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Feb 13 at 3:13 comment added kiran I was thinking: filter X as a colimit of finite spectra X_n such that the first stage is F. Then all other stages have their identity map factoring through F so theyre summands of F, so X is also.
Feb 12 at 17:33 comment added Tim Campion @kiran I like that argument -- if you see how to complete it, I'd be interested!
Feb 12 at 15:52 comment added kiran You could also argue via the fiber, which would be a finite F-->X through which all maps finite-->X factor. I think that forces X to be finite already?
Feb 11 at 4:55 vote accept Tim Campion
Feb 6 at 23:59 history answered Tim Campion CC BY-SA 4.0