Loop Spaces and OperadsMARU SARAZOLA

This document serves as an introduction to operads and their algebras,along with basic examples. We review the theory necessary to show May’srecognition principle for loop spaces; namely, that every loop space ΩkX isan algebra for the little k-cubes operad, and every connected little k-cubesalgebra has the weak homotopy type of a k-fold loop space [May72]. Wethen follow Berger and Moerdijk [BM02] and present a model structureon the category of operads which allows us to show that every loop spacecan be rectified to a topological monoid.

Recognizing loop spaces

In algebraic topology, one of the main ways to probe a space X is by lookingat maps from spheres Sn → X. If we fix a point ∗ ∈ X, we can define its nthloop space ΩnX as the space of all maps Sn → X taking the north pole to ∗,with the compact open topology. Loop spaces are of great importance, and soit makes sense to wonder when a given space X is of the homotopy type of ak-fold loop space ΩkY for some other space Y .

Let’s start by looking at ΩY . Some properties are easy to pin down; first onthe list is the fact that we have a binary operation

µ : ΩY × ΩY → ΩY

given by concatenation of loops. As we well know, this operation is not asso-ciative: µ(1 × µ) and µ(µ × 1) are not equal, but instead differ by a change ofparametrization.

µ(µ× 1)

µ(1× µ)

f1 f2 f3

f1 f2 f3

However, there is a homotopy h between these maps, which we can easily rep-resent as

f1 f2 f3

and which makes µ a homotopy associative operation, that is, associative up tohomotopy.

What happens when we try to multiply four maps? Well, if we rememberour choice of homotopy h for multiplying three maps, we get homotopies

f1(f2(f3f4)) f1((f2f3)f4) (f1(f2f3))f4 ((f1f2)f3)f4µ(1×h) h(1×µ×1) µ(h×1)

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f1(f2(f3f4)) (f1f2)(f3f4) ((f1f2)f3)f4h(1×1×µ) h(µ×1×1)

and once again, we can find a homotopy between these homotopies! Iteratingthis process, we see that by remembering our choices of homotopies along theway, we get that loop composition is an operation that’s associative up to ho-motopy, with homotopies between these homotopies, and homotopies betweenthose, and so on.

How can we keep track of all of this structure in a reasonable way? Notethat we can encode each possible way of composing n loops as an embeddingof n copies of the unit interval I into itself, as we did in a previous picture fortwo cases of n = 3. Asking for these higher homotopies that we find in the loopspace scenario essentially boils down to making sure each of these spaces of nembeddings is contractible.

What about iterated loop spaces? Following the same train of thought, wecan encode a composition of n maps Sk → X as an embedding of n copies of thecube Ik into itself. Aside from having all higher homotopies for associativity,we know that a space that can be delooped twice will have a product that’scommutative up to homotopy. Moreover, one can see that the number of timesthe space can be delooped corresponds to how many levels of higher homotopiesone has for commutativity, and this in turn is given by the connectivity of thesespaces of embeddings.

By this point, it looks like there has to be a connection between loop spacesand some adequate spaces encoding embeddings, since these seem to capturemany essential properties. But even if this connection is apparent, it is still farfrom obvious to what extent they can actually be used to characterize (iterated)loop spaces. Our goal will be to show a sketch of the following result:

Theorem 1 (May). There exists an operad Ck suitably encoding embeddings ofcopies of Ik into itself, with the property that every loop space ΩkX is a Ck-algebra. “Conversely”, every connected Ck-algebra has the weak homotopy typeof a k-fold loop space.

Operads and their algebras

A key component in our main theorem is the notion of operad and that ofalgebra for an operad, so we will start by introducing them, assuming basicnotions from category theory.

Conceptually, an operad O consists of a family of sets O(n)n≥0, whereeach O(n) is interpreted as a set of abstract “n-ary operations”, together withstructure maps indicating how composition behaves.

The following insightful representation, found in a review written by JohnBaez[Bae02], is good to keep in mind before moving on to a more formal defini-tion: if we think about these abstract n-ary operations as black boxes with one

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wire coming in for each input and one coming out for the output,

f

we require the following:

• There should be a way to compose an n-ary operation with n other oper-ations, which we can interpret as the grafting of the boxes

f

g1 g2 g3

• Composition should be associative, allowing us to make sense of a picturelike

f

g1 g2 g3

h1 h2 h3 h4 h5

• There must exist a unary operation that acts as an identity for composition

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in the two obvious ways.

1

f

f=

f

1 1 1

f=

• Permuting the inputs of an n-ary operation gives us a new operation

f

in such a way that this constitutes a right action of the permutation groupSn on the set O(n).

• Compositions must be compatible with the actions, as represented by

Let’s formally define all of this!

Definition 2. An operad in a (strict) symmetric monoidal category C is afamily of objects O(n)n≥0, together with morphisms

γn;m1,...,mn : O(n)⊗O(m1)⊗ · · · ⊗ O(mn) −→ O(m1 + · · ·+mn)

satisfying the following axioms:

• Associativity. The following diagram commutes

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O(n)⊗n⊗i=1

O(mi)⊗n⊗i=1

(O(ki,1)⊗ · · · ⊗ O(ki,mi))

O(m1 + · · ·+mn)⊗n⊗i=1

(O(ki,1)⊗ · · · ⊗ O(ki,mi))

O(n)⊗n⊗i=1

(O(mi)⊗O(ki,1)⊗ · · · ⊗ O(ki,mi))

O(n)⊗n⊗i=1

O(ki,1 + · · ·+ ki,mi)

O(k1,1 + · · ·+ kn,mn)

γn;m1,...,mn ⊗ 1

γm1+···+mn;k1,1,...,kn,mn

1⊗n⊗i=1

γmi;ki,1,...,ki,mi

γn;k1,1+···+k1,m1,...,kn,1+···+kn,mn

ρ

where ρ uses the symmetry in C to permute the factors accordingly.

• Unit. If 1 is the unit object of C, then there exists a morphism η : 1→ O(1)such that the compositions

O(n)⊗ 1⊗n O(n)⊗O(1)⊗ · · · ⊗ O(1) O(n)1⊗η⊗n γn;1,...,1

1⊗O(m) O(1)⊗O(m) O(m)η⊗1 γ1;m

are the right and left unit morphisms given by the monoidal structure inC.

• Action. Each O(n) admits a right action by Sn.

• Equivariance. Given σ ∈ Sn, the following diagram commutes

O(n)⊗O(m1)⊗ · · · ⊗ O(mn) O(n)⊗O(mσ−1(1))⊗ · · · ⊗ O(mσ−1(n))

O(m1 + · · ·+mn)

O(n)⊗O(m1)⊗ · · · ⊗ O(mn) O(m1 + · · ·+mn)

1⊗ρ

σ⊗1

γn;mσ−1(1)

,...,mσ−1(n)

σm1,...,mn

γn;m1,...,mn

where σm1,...,mn takes a tuple of m1 + · · · + mn elements, splits it into nblocks of lengths m1, . . . ,mn (in that order) and permutes the blocks byapplying the action of σ.

Operads form a category Op(C), where a morphism of operads is a collectionof morphisms O(n)→ P(n)n≥0 preserving compositions, units, and the actionof the Sn’s.

Let’s take a look at some examples.

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Example 3. When it comes to examples, perhaps the most notable case is theone of the endomorphism operad. For an object X in a symmetric monoidalclosed category C, the operad EndX is defined as

EndX(n) = Hom(X⊗n, X)

where composition

γn,m1,...,mn : EndX(n)⊗EndX(m1)⊗· · ·⊗EndX(mn) −→ EndX(m1 + · · ·+mn)

is given as follows:If C = Set, then

γn,m1,...,mn(f, g1, . . . , gn) = f (g1 × · · · × gn)

More generally, the element

γn,m1,...,mn ∈ C(EndX(n)⊗EndX(m1)⊗· · ·⊗EndX(mn), EndX(m1 + · · ·+mn))

is defined as the transpose of the map

γ ∈ C((EndX(n)⊗ EndX(m1)⊗ · · · ⊗ EndX(mn)

)⊗X⊗m1+···+mn , X)

given by the composition(EndX(n)⊗ EndX(m1)⊗ · · · ⊗ EndX(mn)

)⊗X⊗m1+···+mn

EndX(n)⊗ (EndX(m1)⊗X⊗m1)⊗ · · · ⊗ (EndX(mn)⊗X⊗mn)

EndX(n)⊗X ⊗ · · · ⊗X

X

1⊗ev⊗n

ev

The unit η : 1 → EndX(1) = Hom(X,X) is the transpose of the unit isomor-phism 1⊗X ' X, which in the case of Set turns out to be the identity map, andthe right action of Sn on EndX is given by acting on X⊗n using the monoidalstructure of C.

We will soon be able to appreciate the importance of this operad in thebigger picture.

Example 4. Denote by S the cartesian category whose objects are the naturalnumbers, identified with the sets [n] = 1, . . . , n, and whose only maps are thepermutations Sn.

Permutation groups form an operad over the category S, with

O(n) = Sn

and composition given by

γn,m1,...,mn : Sn × Sm1× · · · × Smn → Sm1+···+mn

(σ, ρ1, . . . , ρn) 7→ σm1,...,mn (ρ1 × · · · × ρn)

so, split the tuple of m1 + · · ·+mn numbers into n blocks of lengths m1, . . . ,mn,act within the i-th block as prescribed by ρi, and then permute the blocksaccording to σ.

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Example 5. The associative operad is defined by

Ass(n) =∐Sn

1

where compositions and Sn actions are induced by the operad structure of thepermutation groups as shown in example 4.

Example 6. The commutative operad is

Com(n) = 1

the monoidal unit, where all compositions consist of the unit isomorphisms

1⊗ (1⊗ · · · ⊗ 1) 1'

The following is a historically relevant example, being the main reason op-erads were defined and what convinced many mathematicians that their studywas worthwhile –not to mention, why we’re doing all this in the first place.

Example 7. Let Ik denote the standard k-dimensional unit cube in Rk. A littlek-cube is a linear embedding c : Ik → Ik with parallel axes, that is, a map

c(t1, . . . , tn) = (c1(t1), . . . , cn(tn)), ti ∈ I

where ci : I → I is a linear function of the form

ci(t) = (1− t)xi + tyi with 0 ≤ xi < yi ≤ 1

This means each function ci takes the interval [0, 1] to an interval [xi, yi] insideof [0, 1].

The little k-cubes operad Ck consists of the spaces

Ck(n) = lEmb(∐n

Ik, Ik)

of linear embeddings , so an element is an n-tuple (c1, . . . , cn) of little k-cubes,such that the interior of the images ci(I

k) are mutually disjoint. As a topolog-ical space, we identify Ck(n) with a subspace of R2kn using the xi, yiki=1 ascoordinates.

An element of Ck(n) looks like a k-cube with n little disjoint k-cubes insideit. The symmetric groups Sn act on Ck(n) by permuting the labels of the nlittle cubes, and composition is defined by iterating embeddings as shown in thefollowing picture:

1

2 1

2

3

12

12

3

45

, =

Note that for every k there exists an injective operad morphism Ck → Ck+1

given by “embedding into the equator”, and it is with respect to these inclusionsthat we define C∞ = colim Ck.

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Operads are not all that interesting by themselves; usually the real purposeis to study their algebras. The following is the definition of algebra over anoperad, which mirrors that of a module over a ring. Recall that one can definea module M over a ring R to be an abelian group along with a morphism

R→ Hom(M,M)

where we consider the Hom functor internal to abelian groups. Similarly;

Definition 8. Let O be an operad in a symmetric monoidal closed category C.An algebra for O is an object X together with an operad morphism

α : O → EndX

that is, a collection of morphisms αn : O(n)→ Hom(X⊗n, X) that is compatiblewith compositions, units, and the actions of Sn.

Equivalently, using the adjunction we can describe an algebra structure onX as a family of morphisms

αn : O(n)⊗X⊗n → X

which are compatible with compositions, the unit, and are Sn-equivariant (herethe action of Sn on the target X is the trivial one).

Given an operad O, its algebras form a category which we denote O-Alg.A morphism of O-algebras is a map ϕ : X → Y in C that makes the followingdiagram commute

O(n)⊗X⊗n X

O(n)⊗ Y ⊗n Y

αXn

1⊗ϕ⊗n ϕ

αYn

Note that a map of operads O → P induces a “change of rings” type of mapP-Alg→ O-Alg given by O → P → EndX .

We can now understand why EndX is so important: every object X has acanonical EndX -algebra structure via id : EndX → EndX , or equivalently, viathe evaluation morphisms Hom(X⊗n, X) ⊗ X⊗n → X, and thus EndX is insome sense the universal operad acting on X, since definition 8 tells us thatevery other operad action must factor through that of EndX .

Also, we can see that the definition of operad algebras is what motivates theidea of operads as abstract n-ary operations, since having an algebra structureon X means that each element of O(n) will be given an interpretation as anelement of Hom(X⊗n, X). With this in mind, it is only natural that EndX , theobject of all n-ary operations on X, plays a major role in the theory.

Example 9. Algebras over Ass are associative unital monoids, explaining thisoperad’s name.

Proof. [Fre17, Prop. 1.1.17] Let M be an associative unital monoid, with mul-tiplication µ : M ⊗M →M and unit η : 1→M . We want to define an algebrastructure on M , that is, a collection of maps

Ass(n)→ EndM (n)

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Given a permutation σ ∈ Sn, we associate to its corresponding element inAss(n) the morphism fσ : M⊗n →M that takes an n-tuple, permutes it via σ,and multiplies the elements by iterating µ. If our category had elements, theexplicit expression for f would be

f(m1, . . . ,mn) = mσ(1) . . .mσ(n)

When n = 0, we associate to the unique (trivial) permutation id0 ∈ S0 theunit map η : 1 = M⊗0 →M .

In the other direction, given an algebra structure on M we can consider themap η : 1 → M associated to the trivial permutation id0 ∈ S0, and the mapµ : M ⊗M → M associated to the trivial permutation id2 ∈ S2. The mapassociated to the trivial permutation id1 ∈ S1, which denotes the unit of theoperad Ass, acts as the identity map on M.

From the identities

id2(id0, id1) = id1 = id2(id1, id0)

andid2(id2, id1) = id3 = id2(id1, id2)

in Ass we (respectively) deduce the unit condition

µ (η ⊗ 1M ) ' 1M ' µ (1M ⊗ η)

and the associativity relation

µ (µ⊗ 1M ) = µ (1M ⊗ µ)

in M .

Note that the associative operad has exactly n! elements in its set on n-ary operations, representing the fact that in an associative monoid we haveprecisely n! ways of multiplying n elements, one for each possible ordering,since parenthesising is rendered useless because of the associativity.

Example 10. Similarly to the previous example, it’s not hard to show thatalgebras over Com are commutative (associative, unital) monoids.

All the same arguments apply, with the following addition: the instance

Com(2)⊗M⊗2 Com(2)⊗M⊗2

Com(2)⊗M⊗2 M

1⊗σ

σ⊗1 α

α

of the equivariance axiom for σ = (1, 2) implies that

α (σ ⊗ 1)(∗ ⊗m1 ⊗m2) = m1m2

is equal to

α (1⊗ σ)(∗ ⊗m1 ⊗m2) = α(∗ ⊗m2 ⊗m1) = m2m1

and then the binary operation must be commutative.

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One of the following sections will be devoted to characterizing algebras forthe little k-cubes operad. But before delving into that, we must establish arelation between operads and another familiar type of structure.

Operads and monads

Let’s start by recalling some basic definitions.

Definition 11. Let C be any category. A monad in C is an endofunctor T :C → C together with a multiplication transformation µ : T 2 ⇒ T and a unittransformation η : 1⇒ T making the diagrams below commute

T T 2 T

T

ηT

idµ

Tη

id

T 3 T 2

T 2 T

Tµ

µT µ

µ

Definition 12. If T : C → C is a monad, then a T -algebra is an object X inC together with an algebra structure map θ : TX → X such that the followingdiagrams are commutative

X TX

X

η

idθ

T 2X TX

TX X

Tθ

µ θ

θ

For any object X in C, it is always the case that (TX, µ : T 2X → TX) is aT -algebra, which we call the free T -algebra generated by X.

We are interested in two ways through which we can define a monad.

Proposition 13 (Monad associated to an adjunction). Let F : C → D andU : D → C be adjoint functors, that is, there exists an isomorphism

D(FC,D) ∼= C(C,UD)

or equivalently, unit and counit natural transformations η : 1C ⇒ UF andε : FU ⇒ 1D such that the compositions

F FUF FFη εF

U UFU UηU Uε

are the identity transformations on F and G.Then UF : C → C is a monad, with multiplication µ = UεF : UFUF ⇒ UF

and unit η : 1⇒ UF .

Example 14. Let Σ denote the reduced suspension functor, and recall the ad-junction

Hom∗(ΣX,Y ) ∼= Hom∗(X,ΩY )

of pointed maps and pointed spaces. Note that

Hom∗(Σ2X,Y ) ∼= Hom∗(ΣX,ΩY ) ∼= Hom∗(X,Ω

2Y )

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and so by iterating this adjunction we get that

Hom∗(ΣkX,Y ) ∼= Hom∗(X,Ω

kY )

Therefore, proposition 13 implies ΩkΣk has a canonical monad structure.Consider the maps

σk : ΩkΣkX −→ Ωk+1Σk+1X

taking a map f ∈ ΩkΣkX = Hom∗(Sk,ΣkX) to the map σk(f) ∈ Ωk+1Σk+1X =

Hom∗(Sk+1,Σk+1X) given by

Sk+1 = Sk ∧ S1 ΣkX ∧ S1 = Σk+1Xf∧1

Note that each σk is an inclusion, and it’s with respect to these that we define

Ω∞Σ∞ = colim ΩkΣk

This will also be a monad, with product µ∞ = colimµk and unit η∞ = colim ηk.

Now, given an operad O, we will show a way to define a monad whosealgebras coincide with those of O. Define maps σi : O(j) → O(j − 1) for0 ≤ i < j by σi(f) = γ(f, si) where

si = 1i × ∗ × 1j−1−i ∈ O(1)i ×O(0)×O(1)j−1−i

Also, let si : Xj−1 → Xj be the map

si(x1, . . . , xj−1) = (x1, . . . , xi, ∗, xi+1, . . . , xj−1)

Proposition 15 (Monad associated to an operad). [May72, Constr. 2.4] Let Obe an operad, and X an object in C. We define O : C → C, the monad associatedto the operad O, as

OX =∐j

O(j)×Xj/ ∼

where(σif, x) ∼ (f, six) and (fσ, x) ∼ (f, σx)

If we denote the image of (f, x) in OX by [f, x], then, given a map g : X →X ′, define Og : OX → OX ′ by Og[f, x] = [f, gjx]. For the monad structure, wedefine the multiplication and unit transformations by

µ([f, [f1, x1], . . . , [f j , xj ]]) = [γ(f, f1, . . . , f j), (x1, . . . , xj)]

η(x) = [η(1), x]

(note that O2X =∐j O(j)× (OX)j/ ∼=

∐j O(j)× (

∐iO(i)×Xi/ ∼)j/ ∼)

The following result shows the relation between the operad O and its asso-ciated monad O.

Theorem 16. [May72, Prop. 2.8] Let O be an operad and O its associatedmonad. Then there is a one-to-one correspondence between O-algebras and O-algebras.

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Proof. Let X be an O-algebra, with structure maps αn : O(n)×Xn → X. Wedefine a map α : OX =

∐j O(j)×Xj/ ∼−→ X by α[f, x] = αn(f, x).

Conversely, given an O-algebra X with algebra map α : OX =∐j O(j) ×

Xj/ ∼−→ X, we can make it into an O-algebra by defining structure mapsαn : O(n)×Xn → X as αn(f, x) = α[f, x].

It’s a routine problem to check that α is well defined, and that all thecorresponding diagrams commute.

The bar construction

Let’s go back to the simplest instance of our problem: given a space X, wewant to know if X is a loop space; that is, if there exists a space Y such thatX ' ΩY . When our space is a topological group G, the answer is affirmative:we can define its classifying space BG, and we know that G ' ΩBG [Hat02,Prop. 4.66]. With this in mind, our approach will be to show that all Ck-algebrasadmit a generalization of a classifying space satisfying this same property.

Let’s start by recalling the construction of the space BG. If we look at Gas a one-object category, then BG is defined as the geometric realization of thenerve of G. Explicitly,

BG = |B∗G|

where B∗G is the simplicial space with components

BnG = Gn,

face maps di : Bn+1G→ BnG given by

• (g0, . . . , gn) 7→ (g1, . . . , gn), for i = 0

• (g0, . . . , gn) 7→ (g0, . . . , gigi+1, . . . , gn), for 0 < i < n

• (g0, . . . , gn) 7→ (g0, . . . , gn−1), for i = n

and degeneracy maps si : BnG→ Bn+1G given by

(g0, . . . , gn−1) 7→ (g0, . . . , gi−1, 1, gi, . . . , gn−1)

In order to generalize this construction, we introduce the following notion.

Definition 17. Given a monad T on C, a functor F : C → D is said to be aT -functor if T acts on F on the right via a natural transformation λ : FT ⇒ F ,which is required to satisfy analogue conditions for those of a T -algebra structuremap. Namely, the following diagrams must commute:

F FT

F

Fη

1λ

FT 2 FT

FT T

Fµ

λT λ

λ

Remark 18. It follows from this definition, and that of a monad, that T is alwaysa T -functor.

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Now, given a triple (F, T,X) where T is a monad, F a T -functor and X aT -algebra, we can construct a simplicial object B∗(F, T,X) where

Bq(F, T,X) = FT qX,

the face maps are given by

• the T -functor structure map FT → F , for i = 0

• the multiplication T 2 → T , for 0 < i < q

• the algebra structure map TX → X, for i = q

and the degeneracy maps are given by η : 1→ T .

Example 19. If G is a topological group, we can use the group product and unitto define a monad G : Top→ Top by G(X) = G×X.

This allows us to recover the usual classifying space of G:

BG = |B∗(∗, G, ∗)|

The main result

We finally have all the tools we need to give a sketch of the proof of thefollowing result:

Theorem 20. [May72, Thm. 13.1] Let Ck denote the little k cubes operad, for1 ≤ k ≤ ∞. Then every loop space ΩkX is a Ck-algebra, and every connectedCk-algebra has the weak homotopy type of a k-fold loop space.

Proof. The easy part: defining an action αk : Ck → EndΩkX .First consider k <∞; for a fixed n, we define

αk,n : Ck(n)⊗ (ΩkX)n −→ ΩkX

as follows: denote an element λ ∈ (ΩkX)n as a tuple (λ1, . . . , λn) of mapsλi : (Ik, ∂Ik) → (X, ∗), and a little k-cube in Ck(n) as (c1 . . . , cn), where ciare the numbered smaller cubes inside Ik. Then, αk,n((c1 . . . , cn), λ) is definedas the map (Ik, ∂Ik) → (X, ∗) which is λi (rescaled) on ci, and which mapseverything in the complement of the n cubes to the basepoint ∗. Staring at theaction for a bit is probably enough to convince yourself that this actually works.

Note that in the case X = ΩY , the actions are coherent

Ck(n)⊗ (ΩkΩY )n ΩkΩY

Ck+1(n)⊗ (ΩkΩY )n

αk,n

σ×1 αk+1,n

where σ : Ck → Ck+1 denotes the embedding, and so they induce an actionC∞(n)⊗ (Ω∞X)n −→ Ω∞X.

A sketch of the sophisticated part:Due to construction 15 and theorem 16, we know there exists a monad Ck

whose algebras are the algebras of the operad Ck. Recalling the monad ΩkΣk

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from example 14, we can define a morphism of monads ρk : Ck → ΩkΣk givenby the composition

CkX CkΩkΣkX ΩkΣkXCkη θk

(recall that an action map of the operad Ck over Y corresponds to an algebra-structure map CkY → Y ).

With considerable effort, it’s possible to show ([May72, Thm. 6.1]) that ifX is connected, then ρk : CkX → ΩkΣkX is a weak homotopy equivalence.

We now use the bar construction to find a “simplicial resolution” of our spaceX. From remark 18, we know that every monad T is a T -functor; in particular,Ck is a Ck-functor and it makes sense to consider the simplicial space

B∗(Ck, Ck, X)

Let X∗ denote the simplicial object with X at every level and 1 : X → X asface and degeneracy maps. The map

B∗(Ck, Ck, X)→ X∗

Cq+1k X CkX X

µq θk

is a homotopy equivalence, with inverse

X∗ → B∗(Ck, Ck, X)

X Cq+1k X

ηq+1

Recalling that, by definition of algebra over a monad, the following diagramsare commutative

X CkX

X

η

1θk

C2kX CkX

CkX X

µ

Ckθk θk

θk

we can see that

X Cq+1k X CkX X

ηq+1 µq θk

is the identity on X. For the other composition, we can define a simplicialhomotopy

hi : Cq+1k X Cq+1−i

k X Cq+2k X

µi ηi+1

and see with some work that everything checks out.For completeness’ sake –and because I find it impossible to remember– we

include the definition of simplicial homotopy. As expected, it can be definedby mimicking the definition of homotopy in Top, or by using the combinatorialstructure of simplicial objects.

14

Definition 21. A homotopy between two morphisms of simplicial objectsf, g : X → Y is a morphism H : X × ∆[1] → Y that makes the followingdiagram commute

X ' X ×∆[0] X ×∆[1] X ×∆[0] ' X

Y

id×δ1

fH

id×δ0

g

Equivalently, a simplicial homotopy is a family of morphisms hi : Xn → Yn+1

for i = 0, . . . , n and for every n, such that

d0h0 = fn, dn+1hn = gn

and

dihj =

hj−1di, i < j

dihi−1, i = j 6= 0

hjdi−1, i > j + 1

sihj =

hj+1si, i ≤ jhjsi−1, i > j

Going back to our proof, so far we have homotopies

X ∼= |X∗| |B∗(Ck, Ck, X)|

Next, we can see that the monad map ρk : Ck → ΩkΣk constructed pre-viously, together with its transpose ρ#

k : ΣkCkX → ΣkX via the adjunctionΣk a Ωk make ΩkΣk and Σk into Ck-functors:

ΩkΣkCk ΩkΣkΩkΣk ΩkΣkΩkΣkρk µ

ΣkCk Σkρ#k

Therefore, we can consider the simplicial spaces

B∗(ΩkΣk, Ck, X) , B∗(Σ

k, Ck, X)

Since ρk : CkX → ΩkΣkX is a weak homotopy equivalence, it induces aweak homotopy equivalence

B∗(Ck, Ck, X) −→ B∗(ΩkΣk, Ck, X)

Finally, using the fact that there exists a weak homotopy equivalence |ΩYn| ∼=Ω|Yn| whenever Yn is a nice enough simplicial object, we get a chain of weakhomotopy equivalences

X ∼= |X∗| |B∗(Ck, Ck, X)| |B∗(ΩkΣk, Ck, X)| Ωk|B∗(Σk, Ck, X)|

which gives us a precise description of X as a k-fold loop space!

15

Remark 22. The use of the monad ΩkΣk is far from coincidental. Note that amap X → X ′ determines and is determined by a loop map ΩkX → ΩkX ′, andso

Hom∗(ΣkX,Y ) ' HomLoops(Ω

kΣkX,ΩkY )

But we also know that

Hom∗(ΣkX,Y ) ' Hom∗(X,Ω

kY )

so we conclude that

Hom∗(X,ΩkY ) ' HomLoops(Ω

kΣkX,ΩkY )

which makes ΩkΣkX the free k-fold loop space generated by X.

Rectifying loop spaces

From now on we will focus on the little intervals operad C1. From theorem 20,we know that every C1-algebra X is weakly equivalent to a loop space, X ' ΩY .We also know from our previous discussion that loop spaces are associative andunital up to coherent homotopy, but never strictly. What is the relation betweenC1-algebras and strict topological monoids?

If we start with a topological monoid M with unit 1 ∈ M and productm : M ×M →M , then we can give M a C1-algebra structure as follows:

C1(0)→ Hom(∗, X), ∗ 7→ 1

C1(1)→ Hom(X,X), c 7→ idX

C1(2)→ Hom(X2, X), c 7→ mσc

C1(3)→ Hom(X3, X), c 7→ m(1×m)σc

...

Where σc denotes the permutation that gives c its numbering, read left to right.We are not making any arbitrary choices here, since the associativity of mensures that all possible ways to multiply n elements will coincide. Our goalwill be to show the closest thing to a converse that we are able to get; namely,that every cofibrant C1-algebra can be rectified to a topological monoid.

Fortunately, we know of an operad whose precise purpose is to model asso-ciative unital monoids in any category: the associative operad (recall examples5 and 9). Thus, our strategy will be to relate the operads C1 and Ass in a waythat gives us the relation that we need between their categories of algebras.

Here is a glimpse of how this will work: we will define a model structure onthe category of operads over a given symmetric monoidal closed model category,and show that both our operads satisfy a weaker version of cofibrancy. Next,we will construct a weak equivalence

C1 Ass∼

and explain how this implies we have a Quillen equivalence

C1-Alg Ass-Alg

for a suitable model structure on the categories of algebras, which will yield ourrectification property. This section follows [BM02].

16

Some preliminaries on model categories

Definition 23. A monoidal model category is a closed symmetric monoidalcategory together with a model structure satisfying the pushout-product axiom:

Given cofibrations f : X → Y and f ′ : X ′ → Y ′, the induced map from thepushout

X ⊗X ′ Y ⊗X ′

X ⊗ Y ′ P

Y ⊗ Y ′

is a cofibration, and it is trivial if either f or f ′ are. Formally we should alsorequire another axiom, which becomes irrelevant if we assume the unit objectis cofibrant.

Remark 24. It’s tedious (but not hard) to show that a category will satisfy thepushout-product axiom if and only if it satisfies its dual, the pull back-Homaxiom:

For a cofibration f : X → Y and a fibration f ′ : X ′ Y ′, the induced mapto the pull back

Hom(Y,X ′)

P Hom(X,X”)

Hom(Y, Y ′) Hom(X,Y ”)

is a fibration, and it is trivial if either f or f ′ are.

We now show a handy result about model categories, from which we deducea lemma we will use on multiple occasions.

Lemma 25. [Ken Brown’s lemma][Hov99, 1.1.12] Let C be a model categoryand D a category with weak equivalences which satisfy the two out of three ax-iom. Suppose F : C → D is a functor which takes trivial cofibrations betweencofibrant objects to weak equivalences. Then F takes all weak equivalences be-tween cofibrant objects to weak equivalences (and the dual version is, of course,also true).

Proof. Suppose f : X → Y is a weak equivalence of cofibrant objects. Factorthe map (f, 1B) : X

∐Y → Y as

X∐Y Z Y

f

∼g

17

The pushout diagram

0 X

Y X∐Y

shows that the inclusion maps i1 : X → X∐Y and i2 : Y → X

∐Y are cofibra-

tions. By the two out of three axiom, both qi1 and qi2 are weak equivalences,hence trivial cofibrations (of cofibrant objects). By hypothesis, we then havethat both F (qi1) and F (qi2) are weak equivalences. Since F (pqi2) = F (1B) isalso a weak equivalence, we conclude from the two out of three axiom that F (p)is a weak equivalence, and hence that F (f) = F (pqi1) is a weak equivalence, asrequired.

Lemma 26. [BM02, 2.3] Let f : X → Y be a map between cofibrant objectsof a monoidal model category. If f is a weak equivalence, then for every fibrantobject Z, the induced map f∗ : ZY → ZX is a weak equivalence.

Proof. If f is a trivial cofibration, we can apply the pull back-Hom axiom to the

maps X Y∼ and Z 1 and conclude that f∗ : ZY → ZX must be aweak equivalence. Ken Brown’s lemma (25) allows us to conclude the proof

The following transfer principle will be crucial in constructing the desiredmodel structures on the categories of operads and operad algebras:

Theorem 27. Let D be a cofibrantly generated model category and let

F : D E : G

be an adjunction with left adjoint F and right adjoint G. Assume that E hassmall colimits and finite limits. Define a map f in E to be a weak equivalence(resp. fibration) iff G(f) is a weak equivalence (resp. fibration). Then thisdefines a cofibrantly generated model structure on E provided

1. the functor F preserves small objects;

2. E has a fibrant replacement functor;

3. E has functorial path-objects for fibrant objects, that is, for every fibrantX there exists a functorial factorization of its diagonal as

X Path(X) X ×X∼

The proof of this theorem can be easily adapted from [SS00, Lemma 2.3],where the authors assume every object to be fibrant. Clearly, the first hypothesisis there so that we can use the small object argument for F (I) and F (J), whereI (resp. J) denote the generating sets of cofibrations (resp. trivial cofibrations)in D. The other two, which are perhaps a bit obscure at a first glance, are usedto obtain some notion of right homotopy, and relate the maps in F (J) to trivialcofibrations in E .

Model structure on operads and their algebras

18

For a discrete group G, denote by EG the category of objects of E with a(right) G-action. If E is a symmetric monoidal closed category, then so is EG,and we have a free-forgetful adjunction

F : E EG : U

where F (X) =∐GX. If we consider the permutation groups, we can define the

category of collections by

Coll(E) =∏n≥0

ESn

If our free-forgetful adjunction satisfies the hypotheses for the transfer princi-ple, then the model structure on EG will induce a product model structure onColl(E), and we can apply our transfer principle once again to the free-forgetfuladjunction

F : Coll(E) Op(E) : U

to obtain a model structure on the category of operads over E . Here U is thefunctor that forgets the unit and compositions, and F is the free operad functor.

Since the model structure on Op(E) relies on it, we take a moment to de-scribe the construction of the free operad functor

F : Coll(E)→ Op(E)

Let T be the category of finite rooted planar trees, and non-planar isomor-phisms. Each edge has an orientation (going downwards), and any tree will havethree kinds of edges: internal edges with a vertex at the beginning as well asat the end, input edges with a vertex only at the end, and one outgoing edge,called the output of the tree, with the root vertex as its beginning and no vertexat its end.

Any tree T with a root having n incoming edges decomposes canonically inton trees T1, . . . , Tn whose outputs are grafted upon the inputs of the tree tn withone vertex and n inputs. We denote this grafting operation byT = tn(T1, . . . , Tn).

Note that the number of vertices of each Ti is strictly less than the numberof vertices of T , which allows for inductive definitions. Any tree isomorphismϕ : T → T ′ decomposes as ϕ = σ(ϕ1, . . . , ϕn) with isomorphisms σ : tn → tnand ϕi : Ti → T ′σ(i).

For any collection C we define a functor C : Top → E by setting inductivelyC(|) = 1 and

C(T ) = C(tn(T1, . . . , Tn)) = C(n)⊗ C(T1)⊗ · · · ⊗ C(Tn)

On morphisms, for a map ϕ : T → T ′, we get again by induction

Cϕ = σC(ϕ1, . . . , ϕn) = σ ⊗ Cϕσ−1(1) ⊗ · · · ⊗ Cϕσ−1(n)

Finally, define T(n) = T ∈ T : T has n inputs. The inputs of a treein T(n) admit n! different numberings, so we associate to it the object

∐Sn

1.Then, an automorphism ϕ : T → T induces a permutation on the factors of∐Sn

1.We use all this information to define the free operad functor,

F : Coll(E)→ Op(E)

19

FC(n) =∐

[T ]∈T(n)/∼

C(T )⊗Aut(T ) qSn1

where T(n)/ ∼ denotes the isomorphism classes of trees.Now that we have a good understanding of the key adjunction involved, let’s

start by proving we can define a model structure on EG by transfer.

Proposition 28. For any cofibrantly generated, symmetric monoidal closedmodel category E with cofibrant unit, the adjunction F : E EG : U satisfiesthe hypotheses of the transfer principle.

Proof. Since limits and colimits in EG are computed pointwise, we trivially haveU(colimXi) = colimU(Xi) and so U preserves (filtered) colimits.

Now, E is cofibrantly generated and therefore has a fibrant replacementfunctor (the one given by the small object argument, if nothing else). Given,X ∈ EG, consider its fibrant replacement in E ,

X X∼ 1∼

Then, X∼ ∈ EG by composing the functors

G E E

∗ X X∼

( )∼

and X X∼∼ is a map in EG due to the fact that ( )∼ is functorial.

Let X be a fibrant object in EG. Consider the folding map 1∐

1 → 1 andfactor it using the model structure in E as

1∐

1 J 1∼

(note that 1 is also in EG, with the trivial action). Then, we have

X ' X1 −→ XJ −→ X1∐

1 ' X ×X

where, like before, one can easily see these to be G-maps between G-objects. Itremains to show that this composition consists of a weak equivalence followedby a fibration (since it was already constructed in a functorial manner).

For the first map, note that the fact that 1 is cofibrant implies 1∐

1 is too,and so the cofibration 1

∐1 → J makes J cofibrant as well. Then, since X

is fibrant we can use lemma 26 for the weak equivalence J 1∼ to get that

X1 −→ XJ is a weak equivalence.Finally, we apply the pull back-Hom axiom to the maps 1

∐1 → J and

X 1 and conclude that XJ −→ P = X1∐

1 is a fibration.

Now we move on to the second part of our proof: we have shown thatColl(E) is a (cofibrantly generated) model category, and so we apply the transferprinciple to the adjunction

F : Coll(E) Op(E) : U

to define a model structure on the category of operads.

20

Theorem 29. [BM02, Thm. 3.2] Let E be a cofibrantly generated, cartesianclosed model category with cofibrant unit, possessing a symmetric monoidal fi-brant replacement functor. Then, there exists a cofibrantly generated modelstructure on Op(E) where a map P → Q is a weak equivalence (resp. fibration)if and only if the map P(n) → Q(n) is a weak equivalence (resp. fibration) inE, for every n.

Proof. Once again, we will check the hypotheses for the transfer principle. First,note that the forgetful functor trivially preserves (filtered) colimits.

Let’s define a fibrant replacement functor in Op(E). For an operad P, wedefine its fibrant replacement P∼ as the operad P∼(n) = (P(n))∼. Here the Snaction is given by

P∼(n) P∼(n)σ∼

using the fact that ( )∼ is a functor, and the operad structure maps are definedas

P∼(n)× P∼(m1)× · · · × P∼(mn)

(P(n)× P(m1)× · · · × P(mn))∼

(P(m1 + · · ·+mn))∼

'

γ∼

‖P∼(m1 + · · ·+mn)

where the first isomorphism is given by ( )∼ being symmetric monoidal. Notethat this is functorial because ( )∼ is, and that it was defined to be an operadmap.

Now, given a fibrant operad P (that is, an operad such that each P(n) isfibrant in E), we will define a functorial path object

P Path(P) P × P∼

This will be done just like in Prop. 28: we factor the folding map 1∐

1 → 1using the model structure in E as

1∐

1 J 1∼

and then claim that our path object sequence will be given by

P ' P1 −→ PJ −→ P1∐

1 ' P × P

Here we define PX as the operad PX(n) = Hom(X,P(n)), with Sn actiongiven by

Hom(X,P(n)) Hom(X,P(n))Hom(X,σ)

21

and operad structure given by

Hom(X,P(n))×Hom(X,P(m1))× · · · ×Hom(X,P(mn))

Hom(X,P(n)× P(m1)× · · · × P(mn))

Hom(X,P(m1 + · · ·+mn))

'

Hom(X,γ)

where the isomorphism is due to the fact that, in a cartesian closed category,exponentiation is symmetric monoidal.

It’s easy to see from the definition that all maps in our sequence are operadmaps, so it suffices to show that the composition consists of a weak equivalencefollowed by a fibration (since it was already constructed in a functorial manner).

For the first map, the fact that 1 is cofibrant implies 1∐

1 is too, and sothe cofibration 1

∐1 → J makes J cofibrant as well. Then, since each P(n) is

fibrant, we can use lemma 26 for the weak equivalence J 1∼ to get that each

P(n)1 −→ P(n)J is a weak equivalence, and therefore, P1 −→ PJ is a weakequivalence.

Finally, we apply the pull back-Hom axiom to the maps 1∐

1 → J andP(n) 1 for each n and conclude that each P(n)J −→ P(n)1

∐1 is a fibration,

and then so is PJ −→ P1∐

1.

Example 30. Let Top denote the category of compactly generated Hausdorfftopological spaces. This is a cofibrantly generated, cartesian closed model cat-egory, where every object is fibrant. Therefore, topological operads admit amodel structure.

Now that we have defined a model structure on the category of operads(over certain model categories), we proceed to define a model structure on thecategory of algebras for a (certain type of) operad. Once again, we will do thisby using the transfer principle for the free-forgetful adjunction

FO : E O-Alg : U

where FO(X) =∐j O(j)×Xj/ ∼, with relations as in Prop. 15.

Definition 31. An operad O in Op(E) is called admissible if the categoryO-Alg carries a model structure which is transferred from E along the free-forgetful adjunction.

The following result shows that, in the context with which we are dealing,all operads are admissible.

Theorem 32. Let E be a category under the hypotheses of Theorem 29; thatis, a cofibrantly generated, cartesian closed model category with cofibrant unit,possessing a symmetric monoidal fibrant replacement functor. Then all operadsin Op(E) are admissible.

Proof. We will show that the conditions for the transfer principle are satisfiedfor any operad O. Note that the forgetful functor U : O-Alg → E preservesfiltered colimits.

22

To define a fibrant replacement functor on O-Alg, let X be an O-algebra and

consider its fibrant replacement in E , X X∼∼ . If O O∼∼ denotesthe fibrant replacement that we get from Thm. 29, then X∼ has an O∼-algebrastructure given by

O∼(n)× (X∼)n O(n)∼ × (Xn)∼ (O(n)×Xn)∼ X∼' '

where the existence of the two isomorphisms is ensured by the fact that thefibrant replacement functor in E is monoidal.

Then, the operad map O O∼∼ induces an O-algebra structure on X∼,by

O(n)× (X∼)n → O∼(n)× (X∼)n → X∼

which makes the map X X∼∼ a weak equivalence of O-algebras.Now suppose X is fibrant. In the same way as before, we factor the folding

map 1∐

1→ 1 as

1∐

1 J 1∼

using the model structure in E , and consider

X ' X1 −→ XJ −→ X1∐

1 ' X ×X

The first map is a weak equivalence by lemma 26, and the second map is afibration, as we can see from an application of the pull back-Hom axiom to themaps 1

∐1 → J and X 1.

This construction is clearly functorial, so in order for it to work, we just needto show this factorization lives in the category O-Alg. But XJ can be madeinto an O-algebra by

O(n)×Hom(J,X)n

Hom(Xn, X)×Hom(Jn, Xn)

Hom(Xn, X)×Hom(Jn, Xn)× 1

Hom(Xn, X)×Hom(Jn, Xn)×Hom(J, Jn)

Hom(J,X)

α×ϕ

'

1×1×d

where α : O(n) → Hom(Xn, X) is the algebra map, ϕ : Hom(J,X)n →Hom(Jn, Xn) is the map that defines a morphism coordinate-wise, d : 1 →Hom(J, Jn) is the map that chooses the diagonal, and is composition. It’seasy to check that this is actually an algebra structure map, and that it makesthe maps X → XJ and XJ → X ×X into algebra maps.

Example 33. Since the category of compactly generated Hausdorff spaces sat-isfies the hypotheses of Theorem 29, we conclude that all topological operadsmust be admissible.

23

The category of algebras for a cofibrant operad will behave nicely (homo-topically speaking) but being cofibrant is, in practice, too strong a conditionsometimes. Fortunately, full cofibrancy is not always necessary, and we canweaken it to the following.

Definition 34. An operad O in Op(E) is said to be Σ-cofibrant if its underlyingcollection is cofibrant for the model structure in Coll(E).

Recall that a model category is called left proper if weak equivalences arepreserved by pushouts along cofibrations.

Theorem 35. [BM02, Thm. 4.4] Let E be a left proper monoidal closed modelcategory with cofibrant unit, and

P Q∼

be a weak equivalence between Σ-cofibrant operads. Then the base-change ad-junction

ϕ! : P-Alg Q-Alg : ϕ∗

is a Quillen equivalence.

Proof. First, recall that by the definition of the model structures, an algebramap f : X → Y is a weak equivalence (resp. fibration) if and only if it is so as amap in E , forgetting the algebra structure. Then, since ϕ∗f = f , we see that ϕ∗

preserves weak equivalences and fibrations, and so the base-change adjunctionis a Quillen pair.

Since ϕ∗ also reflects weak equivalences, the derived adjunction will be anequivalence if (and only if) the unit induces a weak equivalence

X → ϕ∗Rϕ!X

for each cofibrant P-algebra X.[Hov99, 1.3.16]This is a very technical condition to verify, that uses the fact that the model

category is left proper, and the operads are Σ-cofibrant.

With these results in hand, we are finally ready to start proving our rectifi-cation property; we will do so by using Theorem 35. First of all, note that theoperads involved, C1 and Ass, are admissible since by Example 33, all topolog-ical operads are. We will start by showing that they are also Σ-cofibrant.

Let’s convince ourselves that every connected component of C1(n) is a CW-complex. For each n, the space of embeddings C1(n) will have one connectedcomponent for each possible ordering of 1, 2, . . . , n. Let’s look at one of thosecomponents; say, the one indexed by id ∈ Sn. If the starting point of the ithembedded little interval is denoted by xi, and the endpoint by yi, then thisconnected component can be identified with the subspace of [0, 1]2n of vectors(x1, y1, . . . , xn, yn) such that

x1 < y1 ≤ x2 < y2 ≤ · · · ≤ xn < yn

That is, we consider the hyperplanes x1 = y1, y1 = x2, . . . , xn = yn that divide[0, 1]2n into sections, and our connected component will be the chunk of spacegiven by the correct intersection of sections. This is clearly seen to be a CW-complex (and moreover, contractible, which we will need in the future).

24

Now, consider a commuting diagram of collections and collection maps

∅ C

C1 D

f

g

where f is a trivial fibraton. We want to define a lift h : C1 → C in Coll(Top),that is, a family of maps hn : C1(n) → C(n) that preserve the Sn actions andsuch that

C(n)

C1(n) D(n)

fn

gn

hn

commutes. Let C1(n)id denote the connected component of C1(n) whose littleintervals are numbered by id ∈ Sn. Then a lift exists in the following diagram

C(n)

C1(n)id D(n)

fn

gn

hidn

since fn is a trivial fibration and C1(n)id is a CW-complex (and therefore, cofi-brant).

Noting that any point in another connected component x ∈ C1(n)σ can bewritten as σy for a unique y ∈ C1(n)id, we define hσn : C1(n)σ −→ C(n) byhσn(x) = σhid

n (y). These will be continuous, since the Sn action acts continu-ously, and

fn(hσn(x)) = fn(σhidn (y)) = σfn(hid

n (y)) = σgn(y) = gn(σy) = gn(x)

so each hσn effectively defines a lift on its corresponding component.Putting all these together, we get a function hn : C1(n) → C(n), that will

be continuous since it is so on every connected component. This will be a liftfor our diagram, and a collection map (since we specifically defined it for thispurpose).

This shows C1 is a Σ-cofibrant operad; verifying that Ass is Σ-cofibrant isidentical, since its connected components ∗ are CW-complexes, and the point∗ in the component indexed by σ is obtained as ∗σ = σ∗id.

The only thing left to see in order to use Theorem 35 is that there exists aweak equivalence

C1 Ass∼

For each n, let fn : C1(n) −→ Ass(n) =∐Sn∗ be the map that takes an

embedding of n little intervals to the point ∗ in the coproduct indexed by thepermutation corresponding to the numbering of the embedding, read from leftto right. Do these define an operad map?

It’s clear that the maps will respect the Sn-action and the unit. Drawing apicture for a particular case illustrates how this will also respect compositions.

25

According to the model structure on operads, this map will be a weak equiv-alence if and only if each fn is, so we need to show that all fn’s induce isomor-phisms on the homotopy groups, for all choices of basepoints. But Ass(n) is adiscrete space, so π0(Ass(n)) = Sn, and πi(Ass(n)) = 0 for every n and i ≥ 0.Then, it suffices to show that the maps fn induce bijections on π0 and thatπi(C1(n)) = 0 for every n and i ≥ 0.

For each n, the space of embeddings C1(n) will have one connected compo-nent for each possible ordering of 1, 2, . . . , n, so π0(C1(n)) = Sn and we cansee that fn = id. Furthermore, each of these components will be contractible,so no matter the choice of basepoint we will have πi(C1(n)) = 0 for every n andi ≥ 0.

Therefore, Theorem 35 ensures us that the induced adjunction

ϕ! : C1-Alg Ass-Alg : ϕ∗

is a Quillen equivalence, which implies that for every cofibrant C1-algebra X,the unit map X → ϕ∗ϕ!X is a weak equivalence in C1-Alg. But ϕ!X is anAss-Alg, i.e. a topological monoid, and ϕ∗ϕ!X is equal to ϕ!X as a space, witha C1 action that’s constant in each connected component of C1, and acts likeAss on each of them, that is, ϕ∗ϕ!X is also a monoid as a C1-algebra.

We conclude that cofibrant C1-algebras can be rectified to topological monoids;that is, for every cofibrant C1-algebra X, there exists a topological monoid Mand a weak equivalence

X M∼

that respects the C1-algebra structures. Furthermore, we have an explicit de-scription of M as M = ϕ∗ϕ!X.

References

[Bae02] John Baez. Review of operads in algebra, topology and physics byMartin Markl, Steve Schnider and Jim Stasheff, 2002.

[BM02] Clemens Berger and Ieke Moerdijk. Axiomatic homotopy theory foroperads. Commentarii Mathematici Helvetici, 2002.

[Fre17] Benoit Fresse. Homotopy of operads and Grothendieck-Teichmullergroups - Part I. Mathematical Surveys and Monographs, AmericanMathematical Society, 2017.

[Hat02] Allen Hatcher. Algebraic Topology. Cambridge University Press, 2002.

[Hov99] Mark Hovey. Model categories. Surveys and Monographs of the Amer.Math. Soc., 1999.

[May72] Peter May. The Geometry of Iterated Loop Spaces. Springer-Verlag,1972.

[MSS02] Martin Markl, Steve Shnider, and Jim Stasheff. Operads in Algebra,Topology and Physics. Mathematical Surveys and Monographs, Amer-ican Mathematical Society, 2002.

[SS00] Stefan Schwede and Brooke Shipley. Algebras and modules in monoidalmodel categories. Proc. London Math. Soc., 2000.

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