I want to synchronize only particular category events (i.e training) from SharePoint 2013 calendar to outlook.
Outlook only knows the standard calendar columns that exist in a basic calendar in SharePoint. Outlook syncs the event content type columns that it has by default and will sync all the events found in the calendar. SharePoint allows you to create multiple calendar styled views that can filter on various things however outlook does not have the ability to do this that I know of.
Creating a separate calendar for Training in SharePoint is the only way you can choose to sync just the contents of that calendar into outlook. If you want to create other calendars and make use of SharePoint overlays this can help you layer the calendar content if need be. Each calendar has its own items and when synced with outlook all of the events will sync.
I tested making a view that only shows items in a specified category, however outlook ignores the view filter all together and syncs everything. SharePoint views, filters, overlays only work in the browser. Hopefully you can find a way to organize your information.
Outlook has filtering for views, which is client based. Just change your view (or create a new view) to filter in only anything regarding training. As far as syncing only those events, that is based on a whole calendar. You will need to create a different calendar for such events and sync only that calendar. To create an Outlook view:
Switch to the view that you want to base the new view on.
On the View menu, point to Arrange By, and then click Custom.
For each type of change that you want to make, click a button, and then select the options that you want. For example, if you want to add or remove columns, click Fields (columns are also known as fields). Then add or remove fields, or create a custom field.
When you finish making changes, close the Customize View: view name dialog box.
On the View menu, point to Current View, and then click Define Views.
In the Views for folder folder name box, click Current view settings.
In the Name of new view box, enter a name.
To change where the new view will be available, click an option under Can be used on.
Change the view to include the filter you want to filter on.