Infinite Loop Spaces

Daniel Feuntes-Keuthan

1 A Good Notion of Coherence

The theorem of Brown Represntability gives us a correspondence between !- spectra and (reduced) cohomology theories. Likewise, an infinite loop space, along with its deloopings, gives us an !-spectrum and hence a cohomology theory. Because of this we wish to recognize infinite loop spaces and construct explicit deloopings. If we start with an Abelian topological monoid A, we can form the classifying space BA as the geometric realization of the bar construction B.A. Recall n t that simplicial space is defined to have BnA = A as its n h space, that the face maps multiply group elements (except at the endpoints where they leave o↵elements), and that the degeneracies insert identity morphisms. We can also form the (contractible) total space, E.A which is the realization of the n+1 space that has EnA = A and a similar simplicial structure as the classifying space. There is a right action of A on the total space, and a fibration sequence A EA ⇣ BA. The space BA is a topological Abelian group, so we can iterate the! classifying space construction to obtain topological spaces BnA together n 1 n with weak equivalences B A ' B A for n>1. When A is an Abelian ! topological group, the map A ' ⌃BA is also a weak equivalence. Hence in this case A is an infinite loop space,! and the spaces BnA are the explicit deloopings. Unfortunately this turns out to not be a very useful example due to the following. Theorem. Every Abelian topolgoical group is a product of Eilenberg-Maclane spaces. So really the cohomology theories that we get from this construction are nothing more than products of shifted reduced singular cohomology theory. The problem was that we were to strict in requiring the monoidal operations of our space to be associative, commutative, and unital on the nose. Since we are interested in homotopy theory, we will instead look at spaces where these conditions are weakened to hold only up to homotopy. Definition. An H-space is a topological space X equipped with a map µ: X ⇥ X X and a morphism e: X, together with homotopies µ(e, ) idX µ( !,e).WecalltheH-space⇤!X homotopy associative if the two maps ' ' µ( µ( , )),µ(µ( , ), ): X3 X !

1 are homotopic, and homotopy commutative if there is a homotopy between µ and the map which switches the first and second coordinate in X2 and then applies µ.Ofcourseµ is meant to invoke the notion of unital multiplication, with the map e serving as the unit. Note that the definition of H-space makes no reference to inverses, and that for any H-space ⇡0X is a semi-group. We call an H-space group-like if ⇡0X is a group. Every loop space is easily seen to be a homotopy associative group- like H-space, and in addition any k-fold loop space with k 2 is homotopy commutative.. However, while associative H-spaces require some coherence in their multiplication, they lack the higher coherence present in a k-fold loop space, and so cannot on their own serve as a viable option. For simplicity, let µ(x, y)=xy. Note that for any three points x, y, and z in an associative H-space, there is a path connecting (xy)z and x(yz). However, if we add an additional point w X there are two paths between x(y(zw)) and (x(yz))w as shown below. 2

x((yz)w)

x(y(zw)) (x(yz))w

(xy)(zw) ((xy)z)w

In a loop space, we would want these two paths to be homotopic via a higher homotopy. Likewise, if we had more elements, we would ask that higher homotopies exist to fulfill higher coherence in associativity. And this is all before we begin considering the homotopy commutative structure! This quickly gets out of hand, so we need more organized ways to keep track of this coherence information. We will discuss two methods of May and Segal to do this, both of which provide a recognition theorem for infinite loop spaces, as well as a means of delooping using di↵erent two sided bar constructions.

2 Delooping and the Bar Construction

In this section and the next we discuss May’s recognition theorem for k-fold loop spaces. Recall that the suspension ⌃and loop functors ⌦on topological space form an adjunction, so that ⌦⌃is a monad. We recall the definitions of monad and algebra over a monad. Definition. A monad on a category C is a functor T : C C together with nat- ural transformations µ: T 2 T and ⌘ :1 T making! the following diagrams commute. ) )

2 T⌘ ⌘ Tµ T T 2 T T T 3 T 2

µ µT µ µ T T 2 T

We call µ the multiplication of the monad, and ⌘ the unit. Indeed a monad is a monoid object in the category of endomorphisms of C. This definition seems to be said tongue-in-cheek at times, but becomes quite useful as we think of monads acting on algebras and modules. Definition. An algebra over a monad T on a category C is an object A obC together with a morphism TA a A, referred to as the structure map,2 which serves as a left action of the monad! on A. In other words the following diagrams commute.

⌘ µ X X TX T 2X X TX

a Ta a X T 2 a X

Considered as a constant functor at X,aT -algebra is really just a left module over the monoid T . We can consider right monoids as well. These are functors S CC equipped with natural transformations ST S satisfying unit and distribution2 diagrams as above. ) From now on we will restrict to working over Top , though two sided bar constructions exist in much greater generality. ⇤ Definition. Given a monad T , a right module S,andaT algebraX,thetwo- sided simplicial bar construction B.(S, T, X) is a simplicial space with Bn(S, T, X)= STnX. Degeneracies are given using the unit ⌘ of the monad, and face maps are given by the monadic multiplication µ, except for the right and left end points where the module and algebra structure maps are used respectively. The two- sided bar construction B(S, T, X) is the geometric realization of the two-sided simplicial bar construction.

Proposition. B.(T,T,X) is homotopy equivalent to X as simplicial spaces, hence in particular B(S, T, X) is homotopy equivalent to X as spaces. The crux of May’s argument is to show that ⌦k commutes with geometric realization for certain spaces X. Once this is solved, we have the following theorem. Theorem. Let X be a group-like topological space which is an algebra over the monad ⌦k⌃k, then X is homotopy equivalent to a k-fold loop space. Proof. The idea is that we have a chain of homotopy equivalent simplicial ob- jects,

X ' B (⌦k⌃k, ⌦k⌃k,X) ' ⌦kB (⌃k, ⌦k⌃k,X) ! . ! .

3 and so upon taking geometric realization and commuting ⌦k with the bar construction, we get a homotopy equivalence of spaces X ' ⌦kB(⌃k, ⌦k⌃k,X). In particular this gives us an explicit delooping of X. !

3 Operads

The discussion of the previous section provided us with a recognition theorem for infinite loop spaces. However it turns out that finding algebras over the monad ⌦k⌃k is no easy feat either! We remedy this by using the theory of k k operads to define a Ck together with a natural transformation Ck ⌦ ⌃ k k ) which will make every Ck-algebra into a ⌦ ⌃ -algebra, providing another form of the recognition theorem. Definition. An operad Cis an N-indexed collection of topological spaces C(j) together with maps for each set of integers t, j1,...,jt : C(t) C(j ) C(j ) C(⌃ j ) ⇥ 1 ⇥···⇥ t ! i i Which satisfy certain equivariance and associativity properties. In addition we require C(0) ,anda neutral element 1 C(1). '⇤ 2 This definition is perhaps more enlightening if we consider an important example, which gives credence the motto that operads capture j-ary operations on a space.

Definition. The endomorphism operad EX of a topological space X has EX = Map(Xj,X. Given f : Xt X,andg : Xj1 ,...,g : Xjt we define (f,g ,...,g ) ! 1 t 1 t 2 Map(X⌃ji ,X) to be

⇧gi f ⇧t Xji Xt X i=1 ! ! The equivariance properties tell us how we must permute elements when when we permute maps, and vice versa. The neutral element is the identity map on X.

Definition. A morphism of operads is a collection of maps ✓j : C C0 which respect . !

C(t) C(j ) ,..., C(j ) C C(⌃ j ) ⇥ 1 ⇥ ⇥ t i i ⇧✓ ✓ C0 C0(t) C0(j ) ,..., C0(j ) C0(⌃ j ) ⇥ 1 ⇥ ⇥ t i i

Definition. An algebra X over an operad C is a morphism of operads C EX . By adjunction, this is a collection of maps C Xj X, and again we think! of C as encoding the j-ary operations of X. ⇥ ! The name algebra reminds us of algebras over a monad, and indeed we can fold an operad into a monad so that the algebras correspond.

4 Proposition. To any operad C we can associate a monad so that the operad algebras of C and the monad algebras of are in one to oneC correspondence. C j Proof. For a space X,wedefine X = n 0(C(j) ⌃j X ). Unraveling the definitions gives us the correspondenceC we desire. ⇥ ` 4 The Little n-cubes operad

Let Cn(j) be the space of j axes-parallel, labelled, non overlapping n-cubes embedded into an ambient n-cube. We call Cn the little n-cubes operad. is defined, for example, in the case of C2(2) C2(3) C2(1) C2(4) as in the following picture. ⇥ ⇥ !

? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? y

n Every n-fold loop space ⌦ X is an algebra over Cn. The algebra maps C (j) Xj X are defined by collapsing the outside of the embedded cubes to n ⇥ !

5 get a wedge of n-spheres, applying the multiplication of the loop space to each sphere, and then applying the folding map.

The converse is true as well under certain assumptions.

Theorem. Let X be a connected, group-like space, then X is a Cn-algbera if and only if X is weakly equivalent to an n-fold loop space.

Proof. We discusses one direction already. For the converse, it was shown that n n there is a morphism of monads n ⌦ ⌃ which was a weak equivalence for C ! n any connected space X. We can then form the bar construction B(⌃ , n,X), and this is again an n-fold delooping of X. C

In particular, algebras over C1 are referred to as A spaces. Note that 1 there are operad inclusion Cn , Cn+1 by including embedded cubes with one constant coordinate. The colimit! of these arrows is the operad C ,whichis seen to be contractible (ie. each space is contractible) with free ⌃1action. Of course, any operad with such qualities is equivalent, but in some cases perhaps it may make life easier to consider di↵erent models for this operad.

Definition. An operad C is an E operad if it is contractible and has free symmetric group action. An algebra1 over an E operad is called an E space. 1 1 We end this section with an infinite loop space version of the recognition theorem.

Theorem. A connected, group-like space X is an E space if and only if it is weakly equivalent to an infinite loop space. 1

5 Segal’s Method

Segal’s infinite loop space machine is similar in spirit to May’s, but uses a di↵erent version of the two sided bar construction, and captures the loop space structure as a functor rather than an operad algebra. Let’s return back to (strict) topological Abelian monoids for a second, letting A be such an object.

6 op op We associate to A a functor FA : SET,where is the opposite ! S category of finite pointed sets, by defining FA(S)=A . A partially defined f F f map S T defines a map AS A AT by sending a collection ! !

FA(ass)t =⌃as0 . s f 1(t) 02 One can check that this functor satisfies the following two properties.

For any sets S, T , F (S T ) ⇠= F (S) F (T ) • A q ! A ⇥ A F ( ) = • A ; ⇠ ⇤ Conversely, and functor satisfying these two properties F gives rise to an Abelian monoid F (1), with the unit given by the nowhere defined map 1, ;! ⇠= m and multiplication given by FA(1) FA(1) FA(2) FA(1), where m :2 1 is the everywhere defined map. ⇥ ! ! ! In fact, the above can be used to show that in any category C with finite products, monoid objects in C are in correspondence with functors as above. We call such functors -objects. Again we have placed a strong requirement ⇠= on our objects by requiring the maps FA(S T ) FA(S) FA(T )tobe homeomorphisms. We will weaken this by insteadq requiring! them⇥ to be only weak equivalences. Definition. A (very) special -space is a functor F Top so that op ! For all sets S, T , F (S T ) ' F (S) F (T ) • A q ! A ⇥ A The space F ( ) is weakly contractible (this is the “very” part). • ; Note that as in our above discussion, now F (1) becomes a homotopy commu- tative, associative H-space. We would like to produce a simplicial space which looks like

! ...F(1) F (1) ! F (1) ! F ( ) ⇥ ! ! ; '⇤ But this doesn’t quiet work with the simplicial identities, instead we can use the fact that our -space is special to replace it with an equivalent object

! ...F(2) ! F (1) ! F ( ) ! ! ; '⇤ Definition. There is a functor op op which sends an ordinal n to n 0 ,andamapf : n m to the map! n 0 m 0 which sends b { } min a, where! the minimum is defined.{ }! { } 7! a n 0 ,b f(a) 2 { }  Precomposing this functor with a special -space gives us a simplicial object B.F .WeletBF be the geometric realization of this simplicial set. Proposition. If F is a special -space, then BF is a very special -space.

7 Putting this all together and iterating, we obtain our final result. Theorem. Let F be a very special -space, then we obtain a sequence of verty special -spaces BnF ,withB0F = F , and weak equivalences BnF ⌦Bn+1F . In particular, F (1) has the structure of an infinite loop space. '

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