Use \tag{foo}
{{< katex >}}
$$ \tag{1} \overrightarrow{p_i}(\bar{y}) = \begin{cases} p_a(\bar{y})&:i=0 \\ \overrightarrow{p_{\sigma , i}}(\bar{y})\overrightarrow{g_i}(\bar{y})&:otherwise \end{cases}$$
{{< /katex >}}
Please submit further questions of this nature to:
https://github.com/KaTeX/KaTeX/discussions
Thanks.