In higher category theory in mathematics, injective and projective model structures are special model structures on functor categories into a model category. Both model structures "do not have" to exist, but there are conditions guaranteeing their existence. An important application is for the study of limits and colimits, which are functors from a functor category and can therefore be made into Quillen adjunctions.
Definition.
Let formula_1 be a small category and formula_2 be a model category. For two functors formula_3, a natural transformation formula_4 is composed of morphisms formula_5 in formula_6 for all objects formula_7 in formula_8. For those it hence be studied if they are fibrations, cofibrations and weak equivalences, which might lead to a model structure on the functor category formula_9.
For a model structure, the injective trivial cofibrations also have to have the right lifting property with respect to all injective fibrations and the projective trivial fibrations also have to have the left lifting property with respect to all projective cofibrations. Since both doesn't have to be the case, the injective and projective model structure doesn't have to exist.
The functor category formula_9 with the initial and projective model structure is denoted formula_11 and formula_12 respectively.
Quillen adjunctions.
Let formula_2 be a combinatorical model category. Let formula_17 be a functor between small categories, then there is a functor formula_18 by precomposition. Since formula_2 has all small limits and small colimits, this functor has a left adjoint formula_20 with formula_21 known as left Kan extension as well as a right adjoint formula_22 with formula_23 known as right Kan extension. While the former adjunction is a Quillen adjunction between the projective model structures, the latter is a Quillen adjunctions between the injective model structures.