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We study Quillen model categories equipped with a monoidal skew closedstructure that descends to a genuine monoidal closed structure on the homotopycategory. Our examples are 2-categorical and include permutative categories andbicategories. Using the skew framework, we adapt Eilenberg and Kelly's theoremrelating monoidal and closed structure to the homotopical setting. This isapplied to the construction of monoidal bicategories arising from thepseudo-commutative 2-monads of Hyland and Power.

Skew structures in 2-category theory and homotopy theory