THE COALGEBRAIC STRUCTURE OF CELL COMPLEXES

Transcription

1 Theory and pplications o Categories, Vol. 26, No. 11, 2012, pp THE COLGEBRIC STRUCTURE OF CELL COMPLEXES THOMS THORNE bstract. The relative cell complexes with respect to a generating set o coibrations are an important class o morphisms in any model structure. In the particular case o the standard (algebraic) model structure on Top, we give a new expression o these morphisms by deining a category o relative cell complexes, which has a orgetul unctor to the arrow category. This allows us to prove a conjecture o Richard Garner: considering the algebraic weak actorisation system given in that algebraic model structure between coibrations and trivial ibrations, we show that the category o relative cell complexes is equivalent to the category o coalgebras. 1. Introduction The aim o this paper is the proo o a conjecture o Richard Garner, which describes how the recent and categorically motivated concept o an algebraic weak actorisation system is deeply connected, in the speciic case o topological spaces, to the more established and well understood idea o relative cell complexes. Speciically, we will prove that the let map structures (coalgebra structures) o the canonical algebraic weak actorisation system on Top are exactly the relative cell complexes. relative cell complex is a morphism with a property: it can be expressed as a transinite composite o pushouts o coproducts o sphere inclusions. The let map structures, on the other hand, speciy an explicit choice o such an expression. We will spend some o the paper making this notion o a relative cell complex structure precise. The notion o an algebraic weak actorisation system (which we will write as aws both singular and plural) was introduced by Grandis and Tholen [Grandis, Tholen, 2008] as a way to make the notion o a weak actorisation system more amenable to study using the techniques o category theory. Weak actorisation systems are undamental to homotopy theory; they are the main component o model category structures. However, they have some drawbacks: most important or Grandis and Tholen was the act that the classes o let and right maps are not closed under colimits and limits. The deinition o an aws involves a unctorial actorisation o which the let unctor is a comonad and the right unctor is a monad. The notions o let and right maps are naturally replaced with the coalgebras and algebras, meaning that colimits and limits are automatically available. nother possible disadvantage o weak actorisation systems is the standard method Received by the editors and, in revised orm, Transmitted by Martin Hyland. Published on Mathematics Subject Classiication: 1832, 55U35. Key words and phrases: relative cell complexes, algebraic weak actorisation systems, small object argument. Thomas thorne, Permission to copy or private use granted. 304

2 THE COLGEBRIC STRUCTURE OF CELL COMPLEXES 305 o constructing them: the small object argument. This is a transinite induction which does the job well enough or many purposes, but which seems at odds with what should be a very natural concept in category theory; it relies on terminating a sequence at some arbitrary choice o ordinal because it won t converge, it has no universal property and it cannot be considered as an instance o any other transinite categorical construction. In his paper [Garner, 2009], Richard Garner demonstrates that this can be cured; there is a very natural variant o the small object argument or aws which does converge, does have a nice universal property, and can be considered as an instance o generating a ree monoid in a monoidal category. This puts the small object argument irmly in the context o well understood categorical algebra. This discovery o Garner s has demonstrated the value o aws as a much neater structure than weak actorisation systems. Emily Riehl has since gone on to adapt the deinition o model category to that o an algebraic model category, using aws in place o weak actorisation systems, and this work can be ound in [Riehl, 2011a], [Riehl, 2011b] and [Riehl, 2011c]. t the same time Garner has been applying aws in helping to understand higher categories, using them to classiy homomorphisms between weak n-categories see [Garner, 2008]. Throughout the work o Riehl there is an emphasis on the right maps the algebras o a given aws. There was a good reason or this; Garner proved (in [Garner, 2009]) a theorem that characterised the right maps in any coibrantly generated aws a right map structure on a morphism is precisely a choice o solution or every liting problem with a generating let map. Unortunately, there was no such easy description o let map structures. It was clear that any relative cell complex had a let map structure and that any let map was at least a retract o a relative cell complex; Garner s conjecture, which we attack in this paper, was that the let maps are exactly the relative cell complexes (in the speciic case o Top with the standard aws). This result will allow us to access the let maps too. What is more, the let maps as cell complexes are in many ways much more accessible and understandable than the right maps. It is hard to give an example o a right map structure that is neither trivial nor very complex, because o the ininite number o litings that must be speciied. But to give a simple, inite, example o a let map structure is very easy! The let maps have a constructive lavour that makes them, in the author s opinion, easier to work with. The result will also establish an important link between aws and the homotopy theory which is already understood. The relative cell complexes are a class o maps that have been around or a long time, and they are undamental to model categories; in a sense, it is important to check that they are indeed the let maps o the aws in order to make sure that the aws its properly into the existing theory. While this paper s result is restricted to one particular aws, the technique used should easily generalise to many other examples. In this case, there are potential applications to higher category theory: in the case o weak n-categories, the relative cell complexes are very closely connected to the idea o computads. This potential or generalisation will be discussed urther in Section 10.

3 306 THOMS THORNE The approach. We will prove the theorem by irst considering the existing deinition o relative cell complex, which gives a class o morphisms in Top. In Section 4 we adjust this deinition in order to obtain a category CellCx which has a natural orgetul unctor U to Top 2. ter making sure that this category deines a sensible notion o cell complex structure on a map, we exhibit a right adjoint to U, in Section 5. This adjunction can be thought o as a nice concrete expression o the small object argument; the universal property o a ree cell complex is exactly analogous to the liting property the small object argument is designed to obtain. The smallness condition, which we expect to ind somewhere, appears sooner than you might expect it is required or composition to be deined on cell complexes, in Section 4. In Section 6 we demonstrate that the adjunction is comonadic, so cell complexes are coalgebras or the comonad UK. In Section 7 we will see how UK is the let hand side o an aws, and in Section 8 we describe the universal property that CellCx satisies. This allows us, in the remaining Section 9, to prove that UK is isomorphic as a comonad to L, the let hand unctor o the aws we are interested in. This proves our main result: that our notion o cell complex structure is equivalent to the let map structures. cknowledgements. The author would irst o all like to thank his PhD supervisor Nick Gurski, as well as the other members o the Sheield Category Theory Seminar or providing a stimulating environment. He is also grateul or useul discussions he had with both Richard Garner and Emily Riehl in their recent visits to Sheield. Thanks are also due to the EPSRC or unding. 2. Background ny unctorial actorisation on a category C can be described as a pair (L, R) o a copointed endounctor and a pointed endounctor on C 2 (the category o arrows in C), with the ollowing properties: ˆ L is domain preserving, ˆ R is codomain preserving, ˆ the unctors cod L and dom R are equal, ˆ R L = or any. It is useul to give the unctor cod L, or equivalently dom R, a name; we will call it the central unctor o (L, R), and generally write it as M : C 2 C. In an algebraic weak actorisation system, we simply ask that L be a comonad and R be a monad. This turns out to be essentially the same as making a choice o solution or every liting problem between a coalgebra and an algebra. We should note that the original name was natural weak actorisation system; we ollow the name adopted by [Riehl, 2011a].

4 THE COLGEBRIC STRUCTURE OF CELL COMPLEXES Deinition. n algebraic weak actorisation system on a category C is a pair (L, R) where L = (L, ɛ, δ) is a comonad on C 2, R = (R, η, µ) is a monad on C 2, the copointed endounctor (L, ɛ) together with the pointed endounctor (R, η) make up the data o a single unctorial actorisation, and the pair satisies the distributivity axiom, explained below. The inal condition will ensure that the monad and comonad behave properly with respect to one another. It ollows rom the monad laws that δ must have trivial domain component and µ must have trivial codomain component, so their components take the orms (1, δ ) and (µ, 1): MR µ L M δ LL RR µ R δ ML or some δ and µ. Then we can deine a natural transormation : LR RL with components given by (δ, µ ). The distributivity axiom says that this is a distributive law o the comonad over the monad, meaning that it commutes with the unit, counit, multiplication and comultiplication transormations. The best notions o let map and right map are now given to us by the algebraic structure. Let L-Map be the category o coalgebras or the comonad L and let R-Map be the category o algebras or the monad R. What exactly does a let map structure on a morphism in M look like? s always, a coalgebra is an object : B equipped with a structure map L, which appears in this case as map α: α L M R B a kind o partial inverse to. The coalgebra axioms translate into very natural properties or α; in particular α = L and R α = 1 B. They also orce the domain part o the structure map to be the identity on this is why we can represent the let map structure with just one morphism in Top. In his paper [Garner, 2009] Garner introduces a revised version o the small object argument. This allows us to take any category I over C 2 (assuming some smallness conditions similar to those or the original small object argument) and produce an aws or which the category I is naturally a subcategory o the let map category. The argument is a transinite iteration where a single step perormed on : B involves considering the set o commutative squares X i Y B

5 308 THOMS THORNE where i is in the category I, and then orming the pushout o with many copies o each i I, one or each o the squares. This can be visualised as gluing many cells onto one or every way such a cell can be included in B via. When we iterate, we are essentially adding layer ater layer o cells in this way. In Garner s small object argument, there is a mechanism to prevent us rom adding superluous cells, and as a result the sequence converges. We obtain a actorisation o : M B, where any cell complex in B can be lited to M. We now consider the set o morphisms in Top, given by the the inclusion S n 1 D n or all n 0, where S 1 is considered to be the empty space and S 0 the pair o endpoints or D 1. We call this J and will treat it as a discrete category over Top 2. pplying the small object argument to J produces an aws on Top which is arguably the most undamental interesting example or homotopy theory; it is very close to the weak actorisation system between coibrations and trivial Serre ibrations that appears in the standard model category structure on Top. For the rest o this paper we will write it as (L, R); this is the aws or which we prove our result. 3. Strata The class o morphisms in Top which we write as J -cell and call the relative J -cell complexes is usually deined to be the smallest class containing J which is closed under coproducts, pushouts and transinite composition. Starting with this notion, we seek to deine a category o relative J -cell complexes, which we will call CellCx. There will be a orgetul unctor U : CellCx Top 2 whose image is precisely J -cell. In other words, every morphism in J -cell will have one or more cell complex structure. We ll introduce some helpul notation at this point. I is a relative cell complex, we will generally write U as, and we ll call the boundary or base space o, and the body o. In our deinition, cell complexes will be ormed as sequences o layers which we call strata. Each stratum is the pushout o a coproduct o single cells Deinition. stratum (X, S) consists o the ollowing data: a topological space X, a set o cells S, and or each s S a choice o κ s J and a continuous map b s : κ s X. The boundary o the stratum is X, and the body is given by the ollowing pushout square: s S κ s S Uκs s s S κ s b s s S X (X, S) So we can consider a stratum as a special sort o relative cell complex, or which U(X, S) is the bottom arrow o the diagram. We will later deine general relative cell complexes as sequences o strata satisying certain properties; irst we will consider some properties

6 THE COLGEBRIC STRUCTURE OF CELL COMPLEXES 309 o the category o strata. To work with the category Strata we must irst deine the morphisms Deinition. Let (X, S) and (Y, T ) be any two strata. morphism o strata (, p): (X, S) (Y, T ) is a continuous unction : X Y and a unction p: S T, satisying the requirement that or every s S, κ s = κ p(s) and b s = b p(s). Composition is the obvious thing: (, p) (, p ) = (, p p ). The unctor U : Strata Top 2 is deined as one would expect; (, p) is just, and (, p): (X, S) (Y, T ) is the unique map making the various diagrams commute. The irst thing we note about Strata is that each object J o J has a canonical stratastructure, ( J, { }) where κ = J and b = 1 J. Secondly, we note that strata-structures can be transerred along pushout: i : X Y has a strata-structure (X, S) then in the pushout X g Z Y Z X Y the bottom map has a canonical strata-structure, given in the ollowing deinition Deinition. Given any stratum (X, S) and map g : X Z, the pushorward o (X, S) along g, which we write g (X, S), is the stratum (Z, S) with each κ s the same as in the original stratum and each b s given by the original b s composed with g. We can see that Z X Y is the body o (Z, S) by commutativity o pushouts. Next we will consider colimits in Strata Proposition. The category Strata has all small colimits, and the unctor U : Strata Top 2 preserves them. Proo. We check coproducts and then coequalisers. indexed by. We claim that a (X a, S a ) = ( a X a, a Let (X a, S a ) be a set o strata where each κ s is the same as its original on the let, and each b s is given by composition o its original with the inclusion map. Given a set o strata morphisms (λ a, ρ a ): (X a, S a ) (Y, T ) there is a unique pair (, p) making the ollowing diagrams commute: S a ) (X a, S a ) ( a (λ X a, a S a) (,p) a,ρ a) (Y, T )

7 310 THOMS THORNE We must merely check that this (, p) is a strata morphism; given s S a, κ s = κ ρa(s) = κ p(s), and b p(s) = λ a b s = i a b s where i a is the inclusion X a X a. We see that U preserves this coproduct because pushouts and coproducts commute. s or coequalisers, consider a pair o strata morphisms (, p) and (g, q) between (X, S) and (Y, T ). We claim their coequaliser is the strata ormed by the coequaliser o and g and the coequaliser o p and q, which we will write as (Z, U). Each u U is an equivalence class o elements o T, all o which must have the same κ t, giving us κ u. They don t necessarily all have the same b t. However, we know the equivalence relation is generated by p(s) q(s). Since b p(s) = b s and b q(s) = g b s when we compose them with the coequaliser map Y Z we get a unique deinition o b u, which makes the map (l, m) below into a strata morphism. (X, S) (,p) (g,q) (Y, T ) (l,m) (Z, U) (h,r) (k,s) (, V ) To check the coequaliser property, let (h, r) coequalise (, p) and (g, q). We get a unique pair (k, s) making the diagrams commute; as beore we must simply check this is a strata morphism. For u U, κ s(u) = κ r(t) = κ t = κ u, using any t in the equivalence class o u. lso, b s(u) = h b t = k l b t = k b u. Finally, we must check U preserves this coequaliser; clearly κ u is the coequaliser o the two maps κ s κ t given by p and q, so the result ollows because coequalisers commute with pushouts. Finally we prove a useul lemma about morphisms o strata Lemma. [Pullback Lemma or Strata] Let (, p): (X, S) (Y, T ) be any strata morphism. The commutative square deined by U(, p) is a pullback square. Proo. We are considering the square X (X, S) (,p) Y (Y, T ) which we can demonstrate to be a pullback square using our understanding o limits and colimits in Top. Firstly, we know that X is a subspace o (X, S). Given a point x (X, S), assume it is not in X. Then it must be in some cell s S: it is a point in κ s, and not a point o the boundary κ s. So its image under (, p) is in the same position in the corresponding cell p(s) T, and hence not in Y. This demonstrates that as a point set, X is the pullback; since it is a subspace o X, and this determines its open sets, it is also the pullback as a space.

8 4. Cell complexes THE COLGEBRIC STRUCTURE OF CELL COMPLEXES 311 Now we understand the category o strata, we will move on to general cell complexes. These are deined as ininite sequences o strata satisying two important properties. The irst property says that the strata link together correctly, while the second is a kind o normal orm property: it says that every cell appears in the lowest possible stratum Deinition. n ininite sequence o strata is connected i the boundary o each stratum is equal to the body o the previous stratum Deinition. n ininite sequence o strata, (X n, S n ) N, is proper i there is no s S n or any n such that b s can be actored through the boundary o a lower stratum Deinition. relative cell complex is a proper connected sequence o strata. The image under U o such a relative cell complex is the transinite composite in Top o all the U(X n, S n ). gain we will talk about the boundary (X n, S n ) N and the body (X n, S n ) N. relative cell complex has ininite height i no S n is empty. lternatively it has height n i S n+1 is the irst empty set o cells. Clearly (because o the required property o properness) i S n is empty, then so is S m or all m > n. The trivial cell complex on X is the unique height 0 cell complex with boundary X. We deine a morphism o cell complexes by similarly extending the deinition or morphisms o strata Deinition. Given any two relative cell complexes (X n, S n ) N and (Y n, T n ) N, a relative cell complex morphism between them is a sequence o morphisms o strata, ( n, p n ) N : (X n, S n ) N (Y n, T n ) N, satisying the coherence condition that n+1 = ( n, p n ) or all n 0. The image under U appears as X 0 X 1 X 2... (X n, S n ) N ( n,p n) N Y 0 Y 1 Y 2... (Y n, T n ) N where ( n, p n ) N is the unique map that makes the diagram commute. We will write CellCx or the category whose objects are relative cell complexes and whose morphisms are relative cell complex morphisms. Strata embeds in CellCx as the subcategory o complexes with height less than or equal to one. We must check a ew important acts about CellCx. Firstly, we want to extend the result about colimits rom Strata to CellCx. Secondly, we look at some other constructions that can be made in the category. Finally, to make sure it is a good candidate or a category o relative cell complexes, we show that the image o U in Top 2 is exactly the class J -cell Proposition. The category CellCx has all small colimits, and the unctor U preserves them.

9 312 THOMS THORNE Proo. In act, the colimits o CellCx can be computed component-wise; so given a diagram o cell complexes the colimit is given by taking a colimit o strata or each natural number n. This deines a sequence o strata, and a sequence o strata morphisms. We must check these are connected, proper and coherent. We will need some notation; let the diagram consist o (X dn, S dn ) N or d ranging over the objects o the diagram category, let the proposed colimit be (Z n, U n ) N and let the colimit cocone morphisms be ( dn, p dn ) N : (X dn, S dn ) N (Z n, U n ) N. By connectedness, each (X dn, S dn ) = X d(n+1). Since U preserves the colimits o strata, (Z n, U n ) is the colimit o the (X dn, S dn ), meaning that we can choose it to be equal to Z n+1. This shows our proposed colimit cell complex is connected. By the same argument d(n+1) = ( dn, p dn ); so we ve also shown that the ( dn, p dn ) N are all coherent. Properness is harder to show; we will use the Pullback Lemma or strata. ssume the proposed colimit is improper: let u U n with b u actorable through Z n 1 as in the diagram. b s X dn κ u X d(n 1) d(n 1) b b u u Z n 1 Z n Now let s S dn be some cell with the property that p dn (s) = u; such a cell must exist by the deinition o U n as a colimit. Thus the map b u can also be actorised through X dn using b s. Using the pullback square we get a actorisation o b s through X d(n 1), yeilding a contradiction because the objects in the diagram are assumed to be cell complexes, and hence proper. We also need to check the colimit property. Given a cocone o morphisms (X dn, S dn ) N (Y n, T n ) N, there is a unique candidate sequence o strata morphisms (g n, q n ) N : (Z n, U n ) N (Y n, T n ) N given by each individual colimit property in Strata. We just have to check this sequence is coherent; this ollows easily rom the act that each sequence o strata is connected and that U preserves colimits o strata. Finally, since U o a cell complex is deined by transinite composition, we can see that U preserves colimits using the act that transinite composition commutes with other colimits. Now recall the deinition o pushorwards in the category Strata; this can be extended to CellCx. We construct the pushorward o each stratum in turn and because o the connectedness property there is only one way this can be done Deinition. Given any cell complex (X n, S n ) N and any map g : X 0 Z, the pushorward o (X n, S n ) N along g, which we write as g (X n, S n ) N, is the ollowing (Z n, S n ) N. Firstly, Z 0 = Z and (Z 0, S 0 ) is the stratum g (X 0, S 0 ). There is then a map (X 0, S 0 ) (Z 0, S 0 ) which we call g 1 ; (Z 1, S 1 ) is deined to be the stratum (g 1 ) (X 1, S 1 ). Continuing in this manner we construct each stratum o (Z n, S n ) N, and we obtain a morphism o cell complexes where each commutative square in the sequence is a pushout square. It dn

10 THE COLGEBRIC STRUCTURE OF CELL COMPLEXES 313 is a trivial equivalence o two colimits to see that then Ug (X n, S n ) N is the pushout o U(X n, S n ) N along g. new construction that we can perorm in CellCx, which was not possible in Strata, is that o composition. Suppose we are given two cell complexes, with the boundary o the second equal to the body o the irst. Because o some smallness conditions satisied by the maps o J in Top, we can combine the two into a single complex, whose underlying map is the composite in Top o the underlying maps o the two original complexes. This construction expresses the intuition that cell complexes can be glued onto one another to make larger complexes. To deine the composite o two cell complexes in general, we will start with a simple case. Let (X n, S n ) N be any cell complex and (Y, T ) be a stratum, which we consider as a height one cell complex. lso let Y = (X n, S n ) N, so that composition makes sense. We deine the composite, which we will write as (Z n, U n ) N, as ollows. First, we use the standard result (see, or example, Proposition in [Hovey, 1999]) that compact spaces are inite relative to closed T 1 inclusions. Each map X n X n+1 is a closed T 1 inclusion, and the boundary o every cell is compact hence or every t T there is a smallest n t such that b t actors through X nt. This partitions T into a sequence o sets, (T n ) N. Now let Z 0 = X 0 and let U 0 = S 0 + T 0. Thereater, we let each Z n = (Z n 1, U n 1 ) and each U n = S n + T n, where all the b s and b t are deined in the obvious way. straightorward equivalence o two colimits shows that the underlying map o the composite is the composite o the underlying maps. It is also important to note that there is a canonical cell complex morphism (X n, S n ) N (Z n, U n ) N, with (1 X0, U(Y, T )) as its underlying morphism in Top 2 : X 0 U(X n,s n) N Y U(Y,T ) X 0 U(Z n,u n) N (Y, T ) Deinition. Given any two cell complexes and B, such that the body o is the boundary o B, we deine the composite B by repeating the above construction or each stratum in B. This gives a sequence o cell complexes n, where each n is with the irst n strata o B composed onto it. Because we have all small colimits in CellCx, we can deine B as the colimit o this sequence, and because U preserves colimits, this has the correct composite as its underlying map. We note here that this deinition o composite can be extended easily to composing a transinite sequence o cell complexes, using exactly the same technique any ordinal sequence o cell complexes gives an ordinal sequence o strata which we add one by one, taking the colimit at each limit ordinal. nother very important observation is the ollowing: 4.8. Proposition. Given a pair o cell complex morphisms φ: and ψ : B B, i and B are composable, and B are composable, and ψ = φ, then they give rise

11 314 THOMS THORNE to a new cell complex morphism ψ φ: B B, such that U(ψ φ) = ( φ, ψ). We call it the horizontal composite o the two cell complex morphisms. Proo. Consider the case where B and B are height one, write B = (Y, T ) and B = (Y, T ), and let ψ = (g, q): we must check that q respects the partitioning o T and T into (T n ) N and (T n) N ; that is, or each u U, we want n u = n q(u). But n q(u) n u ollows rom the act that (g, q) is a strata morphism, and the pullback lemma ensures that n u n q(u). Now an induction argument on the height o B will show that ψ φ is well deined. It is worth noting that we have now deined a double category whose objects are spaces, whose vertical morphisms are continuous unctions, whose horizontal morphisms are cell complexes and whose 2-cells are cell complex morphisms Proposition. The image o the unctor U : CellCx Top 2 is exactly the class o morphisms J -cell. Proo. Firstly, the deinition o U(X n, S n ) N is as a transinite composite o pushouts o coproducts o elements o J, so the image is certainly a subclass o J -cell. To show the opposite inclusion, since it is clear that each element o J has a CellCx structure, we need only check that the image o U is closed under coproducts, pushouts and transinite composites. We have just proved that all colimits exist in CellCx and are preserved by U, and we have just deined pushorwards o cell complexes. We have also just deined composites, and as we pointed out these are easily extended to transinite composites. Finally, there s also a pullback lemma or cell complexes Lemma. [Pullback Lemma or Cell Complexes] Given any morphism o cell complexes, its image under U, when viewed as a commutative square in Top, is a pullback square. Proo. Let ( n, p n ) N : (X n, S n ) N (Y n, T n ) N be a cell complex morphism. ny map rom the one point space to (X n, S n ) N must actor through some X n, because the one point space is compact. Thus, given a point in Y 0 and a point in (X n, S n ) N with the same image in (Y n, T n ) N, a inite number o applications o the pullback lemma or strata will give a unique point in X 0, and this shows that as a set at least, X 0 is the pullback we want it to be. But its open subsets are determined by the subspace inclusion into (X n, S n ) N, and this shows that it is indeed the pullback Remark. This result is really a little stronger than stated; it implies that given any morphism o cell complexes one can remove any inite number o strata rom the beginning and the remaining square is also a pullback in Top simply because it is also a morphism o cell complexes. This act will be vital in Section 6.

12 5. The adjunction THE COLGEBRIC STRUCTURE OF CELL COMPLEXES 315 We now examine some more properties o the category o cell complexes; they will let us see that it is a category o let maps, and in act one with a useul universal property with respect to J. First we show that there is a right adjoint to U, which makes an adjunction that will turn out to be comonadic. In order to construct this right adjoint K, we will irst restrict our attention to Strata, and then consider the whole o CellCx. It is worth bearing in mind that the construction in this section is very closely analogous to the small object argument; the irst proposition gives a single step, the second proposition iterates it Proposition. The unctor U : Strata Top 2 has a right adjoint K 1. Proo. Let : B be any continuous unction between topological spaces; in other words, any object o Top 2. Let S be the set o all morphisms to o the orm j Uj j B in Top 2, or any j J. Notice that any element s S comes with a canonical choice o κ s J and b s : κ s. This means that (, S) is a stratum. It also comes with a canonical morphism (1, E 1 ): U(, S) in Top 2, whose codomain part E 1 : (, S) B is determined by the pushout property o (, S), using and the codomain part o each s S. We claim that we have just constructed K 1, and that (1, E 1 ) is the counit o the adjunction. Suppose (X, T ) is any stratum and (g, h): U(X, T ) a morphism o Top 2. Because o the pushout deinition o (X, T ), the unction h is determined by g and a morphism h t : Uκ t in Top 2 or each t T ; this is all the inormation that makes up (g, h). But each h t gives an element s S, so this inormation also exactly deines a morphism o strata, (g, t h t ): (X, T ) (, S), and actors (g, h) through (1, E 1 ). The actorisation is unique because E 1 is epic; we have demonstrated the correspondence necessary or K 1 to be the right adjoint o U. The construction in this proo o E 1 and UK 1 is a unctorial actorisation o exactly the actorisation produced by the irst step o either small object argument (Garner s or Quillen s). In the next proposition we extend the right adjoint to CellCx, iterating in precisely the manner o Garner s small object argument Proposition. The unctor U : CellCx Top 2 has a right adjoint K. Proo. In the proo o the previous proposition, we constructed a right adjoint to K 1 : Top 2 Strata or U in the case o strata. We also deined a unctor E 1 : Top 2 Top 2 which appeared in the counit o the adjunction and which will prove rather useul. gain, consider : B, any object o Top 2. pply K 1 to to obtain the stratum

13 316 THOMS THORNE (, S) and the unction E 1 : (, S) B. Then apply K 1 again, this time to E 1, to get another stratum whose boundary is (, S). This also gives another new unction E 1 E 1, to which we apply K 1 in turn. Continuing this process gives a connected sequence o strata, which would be a good candidate or K, except or one problem: the sequence is not proper. similar approach does work, however (with a little bit more work) i we deliberately omit the improper cells at each stage. Let 0 = and S 0 = S as deined above; there are no improper cells in the irst stratum, so ( 0, S 0 ) is the irst stratum o K. We let 1 = ( 0, S 0 ) and deine S 1 to be the set o morphisms Uj E 1 in Top 2, or any j J, satisying the additional condition that their boundary maps do not actor through 0. s beore, ( 1, S 1 ) is clearly a stratum and we get a new morphism E 2 : ( 1, S 1 ) B. We can continue in this ashion, deining n to be ( n 1, S n 1 ) and S n to be the set o morphisms Uj E n whose boundary maps do not actor through n 1. This produces a connected sequence o strata which is this time proper by construction. Furthermore, there is an unique map E : ( n, S n ) N B which commutes with all the E n E 2 B ( n, S n ) N E 1 E Let (X n, T n ) N be any cell complex and (g, h): U(X n, T n ) N a map o Top 2. The irst thing we note is that the map h corresponds to an ininite sequence o maps, one rom each X n ; we call this h n : X n B. Now (g, h 1 ) is a map in Top 2 rom U(X 0, T 0 ) to, so by the adjunction between Strata and Top 2 we obtain a strata morphism (g, p 0 ): (X 0, T 0 ) ( 0, S 0 ). Say g 0 = g, and g 1 = (g 0, p 0 ). Then (g 1, h 2 ) is a map in Top 2 rom U(X 1, T 1 ) to E 1 ; this induces a strata morphism (g 1, p 1 ): (X 1, T 1 ) ( 1, S 1 ) use the same argument as to construct the morphism (X 1, T 1 ) K 1 E 1, and note that each o the cells in the image is proper. I some cell t T 1 were to have b p1 (t) that could actor through X 0, then by the Pullback Lemma b t would actor through X 0 which is impossible. Now we repeat the construction o (g 1, p 1 ) to deine a sequence o strata morphisms (g n, p n ) N : (X n, T n ) N ( n, S n ) N which is automatically coherent. Since E n g n = h n or each n, we have E (g n, p n ) N = h and we have actored (g, h) through (1, E). The actorisation is unique, again because E can be seen to be epimorphic. This demonstrates the correspondence that makes K the right adjoint o U. It is useul to note that the construction we made in the irst paragraph o the proo, beore we insisted that the sequence be proper, is eectively Quillen s small object argument. When we altered the construction to ensure a proper sequence, we missed out the superluous cells; the distinction between the improper sequence construction and the proper sequence construction that ollows it is precisely the distinction between Quillen s and Garner s small object arguments.

14 6. Comonadicity THE COLGEBRIC STRUCTURE OF CELL COMPLEXES 317 Two lemmas will be suicient or us to prove that the adjunction is comonadic; we ll use the standard result known as Beck s Monadicity Theorem, which can be ound in [Mac Lane, 1998] and many other places besides Lemma. The category CellCx has all equalisers, and the unctor U preserves all equalisers. Proo. We start by showing this result or Strata. Let (, p) and (g, q) be two strata morphisms rom (X, S) to (Y, T ). Now let e: E X be the equaliser o and g in Top, and let r : L S be the equaliser o p and q in Set. We claim that (e, r): (E, L) (X, S) is the equaliser we are looking or in Strata. Firstly, we must check it is actually a stratum; given l L, because p(r(l)) = q(r(l)) we have b r(l) = g b r(l) so there s a unique map κ r(l) E which we use to deine b l. This deinition o κ l and b l makes (e, r) automatically a strata morphism; we must just check the limit property. Given a stratum morphism (h, m): (Z, W ) (X, S) which also equalises (, p) and (g, q), we get a unique pair (k, n): (Z, W ) (E, L), shown by the dotted arrow: (Z, W ) (h,m) (E, L) (e,r) (X, S) (k,n) (,p) (g,q) (Y, T ) This (k, n) is a strata morphism: it is clear that or any w W, κ w = κ n(w) and e k b w = h b w = b m(w) = e b n (w), which implies k b w = b n(w) because e is monic. Furthermore, U preserves this equaliser: consider its image under U irstly, the boundary E is by deinition the equaliser we want. Secondly, a point in (X, S) has the same image under and g i it is either in E X or in a cell s S such that p(s) = q(s); hence as a point set, (E, U) is the equaliser. s a space, its topology is determined by it being a subspace o (X, S), so we are done. To extend to CellCx, we use a very similar argument to that in Proposition 4.5; we claim the equaliser o a pair o cell complex morphisms is given as the sequence o equalisers o strata. This sequence is connected, and the sequence o morphisms is coherent, by exactly the same reasoning as in Proposition 4.5. To show it is proper is in act much easier here, because the equaliser is a subcomplex o the irst cell complex the equaliser map is a sequence o strata inclusions. We also check the limit property; this ollows rom the same argument as in Proposition 4.5. Finally, consider the image under U. Using the result or Strata, the boundary o each stratum in the equaliser is the correct equaliser in Top. Then, because every point in the body appears in the boundary o some stratum (since the one point space is compact) the image under U is correct as a unction o sets. It then ollows it is correct as a continuous unction between spaces, again by considering subspace inclusions which determine its topology.

15 318 THOMS THORNE 6.2. Lemma. The unctor U is conservative. Proo. s usual, we prove this or Strata and then extend the result to CellCx. Let (, p): (X, S) (Y, T ) be a strata morphism and assume that U(, p) is an isomorphism in Top 2. We consider the inverse o U(, p) in Top 2, which we will write (g, h). The unction h is determined by g and a morphism h t : κ t U(X, S) or each t T ; and U(, p) h t is the canonical inclusion o κ t into U(Y, T ). This means that each h t makes a choice o h (t) S such that p(h (t)) = t and h (p(s)) = s. This shows that (g, h) has a strata morphism structure given by (g, h ), and this strata morphism is an inverse to (, p), showing that it is an isomorphism, and hence that U : Strata Top 2 is conservative. Consider a morphism o cell complexes, ( n, p n ) N : (X n, S n ) N (Y n, T n ) N, and assume its image under U is an isomorphism. This immediately shows that 0 is an isomorphism. Using the remark ollowing the pullback lemma or cell complexes, each n is a pullback o ( n, p n ) N, and since the pullback o an isomorphism is an isomorphism, all the n are isomorphisms. Now use the result on Strata to see that all the strata morphisms ( n, p n ) are individually isomorphisms; hence, ( n, p n ) N is an isomorphism and we are done Corollary. Since U is conservative, CellCx has and U preserves all equalisers, the dual o Beck s monadicity theorem implies that the adjunction between U and K is comonadic. 7. The aws t this stage it ollows directly rom a result o Garner (the dual o Theorem 4.9 in [Barthel, Riehl, 2012]) that, since we have a comonad U K whose category o coalgebras admits a composition law, it must appear as part o an aws whose monad part is given by the counit. However, in the interests o clarity we will spend some time proving this act explicitly. From now on we will observe the notational convention o writing morphisms in Top 2 and CellCx vertically (as squares where the source and target morphisms are horizontal). This seems to make the diagrams clearer, and is appropriate i one considers the double category point o view. In some o the diagrams we will also use a notational shorthand where instead o explicitly writing a map and its actorisation, we draw arrows going to and rom the middle o an arrow to mean morphisms to and rom the central object o that arrow s actorisation. Thus a let map will be drawn as and the image o a morphism (a, b) in Top 2 under the actorisation will be drawn as α B a C g K(a,b) B b D.

16 THE COLGEBRIC STRUCTURE OF CELL COMPLEXES 319 rrows to and rom the one quarter point or three quarters point o an arrow mean the obvious thing, where the let or right part o a actorisation has been actorised again Proposition. The endounctor E on Top 2, which was deined in Proposition 5.2, is a monad. Proo. In Deinition 4.7 and Proposition 4.8 we showed how cell complex structures can be composed; this will provide us with the multiplication or the monad E. Firstly, the unit η : 1 E is given by η = (UK, 1 B ) where : B. Now, KE is a cell complex which can be composed with K; we write the composite KE K. There is a morphism in Top 2 given by (1, EE): U(KE K), hence by the adjunction there is a cell complex morphism φ: KE K K. We deine µ to be the codomain part o Uφ, and claim that η and µ = (µ, 1) make E into a monad. First we check that µ is a natural transormation (this is clear in the case o η). Consider (a, b): g any morphism o Top 2 ; in the diagram µ B a K(a,b) K(K(a,b),b) b C g D µ g we wish to compare the two sides o the naturality square which are K(a, b) µ and µ g K(K(a, b), b). Because these are both the body maps o cell complex morphisms whose boundary maps are a, we can use the adjunction between U and K. It s a quick diagram chase to see that either side when composed with Eg gives b EE, which means, by the adjunction, that they are equal and µ is natural. To check the monad laws, we use a similar method: E µ UKE = EE UKE = E immediately shows that µ UKη = 1, and the other unit law ollows rom E µ K(UK, 1 B ) = EE K(UK, 1 B ) = E. Finally, to demonstrate the multiplication law, we wish to show that the diagram µ µ µ E K(µ,1) B B

17 320 THOMS THORNE commutes. By the diagram chase E µ µ E = EEE = EE K(µ, 1) = E µ K(µ, 1) and the act that the two maps we are comparing are both the body maps o cell complex morphisms, we can use the adjunction again and the multiplication law holds Proposition. The pair (U K, E) is an algebraic weak actorisation system. Proo. We have seen already that UK is a comonad, E is a monad and that they it together to orm a unctorial actorisation system. This is almost all that is required to make (UK, E) an aws; the only remaining thing to check is the distributivity axiom. There is a natural transormation : UKE EUK with components given by the square K δ KU K UKE EUK KE where δ is the codomain part o the comultiplication o UK. This is required to be what is called a distributive law o UK over E; this means it must satisy our commutative diagrams, basically saying it commutes with the unit, counit, multiplication and comultiplication o UK and E. When we translate these commutative diagrams into components in Top 2, they become eight identities in Top. Upon examination, six o these identities are immediately true: our o them rom the comonad and monad laws, and two o them simply by deinition o µ and δ. The inal two identities are in act the same, and this single identity is shown in the diagram µ UK δ K(δ,µ ) µ µ K B. We ll use a similar argument to those in Proposition 7.1. We have δ E EUK µ UK K(δ, µ ) δ E = EEUK K(δ, µ ) δ E = µ EUKE δ E = µ = EUK δ µ and the maps we are comparing appear as cell complex morphisms, so we are done.

18 THE COLGEBRIC STRUCTURE OF CELL COMPLEXES 321 Simply knowing how UK appears as the comonad part o an aws is not enough; we have also deined pushorward and composition structures on CellCx and we need to check that these are compatible with the aws (UK, E). First, we note the general deinition o pushorward and composition or the let maps o any aws; then we will check they agree on CellCx. First o all, composition o let maps has been deined by Riehl (see [Riehl, 2011a]) as ollows: 7.3. Deinition. Given a pair o let maps (, α) and (g, β) where and g are composable. Then g has the composite let map structure shown by the dotted arrows in the ollowing diagram g µ g C M(1,g) M(M(1,g) α,1) α B where M is the central unctor o the aws. We will write the composite let map structure as (g, β α). It is straightorward to check that this (β α) satisies the coalgebra axioms; more details can be ound in [Riehl, 2011a]. There is also the ollowing natural deinition o pushorward. Note that we will begin using the notation [a, b]: +B X or the unique map satisying [a, b] i = a and [a, b] i B = b, where i and i B are the inclusion maps o the coproduct, and similarly or maps out o pushout objects Deinition. Given a map : B with a let map structure α or some aws (L, R), and a map g : C. Then the pushout o along g, which we write as g, has a canonical let map structure called the pushorward o α along g and written as g α. It is given by considering g C g α M(g, g) g α g B β g B C and speciying the structure map g α as [Lg, M(g, g) α]. gain, checking the coalgebra axioms is very straightorward. Now, as a rather important sanity check beore we continue, we check the two deinitions o composites and pushorwards agree in CellCx Proposition. The deinition o composition in CellCx given in Deinition 4.7 is the same as the general deinition applied to CellCx as the category o let maps or the aws (UK, E). C

19 322 THOMS THORNE Proo. Given two cell complexes, considered as let maps (, α) and (g, β), it is enough to check that (1, β α) is a cell complex morphism (g, β) (, α) K(g). Then by the adjunction between U and K, the act that E(g) (β α) = 1 C (which is one o the coalgebra axioms) implies that the let map structure is the correct one. To show it is a cell complex morphism, actorise it as B g C B UKg β Kg UK α K K(K(1,g) α,1) UK(g) K(1,g) K(g) UKE(g) KE(g) UK(g) µ g K(g), where every square is a cell complex morphism: some are images under K o morphisms in Top 2, others like (1, α) and (1, β) are cell complex morphisms by deinition. The very bottom square is the cell complex morphism reerred to as φ in Proposition 7.1. We can compose the whole diagram together, using both vertical and horizontal composition o cell complex morphisms (see Proposition 4.8), to demonstrate that (1, β α) does have the structure o a cell complex morphism Proposition. The deinition o pushorward given in Deinition 4.6 is the same as the general deinition applied to CellCx as the category o let maps or the aws (UK, E). Proo. Let (, α) be a cell complex, and write (g, g α) or the pushorward given by the general deinition. Write β or the structure map o the cell complex given by Deinition 4.6; we need to show that β = g α. Firstly, it is clear that g α g = β g this is one hal o the neccessary identity. lso, it is clear rom the deinition o β that (g, g) has a cell complex morphism structure. This means that which is the other hal o the identity. g α g = K(g, g) α = β g, 8. The universal property We now have an aws (UK, E) which we wish to compare with the aws produced by the small object argument. One could take the approach o directly examining the two

20 THE COLGEBRIC STRUCTURE OF CELL COMPLEXES 323 comonads; i you draw pictures o both it becomes clear they are essentially the same. However, in the remainder o this paper we will use a dierent approach in which we exhibit a universal property o CellCx and compare it to the universal property established by Garner or the small object argument. The author eels that this approach while less eicient is more illuminating, as it expresses the universality o the cell complex category construction and thus helps us to see why it is a natural way to build an aws. There is a canonical unctor η : J CellCx over Top 2, given by assigning each map in J its canonical height one, single-cell complex structure. The pair o CellCx and η is universal among unctors rom J to categories o let maps over Top 2, with respect to the composition preserving unctors between let map categories (which are exactly morphisms o aws see [Garner, 2009]). To make this work, we need to be sure that composition in an arbitrary let map category is suiciently well behaved; we begin by proving two lemmas that express this Lemma. For any aws (L, R), the composition rule in L-Map is strictly associative. Proo. Given three composable let maps, (, α), (g, β) and (h, γ), we obtain two let map structures on hg given by the two ways o composing, namely γ (β α) and (γ β) α. The structure maps or these are both shown, using dotted arrows, in the ollowing diagram: hg µ hg µ R(hg) D g M(1,h) M(M(1,h),h) ψ M(1,g) α M(1,hg) hg µ g g B h C D β hg M(1,h) M(M(1,h) β,1) µ hg M(M(1,hg) α,1) D µ R(hg) µ hg γ Using two naturality squares or µ (marked by the little squares in the diagram) and the multiplication law, we can actor both structure maps through MRR(hg), and hence reduce the problem to that o comparing the map M(M(M(1, h), h) M(M(1, g) α, 1) β, 1)

21 324 THOMS THORNE (which is marked in the diagram as ψ) with the composite M(M(M(1, hg) α, 1), 1) M(M(1, h) β, 1). By the unctoriality o M, this reduces to considering M(M(1, h), h) M(M(1, g) α, 1) = M(M(1, hg) α, h) and the two dotted composites in the diagram are the same. = M(M(1, hg) α, 1) M(M(1, h)) The second lemma will prove what we will call the stacking property o let maps; it is absolutely vital in what ollows because it justiies the requirement or cell complexes to be proper sequences. Stacking allows you to take a let map which is deined as a composite o colimits and move the individual elements o the colimits about without altering the let map structure. Since a cell complex is essentially deined as a composite o colimits, this says we can move cells in between strata reely; hence every potential cell complex can be reordered to make it proper and properness deines a natural normal orm or cell complexes which hugely simpliies the deinition Lemma. For any aws (L, R), the composition rule in L-Map is well behaved with respect to coproducts and pushorwards in the ollowing way: given : and g : B B equipped with let map structures α and β, and maps a: X and b: B X, there is an isomorphism o let maps ([a, b] ( + g), [a, b] (α + β)) = (((a ) b) g, ((a ) b) β) (a, a α) Remark. The proposition basically says that and g can be glued on to X in any order, or simultaneously by taking a coproduct irst, and it makes no dierence to the resulting let map. In the orm o a picture: X + B X a [a,b] = (α ) b Proo. The let map on the let hand side is constructed using the pushout square +g + B + B B [a,b] X [a,b] (+g) Y [a,b]

22 THE COLGEBRIC STRUCTURE OF CELL COMPLEXES 325 while the let map on the right hand side is constructed using the two pushout squares in the diagram a a X Z W (a ) b b B B a ((a ) b) g and then composing. The irst thing to note is that Y and W have exactly the same universal property; we are thus able to choose pushouts in such a way that Y = W. Furthermore, i we make this choice, the underlying maps o the let maps we are comparing are identical. So i we check that the structure maps are equal, with this choice o pushout objects, then or any other choice the let maps we obtain will be isomorphic. We will now write h: X W or the underlying map; we have two structure maps W Mh and we wish to show they are equal. Using the universal property o W, it is suicient to show the maps X Mh, Mh and B Mh that make up these structure maps agree. The X Mh parts are both just Lh, so they are easy. Showing that the other two parts agree can be done with two simple diagram chases. First, to simpliy notation, we will start writing g or ((a ) b) g, and or a ; we will also write α or the pushorward structure map on and β or the one on g. Next, we note that the map [a, b] can be written as [g a, b ], using the universal property o W. Now consider the diagram X h M(1,g ) M(M(1,g ) α,1) X α β g Z W a b a b B g B. α β Two straightorward chases show that (β α ) g a = M(a, g a ) α and that (β α ) b = M(b, b ) β. We then actor M(a, g a ) through M(i, i ) and actor M(b, b ) through M(i B, i B ), the two inclusion maps to the coproduct. Then using the act that, by deinition o (α, β), M(i, i ) α = (α + β) i and M(i B, i B ) β = (α, β) i B, it is clear that (β α ) and [a, b] (α + β) agree on both and B. The universal property o CellCx with respect to J now ollows without too much diiculty. µ h g b W