A categorical version of the Lefschets-Nöbeling-Pontryagin theorem on embedding compacta in Rn
✍ Scribed by Dmitri B. Shakhmatov
- Book ID
- 104295302
- Publisher
- Elsevier Science
- Year
- 1998
- Tongue
- English
- Weight
- 333 KB
- Volume
- 85
- Category
- Article
- ISSN
- 0166-8641
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✦ Synopsis
For a category K we use Ob(lC) to denote the class of all objects of K; if X,Y E Oh(K), then MorK (X, Y) is the set of all K-morphisms from X into Y. Let A and a be subcategories of the category of all topological spaces and their continuous maps. We say that a covariant functor F : A + f3 is an embedding functor if there exists a class {ix: X E Oh(A)} satisfying the following conditions: (i) ix : X -+ F(X) is a homeomorphic embedding for every X E Oh(A), and (ii) if X, Y E Oh(A) and f E MorK(X, Y), then F(f) o ix = iy o f. For a natural number n let C(n) denote the category of all n-dimensional compact metric spaces and their continuous maps. Let G(< m) be the category of all Hausdorff finite-dimensional topological groups and their continuous group homomorphisms.
We prove that there is no embedding covariant functor F : C( 1) + G(< oo), but there exists a covariant embedding functor F: C(0) + E(O), where G(O) is the category consisting of the single (zero-dimensional) compact metric group Zy and all its continuous group homomorphisms into itself, i.e., Ob(G(0)) = {Z;i} and Morg(o) (Z,W, i?!;) is the set of all continuous group homomorphisms from Zy into Zr. 0 1998 Elsevier Science B.V.