Enriched Orthogonality and Equivalences
نویسندگان
چکیده
In this paper, we consider an enriched orthogonality for classes of spaces, with respect to groupoids, simplicial sets and spaces themselves. This point of view allows one to characterize homotopy equivalences, shape and strong shape equivalences. We show that there exists a class of spaces, properly containing ANR-spaces, for which shape and strong shape equivalences coincide. For such a class of spaces homotopy orthogonality implies enriched orthogonality.
منابع مشابه
Orthogonality, Saturation and Shape
The class of shape equivalences for a pair (C,K) of categories is the orthogonal of K, that is Σ = K. Then Σ is internally saturated (Σ = Σ). On the other hand, every internally saturated class of morphisms Σ ⊂ Mor(C), is the class of shape equivalences for some pair (C,K). Moreover, every class of shape equivalences Σ enjoys a calculus of left fractions and such a fact allows one to use techni...
متن کاملThe Euler Characteristic of an Enriched Category
We develop the homotopy theory of Euler characteristic (magnitude) of a category enriched in a monoidal model category. If a monoidal model category V is equipped with an Euler characteristic that is compatible with weak equivalences and fibrations in V, then our Euler characteristic of V-enriched categories is also compatible with weak equivalences and fibrations in the canonical model structu...
متن کاملOn Approximate Birkhoff-James Orthogonality and Approximate $ast$-orthogonality in $C^ast$-algebras
We offer a new definition of $varepsilon$-orthogonality in normed spaces, and we try to explain some properties of which. Also we introduce some types of $varepsilon$-orthogonality in an arbitrary $C^ast$-algebra $mathcal{A}$, as a Hilbert $C^ast$-module over itself, and investigate some of its properties in such spaces. We state some results relating range-kernel orthogonality in $C^*$-algebras.
متن کاملEquivalences in Bicategories
In this paper, we establish some connections between the concept of an equivalence of categories and that of an equivalence in a bicategory. Its main result builds upon the observation that two closely related concepts, which could both play the role of an equivalence in a bicategory, turn out not to coincide. Two counterexamples are provided for that goal, and detailed proofs are given. In par...
متن کاملSemantic Orthogonality of Type Disciplines
We consider a version of PCF, and prove, using both syntactic and semantic means, that the op-erational equivalences of the base language are preserved when the language is extended with sum andproduct types, with polymorphic types, and with recursive types. These theorems show that the additionsto the type systems are orthogonal to the original language.
متن کامل