How to make LOW for sensor, where sensor have HIGH on ready state

I have a problem to emulate one sensor in my project with Arduino. Sensor is HIGH in ready state and LOW in work state (when I meansured is 15kOhm between output-GND in work state). When I removed sensor and connect resistor 10kOhm between GND and INPUT of mainboard( emulated OUTPUT sensor to mainboard) this works fine as with sensor. But this voltage on sensor is from mainboar - No from Arduino. I must to emulate this state of sensor with Arduino,( Arduino have GND together with sensor) This mean Arduino must make resistor cc 10kOhm between Pinout - GND, how I can make, please help.

Use a resistor of 10k for the LOW state and set Arduino pin as OUTPUT and LOW. What should the pin be in high state ?

Did you measure the output of your sensor in the "work" state with an ohmmeter?

A good example of where a schematic would help; I can't picture this at all.....