Variants on model structure axioms (taken from Sevilla lectures): Baues, Brown, Cisinski, Thomason. See also Waldhausen category.
http://nlab.mathforge.org/nlab/show/Thomason+model+structure
http://mathoverflow.net/questions/29635/what-determines-a-model-structure
arXiv:1102.2512 Partial model categories and their simplicial nerves from arXiv Front: math.CT by C. Barwick, D. M. Kan In this note we consider partial model categories, by which we mean relative categories that satisfy a weakened version of the model category axioms involving only the weak equivalences. More precisely, a partial model category will be a relative category that has the two out of six property and admits a 3-arrow calculus
We then show that Charles Rezk’s result that the simplicial space obtained from a simplicial model category by taking a Reedy fibrant replacement of its simplicial nerve is a complete Segal space also holds for these partial model categories
We also note that conversely every complete Segal space is Reedy equivalent to the simplicial nerve of a partial model category and in fact of a homotopically full subcategory of a category of diagrams of simplicial sets.
nLab page on Model category axioms