Made-to-Order Weak Factorization Systems

Transcription

1 Made-to-Order Weak Factorizatio Systems Emily Riehl 1 The Algebraic Small Obect Argumet For a cocomplete category M which satisies certai smalless coditio (such as beig locally presetable), the algebraic small obect argumet deies the uctorial actorizatio ecessary or a made-to-order weak actorizatio system with right class.forow, is a arbitrary set o morphisms o M but later we will use this otatio to represet somethig more sophisticated. The small obect argumet begis by deiig a geeric litig problem,asigle litig problem that characterizes the desired right class: Sq Sq L s R (1) The diagoal map deies a solutio to ay litig problem betwee ad. Takig a pushout trasorms the geeric litig problem ito the step-oe uctorial actorizatio,aothergeericlitigproblemthatalsoactors. This deies a poited edouctor R 1 W M 2! M 2 o the arrow category. A R 1 -algebra is a pair. ; s/ as displayed. By costructio, L 1 2. /.However, there is o reaso to expect that R 1 2 :mapsitheimageor 1 eed ot be R 1 - algebras uless R 1 is a moad. The idea o the algebraic small obect argumet, due to Garer [5], is to reely replace the poited edouctor R 1 by a moad. (Whe all maps i the let class are moomorphisms, the ree moad is deied by iteratively attachig o-redudat cells util this process coverges.) E. Riehl ( ) Departmet o Mathematics, Harvard Uiversity, Cambridge, MA, USA eriehl@math.harvard.edu SprigerIteratioalPublishigSwitzerlad2015 M. del Mar Gozález et al. (eds.), Exteded Abstracts Fall 2013, Treds i Mathematics, DOI / _17 87

2 88 E. Riehl Followig Kelly [6], ad assumig certai smalless or boudedess coditios, it is possible to costruct the ree moad R o a poited edouctor R 1 i such a way that the categories o algebras areisomorphic.garershowsthatwith suiciet care, Kelly s costructio ca be perormed i a way that preserves the act that the edouctor R 1 is the right actor o a uctorial actorizatio whose let actor L 1 is already a comoad. I this way, the algebraic small obect produces auctorialactorizatio D R L i which L is a comoad, R is a moad, ad R-Alg Š R 1 -Alg Š. Example 1 Cosider ;! g o the category o sets. The algebraic small obect argumet produces the geeric litig problem displayed o the let ad the step-oe uctorial actorizatio displayed o the right: Every litig problem ater step oe is redudat. Ideed, R D ` 1 is already a moad ad the costructio coverges i oe step to deie the actorizatio D ` 1 icl 1. Example 2 g 0 o the category o simplicial sets. Here we may cosider litig problems agaist a sigle geerator at a time, iductively by dimesio. The step-oe actorizatio o X! Y attaches the 0-skeleto o Y to X. Thereareoo-redudatlitigproblemsivolvigthegeerator;,! 0, so we move up a dimesio. The step-two actorizatio o X! Y ow attaches 1-simplices o Y to all possible boudaries i X [ sk 0 Y.Aterdoigso,thereareo o-redudat litig problems 1,! 1.Thecostructiocoverges at step!. The algebraic small obect argumet produces a algebraic weak actorizatio system.l; R/,auctorialactorizatiothatuderliesacomoadL ad a moad R, ad i which the caoical map LR ) RL deies a distributive law. The uctorial actorizatio D R L characterizes the uderlyig weak actorizatio system.l; R/:

3 Made-to-Order Weak Factorizatio Systems 89 because the speciied lits assemble ito a caoical solutio to ay litig problem: u u L t Lg g R s Rg 2 Geeralizatios o the Algebraic Small Obect Argumet The costructio o the geeric litig problem admits a more categorical descriptio which makes it evidet that it ca be geeralized i a umber o ways, expadig the class o weak actorizatio systems whose uctorial actorizatios ca be made-to-order. Step zero o the algebraic small obect argumet orms the desity comoad,i.e., the let Ka extesio alog itsel, o the iclusio o the geeratig set o arrows: M 2 La L M 2 L Sq Whe M is cocomplete, this costructio makes sese or ay small category o arrows.thecouitothedesitycomoaddeiesthegeericlitigproblem(1), admittig a solutio i ad oly i 2 butow deotes the category i which a obect is a map together with a choice o solutio to ay litig problem agaist that is coheret with respect to (i.e., commutes with) morphisms i. Proceedig as beore, the algebraic small obect argumet produces a algebraic weak actorizatio system.l; R/ so that R-Alg Š over M 2,adL-coalgebras lit agaist R-algebras. Example 3 I the category o cubical sets, let u; A; suggestively deote our subuctors o the 2-dimesioal represetable 2. For > 2 ad 1; : : : ; g with D 2, deie u to be u, ad similarly or the other three shapes. Cosider the category whose obects are

4 90 E. Riehl the iclusios u,! or each shape ad whose morphisms are geerated by the proectios or each,ad the iclusios embeddig as the ace or i This geerates the ibrat replacemet uctor, see Bezem Coquad Huber [3]. Example 4 ([8, 4.2]) Ay algebraic weak actorizatio system.l; R/ o M iduces a poitwise-deied algebraic weak actorizatio system.l A ; R A / o the category M A o diagrams. Moreover, whe.l; R/ is geerated by,.l A ; R A / is geerated by the category A op,whoseobectsaretesorsoarrowso with covariat represetables. I M is tesored, cotesored, ad eriched over a closed mooidal category V, we may choose to deie the geeric litig problem usig the V-eriched let Ka extesio Z 2 L 0 D Sq.; / ; where Sq.; / 2 V is the obect o commutative squares. The eriched algebraic small obect argumet produces a algebraic weak actorizatio system whose uderlyig let ad right classes satisy a eriched litig property, deied iterally to V. The classes o a ordiary weak actorizatio system satisy this eriched litig property i ad oly i tesorig with obects rom V preserves the morphisms i the let class [9, 13]. Example 5 Cosider 0! Rg i the category o modules over a commutative rig R with idetity. I aalogy with Example 1, theuerichedalgebraicsmall obect argumet produces the let-had uctorial actorizatio, while the eriched algebraic small obect argumet produces the actorizatio o the right: X icl! X. YR/ ev! Y; X icl! X Y 1! Y: Example 6 (Barthel May Riehl [1]) O the category o ubouded chai complexes o R-modules, cosider the sets 0! D g 2Z ad S 1,! D g 2Z,where D is the chai complex with R i degrees ad 1 ad idetity dieretial,

5 Made-to-Order Weak Factorizatio Systems 91 ad where S has R i degree ad zeroes elsewhere. The eriched algebraic small obect argumet coverges at step oe i the ormer case ad at step two i the latter case to produce the atural actorizatios through the mappig cocylider ad the mappig cylider, respectively (see Mappig (co)cylider actorizatios via the small obect argumet o the -Category Caé). The algebraic weak actorizatio systems costructed i Examples 4 ad 6 are ot coibratly geerated (i the usual sese) [4, 7]. Example 7 (Barthel Riehl [2]) There are two algebraic weak actorizatio systems o topological spaces whose right class is the class o Hurewicz ibratios. A map is a Hurewicz ibratio i it has the homotopy litig property, i.e., solutios to litig problems A X icl A I Y (2) deied or every topological space A.Asthereisproperclassogeerators,itisot possible to orm the coproduct i (1). However, the uctor Top op! Set sedig A to the set o litig problems (2)isrepresetedbythemappigcocyliderN : N Y I N X ev icl X Y N I Y It ollows that ay litig problem (2) actors uiquely through the geeric litig problem displayed o the right. The algebraic small obect argumet proceeds as usual, though there are some subtleties i the proo o its covergece. There is aother algebraic weak actorizatio system oud i the wild : the actorizatio through the space o Moore paths. The category o algebras or the Moore paths moad admits the structure o a double category i such a way that the orgetul uctor to the arrow category becomes a double uctor. A recogitio criterio due to Garer implies that this deies a algebraic weak actorizatio system. Ackowledgemets IwouldliketothakSteveAwodeyorsparkigmyiterestihomotopy type theory ad ecouragig me to keep i touch with recet progress.

6 92 E. Riehl Reereces 1. T. Barthel,.P. May, E. Riehl, Six model structures or DG-modules over DGAs: model category theory i homological actio (2013, preprit). arxiv: T. Barthel, E. Riehl, O the costructio o uctorial actorizatios or model categories. Algebr. Geom. Topol. 13, (2013) 3. M. Bezem, T. Coquad, S. Huber, A model o type theory i cubical sets (2014, preprit) D. Christese, M. Hovey, Quille model structures or relative homological algebra. Math. Proc. Camb. Phil. Soc. 133, (2002) 5. R. Garer, Uderstadig the small obect argumet. Appl. Categ. Struct. 17, (2009) 6. G.M. Kelly, A uiied treatmet o trasiite costructios or ree algebras, ree mooids, colimits, associated sheaves, ad so o. Bull. Aust. Math. Soc. 22(1), 1 83 (1980) 7. S. Lack, Homotopy-theoretic aspects o 2-moads.. Homotopy Relat. Struct. 2(2), (2007) 8. E. Riehl, Algebraic model structures. N. Y.. Math. 17, (2011) 9. E. Riehl, Categorical homotopy theory. New Mathematical Moographs (to appear i, Cambridge Uiversity Press, 2013).