In homotopy theory, a branch of mathematics, a **Quillen adjunction** between two closed model categories **C** and **D** is a special kind of adjunction between categories that induces an adjunction between the homotopy categories Ho(**C**) and Ho(**D**) via the total derived functor construction. Quillen adjunctions are named in honor of the mathematician Daniel Quillen.

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**Quillen Adjunction**- Properties

... It is a consequence of the axioms that a left (right)

**Quillen**functor preserves weak equivalences between cofibrant (fibrant) objects ... The total derived functor theorem of

**Quillen**says that the total left derived functor LF Ho(C) → Ho(D) is a left adjoint to the total right derived functor RG Ho(D) → Ho(C) ... This

**adjunction**(LF, RG) is called the derived

**adjunction**...

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