MADE-TO-ORDER WEAK FACTORIZATION SYSTEMS EMILY RIEHL The aim o this note is to briely summarize techniques or building weak actorization systems whose right class is characterized by a particular liting property. Weak actorization systems Weak actorization systems are o paramount importance to homotopical algebra. This connection is best illustrated by the ollowing deinition, due to Joyal and Tierney [JT07]. Deinition. A Quillen model structure on a category M, with a class o maps W called weak equivalences satisying the 2-o-3 property, consists o two classes o maps C and F so that (C W, F) and (C, F W) are weak actorization systems. Deinition 2. A weak actorization system (L, R) on a category M consists o two classes o maps so that Any map M can be actored as = r l with l L and r R. Any liting problem, i.e., any commutative square () L l r R has a solution, i.e., a diagonal arrow making both triangles commute. The classes L and R are closed under retracts. Remark 3. () By the retract argument: L l r s s the class L consists o retracts o maps appearing as let actors. (2) Consequently, the let and right classes determine each other. More precisely: L = R and R = L meaning L and R are maximal with the liting property (). l r I would like to thank Steve Awodey or sparking my interest in homotopy type theory and encouraging me to keep in touch with recent progress.
2 EMILY RIEHL (3) Any class o maps o the orm R is closed under retracts, coproducts, pushouts, transinite composition, and contains the isomorphisms. Such classes are called weakly saturated. A class L is closed under the dual limit constructions. (4) Given any set o maps J, there is a coibrantly generated weak actorization system ( (J ), J ) provided that it is possible to construct actorizations. This is accomplished by means o the small obect argument. The algebraic small obect argument Assuming the category M is cocomplete and satisies a certain smallness condition (such as being locally presentable), the algebraic small obect argument deines the unctorial actorization necessary or a made-to-order weak actorization system with R = J. For now, J is an arbitrary set o morphisms o M but later we will use this notation to represent something more sophisticated. Generic liting problems. The small obect argument begins by deining a generic liting problem, a single liting problem that characterizes the desired right class: (2) J J Sq(,) The diagonal map deines a solution to any liting problem between J and. Step-one unctorial actorization. Taking a pushout transorms the generic liting problem (2) into the step-one unctorial actorization, another generic liting problem that also actors. (3) J J Sq(,) L The step-one unctorial actorization deines a pointed endounctor R : M 2 M 2 o the arrow category. An R -algebra is a pair (, s) satisying (3). A map admits the structure o an R -algebra i and only i it is in the class J. The ree monad construction. By the closure properties enumerated in Remark 3(3), L L = (J ). However, there is no reason to expect that R J : maps in the image o R need not be R -algebras unless R is a monad. The idea o the algebraic small obect argument, due to Garner [Gar09], is to reely replace the pointed endounctor R by a monad. 2 Following Kelly [Kel80], assuming certain smallness or boundedness conditions, it is possible to construct the ree monad R on a pointed endounctor R in such a way that the categories o algebras are isomorphic. Garner shows that with suicient care, Kelly s construction can be perormed in a way that preserves the Dual ibrantly generated weak actorization systems are much rarer because the technical conditions necessary to permit the small obect argument are satisied by Set but not by Set op. 2 When all maps in the let class are monomorphisms, the ree monad is deined by iteratively attaching non-redundant cells until this process converges. s R
MADE-TO-ORDER WEAK FACTORIZATION SYSTEMS 3 act that the endounctor R is the right actor o a unctorial actorization whose let actor L is already a comonad. In this way, the algebraic small obect produces a unctorial actorization = R L in which L is a comonad, R is a monad, and R-Alg = R -Alg = J. Remark 4. When the weak saturation o the generating set J is contained within the monomorphisms, the algebraic small obect argument has the ollowing simpliied description: Iteratively apply the construction o the step-one unctorial actorization to the right actor constructed in the previous stage, but avoid attaching solutions to redundant liting problems. This means that any liting problem whose attaching map actors through a previous stage L should be omitted rom the coproduct deining the let-hand side o (2). Example 5. Consider { } on the category Set. The algebraic small obect argument produces the generic liting problem displayed on the let and the step-one unctorial actorization displayed on the right: R X X X 8 8 incl 8 8 Y Y Y X Y Y Every liting problem ater step one is redundant. Indeed, R = is already a monad and the algebraic small obect argument converges in one step to deine the unctorial actorization (L = incl, R). Example 6. Consider { n n } n 0 on the category o simplicial sets. Here we may consider liting problems against a single generator at a time inductively by dimension. The step-one actorization o X Y attaches the 0-skeleton o Y to X. There are no non-redundant liting problems involving the generator 0, so we move up a dimension. The step-two actorization o X Y now attaches -simplices o Y to all possible boundaries in X sk 0 Y. Ater doing so, there are no non-redundant liting problems involving. The construction converges at step ω. Algebraic weak actorization systems Functorial actorizations and liting properties. A unctorial actorization = R L or a weak actorization system (L, R) characterizes that weak actorization system: L L s R g R Lg t Rg g
4 EMILY RIEHL because the speciied lits assemble into a canonical solution to any liting problem u v g L R u t s v Lg Rg A natural question is whether any unctorial actorization deines a weak actorization system. In order to have L L and R R we must have maps L (4) 2 L RL LR These need not exist in general but do when the pointed endounctors L and R underlie a comonad and monad respectively. 3 So, in particular, the unctorial actorization produced by the algebraic small obect argument determines its weak actorization system. Algebraic weak actorization systems. Put another way, the algebraic small obect argument produces an algebraic weak actorization system (L, R), a unctorial actorization that underlies a comonad L and a monad R and in which the natural transormation deined using the canonical lits (4) deines a distributive law LR RL [GT06]. In act, either o the categories L-coalg or R-alg determine the other; see [Rie, 2.5], [Rie3c, 5.]. Example 7. In Example 5, L-coalg is the category o monomorphisms and pullback squares, while R-alg = { } is the category o split epimorphisms and commutative squares preserving the splittings. Example 8. In Example 6, L-coalg is the category o monomorphisms and commutative squares that induce pullback squares between the relative latching maps, while R-alg = { n n } n 0 is the category o algebraic acyclic Kan ibrations and maps preserving the chosen litings. This is the Reedy algebraic weak actorization system deined with respect to the algebraic weak actorization system o Example 7 [Rie3d]. R 2 R Generalizations o the algebraic small obect argument The construction o the generic liting problem admits a more categorical description which makes it evident that it can be generalized in a number o ways, expanding the class o weak actorization systems whose unctorial actorizations can be made-to-order. 3 This is why L L = (J ).
MADE-TO-ORDER WEAK FACTORIZATION SYSTEMS 5 Coherence. Step zero o the algebraic small obect argument orms the density comonad, i.e., the let Kan extension along itsel, o the inclusion o the generating set o arrows: J 5 5555555 M 2 Lan=L 0 M 2 L 0 = J Sq(,) When M is cocomplete, this construction makes sense or any small category o arrows J. The counit o the density comonad deines the generic liting problem (2), admitting a solution i and only i J but now J denotes the category in which an obect is a map together with a choice o solution to any liting problem against J that is coherent with respect to (i.e., commutes with) morphisms in J. Proceeding as beore, the algebraic small obect argument produces an algebraic weak actorization system (L, R) so that R-alg = J over M 2 and L-coalgebras lit against R-algebras. Example 9. In the category o cubical sets, let,,, suggestively denote our subunctors o the 2-dimensional representable 2. For n > 2 and J {,..., n} with J = n 2, deine J n to be J and similarly or the three other shapes. Consider the category whose obects are the inclusions J n or each shape and whose morphisms are generated by the proections the inclusions J J\{} n π n or each J, and J J {i} embedding n as the ace i = 0 or i =. n n+ This category generates the ibrant replacement unctor described by Simon Huber. Example 0 ([Rie, 4.2]). Any algebraic weak actorization system (L, R) on M induces a pointwise-deined algebraic weak actorization system (L A, R A ) on the category M A o diagrams. Moreover, when (L, R) is generated by J, (L A, R A ) is generated by the category A op J, whose obects are tensors o arrows o J with covariant representables. Enrichment. Now suppose that M is tensored, cotensored, and enriched over a closed symmetric monoidal category V. In this context, we may choose to deine the generic liting problem using the V-enriched let Kan extension L 0 = J Sq(, ), where Sq(, ) V is the obect o commutative squares rom to deined by the evident pullback involving the hom-obects o M. The enriched algebraic small obect argument produces an algebraic weak actorization system whose underlying let and right classes satisy an enriched liting property, deined internally to V. The classes o an ordinary weak actorization
6 EMILY RIEHL system satisy this enriched liting property i and only i tensoring with obects in V preserves the morphisms in the let class. See [Rie3a, 3] or more details. Example. Consider {0 R} in the category o modules over a commutative ring R with identity. In analogy with Example 5, the unenriched algebraic small obect argument produces the unctorial actorization X incl X ( Y R) ev Y whereas the enriched algebraic small obect argument produces the actorization X incl X Y Y. Example 2 ([Rie3b]). On the category o unbounded chain complexes o R- modules, consider the sets {0 D n } n Z and {S n D n } n Z where D n is the chain complex with R in degrees n and n and identity dierential and S n has R in degree n and zeros elsewhere. The enriched algebraic small obect argument converges at step one in the ormer case and at step two in the latter case to produce the natural actorizations through the mapping cocylinder and the mapping cylinder respectively [BMR3]. Class coibrantly generated. The algebraic weak actorization systems constructed in Examples 0 and 2 are not coibrantly generated (in the usual sense) [CH02, Lac07]. The ollowing examples are not coibrantly generated even in the expanded sense introduced here. Example 3 ([BR3]). There are two algebraic weak actorization systems on topological spaces whose right class is the class o Hurewicz ibrations. A map is a Hurewicz ibration i it has the homotopy liting property, i.e., solutions to liting problems (5) A X incl 0 A I Y deined or every topological space A. As there is proper class o generators, it is not possible to orm the coproduct in (2). However, the unctor Top op Set that sends A to the set o liting problems (5) is represented by the mapping cocylinder N: N Y I N X ev 0 incl 0 X Y N I Y It ollows that any liting problem (5) actors uniquely through the generic liting problem displayed on the right. The algebraic small obect argument proceeds as usual, though there are some subtleties to the proo that it converges. There is another algebraic weak actorization system ound in the wild : the actorization through the space o Moore paths. The category o algebras or the Moore paths monad admits the structure o a double category in such a way that the orgetul unctor to the arrow category becomes a double unctor. A recognition
MADE-TO-ORDER WEAK FACTORIZATION SYSTEMS 7 criterion due to Garner implies that this deines an algebraic weak actorization system. Reerences [BMR3] Barthel, T.; May, J.P.; Riehl, E. Six model structures or DG-modules over DGAs: Model category theory in homological action, (203), arxiv:30.59. [BR3] Barthel, T. and Riehl, E. On the construction o unctorial actorizations or model categories, Algebraic and Geometric Topology, 3 (203) 089 24. [CH02] Christensen, J.D.; Hovey, M. Quillen model structures or relative homological algebra. Math. Proc. Camb. Phil. Soc. 33 (2002), 26 293. [Gar07] Garner, R. Coibrantly generated natural weak actorisation systems, (2007), arxiv:0702.290. [Gar09] Garner, R. Understanding the small obect argument. Appl. Categ. Structures. 7(3) (2009) 247 285. [GT06] Grandis, M.; Tholen, W. Natural weak actorization systems. Arch. Math. (Brno) 42(4) (2006) 397 408. [JT07] Joyal, A.; Tierney, M. Quasi-categories vs Segal spaces. In Categories in algebra, geometry and mathematical physics, Contemp. Math. 43, Amer. Math. Soc., Providence, RI (2007) 277 326. [Kel80] Kelly, G.M. A uniied treatment o transinite constructions or ree algebras, ree monoids, colimits, associated sheaves, and so on. Bull. Austral. Math. Soc. 22() (980) 83. [Lac07] Lack, S. Homotopy-theoretic aspects o 2-monads. J. Homotopy Relat. Struct. 2(2) (2007) 229 260. [Rie] Riehl, E. Algebraic model structures. New York J. Math. 7 (20) 73-23. [Rie3a] Riehl, E. Categorical homotopy theory, to appear in New Mathematical Monographs, Cambridge University Press, (203), available rom www.math.harvard.edu/ eriehl/cathtpy.pd [Rie3b] Riehl, E. Mapping (co)cylinder actorizations via the small obect argument, (203), http://golem.ph.utexas.edu/category/203/09/mapping cocylinder actorizations via the small obect argument.html [Rie3c] Riehl, E. Monoidal algebraic model structures J. Pure Appl. Algebra, 27(6) (203) 069 04. [Rie3d] Riehl, E. On the notion o a generalized Reedy category, in preparation. Dept. o Mathematics, Harvard University, Oxord Street, Cambridge, MA 0238 E-mail address: eriehl@math.harvard.edu