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 2adjunctions, and extend the CassidyHébertKelly 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 LópezFranco.)[1] F. Cagliari, M.M. Clementino, S. Mantovani, Fibrewise injectivity and KockZoberlein monads. J. Pure Appl. Algebra 216 (2012) 24112424. [2] C. Cassidy, M. Hébert, G.M. Kelly, Reflective subcategories, localizations and factorization systems. J. Austral. Math. Soc. Ser. A 38 (1985) 287329. [3] R. Garner, Understanding the small object argument. Appl. Categ. Structures 17 (2009) 247285. [4] M. Grandis, W. Tholen, Natural weak factorization systems. Arch. Math. (Brno) 42 (2006) 397408.
