  We introduce a notion of "weak model category" which is a weakening of the
notion of Quillen model category, still sufficient to define a homotopy
category, Quillen adjunctions, Quillen equivalences and most of the usual
construction of categorical homotopy theory. Both left and right semi-model
categories are weak model categories, and the opposite of a weak model category
is again a weak model category. The main advantages of weak model categories is
that they are easier to construct than Quillen model categories. In particular
we give some simple criteria on two weak factorization systems for them to form
a weak model category. The theory is developed in a very weak constructive
framework and we use it to produce, completely constructively (even
predicatively), weak versions of various standard model categories, including
the Kan-Quillen model structure, the variant of the Joyal model structure on
marked simplicial sets, and the Verity model structure for weak complicial
sets. We also construct semi-simplicial versions of all these.
