Monoidal algebraic model structures

TL;DR

This paper introduces monoidal algebraic model structures, emphasizing algebraic Quillen two-variable adjunctions and cellularity, providing criteria for their characterization and showing their presence in familiar monoidal model categories.

Contribution

It defines monoidal algebraic model structures, develops the theory of algebraic Quillen two-variable adjunctions, and establishes criteria linking them to cellular structures on pushout-products.

Findings

Algebraic Quillen two-variable adjunctions correspond to cell structures on pushout-products.

Familiar monoidal model structures on categories and simplicial sets admit this algebraic structure.

Cellularity plays a key role in characterizing algebraic structures in model categories.

Abstract

Extending previous work, we define monoidal algebraic model structures and give examples. The main structural component is what we call an algebraic Quillen two-variable adjunction; the principal technical work is to develop the category theory necessary to characterize them. Our investigations reveal an important role played by "cellularity" - loosely, the property of a cofibration being a relative cell complex, not simply a retract of such - which we particularly emphasize. A main result is a simple criterion which shows that algebraic Quillen two-variable adjunctions correspond precisely to cell structures on the pushout-products of generating (trivial) cofibrations. As a corollary, we discover that the familiar monoidal model structures on categories and simplicial sets admit this extra algebraic structure.