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Concrete dualities. (English) Zbl 0761.18001

Category theory at work, Proc. Workshop, Bremen/Ger. 1991, Res. Expo. Math. 18, 111-136 (1991).
[For the entire collection see Zbl 0732.00006.]
Let \(({\mathbf A},U)\) and \(({\mathbf B},V)\) be a pair of concrete categories with representable faithful underlying functors \(U\simeq\operatorname{Hom}_{{\mathbf A}}(A_ 0,-): {\mathbf A}\to\mathbf{Set}\) and \(V\simeq\operatorname{Hom}_{{\mathbf B}}(B_ 0,-): {\mathbf B}\to\mathbf{Set}\), respectively. The paper gives sufficient conditions under which there exists a pair of contravariant functors \(T: {\mathbf A}\to{\mathbf B}\) and \(S: {\mathbf B}\to{\mathbf A}\), and a pair of natural morphisms \(\eta: 1_{{\mathbf B}}\to TS\) and \(\varepsilon: 1_{{\mathbf A}}\to ST\) such that \(T\varepsilon\circ\eta T=1_ T\) and \(S\eta\circ\varepsilon S=1_ S\). This situation is called a concrete dual adjunction. In the case where \(\eta\) and \(\varepsilon\) are isomorphisms, it is called a concrete duality. Any concrete dual adjunction induces a maximal concrete duality between some full subcategories. The generators \(A_ 0\) in \({\mathbf A}\) and \(B_ 0\) in \({\mathbf B}\) look like two faces of a unique object sitting in both the categories \({\mathbf A}\) and \({\mathbf B}\), and called a schizophrenic object, of particular importance. Many examples illustrate the results.

MSC:

18A40 Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.)
18A23 Natural morphisms, dinatural morphisms

Citations:

Zbl 0732.00006
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