I have a dedicated walk-in insulated fermentation closet in my basement. I use 2 ink bird controllers. My 5,000 btu window AC control is electronic so it does not automatically turn on when power is restored so I needed to get creative I moved the A/C temp probe and taped it to the side of the AC. I also taped an incandescent nightlight near the probe.
The Inkbird 1 probe is in a thermo well centered in the fermenter. When the wort temp is above the set temp, it turns on the night light which tricks the AC to begin cooling. The problem comes if the AC ices up trying to get to the low Inkbird 1 set temperature. Inkbird 2 takes care of that. The probe is placed in the AC coil. The set temp is above freezing (I use 38F). At 38F, Inkbird 1 turns off the power to Inkbird 1. This turns off the night light which in short order tells the AC that set temp has been achieved. The compressor shuts off while the fan continues to run and warms the coils with room temp air. The AC has built in short cycle prevention which can be extended with Inkbird 2 setting if needed when the coil temp rises above 38F. At that point, Inkbird 2 turns on Inkbird 1 and the cycle repeats. You'll have to play around with Inkbird and AC temp settings and the distance between the AC probe to balance the process. I also use a small space heater if needed to raise the wort temp toward the end of fermentation (Inkbird 2 is not used). The heater wattage should not exceed the inkbird rating. Most small space heaters have a somewhat accurate thermostat. But, it should be set to somewhere near the Inkbird setting as a fail safe. On the cooling process this is not an issue since the controlled device is only a 7 watt light bulb.
I use the closet to store grain and other brewing gear which provides thermal mass so temp fluctuations are gradual when measuring wort temp. I use a small fan to constantly circulate the air
There is a point where this set up becomes impractical for a walk in closet. If you are doing lagers you'll likely need a real refrigerator with external temp control with the probe in a thermowell in the fermenter.
Too complex? A Coolbot will do the same thing but it will cost you $375. Inkbirds are regularly on sale on Amazon around $20.