What version of macOS are you trying upgrading from? (Sierra, High Sierra or Mojave)
Can I upgrade Yosemite 10.10.5 OS to Catalina? MacBook Air 13', OS X 10.10 Posted on Jan 1, 2020 8:02 PM. MacOS 10.15 “Catalina” Upgrade Process. Open the Self Service application from the Applications folder, dock, Spotlight or via Launchpad. From the Self-Service application search for “Catalina” and select the “Upgrade to Catalina” policy. After El Capitan is installed then you can upgrade to Catalina, but if you are not aware yet Catalina does not run 32bit apps. So I would suggest you install Mojave again and all the apps you would use, Upgrade to macOS Mojave - Official Apple Support.
What if any messages do you get, when trying to upgrade to Catalina?
Have you done everything?
at > How to upgrade to macOS Catalina - Apple Support
If it will not startup after attempting to upgrade to Catalina?
try Safe Mode as per > How to use safe mode on your Mac - Apple Support
If that doesn't help and your Mac Mini still has the original Hard Drive, then your Hard Drive may have failed during the installation process and need to be replaced with an SSD.
Upgrade Catalina To Big Sur Error
Mar 31, 2020 9:47 AM