Does SharePoint 2013 support Co-Authoring?

Yes. You can use the co-authoring function in SharePoint Server 2013 or SharePoint Online to enable multiple users to work on a document without interfering with each other's changes.

Co-authoring eliminates barriers to server-based document collaboration and helps organisations reduce the overheads associated with traditional document sharing through attachments. This capability requires no additional server setup and is the default for documents stored in SharePoint 2013 and SharePoint Online.