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March 21, 2017

On the Operads of J. P. May

Posted by Emily Riehl

Guest post by Simon Cho

We continue the Kan Extension Seminar II with Max Kelly’s On the operads of J. P. May. As we will see, the main message of the paper is that (symmetric) operads enriched in a suitably nice category 𝒱\mathcal{V} arise naturally as monoids for a “substitution product” in the monoidal category [P,𝒱][\mathbf{P}, \mathcal{V}] (where P\mathbf{P} is a category that keeps track of the symmetry). Before we begin, I want to thank the organizers and participants of the Kan Extension Seminar (II) for the opportunity to read and discuss these nice papers with them.

Some time ago, in her excellent post about Hyland and Power’s paper, Evangelia described what Lawvere theories are about. We might think of Lawvere theories as a way to frame algebraic structure by stratifying the different components of an algebraic structure into roughly three ascending levels of specificity: the product structure, the specific algebraic operations (meaning, other than projections, etc.), and the models of that algebraic structure. These structures are manifested categorically through (respectively) the category 0 op\aleph_0^{\text{op}} of finite sets and (the duals of) maps between them, a category \mathcal{L} with finite products that has the same objects as 0\aleph_0, and some other category 𝒞\mathcal{C} with finite products. Then a Lawvere theory is just a strict product preserving functor I: 0 opI: \aleph_0^{\text{op}} \rightarrow \mathcal{L}, and a model or interpretation of a Lawvere theory is a (non-strict) product preserving functor M:𝒞M: \mathcal{L} \rightarrow \mathcal{C}.

Thus 0 op\aleph_0^{\text{op}} specifies the bare product structure (with the attendant projections, etc.) which gives us a notion of what it means to be “nn-ary” for some given nn; II then transfers this notion of arity to the category \mathcal{L}, whose shape describes the specific algebraic structure in question (think of the diagrams one uses to categorically define the group axioms, for example); MM then gives a particular manifestation of the algebraic structure \mathcal{L} on an object MI(1)𝒞M \circ I (1) \in \mathcal{C}.

The reason I bring this up is that I like to think of operads as what results when we make the following change of perspective on Lawvere theories: whereas models of Lawvere theories are essentially given by specifying a “ground set of elements” A𝒞A \in \mathcal{C} and taking as the nn-ary operations morphisms A nAA^n \rightarrow A, we now consider a hypothetical category whose (nn-indexed) objects themselves are the homsets 𝒞(A n,A)\mathcal{C}(A^n, A), along with some machinery that keeps track of what happens when we permute the argument slots.

Cosmos structure on [P,𝒱][\mathbf{P}, \mathcal{V}]

More precisely, consider the category P\mathbf{P} with objects the natural numbers, and morphisms P(m,n)\mathbf{P}(m,n) given by P(n,n)=Σ n\mathbf{P}(n,n) = \Sigma_n (the symmetric group on nn letters) and P(m,n)=\mathbf{P}(m,n) = \emptyset for mnm \neq n.

Let 𝒱\mathcal{V} be a cosmos, that is, a complete and cocomplete symmetric monoidal closed category with identity II and internal hom [,][-,-].

Fix A𝒱A \in \mathcal{V}. The assignment n[A n,A]n \mapsto [A^{\otimes n}, A] defines a functor P𝒱\mathbf{P} \rightarrow \mathcal{V} (where functoriality in P\mathbf{P} comes from the symmetry of the tensor product in 𝒱\mathcal{V}). This turns out to be a typical example of a 𝒱\mathcal{V}-operad, which we call the “endomorphism operad” on AA. In order to actually define what an operad is, we need to lay some groundwork.

(A point of notation: we will henceforth denote A nA^{\otimes n} by A nA^n.)

We’ll need the fact that the functor 𝒱(I,):𝒱textbfSets\mathcal{V}(I, -): \mathcal{V} \rightarrow \textbf{Sets} has a left adjoint FF given by FX= XIFX = \coprod_X I. FF takes the product to the tensor product (since it’s a left adjoint and tensor products in 𝒱\mathcal{V} distributes over coproducts), and in fact we can assume that it does so strictly. Henceforth for XtextbfSetsX \in \textbf{Sets} and A𝒱A \in \mathcal{V} we write XAX \otimes A to actually mean FXAFX \otimes A.

We then get a cosmos structure on \mathcal{F} which is given by Day convolution: for T,ST,S \in \mathcal{F} we have TS= m,nP(m+n,)TmSnT \otimes S = \int^{m,n} \mathbf{P}(m+n, - ) \otimes Tm \otimes Sn Since we are thinking of a given TT \in \mathcal{F} as a collection of operations (indexed by arity) on which we can act by permuting the argument slots, we can think of (TS)k(T \otimes S) k as a collection of the kk-ary operations that we obtain by freely permuting mm argument slots of type TT and nn argument slots of type SS (where m,nm,n range over all pairs such that m+n=km+n = k), modulo respecting the previously given actions of Σ m\Sigma_m (resp. Σ n\Sigma_n) on TmTm (resp. SnSn).

The identity is then given by P(0,)I\mathbf{P}(0,-) \otimes I.

Associativity and symmetry of the cosmos structure. Now let T,S,RT,S, R \in \mathcal{F}. If we unpack the definition, draw out some diagrams, and apply some abstract nonsense, we find that T(SR)(TS)R m+n+kP(m+n+k,)TmSnRkT \otimes (S \otimes R) \simeq (T \otimes S) \otimes R \simeq \int^{m+n+k} \mathbf{P}(m+n+k, - ) \otimes Tm \otimes Sn \otimes Rk which we can again assume are actually equalities.

Before we address the symmetry of this monoidal structure, we make a technical point. P\mathbf{P} itself has a symmetric monoidal structure, given by addition. Thus for n 1,,n mPn_1, \dots, n_m \in \mathbf{P} we have n 1++n mPn_1 + \cdots + n_m \in \mathbf{P}. There is evidently an action of Σ m\Sigma_m on this term, which we require to be in the “wrong” direction, so that ξΣ m\xi \in \Sigma_m induces ξ:n ξ1++n ξmn 1++n m\langle \xi \rangle: n_{\xi 1} + \cdots + n_{\xi m} \rightarrow n_1 + \cdots + n_m rather than the other way around.

(However, for the symmetry of the monoidal structure on 𝒱\mathcal{V}, given a product A 1A mA_1 \otimes \cdots \otimes A_m we require that the action of Σ m\Sigma_m on this term is in the “correct” direction, i.e. ξΣ m\xi \in \Sigma_m induces ξ:A 1A mA ξ1A ξm\langle \xi \rangle: A_1 \otimes \cdots \otimes A_m \rightarrow A_{\xi 1} \otimes \cdots \otimes A_{\xi m}.)

We thus have:

T 1T m = n 1,,n mP(n 1+n m,)T 1n 1T mn m ξ P(ξ,)ξ T ξ1T ξm = n 1,,n mP(n ξ1+n ξm,)T ξ1n ξ1T ξmn ξm \begin{matrix} T_1 \otimes \cdots \otimes T_m &=& \int^{n_1, \dots, n_m} \mathbf{P}(n_1 + \cdots n_m, - ) \otimes T_1 n_1 \otimes \cdots \otimes T_{m} n_m\\ &&\\ {\langle \xi \rangle} \Big \downarrow && \Big \downarrow {\mathbf{P}(\langle \xi \rangle, -) \otimes \langle \xi \rangle}\\ &&\\ T_{\xi 1} \otimes \cdots \otimes T_{\xi m} &=& \int^{n_1, \dots, n_m} \mathbf{P}(n_{\xi 1} + \cdots n_{\xi m}, - ) \otimes T_{\xi 1} n_{\xi 1} \otimes \cdots \otimes T_{\xi m} n_{\xi m}\\ \end{matrix}

Now ξ:n ξ1++n ξmn 1++n m\langle \xi \rangle: n_{\xi 1} + \cdots + n_{\xi m} \rightarrow n_1 + \cdots + n_m extends to an action ξ:T 1T mT ξ1T ξm\langle \xi \rangle: T_1 \otimes \cdots \otimes T_m \rightarrow T_{\xi 1} \otimes \cdots \otimes T_{\xi m} as we saw previously. Therefore we now have a functor P op×\mathbf{P}^{\text{op}} \times \mathcal{F} \rightarrow \mathcal{F} given by (m,T)T m(m, T) \mapsto T^m, a fact which we will later use.

\mathcal{F} as a 𝒱\mathcal{V}-category. There is a way in which we can regard 𝒱\mathcal{V} as a full coreflective subcategory of \mathcal{F}: consider the functor ϕ:𝒱\phi: \mathcal{F} \rightarrow \mathcal{V} given by ϕT=T0\phi T = T0. This has a right adjoint ψ:𝒱\psi: \mathcal{V} \rightarrow \mathcal{F} given by ψA=P(0,)A\psi A = \mathbf{P}(0, -) \otimes A.

The inclusion ψ\psi preserves all of the relevant monoidal structure, so we are justified in considering A𝒱A \in \mathcal{V} as either an object of 𝒱\mathcal{V} or of \mathcal{F} (via the inclusion ψ\psi). With this notation we can write, for A𝒱A \in \mathcal{V} and T,ST,S \in \mathcal{F}: (AT,S)𝒱(A,[T,S])\mathcal{F}(A \otimes T, S) \simeq \mathcal{V}(A, [T,S]) If T,ST, S \in \mathcal{F} then their \mathcal{F}-valued hom is given by [[T,S]][[T,S]], where for kPk \in \mathbf{P} we have [[T,S]]k= n[Tn,S(n+k)][[T,S]]k = \int_n [Tn, S(n+k)] and their 𝒱\mathcal{V}-valued hom, which makes \mathcal{F} into a 𝒱\mathcal{V}-category, is given by [T,S]=ϕ[[T,S]]= n[Tn,Sn][T,S] = \phi [[T,S]] = \int_n [Tn, Sn]

The substitution product

Let us return to our motivating example of the endomorphism operad (which we denote by {A,A}\{A,A\}) on AA, for a fixed A𝒱A \in \mathcal{V}. For now it’s just an object {A,A}\{A, A\} \in \mathcal{F}; but it contains more structure than we’re currently using. Namely, for each m,n 1,,n mPm, n_1, \dots, n_m \in \mathbf{P} we can give a morphism [A m,A]([A n 1,A][A n m,A])[A n 1++n m,A][A^m, A] \otimes \left ( [A^{n_1}, A] \otimes \cdots \otimes [A^{n_m}, A] \right ) \rightarrow [A^{n_1 + \cdots + n_m}, A] coming from evaluation (see the section below about the little nn-disks operad for details). We would like a general framework for expressing such a notion of composing operations.

Definition of an operad. Recall from the previous section that, for given TT \in \mathcal{F}, we can consider nT nn \mapsto T^n as a functor P op\mathbf{P}^{\text{op}} \rightarrow \mathcal{F}. We can thus define a (non-symmetric!) product TS= nTnS nT \circ S = \int^n Tn \otimes S^n. It is easy to check that if S𝒱S \in \mathcal{V} then in fact TS𝒱T \circ S \in \mathcal{V}, so that \circ can be considered as a functor either of type ×\mathcal{F} \times \mathcal{F} \rightarrow \mathcal{F} or of type ×𝒱𝒱\mathcal{F} \times \mathcal{V} \rightarrow \mathcal{V}.

The clarity with which Kelly’s paper demonstrates the various important properties of this substitution product would be difficult for me to improve upon, so I simply list here the punchlines, and refer the reader to the original paper for their proofs:

  • For T,ST,S \in \mathcal{F} and nPn \in \mathbf{P}, we have (TS) nT nS(T \circ S)^n \simeq T^n \circ S which is natural in T,S,nT, S, n. Using this and a Fubini style argument we get associativity of \circ.

  • J=P(1,)IJ = \mathbf{P}(1, - )\otimes I is the identity for \circ.

  • For SS \in \mathcal{F}, S:- \circ S: \mathcal{F} \rightarrow \mathcal{F} has the right adjoint {S,}\{S, -\} given by {S,R}m=[S m,R]\{S, R\}m = [S^m, R]. Moreover if A𝒱A \in \mathcal{V} then we in fact have 𝒱(TA,B)(T,{A,B})\mathcal{V}(T \circ A, B) \simeq \mathcal{F} (T, \{A, B\}).

We can now define an operad as a monoid for \circ, i.e. some TT \in \mathcal{F} equipped with μ:TTT\mu: T \circ T \rightarrow T and η:JT\eta: J \rightarrow T satisfying the monoid axioms. Operad morphisms are morphisms TT T \rightarrow T^\prime that respect μ\mu and η\eta.

{A,A}\{A, A\} as an operad. Once again we turn back to the example of {A,A}\{A, A\} \in \mathcal{F}. Note that our choice to denote the endomorphism operad (n[A n,A])(n \mapsto [A^n, A]) by {A,A}\{A, A\} agrees with the construction of {A,}\{A, -\} as the right adjoint to A- \circ A.

There is an evident evaluation map {A,A}AeA\{A, A\} \circ A \xrightarrow{e} A, so that we have the composition {A,A}{A,A}A1e{A,A}AeA\{A, A\} \circ \{A, A\} \circ A \xrightarrow{1 \circ e} \{A,A\} \circ A \xrightarrow{e} A which by adjunction gives us μ:{A,A}{A,A}{A,A}\mu:\{A,A\} \circ \{A,A\} \rightarrow \{A,A\} which we take as our monoid multiplication. Similarly JAAJ \circ A \simeq A corresponds by adjunction to η:J{A,A}\eta: J \rightarrow \{A, A\}. We thus have that {A,A}\{A,A\} is an operad. In fact it is the “universal” operad, in the following sense:

Every operad TT \in \mathcal{F} gives a monad TT \circ - on \mathcal{F}, or on 𝒱\mathcal{V} via restriction. Given AA \in \mathcal{F}, algebra structures h :TAAh^{\prime}: T \circ A \rightarrow A for the monad TT \circ - on AA correspond precisely to operad morphisms h:T{A,A}h: T \rightarrow \{A,A\}. In this case we say that hh gives an algebra structure on AA for the operad TT.

The little nn-disks operad

There are some other aspects of operads that the paper looks at, but for this post I will abuse artistic license to talk about something else that isn’t exactly in the paper (although it is indirectly referenced): May’s little nn-disks operad. For a great introduction to the following material I recommend Emily Riehl’s notes on Kathryn Hess’s two-part (I,II) talk on operads in algebraic topology.

Let 𝒱=(Top nice,×,{*})\mathcal{V} = (\mathbf{Top}_{\text{nice}}, \times, \{*\}) where Top nice\mathbf{Top}_{\text{nice}} is one’s favorite cartesian closed category of topological spaces, with ×\times the appropriate product in this category.

Fix some nn \in \mathbb{N}. For kPk \in \mathbf{P}, we let d n(k)=sEmb( kD n,D n)d_n(k) = \text{sEmb}(\coprod_{k} D^n, D^n), the space of standard embeddings of kk copies of the closed unit nn-disk in n\mathbb{R}^n into the closed unit nn-disk in n\mathbb{R}^n. By the space of standard embeddings we mean the subspace of the mapping space consisting of the maps which restrict on each summand to affine maps xλx+cx \mapsto \lambda x + c with 0λ10 \leq \lambda \leq 1.

Given ξP(k,k)\xi \in \mathbf{P}(k, k) we have the evident action ξ:sEmb( kD n,D n)sEmb( ξkD n,D n)\langle \xi \rangle: \text{sEmb}(\coprod_{k} D^n, D^n) \rightarrow \text{sEmb}(\coprod_{\xi k} D^n, D^n), which gives us a functor d n:PTop niced_n: \mathbf{P} \rightarrow \mathbf{Top}_{\text{nice}}, so d nd_n \in \mathcal{F}.

Fix some k,lPk,l \in \mathbf{P}; then d n k(l)= m 1,,m kP(m 1++m k,l)d n(m 1)d n(m k)d_n^k(l) = \int^{m_1, \dots, m_k} \mathbf{P}(m_1 + \cdots + m_k, l) \otimes d_n(m_1) \otimes \cdots \otimes d_n(m_k), which we can roughly think of as all the different ways we can partition a total of ll disks into kk blocks, with the i thi^{\text{th}} block having m im_i disks, and then map each block of m im_i disks into a single disk, all the while being able to permute the ll disks amongst themselves (without necessarily having to respect the partitions).

We then get μ:d nd nd n\mu: d_n \circ d_n \rightarrow d_n by composing the disk embeddings. More precisely, for each ll we get a morphism μ l:(d n(k)d n k)ld n(k)(d n k(l))d n(l)\mu_l: (d_n(k) \otimes d_n^k)l \simeq d_n(k) \otimes (d_n^k(l)) \rightarrow d_n(l) from the following considerations:

First we note that d n(k)d n(m 1)d n(m k) =sEmb( kD n,D n)×( 1iksEmb( m iD n,D n)) sEmb(D n,D n) k×( 1iksEmb( m iD n,D n)) 1ik(sEmb( m iD n,D n)×sEmb(D n,D n)). \begin{aligned} d_n(k) \otimes d_n(m_1) \otimes \cdots \otimes d_n(m_k) &= \text{sEmb}(\coprod_k D^n, D^n) \times (\prod_{1 \leq i \leq k} \text{sEmb}(\coprod_{m_i} D^n, D^n))\\ &\simeq \text{sEmb}(D^n, D^n)^k \times (\prod_{1 \leq i \leq k} \text{sEmb}(\coprod_{m_i} D^n, D^n))\\ &\simeq \prod_{1 \leq i \leq k} (\text{sEmb}(\coprod_{m_i} D^n, D^n) \times \text{sEmb}(D^n, D^n)). \end{aligned} Now for each ii there is a map sEmb( m iD n,D n)×sEmb(D n,D n)sEmb( m iD n,D n)\text{sEmb}(\coprod_{m_i} D^n, D^n) \times \text{sEmb}(D^n, D^n) \rightarrow \text{sEmb}(\coprod_{m_i}D^n, D^n) induced from iterated evaluation by adjunction. Then by the above, this gives a morphism d n(k)d n(m 1)d n(m k) 1iksEmb( m iD n,D n) sEmb( m 1++m kD n,D n) =d n(m 1++m k). \begin{aligned} d_n(k) \otimes d_n(m_1) \otimes \cdots \otimes d_n(m_k) &\rightarrow \prod_{1 \leq i \leq k} \text{sEmb} (\coprod_{m_i} D^n, D^n)\\ &\simeq \text{sEmb}(\coprod_{m_1 + \cdots + m_k} D^n, D^n)\\ &= d_n(m_1 + \cdots + m_k). \end{aligned}

A big reason that the little nn-disks operad is relevant to algebraic topology is that there is a big theorem stating that a space is weakly equivalent to an nn-fold loop space if and only if it’s an algebra for d nd_n.

One direction is straightforward: consider a space AA and its nn-fold loop space Ω nA\Omega^n A. Given an element of d n(k)d_n (k) and kk choices of “little maps” (D n,D n)(A,*)(D^n, \partial D^n) \rightarrow (A, \ast), we can stitch together these little maps into one large map (D n,D n)(A,*)(D^n, \partial D^n) \rightarrow (A,\ast) according to the instructions specified by the chosen element of d n(k)d_n(k) (where we map everything in the complement of the kk little disks to the basepoint in AA). Doing this for each kk, we get an operad morphism d n{Ω nA,Ω nA}d_n \rightarrow \{\Omega^n A, \Omega^n A\}.

The other direction is much harder, and Maru gave an absolutely fantastic sketch of the basic story in our group discussions, which I hope she will post in the comments; I refrain from including it in the body of this post, partially for reasons of length and partially because I would just end up repeating verbatim what she said in the discussion.

Posted at March 21, 2017 11:07 AM UTC

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Re: On the Operads of J. P. May

Just to get it out of the way… \mathcal{F} is [P,𝒱][P,\mathcal{V}]?

Posted by: Jesse C. McKeown on March 23, 2017 5:15 AM | Permalink | Reply to this

Re: On the Operads of J. P. May

Yes, that’s right.

Posted by: Maru Sarazola on March 23, 2017 7:27 AM | Permalink | Reply to this

Re: On the Operads of J. P. May

Thanks Simon for spelling out some of the details for us; I got a bit lost in the technicalities when I first tried to read this article.

Here’s a sketch of the proof that every connected d nd_n-algebra has the weak homotopy type of (i.e. its homotopy groups are isomorphic to those of) an nn-fold loop space.

  1. Let SS denote the suspension functor, and recall the adjunction Hom(X,ΩY)Hom(SX,Y)\operatorname{Hom}(X,\Omega Y)\cong\operatorname{Hom}(S X, Y) By iterating this adjunction and reinterpreting it in the category of loop spaces, one can see that Ω nS nX\Omega^n S^n X is the free nn-loop space generated by XX.

  2. That means we can look at Ω nS n\Omega^n S^n as a monad; if we denote by d nd_n the monad whose algebras are the algebras of the operad d nd_n, then we get a morphism of monads α n:d nΩ nS n\alpha_n:d_n\to\Omega^n S^n given by the composition d nXd nηd nΩ nS nXθ nΩ nS nXd_n X\xrightarrow{d_n \eta} d_n\Omega^n S^n X\xrightarrow{\theta_n} \Omega^n S^n X (recall that an action map of the operad d nd_n over YY corresponds to an algebra-structure map d nYYd_n Y\to Y when d nd_n is viewed as a monad).

  3. With considerable effort, it’s possible to show that if XX is connected, then α n:d nXΩ nS nX\alpha_n :d_n X\to\Omega^n S^n X is a weak homotopy equivalence.

  4. For a monad TT on CC, a functor F:CDF:C\to D is said to be a TT-functor if TT acts on FF on the right via a natural transformation λ:FTF\lambda:FT\to F, which is required to satisfy analogue conditions for those of a TT-algebra structure map. It’s easy to see that TT is always a TT-functor.
    For the case of T=d nT=d_n, the map α n:d nΩ nS n\alpha_n:d_n\to\Omega^n S^n from item 2 and its iterated transpose α n #:S nd nXS nX\alpha_n^\#: S^n d_n X\to S^n X via the iterated adjunction mentioned in 1 make Ω nS n\Omega^n S^n and S nS^n into d nd_n-functors: Ω nS nd nΩ nS nα nΩ nS nΩ nS nμΩ nS n\Omega^n S^n d_n\xrightarrow{\Omega^n S^n\alpha_n} \Omega^n S^n \Omega^n S^n \xrightarrow{\mu} \Omega^n S^n S nd nα n #S nS^n d_n\xrightarrow{\alpha_n^\#} S^n

  5. The Bar construction: Given a triple (F,T,X)(F,T,X) where TT is a monad, FF a TT-functor and XX a TT-algebra, we can construct a simplicial object B *(F,T,X)B_{\ast}(F,T,X) where B *(F,T,X) q=FT qX,B_{\ast} (F,T,X)_q=FT^q X, the face maps are given by

    • the TT-functor structure map FTFFT\to F, for i=0i=0
    • the multiplication T 2TT^2\to T, for 1iq11\leq i\leq q-1
    • the algebra structure map TXXTX\to X, for i=qi=q

    and the degeneracy maps are given by η:1T\eta:1\to T.
    Thanks to item 4, it makes sense to consider the simplicial objects B *(d n,d n,X)B_{\ast} (d_n,d_n,X), B *(Ω nS n,d n,X)B_{\ast} (\Omega^n S^n ,d_n,X) and B *(S n,d n,X)B_{\ast} (S^n ,d_n,X) for any d nd_n-algebra XX.

  6. Let X *X_{\ast} denote the simplicial object with XX at every level and 1:XX1:X\to X as face maps. The map B *(d n,d n,X)X *B_{\ast}(d_n,d_n,X)\to X_{\ast} defined on the qq-simplex by d n q+1Xμ qd nXθ nXd_n^{q+1}X\xrightarrow{\mu^q} d_n X\xrightarrow{\theta_n} X is a homotopy equivalence, with inverse X *B *(d n,d n,X)X_{\ast}\to B_{\ast}(d_n,d_n,X) defined by Xη q+1d n q+1XX\xrightarrow{\eta^{q+1}} d_n^{q+1}X

  7. Now suppose XX is a connected d nd_n-algebra. Taking geometric realizations, and using (in this order) the facts from 6, 3, and the fact that there exists a weak homotopy equivalence |Ω{Y n}|Ω|{Y n}|\vert\Omega \{ Y_n\}\vert\cong \Omega \vert\{Y_n\}\vert whenever {Y n}\{Y_n\} is a nice enough simplicial object, we get a chain of weak homotopy equivalences X|X *||B *(d n,d n,X)||B *(Ω nS n,d n,X)|Ω n|B *(S n,d n,X)|X\cong \vert X_{\ast}\vert \xleftarrow{} \vert B_{\ast}(d_n,d_n,X)\vert\to \vert B_{\ast} (\Omega^n S^n,d_n,X)\vert\to \Omega^n \vert B_{\ast} (S^n,d_n,X)\vert which gives us a precise description of XX as an nn-fold loop space!

Posted by: Maru Sarazola on March 24, 2017 3:52 AM | Permalink | Reply to this

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