Monoidal algebraic model structures

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.

Cyclic multicategories, multivariable adjunctions and mates

“…In an algebraic model category structure, the weak factorisation systems involved have certain extra linked algebraic/coalgebraic structure; all cofibrantly generated model categories can be made algebraic in this sense. That research was one direct motivation for the work and results in this paper, which are essential to define the central notion studied in [24].…”

“…If we take every n i = 1 or k = 1 this says we can compose an n-variable adjunction with 1-variable adjunctions (pre-or post-) to get a new nvariable adjunction; this example is mentioned for composing 2-ary with 1-ary adjunctions in [24].…”

“…encoding coherence conditions) have "dual" forms. This sort of structure occurs in work of the third author in the context of monoidal and enriched algebraic model category structures [23,24]. In an algebraic model category structure, the weak factorisation systems involved have certain extra linked algebraic/coalgebraic structure; all cofibrantly generated model categories can be made algebraic in this sense.…”

“…The motivation for this work is the third author's work on algebraic monoidal model categories. In the theory of algebraic model categories [23] the double category framework for 1-variable adjunctions and mates plays a crucial role. For the monoidal version [24], multivariable adjunctions and mates are needed, not simply to describe the equivalent forms the definition of a monoidal algebraic model category might take but to state the correct definition at all.…”

“…In the theory of algebraic model categories [23] the double category framework for 1-variable adjunctions and mates plays a crucial role. For the monoidal version [24], multivariable adjunctions and mates are needed, not simply to describe the equivalent forms the definition of a monoidal algebraic model category might take but to state the correct definition at all. Examples that could now be made algebraic using the results of the present paper include the model structures arising from 2-category theory [18], in particular the monoidal model structure on 2-categories with the Gray tensor product [16,17].…”