Add category grouping to sidebar and dashboard

Sidebar now renders uncategorised apps flat and categorised apps in
collapsible accordion sections; active category stays expanded. Dashboard
groups tiles under category section headings.

Co-Authored-By: Claude Sonnet 4.6 <noreply@anthropic.com>

src/types.ts

@@ -5,6 +5,7 @@ export interface App {
   base_path: string
   icon: string
   theme_color: string
+  category: string | null
 }
 
 export interface User {