“κάτσω” comes from the verb “κάθομαι” (= Ι sit).
It is used in future tense or in Subjunctive.
θα κάτσω = I will sit (future).
θέλω να κάτσω = I want to sit ( Subjunctive).
“κάτσω” actually is not an abbreviation. It is a second, more informal form of “καθίσω”, which is the future tense or Subjunctive of “κάθομαι”.
“κάτσω” is only used in oral speech. When we write or speak in a formal way, we always say “καθίσω”.
Example: θέλω να καθίσω. = I want to sit.