The catalog cannot be shared. Your wife will have to create her own catalog under her username and work with photos independently. That images can be shared, but it would be necessary for you to copy the catalog from your user space to hers.
You can not sync Lightroom catalogs.
However what you can is to move your catalog and images on an external drive and then you both can share that external drive, do your processing and the data will be saved on the Lightroom catalog in that external drive.
So if I understand what you are saying I can put everything (photos, and catalog) on an external drive and then have lightroom, either under my login or my wifes, point to that catalog and we can both edit from that catalog (presumably not at the same time) and we will be able to see any imports or edits that the other person made? If that is the case could I not do the same thing just by having the catalog (and photographs) in a shared folder that can be accessed by either person when they are logged in? If so I can't seem to find a way to do so.
Thanks for you your previous answer and any future assistance you can offer.
You can but it is not recommended to so, as you might end up with different issues in LR catalog, it is rather best to use an external drive.
And yes only once person can access it at a time.
If that is the case could I not do the same thing just by having the catalog (and photographs) in a shared folder that can be accessed by either person when they are logged in?
Some people do exactly that successfully, esp. using Cloud services such as Dropbox to fool Lightroom into seeing the catalogue as locally attached which Dropbox then replicates via the Cloud.
The shared folder approach requires discipline from the user to open and close Lightroom at the start and end of every editing session - every time - before the next person accesses it since only one person can access the catalogue at a time in order to prevent database corruption and conflicted copies (automatically created by Dropbox when it gets confused on version control).