CW-complex, Hausdorff space, second-countable space, sober space
connected space, locally connected space, contractible space, locally contractible space
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 $\Omega_C$.
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