installation of big sur on macbook pro late 2013

What happens when you try to update to Big Sur exactly? Does it issue an error? Or can you simply not find the download for Big Sur?

If it issues an error, tell us what it says.

Otherwise see the following support document for a download link for Big Sur.

click here ➜ How to download and install macOS - Apple Support

The above post by HWTech is incorrect. 10.9.5 Mavericks has access to the App Store, and the upgrade to Big Sur is supported on it according to Apple.

click here ➜ macOS - How to Upgrade - Apple

 If your Mac is running OS X Mavericks 10.9 or later, you can upgrade directly to macOS Big Sur. You’ll need the following:

• OS X 10.9 or later
• 4GB of memory
• 35.5GB available storage on macOS Sierra or later*
• Some features require an Apple ID; terms apply.
• Some features require a compatible internet service provider; fees may apply.

You should not need to upgrade to an intermediate version.

You will first need to upgrade to macOS 10.11 El Capitan before you can get access to the App Store to download later versions of macOS 10.13+. Fortunately macOS 10.11 El Capitan can be downloaded from outside of the App Store using any computer or OS which can access the Internet. Just transfer the downloaded DMG file to this Mac where you need to open/mount the DMG, run the .pkg file within to extract the real full installer into the Applications. Finally run the macOS installer found within the Applications folder.

Here is an Apple article with the details.....read it carefully since there are multiple steps involved which I summarized in the previous paragraph:

How to download and install macOS - Apple Support

FYI, you can also try booting into Internet Recovery Mode using Command + Option + R to attempt to access the online macOS 11.x Big Sur installer assuming this laptop at some point in the past actually had macOS 10.13+ installed on it. Unfortunately some Macs may only boot to the online installer for the OS which originally shipped with the Mac from the factory regardless of the keys used for booting. IIRC, the Big Sur installer should automatically convert the HFS+ Mavericks file system to APSF, however, if it does not, then you will need to erase the drive as GUID partition and APFS (top option) before selecting the "Install macOS Big Sur" option.