A recognition principle for iterated suspensions as coalgebras over the little cubes operad Oisín Flynn-Connolly José M. Moreno-Fernández Felix Wierstra

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A recognition principle for iterated suspensions as coalgebras

over the little cubes operad

Oisín Flynn-Connolly, José M. Moreno-Fernández, Felix Wierstra

Abstract

Our main result is a recognition principle for iterated suspensions as coalgebras over the little

cubes operads. Given a topological operad, we construct a comonad in pointed topological spaces

endowed with the wedge product. We then prove an approximation theorem that shows that the

comonad associated to the little

-cubes operad is weakly equivalent to the comonad

ΣnΩn

arising

from the suspension-loop space adjunction. Finally, our recognition theorem states that every

little

-cubes coalgebra is homotopy equivalent to an

-fold suspension. These results are the

Eckmann–Hilton dual of May’s foundational results on iterated loop spaces.

1 Introduction

Since the invention of operads by May, they have played an important role in many parts of

mathematics and physics. The ﬁrst application and the original motivation for their invention

was for the study of iterated loop spaces (see [

] and [

]). Operads provide a way of, and a

coherent framework for, studying objects equipped with many "multiplications", i.e. operations

with multiple inputs and one output, satisfying certain homotopical coherences. An important

class of such objects are

-fold loop spaces, which are algebras over the little

-cubes operad.

May showed in his recognition principle a homotopical converse, namely that every little

-cubes

algebra is weakly equivalent to an

-fold loop space; and further proved an approximation theorem

which asserts that the monad associated to the little

-cubes operad is weakly equivalent to the

monad

ΩnΣn

. This approximation theorem reduced the study of operations on the homology of

iterated loop spaces to the combinatorics of the little cubes operads, a perspective which unravelled

their complete algebraic structure (see [7]).

The goal of this paper is to prove the Eckmann–Hilton dual results of May’s work on iterated

loop spaces. First of all, we construct a comonad in the category of pointed spaces associated to an

operad. Next, we show that

-fold suspensions are coalgebras over the little

-cubes operad

More precisely we prove the following theorem.

Theorem A.

The

-fold reduced suspension of a pointed space

is a

-coalgebra. More precisely,

there is a natural and explicit operad map

∇:Cn→CoEndΣnX,

where

CoEndΣnX

is the coendomorphism operad of

ΣnX

. The map

∇

encodes the homotopy coas-

sociativity and homotopy cocommutativity of the classical pinch map

ΣnX→ΣnX∨ΣnX

. In

particular, the pinch map is an operation associated to an element of

(2). Furthermore, for any

based map X →Y , the induced map ΣnX→ΣnY extends to a morphism of Cn-coalgebras.

All details will be explained later on. Bearing the above result in mind, it is natural to wonder if

the Eckmann–Hilton dual of May’s celebrated recognition of iterated loop spaces is true in this new

setting. This is indeed the case, as the following result shows.

Theorem B.

Let

be a

-coalgebra. Then there is a pointed space

Γn

(

), naturally associated to

X , together with a weak equivalence of Cn-coalgebras

arXiv:2210.00839v2 [math.AT] 2 Feb 2023

ΣnΓn(X)X,

which is a retract in the category of pointed spaces. Therefore, every

-coalgebra has the homotopy

type of an n-fold reduced suspension.

Together, our theorems Aand Bprovide the following intrinsic characterization of

-fold

reduced suspensions as Cn-coalgebras.

Corollary.

Every

-fold suspension is a

-coalgebra, and if a pointed space is a

-coalgebra then

it is homotopy equivalent to an n-fold suspension.

It is worth noting that this result already exists at the level of

ΣnΩn

coalgebras , see Theorem 4.9.

Another celebrated result in [

] is the approximation theorem. It constitutes an essential step for

proving the recognition principle for

-fold loop spaces, and it is also the key for unlocking certain

computations on the homology of iterated loop spaces. Roughly speaking, the approximation

theorem for loop spaces asserts that the free

-algebra on a pointed space

is weakly equivalent

to ΩnΣnX. We also prove the Eckmann–Hilton dual of this result. It reads as follows.

Theorem C. For every n ≥1, there is a natural morphism of comonads

αn:ΣnΩn−→ Cn.

Furthermore, for every pointed space

, there is an explicit natural homotopy retract of pointed

spaces

ΣnΩnX Cn(X)

In particular, αn(X)is a weak equivalence.

The comonad

in the statement above is constructed in a natural way from the little

-cubes

operad. Essentially, it is an Eckmann–Hilton dualization of May’s monad associated to

. To

our knowledge, this comonad has not been studied elsewhere, and it seems to be an exciting new

object that might shed light on further understanding

-fold reduced suspensions, as well as on

other objects that support a coaction of the little n-cubes operad.

Let us place our work in historical context. It has been known for a long time that any (

n−

1)-

connected CW complex of dimension less than or equal to (2

n−

1) has the homotopy type of a

(1-fold) suspension. In [

], [

] and ﬁnally [

], this result was successively improved on. In

modern language, these authors showed that an (

n−

1)-connected co-

-space equipped with an

comultiplication which is of dimension less than or equal to

(

n−

3 is a suspension. The

case of

k= ∞

in [

] can be thought of as the

-version of Theorem B, although our proof strategy

is very different. From a different angle, the case of iterated suspensions considered as coalgebras

over (a homotopical version of) the

ΣnΩn

-comonad was recently treated in [

], where the authors

obtained a recognition principle for (

1)-connected,

-fold (simplicial) suspensions. This last

result differs from our Theorem Bin several key respects. Firstly; our notions of coalgebra differ

as they pass to a derived functor in the homotopy category of pointed spaces, while we consider

only

ΣnΩn

-coalgebras in the classical sense of coalgebras over comonads. Secondly; our result

has the sharpest possible connectivity requirement. The most striking difference with all previous

scholarship is that we make heavy use of the little

-cubes operad and the comonad

; whereas

these objects do not seem to have appeared in previous literature on the homotopy theory of

iterated suspensions (with the exception of [

] in a very different context). In particular, there is

no approximation theorem in [4].

One of the main contributions of this article is therefore providing a link between iterated

suspensions and the combinatorics of the little cubes operads. Potentially, this connection can be

exploited further.

To conclude, a few remarks are in order. The ﬁrst remark is that to prove our theorems

and

, we do not follow an Eckmann–Hilton dual approach to May’s proof in the case of iterated loop

spaces. While we believe it may be possible to pursue this approach, we have found a framework

and proof which depends on explicit homotopies and hence avoids the use of quasi-ﬁbrations

and the construction of auxiliary spaces. In this sense, our approach is technically simpler. The

approximation of suspensions is an independent result that we believe might have potential side

applications. Finally, most of the results of this paper could have been stated using little

-disks

instead of little

-cubes. However, using cubes signiﬁcantly simplify many of the explicit formulae

that appear when proving our results, and therefore we choose to present things this way.

1.1 Notation and conventions

All topological spaces are compactly generated and Hausdorff. We denote by

the unit interval in

Rand by Jits interior:

J=(0,1) ⊆[0,1] =I.

The symmetric group on nletters is denoted Sn.

For

(

X,∗

) a pointed space, it will be convenient to identify the

-fold wedge

X∨r

as a

subspace of the cartesian product X×r. To do so, consider

X∨r=

[

i=1

{∗}× · · · × X

|{z}

× · · · × {∗}⊂X×r.

A point

in the

-th factor of the wedge

X∨r

is therefore identiﬁed with the point (

∗,...,∗,x,∗,...,∗

)

having

at its

-th component and the base point at all others. We further use the convention that

both

X∨0

and

X×0

are equal to the base point. Given pointed maps

ϕ1,...,ϕr

X→Y

, we denote by

¡ϕ1,...,ϕr¢

the induced map

X→Y×r

to the product. Here, we implicitly used the diagonal map

X→X×r

given by

(

)

(

x,..., x

). To simplify the notation we will omit the diagonal from the

notation when this is clear from the context. If the image of this map lands in the wedge subspace

Y∨r

, we denote the corresponding restriction by

©ϕ1,...,ϕrª

. Thus, the curly brackets notation

emphasizes that the map lands in the wedge rather than the product. We reserve the notation

ϕ1∨ · · · ∨ ϕrfor the induced map X∨r→Y∨rgiven by

¡ϕ1∨ · · · ∨ ϕr¢(∗,...,∗,xi,∗,...,∗)=¡∗,...,∗,ϕi(xi),∗,...,∗¢.

We frequently use the identiﬁcation

ΣnX=Sn∧X

for the

-fold reduced suspension of a pointed

space

. Thus, points in

ΣnX

will be denoted [

t,x

], where

t∈Sn

and

x∈X

. Since points in

the suspensions are equivalence classes, we use the square brackets notation. From now on, we

implicitly assume all suspensions are reduced.

We assume the reader is familiar with operad theory, especially in topological spaces, and we

refer to [

]. We use the following conventions. An operad

in a symmetric monoidal category

M=(M,⊗,1)

is unitary if

(0)

, and non-unitary if

(0) is not deﬁned (i.e., the underlying

symmetric sequence of

starts in arity 1). We borrow this nomenclature from [

, Section 2.2].

We will make heavy use of the operad of little

-cubes

, considered as a unitary operad where

Cn(0) = ∗ is a single point.

Acknowledgments:

The authors would like to thank Sergey Mozgovoy and Jim Stasheff for useful

conversations and comments. The second author has been partially supported by the MICINN

grant PID2020-118753GB-I00. The third author was supported by the Dutch Research Organisation

(NWO) grant number VI.Veni.202.046. This project has received funding from the European

Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie

grant agreement No 945322.

MSC 2020: 18M75, 55P40, 55P48

Key words and phrases: Little cubes operad, Suspensions, Coalgebras, Recognition principle.

2 Coalgebras over topological operads

Given a unitary topological operad

, we construct an explicit comonad

in pointed spaces.

In Section 2.1, we carefully construct this comonad and study some of its basic properties. The

comonad CPgives rise to the category of coalgebras over P, also called P-coalgebras. There is a

second way of deﬁning

-coalgebras by using the coendomorphism operad that does not require

the explicit construction of the comonad

. This alternative construction has the advantage that

it can be deﬁned for all operads even when they are not necessarily unitary. The disadvantage is

that it is not clear how to get an explicit comonad out of this deﬁnition. We explain this alternative

construction and show that in the case of unitary operads it gives an equivalent notion of

coalgebras in Section 2.2. We specialize to the case in which

is the operad

of little

-cubes in

Section 2.3, producing the central comonad of this paper. Finally, we prove Theorem Ain Section

2.4 - that the

-fold reduced suspension of a pointed space is naturally a

coalgebra. Therefore,

the n-fold reduced suspensions are the paradigmatic examples of Cn-coalgebras.

Remark 2.1.

In our constructions of coalgebras, we are mixing pointed and unpointed spaces. All

our operads live in the category of unpointed spaces while the coalgebras over the operads and

associated comonads live in the category of pointed spaces.

2.1 Construction of topological comonads

In this section, we construct the mentioned comonad

in pointed spaces out of a unitary operad

Pin unpointed spaces.

Let us ﬁrst establish some preliminary notation. Denote

Top =¡Top,×,{∗}¢and Top∗=¡Top∗,∨,{∗}¢

the symmetric monoidal categories of spaces endowed with the cartesian product

, and pointed

spaces endowed with the wedge product

∨

, respectively. Let

be a unitary operad in

Top

with

composition map

and denote the unitary operation by

∗ ∈ P

(0). Deﬁne the restriction operators,

for all n≥1 and 1 ≤i≤n, by inserting the unique point ∗ ∈ P(0) at the i-th component:

P(n)P(n−1)

θ γ(θ;id,...,∗,...,id).

Let X∈Top∗. The wedge collapse maps, deﬁned for all n≥1 and 1 ≤i≤n, are given by collapsing

the i-th factor in the wedge as follows:

X∨nX∨(n−1)

(x1,..., xn) (x1,..., b

xi,..., xn).

πi

Here, the

-fold wedge is seen inside the

-fold cartesian product, and the notation

means that

we are sending the i-th component to the basepoint.

Notation 2.2. If Pis a unitary operad and Xis a pointed space, we denote

Tot(P,X):=Y

n≥0

MapSn¡P(n), X∨n¢.

Each space

MapSn¡P(n), X∨n¢

consists of the equivariant maps from the arity

component

equipped with its usual

-action to the

-fold wedge of

with itself endowed with the

-action that permutes the coordinates of its points by

σ·(x1,..., xn)=¡xσ(1),...,xσ(n)¢

. We fre-

quently disregard the 0-th component in the inﬁnite product above, since the mapping space

Map

(

(0)

,X∨0

) is just a point. It can therefore be ignored in all computations that follow. Thus,

the point

¡f0,f1,f2,...¢∈Tot(P,X)

will be denoted

¡f1,f2,...¢

. The topology on the space

Tot(P,X)

is the usual product topology.

We are ready to deﬁne the underlying endofunctor of our comonad CP.

Deﬁnition 2.3. Let Pbe a unitary operad in Top. Deﬁne the endofunctor in pointed spaces

CP:Top∗Top∗

X CP(X),

where

is the subspace of

Tot(P,X)

formed by those sequences

¡f1,f2,...¢

that commute with the restric-

tion operators and wedge collapse maps. That is, for all

n≥

2 and 1

≤i≤n

, the following diagram

commutes:

P(n)X∨n

P(n−1) X∨(n−1)

πi

fn−1

The base point of

CP(X)

is the sequence

α=¡f1,f2,...¢

where each

has image the base point of

X∨r

. Since the base point of the wedge

X∨r

is ﬁxed by the

-action, the base point is well-deﬁned.

If f:X→Yis a pointed map, then CP¡f¢:CP(X)→CP(Y)is deﬁned by

CP¡f¢(α)=¡f◦f1,¡f∨f¢◦f2,...,¡f∨...∨f¢◦fn,...¢.

The nth term in the sequence above is given by

¡f∨...∨f¢◦fn:P(n)fn

−−→ X∨nf∨...∨f

−−−−−−→ Y∨n.

Remarks 2.4.

The idea of deﬁning

above as a subspace of

Tot(P,X)

arises from an Eckmann–Hilton

dualization of May’s deﬁnition of the monad associated to an operad [

]. Recall that the

monad

in pointed spaces deﬁned in loc. cit. by using the little

-cubes operad is given by

Mn(X)=µa

r≥0

Cn(r)×X×r¶/∼,

where

∼

is the equivalence relation that glues level

to level

1 by combining the restric-

tion operators with the insertion of the base point, (

(

)

∼

(

c,si

(

)), and imposing the

compatibility with the group action, (c·σ,y)∼(c,σ·y). 1

The compatibility condition of a sequence

α∈Tot(P,X)

with the restriction operators and

wedge collapse maps,

πifn=fn−1di, for all n≥1 and 1 ≤i≤n(1)

is the precise condition needed to incorporate a counit to the coalgebras in pointed spaces

that result from the comonad CP. See Remark 2.17 for further details.

The comonad

can be constructed in more general symmetric monoidal categories. For

the applications that we give in this paper, we are only interested in the category of topological

spaces.

Our next goal is to endow the endofunctor

with a comonad structure. Before doing so, we

make two elementary observations that will simplify some of our proofs later on. We will use the

following notation: if h1,...,hris a family of maps such that the composition

h1◦ · ·· ◦ hi−1◦hi+1◦ ·· · ◦ hr

makes sense, then we denote the expression above by

h1· · · c

hi· · · hr.

That is, the hat

(−)

on top of the

-th map indicates that this component is removed from the

composition. The ﬁrst observation is the following.

Here, (

c,y

)

∈Cn

(

)

×X×(r−1)

(

) is the point of

X×r

where we insert the base point at the

-th component, and

σ∈Sr.

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ArecognitionprincipleforiteratedsuspensionsascoalgebrasoverthelittlecubesoperadOisínFlynn-Connolly,JoséM.Moreno-Fernández,FelixWierstraAbstractOurmainresultisarecognitionprincipleforiteratedsuspensionsascoalgebrasoverthelittlecubesoperads.Givenatopologicaloperad,weconstructacomonadinpointedtopologic...

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A recognition principle for iterated suspensions as coalgebras over the little cubes operad Oisín Flynn-Connolly José M. Moreno-Fernández Felix Wierstra

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