In higher category theory in mathematics, co- and contravariant model structures are special model structures on slice categories of the category of simplicial sets. On them, postcomposition and pullbacks (due to its application in algebraic geometry also known as base change) induce adjoint functors, which with the model structures can even become Quillen adjunctions.
Definition.
Let formula_1 be a simplicial set, then there is a slice category formula_2. With the choice of a model structure on formula_3, for example the Joyal or Kan–Quillen model structure, it induces a model structure on formula_2.
The slice category formula_2 with the co- and contravariant model structure is denoted formula_10 and formula_11 respectively.
Homotopy categories.
For any model category, there is a homotopy category associated to it by formally inverting all weak equivalences. In homotopical algebra, the co- and contravariant model structures of the Kan–Quillen model structure with weak homotopy equivalences as weak equivalences are of particular interest. For a simplicial set formula_1, let:
 formula_13
 formula_14
Since formula_15 is the terminal object of formula_3, one in particular has:
 formula_17
Since the functor of the opposite simplicial set is a Quillen equivalence between the co- and contravariant model structure, one has:
 formula_18
Quillen adjunctions.
Let formula_19 be a morphism of simplicial sets, then there is a functor formula_20 by postcomposition and a functor formula_21 by pullback with an adjunction formula_22. Since the latter commutes with all colimits, it also has a right adjoint formula_23 with formula_24. For the contravariant model structure (of the Kan–Quillen model structure), the former adjunction is always a Quillen adjunction, while the latter is for formula_25 proper. This results in derived adjunctions:
 formula_26
 formula_27