Skew structures in 2-category theory and homotopy theory

TL;DR

This paper explores skew monoidal structures in 2-category theory and homotopy theory, demonstrating how they induce genuine monoidal closed structures on homotopy categories, with applications to bicategories and 2-monads.

Contribution

It introduces a framework for skew monoidal structures in model categories that induce monoidal closed structures on homotopy categories, extending classical theorems to a homotopical context.

Findings

Skew structures can be used to derive genuine monoidal closed structures on homotopy categories.

Examples include permutative categories and bicategories.

Application to monoidal bicategories from pseudo-commutative 2-monads.

Abstract

We study Quillen model categories equipped with a monoidal skew closed structure that descends to a genuine monoidal closed structure on the homotopy category. Our examples are 2-categorical and include permutative categories and bicategories. Using the skew framework, we adapt Eilenberg and Kelly's theorem relating monoidal and closed structure to the homotopical setting. This is applied to the construction of monoidal bicategories arising from the pseudo-commutative 2-monads of Hyland and Power.