The proper term for that is QHD or 1440p. But chances are you'll end up worse off than that, especially if you want "2K" resolution, by which I assume you mean 2560x1440, even though calling that "2K" is a misnomer. Best case you wasted money on technology you can't properly utilize. That said, I really wouldn't recommend using a 4K 60 Hz display with a system that can't drive it properly. If you want to try to achieve that, try using SwitchResX, which gives you a lot more control over display setups in macOS.
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But it doesn't mean that the system is actually sending a 1080p signal to the display. Those options are there primarily for Retina displays so that you can choose where you want to land on the physical size vs. I suspect what's happening when you run 1080p is that macOS is still sending a 4K signal across the wire and only rendering the desktop at 1080p size. But a 2560x1440 signal does not map cleanly onto a 4K pixel According to the Tech Specs page for that system here, the MBP 13" Late 2013 only supports up to 2560x1600 at 60 Hz, which suggests that it only supports DisplayPort 1.1 and therefore if you want 4K, it would only run that resolution at 30 Hz. Using 1080p if you can get SwitchResX to generate that signal will be less of a problem because at least in that case, each pixel in a 1080p source signal maps perfectly onto a 2x2 pixel grid of a 4K display, so the scaling artifacts won't be nearly as bad - though you may still see some issues due to artifacts from subpixel rendering, which is used to sharpen text. So you'll have essentially have spent more money to get a worse result. The reason is that LCD displays don't look good when driven below their native resolution, and this is especially noticeable with text. In that scenario, you'll actually end up with noticeably worse image quality than you would have gotten if you'd simply bought a 27" 2560x1440 display in the first place, such as the U2719D.
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This case fits MacBook Pro 13 inch Model: A2159 Touch Bar, A1989 Touch Bar, A1706 Touch Bar, A1708 Non-Touch Bar Models released in 2020, 2019, 2018, 2017, and 2016.According to the Tech Specs page for that system here, the MBP 13" Late 2013 only supports up to 2560x1600 at 60 Hz, which suggests that it only supports DisplayPort 1.1 and therefore if you want 4K, it would only run that resolution at 30 Hz. Note: Buy this MacBook Pro clear case after checking the compatibility, as it is not compatible with some MacBook Pro 13 models including MacBook Pro 13 with CD-Rom.
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