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Aug 13, 2022 at 20:24 comment added user420620 Thanks! While your references are very relevant, is there a reference explicitly defining homotopy in terms of endofunctors ? Perhaps formulating and proving properties of homotopy in terms of endofunctors.
Aug 13, 2022 at 18:45 comment added John Rognes Yes, using Waldhausen's formulation of the result.
Aug 13, 2022 at 15:40 comment added user420620 Is it enough to assume that $X$ connected ?
Aug 13, 2022 at 15:33 history answered John Rognes CC BY-SA 4.0