Any functor Q equipped with a natural weak equivalence to or from the identity functor is homotopical by the 2-of-3 property: if any two of gf, g and f are in W then so is the third
WHY?
Homotopical functor is a functor preserving weak equivalences.
2026-03-27 21:20:28.1774646428
Any functor F equipped with a natural weak equivalence to or from the identity functor is homotopical by the 2-of-3 property.
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Easy. If $f:x\to y$ is a weak equivalence and $\eta_y: y\to Qy$ is a weak equivalence then $\eta_y \circ f$ is a weak equivalence. So $Qf: Qx\to Qy$ is a weak equivalence.