How to define 0-sphere in a category with zero object? The 0-sphere $S^0$ is the coproduct of two points,
$$S^0 \simeq \ast \coprod \ast$$
How to define 0-sphere in a category with zero object?
Let $\mathcal{C}$ be a category. A cylinder, $\mathbf{I}$, on $\mathcal{C}$ is a functor (cylinder functor)
$$I:\mathcal{C} \longrightarrow \mathcal{C}$$
together with three natural transformations
$$e^{0}: 1_{\mathcal{C}} \Longrightarrow I , e^{1}: 1_{\mathcal{C}} \Longrightarrow I,  \sigma: I \Longrightarrow 1_{\mathcal{C,}}$$
such that $\sigma  e^{0}= \sigma  e^{1}= 1,$ with $1: 1_{\mathcal{C}} \Longrightarrow 1_{\mathcal{C}}.$
Definition 1. Let $\mathcal{C}$ be a category with a terminal object other than the initial one.  The 0-sphere $S^0$ is the pushout in $\mathcal{C}:$ 
$\require{AMScd}$
\begin{CD}
      \emptyset @>>>\ast \\
    @V  V V @VV  V\\
    \ast @>> > S^0 :=\ast \coprod \ast
\end{CD}
Definition 2. Let $\mathcal{C}$ be a category with a terminal object other than the initial one.  
1. The cone $C^0(X)$ is the pushout in $\mathcal{C}:$ 
$\require{AMScd}$
\begin{CD}
      X @>>^{e^{0}_X}> I(X)  \\
    @V  V V @VV^{\pi_{0}}  V\\
    \ast @>> > C^0(X):=\ast \underset{X}{\sqcup} I(X)
\end{CD}
2. The cone $C^1(X)$ is the pushout in $\mathcal{C}:$ 
$\require{AMScd}$
\begin{CD}
      X @>>^{e^{1}_X}> I(X)  \\
    @V  V V @VV^{\pi_{1}} V\\
    \ast @>> > C^1(X):=I(X) \underset{X}{\sqcup} \ast
\end{CD}
3. The Suspension $\Sigma(X)$ is the pushout in $\mathcal{C}:$ 
$\require{AMScd}$
\begin{CD}
      X @>>^{\pi_{1}\circ e^{0}_X}>  C^1(X)  \\
    @V ^{\pi_{0}\circ e^{1}_X} V V @VV V\\
     C^0(X) @>> > \Sigma(X):=C^0(X)\underset{X}{\sqcup}C^1(X)
\end{CD}
4. The $n$-sphere $S^n:=\Sigma(S^0)$  for $n>0$.
 A: $\require{AMScd}$I guess every sensible definition of "sphere" in a pointed category $\mathcal C$ depends on a notion of homotopy pushout, or homotopy cofiber in $\mathcal C$; in order for this to be defined you might want to impose at least the structure of a cofibration category on $\mathcal C$. Once you have defined it, the zero map $S^0 \to *$ can be factored as $S^0 \to CS^0 \to *$, where the first arrow is a cofibration, and the suspension of $S^0$ is defined by the pushout
$$
\begin{CD}
S^0 @>>> * \\
@VVV @VVV \\
CS^0 @>>> \Sigma S^0
\end{CD}
$$
(it is the homotopy pushout of the diagram $* \leftarrow S^0\to *$, because you replaced one of its legs with a cofibration, such that $CS^0\simeq *$). Of course, having in mind $\mathcal C = \bf Top$, $S^0 \to CS^0$ s the inclusion of $S^0$ in its mapping cone, and its cofiber is $S^1$; so, you might want to define $\Sigma S^0 := S^1$, and $\Sigma^n S^0 := S^n$. Note that


*

*the factorization of zero maps is not unique; but there exists a weak equivalence between any two $CS^0$ and $C'S^0$.

*you really need that pushout to be homotopic; the colimit of $* \leftarrow S^0\to *$ is always the zero object.

