Homotopy Is Not Concrete

not null-homotopic such that [Z,X] [Z, f ] → [Z, Y ] is constant. For any X ′ ⊂ X with fewer than κ cells there exists X ′ → Z → X ′ = 1X′ , from which we may conclude that [X ′, X] [X ′, f ] → [X ′, Y ] is constant and in particular that f |X ′ is null-homotopic. Let H be the homotopy category obtained from T . Its objects are the objects of T , its maps are homotopy-classes of maps. The theorem says that H may not be faithfully embedded in the category of sets—or in the language of Kurosh— H is not concrete. There is no interpretation of the objects of H so that the maps may be interpreted as functions (in a functorial way, at least). H has always been the best example of an abstract category, historically and philosophically. Now we know that it was of necessity abstract, mathematically. The theorem says a bit more: H has a zero-object, that is, an object 0 such that for any X there is a unique 0 → X and a unique X → 0, and consequently for any X,Y a unique X → 0 → Y , the zero-map from X to Y . We shall shortly restrict our attention to zero-preserving functors between categories with zero. Instead of functors into the category of sets, S we’ll consider functors into the category of base-pointed sets S and only those functors that preserve zero. But first: Proposition: If C is a category with zero and T : C → S any functor, then there exists a zeropreserving functor T : C → S such that