Why does hatch not save the virtual environment in the same directory I am working in? I went through the doc and couldn't find a way to achieve that. Maybe it makes sense that it's that way but I can't really think why. Is there a way to create it in the project directory? Or a good reason to not want to do that?
When you use a tool like hatch, you are basically giving up precise control over the virtual environment in favor of letting hatch manage it for you; this includes letting hatch decide where to keep all such managed environments.
That said, hatch stores virtual environments in a specific data directory that you can specify using the --data-dir option to the hatch command itself.
$ mkdir hatchtest; cd hatchtest
$ ls
$ hatch --data-dir . env create
[...]
$ ls
env
(The default data directory likely varies from platform to platform; it's ~/Library/Application Support/hatch on my macOS box.)