# nLab suspension object

### Context

#### Topology

topology

algebraic topology

## Theorems

#### Stable homotopy theory

stable homotopy theory

# Contents

## 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 $\Omega_C$.

### 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

• suspension object

## 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 February 7, 2014 00:49:35 by Adeel Khan (77.9.240.178)