Photos won't accept my iPhoto library

My iPhoto library is located on an external hard drive I use for Time Machine backup.

That's exactly why you can't open the library. The library has a very complex ecosystem of links via various databases and the TM drive format prevents it from operating. You'll have to copy it to the Pictures folder of your boot drive or to another external HD that's formatted OS X Extended (journaled) with ownership set to be ignored.

Where is your "iPhoto Library Main" stored? Is it on an external volume? If yes, what is the file system format of this volume?

How much free storage is on the volume with the library?

And which system version do you have installed?

Please read the following user tip and restate your question/problem in detail: Writing an Effective Apple Support Communities Question - Apple Community  

Include;

What system and Photos versions are you running?

Where is the library located? If on an external HD how is that drive formatted and connected to your Mac?