The Gmail categories are basically virtual folders/sections to categorize the messages in different sections like Primary, Social, Promotions etc.

Airmail Inbox will not show only the Primary emails (Airmail Inbox will contain all the messages from all the categories) in it and you can not avoid this behavior and to accomplish this - you will have to choose the Primary folder in the sidebar.

To enable these categories to show up in Airmail Sidebar - you can enable them in Airmail preferences > Favorites. 

Enabled option will show up in the Airmail sidebar, please refer to the picture below:

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