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In mathematics, a weak equivalence is a notion from homotopy theory that in some sense identifies objects that have the same "shape". This notion is formalized in the axiomatic definition of a model category. A model category is a category with classes of morphisms called weak equivalences, fibrations, and cofibrations, satisfying several axioms.
The equivalence principle is the hypothesis that the observed equivalence of gravitational and inertial mass is a consequence of nature. The weak form, known for centuries, relates to masses of any composition in free fall taking the same trajectories and landing at identical times.
Weak equivalence principle This page was last edited on 27 May 2024, at 02:43 (UTC). Text is available under the Creative Commons Attribution ...
For example, the category of (reasonable) topological spaces has a structure of a model category where a weak equivalence is a weak homotopy equivalence, a cofibration a certain retract and a fibration a Serre fibration. [20] Another example is the category of non-negatively graded chain complexes over a fixed base ring. [21
An equivalent definition is saying that a surjective map p: E → B is a quasifibration if the inclusion of the fibre p −1 (b) into the homotopy fibre F b of p over b is a weak equivalence for all b ∈ B. To see this, recall that F b is the fibre of q under b where q: E p → B is the usual path fibration construction. Thus, one has
Typically, this map is not a weak equivalence. For example, the homotopy pushout encountered above always maps to the ordinary pushout. This map is not typically a weak equivalence, for example the join is not weakly equivalent to the pushout of , which is a point.
A mapping : between total spaces of two fibrations : and : with the same base space is a fibration homomorphism if the following diagram commutes: . The mapping is a fiber homotopy equivalence if in addition a fibration homomorphism : exists, such that the mappings and are homotopic, by fibration homomorphisms, to the identities and . [2]: 405-406
W (weak) and S (strong) are usually not swapped [15] but have been swapped in the past by some tools. [16] W and S denote "weak" and "strong", respectively, and indicate a number of the hydrogen bonds that a nucleotide uses to pair with its complementing partner. A partner uses the same number of the bonds to make a complementing pair. [17]