l, r) l r (commutativity o 1 ) = F (l, r) F (1 A, l) (commutativity o 2 ) = F (l, 1 A ) (since r l = 1 A ) Hence, l r is an isomorphism, and then both l, r must be isos. r A r 3.3. Remarks. Here are three related open problems. 1. The proo o 3.2 uses idempotency o both players o the natural ws (L, R). We do not know whether idempotency o one o them implies idempotency o the other. 2. In a ws (L, R), one class determines the other. We do not know i, or a natural ws (L, R), L and R determine each other. 3. Are there distinct natural ws inducing the same unctorial ws, that is: do there exist distinct natural ws (L, R), (L, R ) with (in the notation o 2.5) F = F, λ = λ, ρ = ρ, but σ σ or π π? 4. Examples 4.1. In a category K with binary products, every map : X Y has a well-known graph-actorization (10) = ρ λ = p 2 1, : X X Y Y, where p 2 is the second projection o the cartesian product. Dually, in a category K with binary sums, a map : X Y has a cographactorization (11) = ρ.λ = [, 1]i 1 : X X + Y Y, where i 1 is the irst injection o the sum and ρ = [, 1]: X + Y Y has co-components ρ i 1 =, ρ i 2 = 1 Y. Plainly, both actorizations are unctorial. Furthermore, they can be made into natural ws, by dual procedures: below, we describe the second, which, when K is lextensive [CJ], leads to the weak actorization system (coproduct injections, split epimorphisms), recently considered in [RT2]. For the irst, it is well-known that, when K = Set, L F coincides with the class o split monos, which amounts to the injective mappings except the

8 8 MARCO GRANDIS AND WALTER THOLEN empty embeddings in non-empty sets, while R F contains all the surjective mappings and empty inclusions Proposition. [The cograph actorization] Let K be a category with binary sums. The cograph actorization o a map, recalled above in (11), can be made into a natural ws, so that, i K is lextensive, the maps o L F, R F can be characterised as coproduct injections and split epimorphisms, respectively. Proo. The cograph actorization is unctorial, with (12) F : K 2 K, F ( : X Y ) = X + Y, F ((u, v): g) = u + v, and the natural transormations λ, ρ deined above, in (11). In order to make it into a natural ws, let us deine the ollowing two natural transormations, related with the actorization o L = i 1 and R = [, 1] displayed in the diagram below (13) σ : F F L, σ = [i 1, i 3 ]: X + Y X + X + Y, π : F R F, π = [i 1, i 2, i 2 ]: X + Y + Y X + Y, X X X + Y X + Y i 1 X + Y σ i 1 X + X + Y [i 1,i 2 ] X + Y + Y π 1 X + Y 1 [i 1,i 1,i 2 ] [,1,1] X + Y X + Y Y Y The last axiom (9) is easily veriied. Let now K be lextensive [CJ], and let us proceed to characterise the sets L F, R F. In the let diagram below [,1] (14) X Y X g Y 1 Y 2 s s 1 s 2 X i1 X + Y Y X [,1] g Y Y Y j 2 the morphism s decomposes as a sum s 1 + s 2 : Y 1 + Y 2 X + Y, and is the composition o a map g : X Y 1 with the injection; but this g is an isomorphism, since the previous diagram restricts to the central diagram above; thus, : X Y is a coproduct-injection. One easily sees that an L-coalgebra is precisely a pair (, s) as above, since the last condition, σ s = F (1 X, s) s, is automatically satisied. Moreover, taking into account the right diagram above, s is determined by the injections : X Y and j 2 : Y 2 Y. Thereore, an L-coalgebra can be equivalently described as a pair o maps ( : X Y, : X Y ) which are the injections o a sumdecomposition o Y.

9 Finally, in the diagram NATURAL WEAK FACTORIZATION SYSTEMS 9 (15) X t X i1 X + Y [,1] Y Y the map t must be o the orm [1 X, t ], with t = 1 Y, whence R F coincides with the set o split epis (and this holds in an arbitrary category K). Again, an R-algebra is just such a pair (, t), which amounts to a splitting t = 1 Y We consider now two dual actorizations o a unctor, or the category CAT (c. [Gy], I, 1.11) (a) First, we can actor an arbitrary unctor : X Y through the comma category F = ( Y ), via a let adjoint right inverse i and a unctor q (16) X i p ( Y ) q Y qi = (i p), (17) i(x) = (x, x; 1: x x), q(x, y; b: x y) = y, (18) p(x, y; b: x y) = x, η : 1 X = pi, ε: ip 1 F, ε (x,y; b) = (1 x, b): (x, x; 1 x ) (x, y; b). (b) Dually, we can also actor an arbitrary unctor g : Y X through the comma category Gg = (X g), via a right adjoint right inverse j and a unctor p (19) Y j q (X g) p X pj = g (q j), (20) j(y) = (gy, y; 1: gy gy), p(x, y; b: x gy) = x, (21) q(x, y; b: x gy) = y, ε: qj = 1 Y, η : 1 Gg jq, η (x,y; b) = (b, 1 y ): (x, y; b) (gy, y; 1 gy ). We prove below that these actorizations can be made into natural ws. I g, then we can identiy ( Y ) with (X g), and ind a actorization o adjunctions; the latter is not unctorial on the category o adjunctions, but on a suitable double category o unctors and adjunctions (see [GP], 3.5) in a sense which will be dealt with in a sequel.

10 10 MARCO GRANDIS AND WALTER THOLEN 4.4. Let us construct a natural ws or the actorization 4.3(a), through ( Y ) (the other can be obtained by duality). With the previous notation, we have a unctor (22) F : CAT 2 CAT, F ( : X Y ) = ( Y ), F ((u, v): g): F F g, F (u, v)(x, y; b: x y) = (ux, vy; vb: vx = gux vy), and a unctorial actorization, deined by the natural transormations: (23) λ: dom F, λ = i: X ( Y ), ρ: F cod, ρ = q : ( Y ) Y, ρλ = κ: dom cod. First, the unctor L = i: X ( Y ) actors as ollows through the comma category F L = (L ( Y )), whose general object is o type (x, x, y; a: x x; b: x y) (24) L = (RL) (LL): X (L ( Y )) ( Y ), LL(x) = (x, x, x; 1 x, 1 x ), RL(x, x, y; a: x x; b: x y) = (x, y; b: x y). The natural transormation σ, related with the previous actorization o L, is deined as: (25) σ : F F L, σ : ( Y ) (L ( Y )), σ (x, y; b: x y) = (x, x, y; 1 x, b: x y), X L=i ( Y ) σ X LL (L ( Y )) 1 RL ( Y ) ( Y ) Second, we actor R = q : ( Y ) Y through F R = (q Y ), whose general object is o type (x, y, b: x y; y ; b : y y ), (26) q = R = (RR) (LR): ( Y ) (q Y ) Y, (LR)(x, y; b: x y) = (x, y, b: x y; y, 1: y y), (RR) = Rq : (q Y ) Y, (RR)(x, y, b: x y; y, b : y y ) = y, and we deine the natural transormation π, related with the previous actorization o R (27) π : F R F, π : (q Y ) ( Y ), (π )(x, y, b: x y; y, b : y y ) = (x, y ; b b: x y ),

11 NATURAL WEAK FACTORIZATION SYSTEMS 11 ( Y ) ( Y ) LR (q Y ) RR π 1 ( Y ) Veriying the remaining axioms is straightorward, i long Proposition. This structure deines a natural ws. Proo. We will use various comma categories, among which (28) writing their projections as ollows (29) Y C = ( Y ), C = (g Y ), C = (L C), C = (LL C ), p: C X, q : C Y, ω : C Y 2, p : C X, q : C Y, ω : C Y 2, p : C X, q : C C, ω : C C 2, P : C X, Q: C C, Ω: C C 2. Computing the components o the transormations introduced above (in 4.4) quickly becomes heavy and conusing. Thereore, let us note that the unctor F deined above is determined as ollows by the projections o C = (g Y ): (30) p F (u, v) = up: ( Y ) X, q F (u, v) = vq : ( Y ) Y, ω F (u, v) = vω : vp vq (vp = gup). Again, the natural transormation σ is determined by the projections o C = (L ( Y )): (31) p σ = p: ( Y ) X, q σ = 1: ( Y ) ( Y ), (ω σ )(x, y; b: x y) = (1 x, b): (x, x; 1) (x, y; b), and also the last equation can be made ree o components, rewriting it as: (32) p ω σ = 1 p, q ω σ = ω : p q, (which amounts to using the 2-dimensional universal property o the comma ( Y )). Now, to test the condition F (1 X, ρ ).σ = 1 F we use the projections p, q, ω o ( Y ), together with the characterisation (30) o the unctors o type F (u, v) p F (1 X, ρ )σ = p σ = p = p 1 F, q F (1 X, ρ )σ = ρ q σ = q = q 1 F, ω F (1 X, ρ )σ = ρ ω σ = qω σ = ω = ω 1 F. Y q

12 12 MARCO GRANDIS AND WALTER THOLEN Similarly, to veriy that σ L σ = F (1 X, σ ) σ : F F LL, we use the projections P, Q, Ω o C = F LL = (LL (L ( Y ))), urther replacing Ω with its projections p Ω, q Ω (as in (32)) P F (1 X, σ )σ = p σ = P σ L σ, Q F (1 X, σ )σ = σ q σ = σ = Q σ L σ, p Ω F (1 X, σ )σ = p σ ω σ = pω σ = 1 p = 1 p σ = p Ω σ L σ, q Ω F (1 X, σ )σ = q σ ω σ = ω σ = q Ω σ L σ. We end with veriying the remaining two conditions o (9) on π. For the irst: π (F (λ, 1 Y )(x, y; b) = (π )(λ (x), y, b) = (π )(x, x, 1 x, y, b) = (x, y; b). For the second, ater computing π q : (Rq Y ) (q Y ), (π q )(x, y, b: x y; y, b : y y ; y, b : y y ) = (x, y, b: x y; y, b b : y y ), we have (working on components): (π π q )(x, y, b; y, b ; y, b ) = (π )(x, y, b; y, b b ) = (x, y ; b b b: x y ), π F (π, 1 Y )(x, y, b; y, b ; y, b ) = (π )(π (x, y, b; y, b ), y, b ) = (π )(x, y ; b b, y, b ) = (x, y ; b b b: x y ). Reerences [AHRT] J. Adámek, H.Herrlich, J.Rosický and W.Tholen, Weak actorization systems and topological unctors. Appl. Categorical Structures 10 (2002) [AHS] J.Adámek, H. Herrlich, G.E.Strecker, Abstract and Concrete Categories, Wiley (New York 1990). [CJ] A. Carboni and G. Janelidze, Decidable (= separable) objects and morphisms in lextensive categories, J. Pure Appl. Algebra 110 (1996) [C] L. Coppey, Algèbres de decompositions et précatégories, Diagrammes 4 (Suppl.) (1980). [GP] M. Grandis and R. Paré, Limits in double categories, Cah. Topol. Géom. Diér. Catég. 40 (1999) [Gy] J.W. Gray, Formal category theory: adjointness or 2-categories, Lecture Notes in Mathematics Vol. 391, Springer-Verlag (Berlin 1974). [KT] M. Korostenski, W. Tholen, Factorization systems as Eilenberg Moore algebras, J. Pure Appl. Algebra 85 (1993) [RT1] J. Rosický, W. Tholen, Lax actorization algebras, J. Pure Appl. Algebra 175 (2002) [RT2] J. Rosický, W. Tholen, Factorization, ibration and torsion, preprint (York University 2006). [RW] R. Rosebrugh and R.J. Wood, Coherence or actorization algebras, Theory Appl. Categories 10 (2002)

13 NATURAL WEAK FACTORIZATION SYSTEMS 13 Dipartimento di Matematica, Università di Genova, Via Dodecaneso 35, Genova, Italy address: Department o Mathematics and Statistics, York University, Toronto, ON M3J 1P3, Canada address: tholen@mathstat.yorku.ca