## Natural weak factorization systems in model structures

### Emily Riehl – 2 December 2009

Factorization systems abound in mathematics: orthogonal factorization systems (ofs) have nice categorical properties but are too strict to describe many homotopical situations, while weak factorization systems (wfs) do figure prominently, especially in the definition of a model structure on a category, but aren't as well behaved. Natural weak factorization systems (nwfs) "algebraicize" the notion of wfs and sit somewhere in between the two, and thanks to Richard Garner's small object argument, it's possible to construct a plethora of examples.
My work explores the consequences of incorporating nwfs into model categories: a natural model structure will consist of a pair of nwfs together with a comparison map between them such that the underlying classes of maps form a model structure in the ordinary sense. After giving the basic definitions and exploring some of the properties of nwfs, I'll devote the remainder of the talk to persuading my audience that this was a good idea. The most significant results build toward an algebraization of the notion of a Quillen adjunction, each of which include five adjunctions of nwfs. In addition to supplying the definition, I am able to show that this extra algebraic structure can be found for a large family of known examples of Quillen adjunctions.

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