ON THE CONSTRUCTION OF FUNCTORIAL FACTORIZATIONS FOR MODEL CATEGORIES

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1 ON THE CONSTRUCTION OF FUNCTORIAL FACTORIZATIONS FOR MODEL CATEGORIES TOBIAS BARTHEL AND EMIL RIEHL Abstract. We present general techniques or constructing unctorial actorizations appropriate or model structures that are not known to be coibrantly generated. Our methods use algebraic characterizations o ibrations to produce actorizations that have the desired liting properties in a completely categorical ashion. We illustrate these methods in the case o categories enriched, tensored, and cotensored in spaces, proving the existence o Hurewicz-type model structures, thereby correcting an error in earlier attempts by others. Examples include the categories o (based) spaces, (based) G-spaces, and diagram spectra among others. 1. Introduction In the late 1960s, Quillen introduced model categories, which axiomatize and thereby vastly generalize a number o classical constructions in algebraic topology and homological algebra. Somewhat ironically, a model category o spaces whose coibrations were the classical, meaning Hurewicz, coibrations and whose ibrations were the Hurewicz ibrations, established in [Str72], is somewhat diicult to obtain. The source o diiculties is two-old. One has to do with subtleties involving point-set topology. The other obstacle is due to the act that this model structure is not known to be coibrantly generated: while its ibrations are certainly deined by a liting property, this liting property is against a proper class o maps, and not simply a set. In the absence o this set-theoretical condition, there is no general procedure or constructing actorizations whose let and right actors satisy the desired liting properties. In particular, while there exist natural notions o Hurewicz coibrations and ibrations, Strøm s ideas seem to be conined to the category o spaces. Only in the last decade has there been progress toward Hurewicz-type model structures in one o their most natural settings: categories enriched, tensored, and cotensored over spaces [SV02, Col06b]. Natural examples include based and unbased spaces, G-spaces, and diagram spectra. In the presence o a Quillen-type model structure, a Hurewicz-type model structure gives rise to a mixed model structure by an observation o Cole [Col06a]. May and Ponto have advertized mixed model structures on topological spaces and categories o spectra [MP12] which combine Quillen- and Hurewicz-type model structures. The weak equivalences and ibrations are the weak homotopy equivalences and the Hurewicz ibrations; the coibrant objects on spaces are the spaces o the homotopy types o CW complexes. May and Ponto argue that Date: January 3, The irst author was supported in part by a scholarship rom Worcester College, Oxord, by the EPSRC, and by the Studienstitung des deutschen Volkes. The second author was supported in part by an NSF postdoctoral ellowship. 1

2 2 TOBIAS BARTHEL AND EMIL RIEHL this is the model structure in which homotopy theory has always implicitly worked. For example, in the parametrized world [MS06], actual cell complexes are subtle and working in the mixed model structure promises a real simpliication. However, the diiculties inherent in this topic resurace in a mistake, recently noticed by Richard Williamson, in a crucial proo in [Col06b], throwing the existence o such model structures once more into doubt. The result claimed by Cole and proven here allows this philosophy to be applied to topological categories satisying a smallness condition. In this paper, we present general techniques or producing actorizations or non-coibrantly generated model categories that make use o the algebraic perspective on ibrations, explained below. We impose algebraic structures in order to replace point-set level arguments step-by-step with categorical ones, ormulating a proo that is not speciic to the category at hand. An interesting eature o this perspective is that it precisely identiies the law in Cole s proo and simultaneously suggests its solution. A test case, spelled out in Section 3 and 4, illustrates how we might use the algebraic perspective to circumvent certain point-set level arguments in the construction o actorizations. Malraison and May [Mal73, May75] observed that the Moore path space allows or an algebraic characterization o Hurewicz ibrations. Based on their results, we present a new actorization or the Strøm model structure on topological spaces, which in particular avoids Strøm s work on Hurewicz coibrations [Str66, Str68]. In act, the construction o this actorization generalizes to any topologically bicomplete category, and we suspect that our arguments could also be used to establish the existence o Hurewicz-type model structures there. However, we preer an alternative approach which can be more easily adapted to other (non-topological) contexts. This construction, outlined below, takes the ordinary path space as its point o departure but requires more elaborate algebraic machinery. We explain our methods in analogy with the coibrantly generated case. The starting point is an observation about Quillen s small object argument, due to Richard Garner [Gar07, Gar09]. In a coibrantly generated model category, a map is a ibration i and only i it has the right liting property against a particular set o arrows; this is the case just when one can choose a solution to each such liting problem. These chosen solutions are encoded as a solution to a single liting problem involving the step-one actorization o Quillen s small object argument, which actors a map as a trivial coibration ollowed by a map that is not typically a ibration. Put another way, a map is a ibration i and only i it admits the structure o an algebra or the (pointed) endounctor that sends a map to its step-one right actor. In this way, the Quillen s step-one actorization gives rise to an algebraic characterization o the ibrations in any coibrantly generated model category. By contrast, the Hurewicz ibrations in a topologically bicomplete category are not characterized by a liting property against a set o maps. Nonetheless, we show that the step-one actorization produced by Cole, while not actoring a map into a trivial coibration ollowed by a ibration, nonetheless provides a precise algebraic characterization o the ibrations. As above, the right actor o this actorization is a pointed endounctor whose algebras are precisely the ibrations. A general categorical construction replaces this unctorial actorization with another whose right unctor is a monad whose algebras are again precisely the ibrations. In

3 ON THE CONSTRUCTION OF FUNCTORIAL FACTORIZATIONS 3 particular, the right actor is itsel a ree algebra and thus a ibration or entirely ormal reasons: no point-set topology is necessary or this proo. It remains to show that the let actor is a trivial coibration or the model structure; again, on account o the algebraic perspective, no point-set topology is required. Instead, we use a composition criterion to show that the let unctor so-constructed is a comonad. In particular, the let actor is a coalgebra or said comonad. For easy ormal reasons, such coalgebras lit (canonically) against algebras or the right unctor, which proves that the let actor is a trivial coibration. This yields our main theorem. Theorem. On any category C enriched, tensored, and cotensored over spaces and satisying a mild set-theoretical condition there exists a model structure whose ibrations, coibrations, and weak equivalences are the h-ibrations, strong h-coibrations, and homotopy equivalences respectively. Abstractly our approach can be described as ollows. Suppose given a category with two distinguished classes o morphisms: a class o ibrations that are characterized as algebras or a pointed endounctor and a class o trivial coibrations that are determined by a liting property against the ibrations. In practice, the ormer oten arise as maps admitting solutions or a certain unctorially constructed generic liting problem ; the pointed endounctor is obtained by pushing out this square. I the pointed endounctor obtained in this manner satisies a certain smallness condition, we can then construct a candidate unctorial actorization by reely replacing it by a monad. I either o the ollowing conditions hold (a) the category o algebras or the pointed endounctor admits a vertical composition law, deined in Section 4 below, or (b) the let actor in the actorization associated to the pointed endounctor is a comonad then these unctors actor a map as a trivial coibration ollowed by a ibration. By work o Garner [Gar09], both (a) and (b) hold automatically in the coibrantly generated case, which is thereby subsumed. In act, the methods o this paper generalize eortlessly to produce algebraic actorizations or any enriched bicomplete category equipped with an interval, i.e., a bipointed object, which satisies a certain smallness condition. This can be used, or instance, to construct model structures on categories enriched in chain complexes. More details will appear in a orthcoming paper with Peter May. Let us briely compare this with previous work extending the small object argument to non-coibrantly generated model categories. The main theorem o [Cho06] states that i (i) there is a cardinal κ such that the domains o the arrows in the generating class o trivial coibrations are κ-small and (ii) there exists a unctorial construction o a generic liting problem in the sense o Remark 5.11, then an analogue o Quillen s small object argument can be used to construct appropriate unctorial actorizations. In spaces, the Hurewicz ibrations are generated by the class {A A I} o cylinder inclusions. There are examples o spaces that are κ-small only i κ exceeds the cardinality o their underlying set. Hence, or

4 4 TOBIAS BARTHEL AND EMIL RIEHL topological categories, the condition (i) is unreasonable. By contrast, our conditions (a) and (b) provide suicient control over the let actor to allow us to weaken the smallness condition. The structure o this paper parallels the gradual removal o point-set topology; in particular, we introduce categorical notions and results along the way as needed. In Section 2, we review Strøm s construction o a unctorial actorization or the Hurewicz-type model structure on spaces and indicate why it is not suitable or generalization. In Sections 3 and 4, we introduce the algebraic perspective on ibrations by considering the Moore path unctorial actorization. In Section 5, we discuss Hurewicz type model structures on any complete and cocomplete category that is tensored, cotensored, and enriched in topological spaces, and in Section 6, we prove that our new unctorial actorization establishes their existence. Finally, in the appendix, we explain the problem with Cole s actorization and present a ew more details about our construction. Acknowledgments. We would like to thank Richard Williamson or bringing this problem to our attention and Richard Garner, whose work inspired much o this paper. This work also beneitted rom conversations with Richard Garner, Peter May, Bill Richter, Mike Shulman, and Richard Williamson. We would like to thank the reeree or many helpul comments on earlier versions o this paper. The irst author would also like to thank Harvard University or its hospitality. 2. Strøm s model structure on spaces In this short section, we review a ew select details o Strøm s construction o a model structure on the category o topological spaces and continuous maps whose weak equivalences are homotopy equivalences, ibrations are Hurewicz ibrations, and coibrations are closed Hurewicz coibrations. Remark 2.1. Even though Strøm works in the category o all topological spaces, we restrict ourselves to a convenient category o spaces, denoted Top, which in particular should be cartesian closed. The two most prominent examples are k- spaces and compactly generated weak Hausdor spaces. For a detailed discussion o these point-set issues, we reer the interested reader to [MS06, Ch. 1]. Write I or the unit interval, topologized in the standard way with endpoints 0, 1. Recall that Hurewicz ibrations are those maps in Top that have the homotopy liting property, i.e., the right liiting property with respect to all inclusions A i0 A I. Dually, the Hurewicz coibrations are those maps with the homotopy extension property, i.e., the let liting property against all projections I p0. There is a subtle, but important point here: In order to organize the aorementioned classes o maps into a model structure on Top, we need the (model structure) coibrations and trivial coibrations, i.e., those coibrations that are also homotopy equivalences, to be precisely the maps that lit against the (model structure) trivial ibrations and ibrations, respectively. In general, however, the Hurewicz coibrations and ibrations don t have this property, but this can be ixed by requiring the (model structure) coibrations to be closed Hurewicz coibrations. Remark 2.2. Coibrations in the category o compactly generated spaces are automatically closed, but interestingly this is not the case in the category o k-spaces;

5 ON THE CONSTRUCTION OF FUNCTORIAL FACTORIZATIONS 5 c. [MS06, 1.6.4]. Compare with the notion o strong coibrations, introduced by [SV02], which we discuss in Section 5. The actorization axiom (CM5) is the most diicult one to establish or this model structure, and here it suices to construct the trivial coibration ibration actorization. The main point-set level input to demonstrate this actorization is the ollowing result o Strøm s [Str66, Thm. 3]: Proposition 2.3. I i: A is an inclusion o a strong deormation retract such that there exists a map q : I with q 1 (0) = A, then i is a closed Hurewicz coibration as well as a homotopy equivalence. This enables Strøm to prove: Proposition 2.4. Every continuous map : can be actored as a homotopy equivalence and closed Hurewicz coibration i ollowed by a Hurewicz ibration p. The proo can be ound in [Str72, Prop. 2], building on earlier work [Str68]. Here, we merely describe the construction so as to highlight the diiculties o naïvely extending it to topologically enriched categories. Strøm s actorization makes use o Deinition 2.5. The mapping path space N o : is deined to be the pullback N χ I φ p 0 Strøm s construction 2.6. Any map can be actored as = π j, with j : N the map that sends a point o to the constant path at its image under and π : N evaluation o paths at their endpoint. This map j is not necessarily a coibration, so Strøm actors it through the space E ormed by gluing E = I N (0, 1] (0,1] along j and the inclusion o the hal open interval. The map j actors as i: x (x, 0) ollowed by the natural projection π obtained by including E into N I and projecting to the mapping space. The result is a commutative diagram j i E π π N Using Proposition 2.3 and [Str68, Thm. 8 and 9], Strøm checks that i is a trivial coibration and p := π π is a ibration.

6 6 TOBIAS BARTHEL AND EMIL RIEHL This construction generalizes without problems to any category C enriched, tensored, and cotensored in spaces. In particular, we might deine E to be the pushout (0, 1] i I j (0,1] N (0, 1] i being the map induced by the inclusion (0, 1] I. However, Strøm s characterization o trivial coibrations is not available in the enriched context, so one needs to check directly that, among other things, the ollowing liting problem can be solved or any A: A E A I But E being deined as a colimit, it seems very diicult i not impossible to check that a lit exists and thus that p is indeed a ibration. We will come back to this point in 5.1. p E 3. The Moore paths actorization I We now present a second construction o the trivial coibration ibration actorization or the h-model structure on Top in order to illustrate some o the key ideas involved in the algebraic perspective on homotopy theory. Following [May75], we introduce a unctorial actorization based on the Moore path space to characterize the Hurewicz ibrations as algebras or a pointed endounctor. We use this characterization to prove that the right actor is a Hurewicz ibration and then apply Proposition 2.3 to show that the let actor is a closed Hurewicz coibration and homotopy equivalence. Interestingly, because this unctorial actorization is particularly nice, the pointset topology input provided by Proposition 2.3 is not necessary to show that the let actor is a trivial coibration. We will explain how this works in Section 4, introducing ideas that will be essential or our construction o a suitable unctorial actorization or a general topologically bicomplete category in Section The Moore path space. Let be a space and let R + = [0, ). The space Π o Moore paths is deined to be the pullback (3.1) Π R+ R + π end const R+ The map shit is adjunct to the map given by precomposing with the addition map R + R + + R +. It has the eect o reindexing a path so that it starts at the indicated time. Unpacking this deinition, Π can be identiied with the set o pairs (p, t), where t R + and p: [0, t] is a path in o length t, topologized as a subspace o R+ R +. The map π end projects onto the end point o the Moore path. shit

7 ON THE CONSTRUCTION OF FUNCTORIAL FACTORIZATIONS 7 Following [May75]: Deinition 3.2. The Moore path space Γ o : is deined to be the pullback (3.3) Γ Π where π 0 : (p, t) p(0) is the evaluation o paths at 0. In other words, Γ is the set o triples (p, t, x) where (p, t) is a Moore path in and x is a point in the iber over p(0). As with N above, we use the space Γ to deine a actorization (3.4) I Γ The let actor I : Γ sends a point x to the length-zero path at (x). The right actor M : Γ is the endpoint-evaluation map, obtained by composing the top map o (3.3) with π end : Π. Unlike the case or the actorization constructed using the ordinary mapping space N, the Moore path space actors a map into a trivial coibration I ollowed by a ibration M. The proos o these acts make use o the algebraic perspective on homotopy theory. To proceed, we need a ew deinitions Functorial actorizations. For the reader s convenience, we briely review the notion o a unctorial actorization. Let C be any category. Write dom, cod: C 2 C or the evident orgetul unctors, deined respectively by precomposing with the domain and codomain inclusions 1 2 o the terminal category into the category. Deinition 3.5. A unctorial actorization consists o a pair o unctors L, R: C 2 C 2 such that π 0 M doml = dom, codr = cod, codl = domr, and with the property that or any C 2, the composite (in C) o L ollowed by R is. It is convenient to assign a name, say E, to the common unctor codl = domr: C 2 C that sends an arrow to the object through which it actors. A unctorial actorizations actors a commutative square (3.6) u W v Z g as u W L Lg E E(u,v) Eg R v Rg Z g

8 8 TOBIAS BARTHEL AND EMIL RIEHL The unctors L and R are equipped with canonical natural transormations to and rom the identity on C 2 respectively, which we denote by ɛ: L id and η : id R. The components o these natural transormations at C 2 are the squares L E L E R R In other words, L and R are pointed endounctors o C 2, where we let context indicate in which direction the unctors are pointed. An algebra or the pointed endounctor R is deined analogously to the notion o an algebra or a monad, except o course there is no associativity condition in the absence o a multiplication map µ: R 2 R. Similarly, a coalgebra or the pointed endounctor L is deined analogously to the notation o a coalgebra or a comonad. Unpacking these deinitions we observe: Lemma 3.7. C 2 is an R-algebra just when there exists a lit (3.8) L t E R Furthermore any choice o lit uniquely determines an R-algebra structure or. Dually, i C 2 is a L-coalgebra just when there exists a lit A i B Li Furthermore any choice o lit uniquely determines a L-coalgebra structure or i. A key point, which we will make use o later, is that any L-coalgebra lits (canonically) against any R-algebra. Lemma 3.9. Any L-coalgebra (i, s) lits canonically against any R-algebra (, t). s Proo. Given a liting problem, i.e., a commutative square (u, v): i, the unctorial actorization together with the coalgebra and algebra structures deine a solution, namely the composite o the dashed arrows: E B Ri A Li u Ei E(u,v) E Ri s B v t L R

9 ON THE CONSTRUCTION OF FUNCTORIAL FACTORIZATIONS The Moore paths unctorial actorization. The construction (3.4) above deines a unctorial actorization I, M : Top 2 Top 2 through the Moore path space. Furthermore, a classical result o May [May75, 3.4] can be stated as ollows: Proposition A map is a Hurewicz ibration i and only i it admits the structure o an M-algebra. Furthermore, as is noted in [Mal73] and [May75], the pointed endounctor (M, η) extends to a monad M = (M, η, µ). This is the point at which Moore paths make their key contribution: composition o paths o variable lengths is strictly associative. In particular, the arrows M are themselves (ree) M-algebras, and are hence ibrations. Lemma The Moore paths unctorial actorization extends to a monad M = (M, η, µ) over cod on the arrow category Top 2. Proo. We need only deine µ: ΓM Γ, the domain component o the multiplication natural transormation M 2 M. A point in ΓM is a Moore path (p, t) in together with a point in Γ this being itsel a Moore path (p, t ) in together with a point x in the iber o p (0) such that p(0) = p (t ). The map µ sends this data to the concatenated path pp o length t + t together with the chosen point x in the iber over pp (0) = p (0). The remaining details are let to the reader. These results allow or an easy proo o Proposition 2.4. Corollary The actorization (3.4) actors into a trivial coibration I and a ibration M. Proo. By Proposition 3.10 and Lemma 3.11, M is a (ree) M-algebra and hence a Hurewicz ibration, so the only thing to check is that I is a trivial coibration. But this ollows immediately rom Proposition 2.3, using the map q : Γ [0, 1] given by sending a Moore path (p, t, x) o length t to min(t, 1). An alternate proo that I is a trivial coibration, which avoids Strøm s characterization 2.3, was suggested by the reeree. By Proposition 3.10 and Lemma 3.9, it suices to show that I is an I-coalgebra. Lemma 3.7 says that a map i: A B is an I-coalgebra i and only i there is a lit A i B Ii This is the case, by the universal property o Γi, i and only i i extends to a Moore strong deormation retract: a retraction p o i together with a Moore homotopy h rom ip to 1 B whose components have length zero when restricted along i. A const Γi B Mi i B h ΠB π end B p A i B π 0

10 10 TOBIAS BARTHEL AND EMIL RIEHL Taking i to be I, it is easy to check that the maps in the pullback (3.3) deine a Moore strong deormation retract, making I an I-coalgebra and hence a trivial coibration. Remark Note, in general the notions o M-algebras (algebras or the ull monad) and M-algebras (algebras or the pointed endounctor) are distinct; the ormer is more restrictive. We will always take care to use a blackboard bold letter to distinguish algebras or the monad rom algebras or the pointed endounctor o the same name. But in act, because these unctors arise in unctorial actorizations, every M-algebra is a retract o an M-algebra, namely, its right actor. In particular, a map has the let liting property with respect to the M-algebras i and only i it has the let liting property with respect to the M-algebras. For the Moore paths actorization, M-algebras are those Hurewicz ibrations that admit a transitive path liting unction in the terminology o [May75]. The ree algebras M are both M-algebras and M-algebras. 4. The Moore paths actorization II In act, by urther developing the algebraic perspective, we can give an alternative proo o Corollary 3.12 that does not rely upon the characterization o closed Hurewicz coibrations rom Proposition 2.3. The key observation is that the Moore paths unctorial actorization is an algebraic weak actorization system, deined below. In Section 5, by extending the methods introduced here, we will be able to construct unctorial actorizations appropriate or categories that are enriched, tensored, and cotensored over topological spaces, but where point-set level characterizations o classes o maps in the ambient category are not generally available. In an algebraic weak actorization system (I, M), the pointed endounctor M extends to a monad and the let actor I extends to a comonad. In particular, the maps I are (ree) I-coalgebras. By Lemma 3.9 the let actor thereore lits against any algebra or the right actor. Our proo that the Moore paths unctorial actorization deines an algebraic weak actorization system uses a simple characterization, due to Richard Garner, that allows us to identiy categories o algebras or the monad o an algebraic weak actorization system existing in the wild. For this, we must irst explore: 4.1. Composition o algebras. Let L, R: C 2 C 2 deine a unctorial actorization. To simpliy the ollowing discussion, we consider only algebras or the right actor R; dual results apply to the case o coalgebras or the let actor L. Deinition 4.1. A morphism (u, v): g in C 2, i.e., a commutative square (3.6) is a map o R-algebras i the square o lits displayed in the interior o the cube L E s R u E(u,v) W Lg Eg v u Rg t W g Z

11 ON THE CONSTRUCTION OF FUNCTORIAL FACTORIZATIONS 11 commutes, i.e., i u s = t E(u, v). Example 4.2. The identity arrow at any object is always an R-algebra with a unique R-algebra structure given by its right actor. Furthermore, or any R-algebra (, s), the map (, 1 ): (, s) (1, R1 ) is an R-algebra map. The proo is a one line diagram chase: by (3.6) and (3.8). R1 E(, 1 ) = 1 R = R = s Write Alg R or the category o R-algebras and R-algebra maps. R-algebras are in particular arrows in C, which can be composed vertically, so the category C 2 is a double category: the composition o morphisms in C 2 gives a notion o horizontal composition o squares. The vertical composition o squares provides a new operation, not visible in the category structure on C 2. Deinition 4.3. The category Alg R admits a vertical composition law i (i) whenever (, s) and (g, t) are R-algebras such that cod = domg, we can speciy an R-algebra structure t s or g (ii) urthermore, or any maps (u, v): (, s) (, s ) and (v, w): (g, t) (g, t ) o R-algebras between composable pairs (, s), (g, t) and (, s ), (g, t ), then (u, w): (g, t s) (g, t s ) is a map o R-algebras. In other words, Alg R admits a vertical composition law i both R-algebras and R-algebra maps can be composed vertically. This latter condition says that the vertical composite o the squares underlying R-algebra maps must again be an R-algebra maps with respect to the composite R-algebras. When this vertical composition law is associative, Alg R is a double category. We will see in Remark 4.11 below that associativity oten comes or ree. Example 4.4. For example, suppose L, R are deined by pushing out rom a particular coproduct o a set o generating trivial coibrations J as in step one o Quillen s small object argument: (4.5) j j J Sq(j,) By the universal property o the deining pushout, an R-algebra structure or is precisely a liting unction φ, i.e., a choice o solution to all liting problems against any j J ; see [Rie11, 2.25]. The category Alg R = J admits a vertical composition law. The R-algebra structure assigned to the composite o (, φ ), (g, φ g ) Alg R is the liting unction that solves u A j L g B v Z R

12 12 TOBIAS BARTHEL AND EMIL RIEHL by irst constructing the dotted lit according to φ g, thereby obtaining a new liting problem against whose dashed solution is chosen according to φ. Note that this composition law is associative. Remark 4.6. For a generic unctorial actorization, there is no reason or there to be a composition law or algebras o the right actor. However, we will see shortly that the existence o such a composition law is characteristic or unctorial actorizations with good liting properties Algebraic weak actorization systems. The ollowing deinition is originally due to [GT06], with a small modiication by Garner [Gar09]. Deinition 4.7. An algebraic weak actorization system on a category C is pair (L, R) with L = (L, ɛ, δ) a comonad on C 2 and R = (R, η, µ) a monad on C 2 such that: (i) (L, ɛ), (R, η) give a unctorial actorization on C, and (ii) The natural transormation : LR RL with components given by the commutative squares δ LR µ RL is a distributivity law, i.e., satisies δ µ = µ L E(δ, µ) δ R. It ollows rom (i) that codr = cod and that the codomain components o both µ and η are the identity; dually, doml = dom and the domain components o δ and ɛ are identities. In other words, R is a monad over the unctor cod, and dually or L. Deinition 4.8. The let class o an algebraic weak actorization system (L, R) is the class o maps that admit an L-coalgebra structure while the right class is the class o maps that admit an R-algebra structure. Equivalently, the let class is the retract closure o the class o L-coalgebras and the right class is the retract closure o the class o R-algebras. Note by Lemma 3.9, each map in the let class lits against every map in the right class. Lemma 4.9 (Garner). I (L, R) is an algebraic weak actorization system, then Alg R has a canonical vertical composition law, which is moreover associative. A proo is given in [Gar09]. We are particularly interested in the converse. Theorem 4.10 (Garner). I R is a monad on C 2 over cod such that its category o algebras Alg R admits a vertical composition law, then there is a canonical algebraic weak actorization system (L, R), with the unctor L: C 2 C 2 deined by the unit. Furthermore, the vertical composition law on Alg R determined by the algebraic weak actorization system (L, R) coincides with the hypothesized one. Partial proos can be ound in [Gar10, Rie11], but on account o our particular interest in associativity o the vertical composition law, we elt that a more leshedout treatment was merited.

13 ON THE CONSTRUCTION OF FUNCTORIAL FACTORIZATIONS 13 Proo. We make requent use o the monadic adjunction C 2 Alg R. The (nontrivial component o the comultiplication) δ : L L 2 is the domain component o the adjunct to the map L 2 EL E RL Explicitly, δ is the composite o E(L 2, 1) with the algebra structure assigned the composite o the ree algebras RL and R. Because arbitrary maps (u, v): g give rise to maps (E(u, v), v): R Rg o ree R-algebras, δ : E EL is a natural transormation. It remains to show that δ gives L the structure o a comonad in such a way that (L, R) is an algebraic weak actorization system. We will check coassociativity and leave the unit and distributivity axioms to the reader. To this end, note that the ollowing rectangles are maps o R-algebras R E δ E(1,δ EL ) EL 2 E δ EL δ L EL 2 R RL E δ EL RL 2 R RL E E RL RL 2 R R RL We will show that the domain components agree by transposing both maps across the monadic adjunction. The domain component o the transpose o the let-hand map is E(1, δ) L 2 = L 3 = δ L L 2, which is the domain component o the transpose o the right-hand map. Hence δ is coassociative. Finally, we veriy that the vertical composition law arising rom the algebraic weak actorization system by Lemma 4.9 agrees with the vertical composition we started with. The key observation is that or any composable pair o R-algebras (, s) and (g, t) we have the ollowing map o R-algebras: R R ELg E(1,E(,1)) E(Lg ) E(1,t) E s RLg Eg E(,1) R(Lg ) Eg t R Rg Rg Z Z Z Z Recall δ g was deined to be µ g µ Lg E(L 2 g, 1), where is the given vertical composition law. By contrast, we write or the vertical composition given by the algebraic weak actorization system; by [Rie11, 2.21], t s is deined to be the g g

14 14 TOBIAS BARTHEL AND EMIL RIEHL composite E(L2 g,1) µg µ Lg E(1,E(,1)) E(1,t) s Because the above pasted rectangle is a map o R-algebras, the composite o the last our arrows is t s E(s E(1, t) E(1, E(, 1)), 1). Precomposing with E(L 2 g, 1), we have a commutative diagram E(L 2 g,1) E(L(Lg ),1) E(E(1,E(,1)),1) E(L,1) E(E(1,t),1) E(s,1) t s Hence t s = t s. Remark Because this composition law agrees with the one determined by the algebraic weak actorization system (L, R), it is necessarily associative, as could also be directly veriied. Morally, this is because vertical composition or an algebraic weak actorization system is deined using the same strategy outlined in Example 4.4. In particular, the category o algebras or the monad o any algebraic weak actorization system is always a double category The Moore paths algebraic weak actorization system. We now use these results to show that the unctorial actorization (3.4) is in act an algebraic weak actorization system. This was noticed independently by Garner. To this end, we must explain how deine a vertical composition law or the category o M-algebras. An M-algebra structure is classically called a path liting unction. The unction ξ : Γ speciying an M-algebra structure or : maps a Moore path (p: [0, t], x p(0) ) to a point ξ(p, t, x) p(t). I ξ is an M-algebra structure, then this assignment must satisy an additional transitivity condition; see Remark 4.14 below. We might hope to use a procedure similar to the one outlined in Example 4.4. Suppose g : Z, ζ : Γg is a second M-algebra. We can use ζ to lit the endpoint o a Moore path (p: [0, t] Z, x p(0) ) to, but we have lost too much inormation to proceed any urther. The key idea is that an M-algebra structure determines a lit, displayed in the lemma below, that might be called a parametrized path liting unction. Lemma There is an isomorphism, over C 2, between the category Alg M and the category o arrows equipped with lits (4.13) Γ Γ R + Π R + Proo. Clearly a parametrized path liting unction determines a path liting unction. For the converse, irst note that or any space A, the map : A A R + admits the structure o an I-coalgebra: The required lit A R + Γ sends a point (a, t) A R + to the path r (a, r) o length t with iber point a. Using this, we deine the parametrized path liting unction to be the canonical lit o the ev

15 ON THE CONSTRUCTION OF FUNCTORIAL FACTORIZATIONS 15 I-coalgebra against the M-algebra obtained rom the unctorial actorization (I, M), as in Lemma 3.9. Explicitly, the diagonal arrow maps a pair consisting o a Moore path (p: [0, t], x p(0) ) together with a parameter s to the value o ξ on the Moore path (p: [0, s], x p(0) ). 1 Remark As detailed in [May75, 3.2], i ξ is an M-algebra, then the associated map (4.13) is a transitive parametrized path liting unction, which means that the lited paths respect concatenation o paths in the ollowing sense. I p and p are composable paths o length t and t, and x is in the iber over p(0), then the lit o the concatenated path agrees with the concatenation o the lit o the irst path ollowed by the lit o the second path starting at ξ(p, t, x). In this way, there is an isomorphism between Alg M and the category o arrows equipped with transitive parametrized path liting unctions. Proposition The category Alg M admits a vertical composition law. Proo. We explain how to compose the transitive path liting unctions associated to M-algebras (, ξ) and (g, ζ) using the construction o Lemma 4.12, i.e., we deine a composite lit Γ(g) ζ ξ Γ(g) R + The dotted lit sends a pair consisting o a Moore path (p: [0, t] Z, x p(0) ) and a parameter s to the value o ζ on the Moore path (p: [0, s] Z, (x) p(0) ). This dotted map now allows us to deine a new Moore path in : ev Z r ζ(p, r, (x)): [0, s]. Call this path ζ(p). The point x lies in the iber over ζ(p)(0). Hence, (ζ(p), x) Γ. We deine the dashed lit to be the map that sends our original Moore path (p, t, x) and parameter s to the point ξ(ζ(p), s, x). The remaining details are straightorward diagram chases, let to the reader. Remark One can wonder whether this proo applies to produce model structures in more general situations, i.e., or a category equipped with some kind o Moore path object. An indication that this is indeed possible is given in [GvdB10], who construct actorizations on so-called path categories. By weakening their axioms, Williamson [Wil] obtains similar results in greater generality. Remark Note that the only non-ormal ingredient in the argument given here is the existence o a well-behaved Moore path object in topological spaces. Work in progress by Bill Richter is aimed at showing that the obvious analogue o (3.3) in a general category enriched, tensored, and cotensored over topological spaces has g 1 I s t, this new p is the restriction o the old one; i s > t, the new p extends the old by remaining constant at p(t) or the necessary duration.

16 16 TOBIAS BARTHEL AND EMIL RIEHL the same good properties. Our proo would then apply verbatim to yield a trivial coibration ibration actorization or a Hurewicz-type model structure. Our motivation or presenting a dierent, more abstract approach in the ollowing sections stems mainly rom its lexibility. We expect that our algebraic methods apply to many contexts in which there exists no obvious analogue o Moore paths. Examples include various model structures on dg-modules over a commutative dierential graded algebra, as investigated by Peter May. 5. Hurewicz model structures on topological categories Let C be a topologically bicomplete category, i.e., a bicomplete category enriched, tensored, and cotensored over some convenient category o spaces Top. We will require one additional condition, akin to the smallness condition or Quillen s small object argument, which we will describe when we explain its purpose below. The tensor and cotensor structure suices to abstract the deinitions o homotopy equivalence, Hurewicz coibration and Hurewicz ibration rom Section Topological categories and Cole s construction. In this section and the next we describe the heart o the construction in [Col06b], set up the notation or the rest o the chapter and state some lemmata that will turn out to be useul in the proo o our main theorem. Deinition 5.1. A homotopy between two maps 0, 1 : in C is a map h: I, or equivalently, a map ĥ: I (its adjunct) such that i 1 I 0 h 1 or equivalently 0 ĥ commutes,, i 1 : I and p 0, p 1 : I being the morphisms induced by the two endpoint inclusions I. In particular, we have a notion o homotopy equivalence in C. Deinition 5.2. A map in C is an h-coibration i it has the let liting property with respect to p 0 : Z I Z or all objects Z C. Dually, is an h-ibration i it has the right liting property with respect to all cylinder inclusions o the orm : Z Z I. Here the h stands or Hurewicz and also or homotopy. We would ideally like to construct a model structure on C whose coibrations are the h-coibrations, whose ibrations are the h-ibrations, and whose weak equivalences are the homotopy equivalences. However, similarly to Section 2, this is not possible because only some o the h-coibrations lit against the class o h-ibrations that are also homotopy equivalences. This motivates the ollowing deinition: 1 I p 0 p 1

17 ON THE CONSTRUCTION OF FUNCTORIAL FACTORIZATIONS 17 Deinition 5.3. The class o strong coibrations is the class o maps that have the let liting property with respect to the h-ibrations that are also homotopy equivalences. Because the maps p 0 : Z I Z are homotopy equivalences and h-ibrations, c. [SV02], strong coibrations are in particular h-coibrations. An immediate corollary o our main theorem, Theorem 5.22 below, establishes a so-called h-model structure, whose weak equivalences are homotopy equivalences, ibrations are h-ibrations, and coibrations are the strong coibrations. Henceorth, we use coibrations and ibrations in the model structure sense, in particular dropping the h. It is possible to describe these right and let liting classes using relative liting properties [MS06, 4.2.2], but all we need is the ollowing result. Lemma 5.4. (i) The natural map : A A I is a trivial coibration or all objects A C. (ii) The class o (trivial) coibrations is closed under retracts, pushouts and sequential colimits. Proo. The proos can be ound in [SV02] and [MS06]. Part (ii) is immediate rom the closure properties o any collection o arrows deined by a liting property. Cole s construction 5.5. Cole s construction attempts to actor an arbitrary map : in C into a trivial coibration ollowed by a ibration. To this end, start by orming the mapping path object N o, in precise analogy with Deinition 2.5. A new object E is constructed by pushing out one o the projections rom the pullback φ : N along the natural map : N N I. Using the morphisms and χ, the adjoint to the other projection χ, we obtain an induced map R : E as shown in the ollowing diagram. (5.6) I p 0 χ N φ N I ψ L E χ In this way, we have actored as R L and urthermore, by Lemma 5.4, the map L : E is a trivial coibration. I the map R were a ibration, we would be done. However, this ails in general, so Cole proposes to iterate this construction, replacing by R, and applying the unctorial actorization (L, R) to the right actor. The eventual right actor o is deined by passing to the colimit R ω = colim(r R 2 R 3 ). The let actor o is then the composite L E LR ER LR2 ER 2 ER ω. Because each map in the image o L is a trivial coibration, the let actor is a trivial coibration. It remains to show that R ω is a ibration, which by [Col06b, 5.2] R

18 18 TOBIAS BARTHEL AND EMIL RIEHL is equivalent to inding a lit in NR ω NR ω I φ R ω χ ER ω To this end, [Col06b] asserts that the required lit is given by ψ Rω ; however, the maps ψ Rn do not glue to induce a map NR ω I ER ω, c. Section 7.1. We will see that there is a natural modiication o the iterative part o Cole s construction that produces an algebraic weak actorization system with the appropriate homotopical properties Algebraic characterization o ibrations. The irst key observation is that, even though the right actor R ails to be an h-ibration, algebras or the pointed endounctor R are precisely h-ibrations. The proo ollows easily once we understand the universal property o the mapping space N. Fix a morphism : and let Sq : C op Set be the unctor that maps an object A to the set o commutative squares o the orm A A I These squares correspond to liting problems that test whether is an h-ibration. Lemma 5.7. The unctor Sq is represented by the mapping path object N. Proo. By the deining universal property o N, a map α : A N classiies a commutative square (5.8) A A I u v A u R ω α φ v N χ I In particular, the identity map at N classiies the right hand square in p 0 (5.9) A α u N φ A I α I N I v χ

19 ON THE CONSTRUCTION OF FUNCTORIAL FACTORIZATIONS 19 which eatures prominently in the construction o the actorization (5.6). By the oneda lemma, or alternatively by adjointness, a square (5.8) actors uniquely as the above diagram (5.9), where α: A N is the classiying map. It is now easy to prove that the h-ibrations are precisely those objects in the image o the orgetul unctor Alg R C 2. Proposition The class o R-algebras coincides with the class o h-ibrations. Proo. By deinition, a ibration has the right liting property against all trivial coibrations, so there exists, in particular, a lit in the ollowing diagram L s E R which makes (, s) an R-algebra. Conversely, suppose (, s) Alg R. To solve a liting problem A A I we irst actor it as displayed in (5.9) and then actor the right hand square in (5.9) through the pushout o (5.6). This yields: A A I N N I u u v v φ ψ χ L s E R The map s deines an evident solution to the original liting problem. Remark This argument shows that is an h-ibration i and only i there is a lit N N I φ as observed in [Col06b, 5.2]. We think o this square as presenting a generic liting problem which detects ibrations. This is analogous to the liting problem given by (4.5) in the coibrantly generated case. At this point we are conronted with a problem: the algebras or the unctor R are precisely the ibrations, but because R is not a monad, the maps R are not themselves R-algebras. One idea is to try and replace the unctor R by its ree monad F, which is characterized by the property that the category o F-algebras χ

20 20 TOBIAS BARTHEL AND EMIL RIEHL is isomorphic over C 2 to the category o R-algebras (so in particular F-algebras are precisely ibrations). There are two obstacles to implementing this idea. The irst is set-theoretical. By an easy application o the monadicity theorem, the ree monad F is equivalently speciied by a let adjoint to the orgetul unctor Alg R C 2. However, it is not quite enough to simply know that an adjoint exists: the resulting monad on C 2 might not be a monad over cod and thus not deine the right actor in a unctorial actorization. A theorem o Kelly, described in the next section, exhibits a certain smallness condition on R under which the ree monad exists constructively ; in this case, a unctorial actorization is produced. A second obstacle remains. Supposing that the ree monad F exists constructively, it is not clear a priori that the let actor will still be a trivial coibration because this construction involves quotienting. However, we can show that the actorization produced by this procedure has the structure o an algebraic weak actorization system; in particular, the let actor is a ree C-coalgebra, thereore lits against the F-algebras and is hence a trivial coibration The ree monad on a pointed endounctor. We now explain what precisely we mean by ree monad and state Kelly s abstract existence result. In the next section, we then veriy that the unctor R satisies his conditions under certain set-theoretical assumptions on the underlying category C. Let R be a pointed endounctor on a category C. The algebraically ree monad on R is a monad F together with an isomorphism Alg F = AlgR over C. When C is locally small and complete, algebraically ree monads coincide with so-called ree monads, which are deined in [Kel80, ]. We use the terminology ree monad because it is shorter. Furthermore, under good conditions, there is a canonical construction that produces the ree monad on R, in which case we say the ree monad exists constructively. The construction is via a colimit deined using transinite induction; the good conditions guarantee that this construction converges. 2 Remark A naïve approach might be to try and deine F to be the colimit o id R R 2 R 3 This works in the case where R is well-pointed, meaning ηr = Rη : R R 2, but not otherwise. Interestingly, the ailure o Cole s unctor R to be well-pointed precisely highlights the subtle point at which his argument breaks down. We ll say more about this in the appendix, c. Section 7.2. The correct construction is due to Kelly, the irst ew stages o which we will describe explicitly in the appendix. We make use o only a special case o his theorem [Kel80, 22.3]. In order to state it, we need to introduce a little bit o terminology. An orthogonal actorization system (E, M) on a category C is a weak actorization system or which both the actorizations and the litings are unique. It is called well-copowered i every object in C has a mere set o E-quotients, up to isomorphism. When C is cocomplete it ollows that the maps in E are epimorphisms [Kel80, 1.3]. 2 Compare with Quillen s small object argument, which never converges, but must be terminated artiicially.

21 ON THE CONSTRUCTION OF FUNCTORIAL FACTORIZATIONS 21 Remark Note that any category that is cocomplete and so that each object has only a sets worth o epimorphism-quotients a condition satisied by all categories one meets in practice has a unctorial actorization where the let actor is an epimorphism and the right actor is a strong monomorphism, see [Bor94, 4.4.3]. The dual hypotheses are equally common in our setting. In practice this means that there are always at least two choices or (E, M): (epimorphisms, strong monomorphisms) and (strong epimorphisms, monomorphisms). A cocone in C all o whose legs are elements o M is called an M-cocone or an M-colimit in the case it is a colimit cocone. It ollows rom the right cancelation property o M that the morphisms in the diagram also lie in M, but our condition is stronger. In what ollows, we will implicitly identiy a regular cardinal α with its initial ordinal, such that α indexes a (transinite) sequence whose objects are β < α. We are now ready to state Kelly s theorem. Theorem 5.14 (Kelly). Suppose C is complete, cocomplete, and locally small. I a pointed endounctor R on C satisies the ollowing smallness condition: ( ) there is a well-copowered orthogonal actorization system (E, M) on C and a regular cardinal α so that R sends α-indexed M-colimits to colimits, then the ree monad F on R exists constructively. I R is a pointed endounctor on C 2 over cod, then each unctor and natural transormation in the ree monad construction is constant on its codomain component. It ollows that F is a monad over cod and hence gives rise to a unctorial actorization. Furthermore, this observation allows us to weaken the smallness condition or such R: It suices to show that R preserves M-colimits o the orm (5.15) 0 m 0 1 m 1 mβ 1 m β β α 1 β See [Gar07, p. 31]. 0 colim β<α β = α 5.4. Smallness. The unctor R o (5.6) is constructed by means o various topologically enriched limits and colimits in C. In this section, we will show that i C satisies a set-theoretical condition, then Cole s unctor R satisies the necessary smallness condition to guarantee convergence o the ree monad sequence. This condition is very similar to Cole s coibration hypothesis [Col06b, 4.1]. Indeed, as explained there, work o Lewis [Lew78] shows that many topologically bicomplete categories o interest satisy our condition. Deinition Suppose (E, M) is a well-copowered orthogonal actorization system on a topologically bicomplete category C. We say C satisies the monomorphism hypothesis i there is some regular cardinal α so that the mapping path space unctor N : C 2 C preserves M-colimits o diagrams o the orm (5.15), in the sense that the natural map is an isomorphism. colim β<α N β N(colim β<α β ) = N α