Holmstrom Homotopical category

See the book on such cats.

http://www.ncatlab.org/nlab/show/homotopical+category

See also the notions:

• Category of fibrant objects nLab, original source is Brown
• Waldhausen category
• Cofibration category nLab
• Model category (various versions, including Thomason’s)

Closed monoidal homotopical cat nLab

Barwick and Kan: Partial model category.

Enriched homotopical cat nLab

nLab page on Homotopical category