I’m under the impression that Katakuri does not want to serve under Big Mom anymore and finds her habit of rampaging through Totto Land when she gets hungry too dangerous for the well being of the people.
However, given how powerful Big Mom is, he has never felt there could be anyone who could defeat her, claim her territory and rule Totto Land justly whilst being capable of keeping the Marines away.
Katakuri himself could never fight Big Mom, putting aside the probability that she is stronger, because he wouldn’t be able to claim Totto Land since the Big Mom Pirates would see him as a traitor.
Then Katakuri meets Luffy, who he assumed was just another rookie. But as his fight with him dragged on he evidently starts to see that Luffy is much more gifted, unique and has greater potential than any other person he has fought.
By the climax of the fight his determination to defeat Luffy has dissipated and he instead wants Luffy to escape in hopes that he will come back to Totto Land stronger and defeat Big Mom, force the Big Mom Pirates to surrender and claim Totto Land under his protection from the Marines.
Katakuri asks Luffy if he will come back to defeat Big Mom and when Luffy resoundingly confirms he will do so Katakuri appears to intentionally fall on his back (a big thing as Katakuri is praised for having never lied on his back for the sake of his siblings) and allows Luffy to flee.
*Theory by AlexanderImAlsoGr8
