Can We Represent Our Categories Like This on a Discourse Hosted Community

You can definitely implement something similar. (That is a hosted Discourse community too)

As a starting point I would use the “Boxes with Subcategories” desktop category page style.

Then your box will look similar to this one
Screenshot_20241012_132300_Firefox

You can also add a category logo in the category settings
Screenshot_20241012_132421_Firefox

Then you can add some CSS to style it the way you like

Tip: you can not only inspect your forum but also your example

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