Is there a way to sync one file gdb across catalog or arcmap, agol, and collector without using a server? What is the best way to track changes to a file gdb when people are editing it in the filed (collector) and remotely (desktop) and it is stored in catalog as well as online?
The only way I found to sync one file GDB across all three platforms was to host it on AGOL as a feature service and edit it in collector from the web map or edit it in Desktop from the "Hosted Services" section in catalog. There was no way, without server, to sync edits across all three platforms.
There are limitations to this: A feature service (even though it was built from a database) cannot handle full editing functions. The feature service can accept adding, changing, or removing features within each layer. However, the feature service cannot handle schema changes directly. The database published to AGOL should be finalized beforehand, but there are workarounds for specific tasks.
In order to update or add a CVD, please reference the following document: http://blogs.esri.com/esri/arcgis/files/2014/10/How-to-Update-Hosted-Feature-Service-Schemas1.pdf
It is not possible to add a field to a specific layer in a feature service after the database has been published. In order to update the schema, the database had to be republished. Republishing from the local database to AGOL will overwrite any edits made in AGOL previously. In order to update the schema post de facto, please reference the following document: http://spatialstudies.redlands.edu/resources/overwriting-a-hosted-service-in-arcgis-online/
This solution is "hacky" and it would be good in the future to have a better way to handle schema edits. The ability to edit the JSON of the service URL is not completely developed and it has a high risk.