I am exposing a single lightning component tab in SF1 and have a profile that limits object exposure to only the relevant objects (no standard objects). This works fine. However, global search in SF1 allows a user to reach custom object records and while they don't have edit rights on those records, they can view them and it makes the UX in SF1 odd. Is there a way for me to disable that component?
It is NOT possible to disable the global search in Salesforce1. You can however control which objects show up in Salesforce using the solution mentioned on this thread.
Quoting the relevant text here
Description When a customer has access to a standard or custom object via the + (tabs listing) ONLY but cannot access that object via an application, they will notice that these objects are not available in the Salesforce1 left menu.
Resolution Many users associate the tabs (objects) in Aloha with the objects available in Stage Left in S1. When a tab is not available in the Aloha UI via one of the available applications, that object will not be shown within the Salesforce1 Stage Left.
Users experiencing this issue should:
1) Click on the + to the right of the visible tabs within any application 2) Choose 'Customize my tabs' 3) Select any appropriate application and ADD the Standard or Custom object to the 'Available' column 4) Save
Now, the added objects should appear under the Salesforce1 Recent Items section.