We show that the homotopy category can be assigned to any category equipped with a weak factorization system. A classical example of this construction is the stable category of modules. We discuss a connection with the open map approach to bisimulations proposed by Joyal, Nielsen and Winskel.
A. Kurz and J. Rosický, “Weak Factorizations, Fractions and Homotopies,” Applied Categorical Structures, vol. 13, no. 2, pp. 141–160, Apr. 2005. DOI: 10.1007/s10485-004-6730-z