In higher category theory in mathematics, a proper model structure is a model structure in which additionally weak equivalences are preserved under pullback (fiber product) along fibrations, called "right proper", and pushouts (cofiber product) along cofibrations, called "left proper". It is helpful to construct weak equivalences and hence to find isomorphic objects in the homotopy theory of the model structure.
Definition.
For every model category, one has:
A model category is then called:
Properties.
For a model category formula_1 and a morphism formula_2 in it, there is a functor formula_3 by precomposition and a functor formula_4 by postcomposition. Furthermore, pushout defines a functor formula_5 and pullback defines a functor formula_6. One has: