Learn Russian: После кризиса люди часто живут в апатии. - After a crisis, people often live in apathy.

The word апатия ends in -ия. Feminine nouns ending in -ия belong to a special group that take -и in the prepositional case instead of the standard -е. Because the sentence describes being inside a state using в, it triggers the prepositional case, resulting in в апатии.

Why do we use the preposition в here instead of another preposition like на?

In Russian, prolonged emotional or psychological states are often treated as metaphorical enclosed spaces. The language imagines you are physically 'inside' the feeling. That is why we use в plus the prepositional case for states like в апатии (in apathy), в депрессии (in depression), or в шоке (in shock).

Could we use the verb быть (to be) instead of живут (live) here?

Yes, but живут в апатии emphasizes that this is a prolonged, ongoing lifestyle or condition after the crisis, rather than just a temporary passing mood. Another very common verb for this structure is находятся (find themselves / are located), which is a highly standard pairing for emotional states.

I know после means 'after', but why exactly does кризис change to кризиса?

The preposition после always requires the genitive case. Since кризис is a masculine noun ending in a hard consonant, it simply adds -а to form the genitive кризиса.