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In a (∞,1)-category $C$ admitting a final object ${*}$, for any object $X$ its suspension object $\Sigma X$ is the homotopy pushout
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.
Let $C$ be a pointed (infinity,1)-category. Write $M^\Sigma$ for the (infinity,1)-category of cocartesian squares of the form
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.
For $X$ a pointed object of a Grothendieck (∞,1)-topos ${\mathcal{H}}$, there is a natural equivalence ${\mathbf{B}}F[X]\simeq \Sigma X$.
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:
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 $H$.
For $X$ a pointed object and $G$ a group object of a Grothendieck (∞,1)-topos ${\mathcal{H}}$, there is a natural equivalence ${\mathbf{B}}(X\otimes G)\simeq X\wedge {\mathbf{B}}G$.
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:
Hence $\Omega (X\wedge {\mathbf{B}}G)$ has the universal property of the tensor product. Delooping gives the required result.
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.
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
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:
Therefore $F[X]$ has the universal property of the tensor product $X\otimes Z$. The required natural equivalence follows by abstract nonsense.
For $X$ a pointed object of a Grothendieck (∞,1)-topos ${\mathcal{H}}$, there is a natural equivalence $\Sigma X\simeq B{\mathbb{Z}}\wedge X$.
Deloop the natural equivalence in Lemma 1 to obtain the natural equivalence ${\mathbf{B}}F[X]\simeq {\mathbf{B}}(X\otimes Z)$. By propositions 1 and 2, this gives the required natural equivalence.
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
which “evaluates” a symmetric spectrum at its $n$th component, admits under these assumptions a left adjoint
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
for all $X,Y \in C$. In particular, $Sus = Sus^0 : C \to \Spect^\Sigma_T(C)$ is a symmetric monoidal functor.
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.
suspension object
A detailed treatment of the 1-categorical case is in the last chapter of