Add the chickpeas, sweet potato, cauliflower, onion, garlic and ginger to a 6-quart slow cooker.
Whisk the coconut milk, tomatoes, broth, curry powder, cumin and salt and pour the mixture over the vegetables.
Cover and cook on low until the sweet potato and cauliflower are tender but still hold their shapes, 4 1/2 to 5 hours.
Stir in the spinach by handfuls, cover and cook on high until wilted, 5 to 8 minutes.
Add the lime juice and stir gently so the sweet potato cubes stay intact.
Taste for salt and serve the curry in bowls with plenty of coconut sauce.