2007/01/16 by Claudio Pisani, Pisani, Claudio
Chemistry · Materials Science · Mathematics · #18Axx #Category Theory (math.CT) #FOS: Mathematics #Mesoporous Materials and Catalysis #Silicone and Siloxane Chemistry #Supramolecular Chemistry and Complexes #math.CT #msc:18Axx
paper · pdf · doi:10.48550/arxiv.math/0701457
59 pages
arxiv created 2007/01/16 · openalex publication_date 2007/01/16 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Some basic features of the simultaneous inclusion of discrete fibrations and discrete opfibrations in categories over a base category X are considered. In particular, we illustrate the formulas (|P)x = ten(x/X,P) ; (P|)x = hom(X/x,P) which give the reflection |P and the coreflection P| of a category P over X in discrete fibrations. The explicit use of the "tensor functor" ten := \comp(- × -) : Cat/X × Cat/X → Set given by the components of products, allows a vast generalization of the corresponding analysis in the two-valued context. For any df A, the functor ten(A,-) : Cat/X → Set has a right adjoint ¬ A valued in dof's (and vice versa); such a complement operator, which in the two-valued case reduces to the classical complementation between lower and upper parts of a poset, turns out to be an effective tool in the set-valued context as well. Various applications of the formulas and of the accompanying conceptual frame are presented.