In [4] Grandis and Tholen introduced natural weak factorisation systems (also called algebraic weak factorisation systems, see [3]) as pairs (L,R), where L is a comonad and R is a monad subject to suitable conditions, and showed that orthogonal factorisation systems are the natural weak factorisation systems with L and R idempotent. In this talk we focus on those natural weak factorisation systems with lax idempotent L and R. Moreover, we introduce simple 2-adjunctions, and extend the Cassidy-Hébert-Kelly construction [2] of factorisation systems for simple adjunctions to this setting. Among our examples we recover the weak factorisation systems given by fibrewise filter monads in topology as studied in [1].
(Joint work with Ignacio López-Franco.)[1] F. Cagliari, M.M. Clementino, S. Mantovani, Fibrewise injectivity and Kock-Zoberlein monads. J. Pure Appl. Algebra 216 (2012) 2411--2424.
[2] C. Cassidy, M. Hébert, G.M. Kelly, Reflective subcategories, localizations and factorization systems. J. Austral. Math. Soc. Ser. A 38 (1985) 287--329.
[3] R. Garner, Understanding the small object argument. Appl. Categ. Structures 17 (2009) 247--285.
[4] M. Grandis, W. Tholen, Natural weak factorization systems. Arch. Math. (Brno) 42 (2006) 397--408.
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