- Weak equivalence
In

mathematics , a**weak equivalence**is a notion fromhomotopy theory which in some sense identifies objects that have the same basic "shape". This notion is formalized in theaxiom atic definition of aclosed model category .**Formal definition**A closed model category by definition contains a class of

morphism s called**weak equivalences**, and these morphisms becomeisomorphism s upon passing to the associatedhomotopy category . In particular, if the weak equivalences of two model categories containing the same objects and morphisms are defined in the same way, the resulting homotopy categories will be the same, regardless of the definitions offibration s andcofibration s in the respective categories.Different model categories define weak equivalences differently. For example, in the category of (bounded)

chain complex es, one might define a model structure where the weak equivalences are those morphisms:$p:\; A\; ightarrow\; B,$

where

:$p\_n:\; H\_n(A)\; ightarrow\; H\_n(B).,$

are isomorphisms for all "n" ≥ 0. However, this is not the only possible choice of weak equivalences for this category: one could also define the class of weak equivalences to be those maps that are

chain homotopy equivalences of complexes.For another example, the category of

CW complex es can be given the structure of a model category where the weak equivalences are theweak homotopy equivalence s "i.e." those morphisms "X" → "Y" that induce isomorphisms inhomotopy group s:$pi\_n(X,\; x)\; cong\; pi\_n(Y,\; y),$

for all choices of basepoints "x" ∈ "X", "y" ∈ "Y", and all "n" ≥ 0.

A

fibration which is also a weak equivalence is also known as a**trivial**(or**acyclic**)**fibration**. Acofibration which is also a weak equivalence is also known as a**trivial**(or**acyclic**)**cofibration**.**Formal Languages**In

formal languages , weak equivalence of two grammars means they generate the same class of strings. If the derivation trees of the languages are also the same, the two grammars are called strongly equivalent.

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