nLab
suspension object

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Topology

Homotopy theory

Stable homotopy theory

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Definition

In a (∞,1)-category CC admitting a final object *{*}, for any object XX its suspension object ΣX\Sigma X is the homotopy pushout

X * * ΣX, \array{ X &\to& {*} \\ \downarrow && \downarrow \\ {*} &\to& \Sigma X } \,,

This is the mapping cone of the terminal map X*X \to {*}. See there for more details.

This concept is dual to that of loop space object.

Suspension functor

As an (infinity,1)-functor

Let CC be a pointed (infinity,1)-category. Write M ΣM^\Sigma for the (infinity,1)-category of cocartesian squares of the form

X * * Y, \array{ X &\to& {*} \\ \downarrow && \downarrow \\ {*} &\to& Y } \,,

where XX and YY are objects of CC. Supposing that CC admits cofibres of all morphisms, then one sees that the functor M ΣCM^\Sigma \to C given by evaluation at the initial vertex (XX) is a trivial fibration. Hence it admits a section s:CM Σs : C \to M^\Sigma. Then the suspension functor Σ C:CC\Sigma_C : C \to C is the composite of ss with the functor M ΣCM^\Sigma \to C given by evaluating at the final vertex (YY).

Σ C\Sigma_C is left adjoint to the loop space functor Ω C\Omega_C.

For XX a pointed object of a Grothendieck (∞,1)-topos {\mathcal{H}}, the suspension object ΣX\Sigma X is homotopy equivalent to BXB{\mathbb{Z}}\wedge X, the smash product by the classifing space of the discrete group of integers.

We outline a proof below. For XX a pointed object of a Grothendieck (∞,1)-topos {\mathcal{H}}, its reduced free group, denoted by F[X]F[X], is the left adjoint to the functor ΩB:Grp() *\Omega {\mathbf{B}}:Grp(\mathcal{H})\to \mathcal{H}_* which sends a group object internal to {\mathcal{H}} to the loop space of its delooping object.

Proposition

For XX a pointed object of a Grothendieck (∞,1)-topos {\mathcal{H}}, there is a natural equivalence BF[X]ΣX{\mathbf{B}}F[X]\simeq \Sigma X.

Proof

This is due to the adjunction (ΣΩ): * *(\Sigma \vdash \Omega):\mathcal{H}_*\leftrightarrows\mathcal{H}_* between suspending and looping and the the adjunction (ΩB):PathConn( *)Grp()(\Omega \vdash {\mathbf{B}}):PathConn(\mathcal{H}_*)\leftrightarrows Grp(\mathcal{H}) between looping and delooping. Indeed, for any group object HH, the above-mentioned adjunctions imply the following natural equivalences:

Grp()(ΩΣX,H) PathConn( *)(ΣX,BH) PathConn( *)(X,ΩBH), \begin{aligned} Grp({\mathcal{H}})(\Omega \Sigma X, H) & \simeq PathConn({\mathcal{H}}_*)(\Sigma X, {\mathbf{B}}H) \\ & \simeq PathConn({\mathcal{H}}_*)(X, \Omega{\mathbf{B}}H) \,, \end{aligned}

Hence ΩΣX\Omega \Sigma X has the universal property of the reduced free group. Delooping gives the required result.

The (∞,1)-category Grp()Grp(\mathcal{H}) of group objects internal {\mathcal{H}} is tensored over *{\mathcal{H}}_*; in particular, for GG a group object and XX a pointed object, we can form the tensor product XGX\otimes G, which is a group object. Explicitly, this tensor product is required to satisfy a homotopy equivalence Grp()(Ω(XG,H)PathConn( *)(X,Grp()(G,H))Grp({\mathcal{H}})(\Omega (X\otimes G, H)\simeq PathConn({\mathcal{H}}_*)(X, Grp({\mathcal{H}})(G, H)), natural in group objects HH.

Proposition

For XX a pointed object and GG a group object of a Grothendieck (∞,1)-topos {\mathcal{H}}, there is a natural equivalence B(XG)XBG{\mathbf{B}}(X\otimes G)\simeq X\wedge {\mathbf{B}}G.

Proof

This is due to the adjunction (ΩB):PathConn( *)Grp()(\Omega \vdash {\mathbf{B}}):PathConn(\mathcal{H}_*)\leftrightarrows Grp(\mathcal{H}) between looping and delooping and the internal hom adjunction. Indeed, for any group object HH, the above-mentioned adjunctions gives the following natural equivalences:

Grp()(Ω(XBG),H) PathConn( *)(XBG,BH) PathConn( *)(X,PathConn( *)(BG,BH)) PathConn( *)(X,Grp()(G,H)), \begin{aligned} Grp({\mathcal{H}})(\Omega (X\wedge {\mathbf{B}}G), H) & \simeq PathConn({\mathcal{H}}_*)(X\wedge {\mathbf{B}}G, {\mathbf{B}}H) \\ & \simeq PathConn({\mathcal{H}}_*)(X, PathConn({\mathcal{H}}_*)({\mathbf{B}}G, {\mathbf{B}}H)) \\ & \simeq PathConn({\mathcal{H}}_*)(X, Grp({\mathcal{H}})(G, H)) \,, \end{aligned}

Hence Ω(XBG)\Omega (X\wedge {\mathbf{B}}G) has the universal property of the tensor product. Delooping gives the required result.

Lemma

For XX a pointed object of a Grothendieck (∞,1)-topos {\mathcal{H}}, there is a natural equivalence F[X]XZF[X]\simeq X\otimes Z, where ZZ is the group object whose delooping object is BB {\mathbb{Z}}, the classifying space of the discrete group of integers.

Proof

Since {\mathcal{H}} is a Grothedieck (,1)(\infty,1)-topos, the (,1)(\infty,1)-functor *B:Group()Func(Δ 1,)*\to {\mathbf{B}}-:Group(\mathcal{H})\to Func(\Delta^1,\mathcal{H}) which sends a group object to the map from the terminal object to its delooping object is a (,1)(\infty,1)-categorial equivalence onto its image, which is the full subcategory of Func(Δ 1,)Func(\Delta^1,\mathcal{H}) spanned by the effective epimorphisms from the terminal object. Hence, for HH a group object, we have

Grp()(Z,H) Func(Δ 1,)(*B,*BH) *(B,BH), \begin{aligned} Grp(\mathcal{H})(Z,H) & \simeq Func(\Delta^1,{\mathcal{H}})(*\to B{\mathbb{Z}},*\to {\mathbf{B}}H) \\ & \simeq {\mathcal{H}}_*(B{\mathbb{Z}},{\mathbf{B}}H) \,, \end{aligned}

This latter based mapping object is equivalent to the based object of deloopable maps from {\mathbb{Z}} to ΩBH\Omega{\mathbf{B}}H, which is just ΩBH\Omega{\mathbf{B}}H, since {\mathbb{Z}} is the discrete free group on one generator.

Hence, there are the following natural equivalences:

Grp()(F[X],H) PathConn( *)(X,ΩBH) PathConn( *)(X,Grp(Z,H), \begin{aligned} Grp({\mathcal{H}})(F[X], H) & \simeq PathConn({\mathcal{H}}_*)(X, \Omega{\mathbf{B}}H) \\ & \simeq PathConn({\mathcal{H}}_*)(X, Grp(Z, H) \,, \end{aligned}

Therefore F[X]F[X] has the universal property of the tensor product XZX\otimes Z. The required natural equivalence follows by abstract nonsense.

Theorem

For XX a pointed object of a Grothendieck (∞,1)-topos {\mathcal{H}}, there is a natural equivalence ΣXBX\Sigma X\simeq B{\mathbb{Z}}\wedge X.

Proof

Deloop the natural equivalence in Lemma 1 to obtain the natural equivalence BF[X]B(XZ){\mathbf{B}}F[X]\simeq {\mathbf{B}}(X\otimes Z). By propositions 1 and 2, this gives the required natural equivalence.

As an ordinary functor

Let CC be a category admitting small colimits. Let Φ\Phi be a graded monoid in the category of groups and F:CCF : C \to C a Φ\Phi-symmetric endofunctor of CC that commutes with small colimits. Let Spect F Φ(C)Spect_F^{\Phi}(C) denote the category of Φ\Phi-symmetric FF-spectrum objects in CC.

Following Ayoub, the evaluation functor

Ev n:Spect F Φ(C)C, Ev^n : Spect_F^{\Phi}(C) \to C,

which “evaluates” a symmetric spectrum at its nnth component, admits under these assumptions a left adjoint

Sus n:CSpect F Φ(C) Sus^n : C \to \Spect_F^\Phi(C)

called the nnth suspension functor, more commonly denoted Σ C n\Sigma_C^{\infty-n}.

When CC is symmetric monoidal, and in the case Φ=Σ\Phi = \Sigma and F=TF = T \otimes - for some object TT, there is an induced symmetric monoidal structure on Spect T Σ(C)Spect^\Sigma_T(C) as described at symmetric monoidal structure on spectrum objects.

Proposition. One has

Sus T p(X)Sus T q(Y)Sus T p+q(XY) Sus^p_T(X) \otimes Sus^q_T(Y) \simeq Sus^{p+q}_T(X \otimes Y)

for all X,YCX,Y \in C. In particular, Sus=Sus 0:CSpect T Σ(C)Sus = Sus^0 : C \to \Spect^\Sigma_T(C) is a symmetric monoidal functor.

Examples

References

A detailed treatment of the 1-categorical case is in the last chapter of

  • Joseph Ayoub, Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique, I. Astérisque, Vol. 314 (2008). Société Mathématique de France. (pdf)

Revised on July 20, 2014 14:03:05 by Colin Tan (220.255.2.22)