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Nope. The catalog must reside locally on each computer. Some people try to use dropbox, but it must be used carefully in order for it to work properly. It's necessary to allow dropbox to synchronize with Lightroom closed before you open the catalog on the other computer. I'm not necessarily recommending this method, but some people do use it.
When you create a Lightroom Catalog (Library), it is stored either on your machine or on external device (depending upon the location specified by you).
It is not stored/synced on Adobe Cloud.
If you want to access the same catalog on both machines, you need to store the catalog and your original images on external drive and plug that external drive on the machine on which you want to work in Lightroom.
Or, you can use online service like dropbox for the same purpose.
Thank you for your answer fast. I will only synchronize the data from Deskop. nice day, Thomas