Monoidal algebraic model structures

Details Monoidal algebraic model structures Monoidal algebraic model structures

2011-09-13

arxiv.org/abs/1109.2883v1

Mathematics

Category Theory

55 pages; a sequel to "Algebraic model structures" New York J. Math. 17 (2011) arXiv:0910.2733v3 [math.CT]

Scientific paper

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.

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