A Quillen equivalence is a Quillen adjunction between model categories that induces an equivalence of their homotopy categories.
For an adjunction 
,
being Quillen requires 
to preserve cofibrations and acyclic cofibrations. It is a
Quillen equivalence iff for every cofibrant 
and fibrant 
, a map 
is a weak equivalence exactly when its adjoint 
is a weak equivalence. The use
of cofibrant and fibrant replacements is essential in passing to the derived adjunction.
Such an equivalence compares homotopy theories
rather than requiring an equivalence of the original categories.
Quillen Equivalence
See also
Adjoint Functor, Model Category, Thomason Model StructureExplore with Wolfram|Alpha
References
Hovey, M. Model Categories. Providence, RI: Amer. Math. Soc., 1999. https://doi.org/10.1090/surv/063.Cite this as:
Weisstein, Eric W. "Quillen Equivalence." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/QuillenEquivalence.html