nLab suspension object

Definition

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

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

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

This concept is dual to that of loop space object.

Suspension functor

As an (infinity,1)-functor

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

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

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

$\Sigma_C$ is left adjoint to the loop space functor $\Omega_C$.

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

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

Proposition

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

Proof

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

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

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

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

Proposition

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

Proof

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

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

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

Lemma

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

Proof

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

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

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

Hence, there are the following natural equivalences:

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

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

Theorem

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

Proof

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

As an ordinary functor

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

Following Ayoub, the evaluation functor

$$Ev^n : Spect_F^{\Phi}(C) \to C,$$

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

$$Sus^n : C \to \Spect_F^\Phi(C)$$

called the $n$th suspension functor, more commonly denoted $\Sigma_C^{\infty-n}$.

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

Proposition. One has

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

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

Examples

• In Top, this is the reduced suspension of a space.

• In a category of chain complexes the suspension of a chain complex is given by shifting the degrees of the chain complex up by one.

Related concepts

• loop space object, free loop space object,

  • delooping

  • loop space, free loop space, derived loop space

• suspension object

  • suspension type, reduced suspension type

  • suspension, reduced suspension

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)