AirTags are not good pet-trackers - unless of course the pet (in this case, your cat) also carries an iPhone...
AirTags have no means by which to independently establish or transmit their location. AirTag is entirely reliant upon other Apple devices detecting the presence of an AirTag in their vicinity - and anonymously relaying their own location (this typically being within 30m/100' of the detected AirTag) to Apple's Find My servers.
iPhone and other Apple devices participating in Apple's Find My network silently "listen" for AirTag beacon signals using BTLE. When detected, the unique AirTag code along with the detecting-device location is relayed to Apple. Recent model iPhones also incorporate a UWB (UltraWideBand) radio - and use this when in close proximity to the owner's AirTag for short-range precision location.
As for your AirTag, ensure that it has a fresh CR2032 "Airtag compatible" battery installed.