Search API group by issue with views
Group by your Category taxonomy and select to Use rendered output to group rows, then in your Filter criteria for your taxonomy turn on autocomplete of the category terms.

Add View


View Settings: Show Content of type Course sorted by Newest first
Page Settings: Create a page selected
Page Display Settings: Display format Unformatted list of fields


note: you may not want to use the pager.

Edit new View


Advanced:  


Relationships: add Relate nodes to taxonomy terms with Vocabularies {Taxonomy} selected and Require this relationship selected
Use aggregation: Selected

Fields: add content: ID, Content: Title, Content: {Taxonomy}


Content: ID: Exclude from display selected, Thousands marker: -None-
Content: ID Aggregation settings: Aggregation type: Count DISTINCT
Content: Title: Formatter Plain text and Link to the Content selected
Content: {Taxonomy}: Exclude from display selected

Format:


Format settings: Grouping field Nr.1 Content: {Taxonomy} and Use rendered output to group rows selected

Filter Criteria: add Content: Title and Content: {taxonomy}


Content: Title: Expose this filter to visitors, to allow them to change it selected and Operator Contains
Content: {Taxonomy}: Expose this filter to visitors, to allow them to change it selected and Operator Is one of
Content: {Taxonomy} settings: Vocabulary {Taxonomy} and Selection type Autocomplete

Sort Criteria: Content: {Taxonomy} (asc), Content: Title (asc), Content: Authored on (desc)


Note: the preview will not display the Grouping titles, but this dose appear on the page.