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Homotopy is not concrete. (English) Zbl 1057.18001

This paper is a reprint of a paper appearing in Steenrod’s Festschrift : “The Steenrod algebra and its applications”, Lect. Notes Math. 168, 25–34 (1970; Zbl 0212.55801)]. A few rewordings and all of the footnotes are new, as well as two addendas.
My favorite definition of a concrete category is a collection of mappings which contains all the identity maps of the domains and ranges and which is closed under composition of maps. A functor \(T\) satisfies the homomorphism property \(T(f \circ g) = T(f) \circ T(g)\). The notion of abstract category replaces the domains and ranges by “objects” and the maps are replaced by abstract morphisms which have an operation satisfying axioms which are the key properties of composition \(\circ\).
Now, historically, the homotopy category of CW complexes and homotopy classes of maps is the most important example of an abstract category. It arises from the concrete category of CW complexes and the homotopy equivalence classes of maps. The paper shows that the homotopy category is more fundamentally abstract by showing there is no faithful functor of the homotopy category into the category of sets and mappings. In other words, the following theorem is shown:
{Theorem.} Let \(\mathcal{T}\) be a category of base-pointed topological spaces which contains all finite-dimensional CW complexes. Let \(T: \mathcal T \rightarrow \mathcal S\) be any set-valued functor that is homotopy invariant. There exists \(f:X \rightarrow Y\) such that \(f\) is not null homotopic, but \(T(f)=T(\star)\) where \(\star\) denotes a null homotopic map.

MSC:

18B30 Categories of topological spaces and continuous mappings (MSC2010)
55P15 Classification of homotopy type

Citations:

Zbl 0212.55801
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