Is it possible to connect a 2014 MacMini to a Thunderbolt 2 display?

Note that there are two old Apple displays that people often confuse with each other.

• The Apple Thunderbolt Display (model A1407) has a hydra cable with two heads: Thunderbolt (Mini DisplayPort connector) and MagSafe. It has FireWire 800 and Gigabit Ethernet hub ports.
• The LED Cinema Display (27-inch) (model A1316) has a hydra cable with three heads: Mini DisplayPort, USB-A, and MagSafe. It has no FireWire 800 or Gigabit Ethernet hub ports.

The Thunderbolt 2 ports on your Late 2014 Mac mini can act either as Thunderbolt ports or as Mini DisplayPorts – depending on what's plugged in. So you could plug either display's video cable directly into your Mac. (Someone with a newer Mac would need to choose an adapter that matched the type of signal required by the display.)

If you have a Thunderbolt Display and the hydra cable is bad, you could try bypassing it by running a cable from its Thunderbolt daisy-chaining port to your Mac mini. In that case, you would want a Thunderbolt 1/2 cable, not "just" a Mini DisplayPort cable.

If you have a LED Cinema Display, you will need to plug in the USB head of the hydra cable if you want to be able to use the USB 2 hub ports on the monitor – or, more importantly, to control the monitor's brightness.

Yes, that is possible.

The 2014 Mac mini supports HDMI AND these video outputs via the Thunderbolt ports:

• Thunderbolt digital video output
  • Native Mini DisplayPort output
  • DVI output using Mini DisplayPort to DVI Adapter (sold separately)
  • VGA output using Mini DisplayPort to VGA Adapter (sold separately)
  • Dual-link DVI output using Mini DisplayPort to Dual-Link DVI Adapter (sold separately)

The full specs for the 2014 Mac mini are here: Mac mini (Late 2014) - Technical Specifications - Apple Support

The 2014 Mac Mini has two Thunderbolt 2/Mini DisplayPort ports and an HDMI port for connecting external displays.

see > Mac mini (Late 2014) - Technical Specifications - Apple Support

Please identify the exact display that you want to connect to the 2014 Mac mini.

If it is an Apple Display..? see > Find the serial number of your Apple display - Apple Support

If it is aftermarket HP, Samsung, LG or other brand.? Please identify the exact brand and model number.