In  Grandis and Tholen introduced natural weak factorisation systems (also called algebraic weak factorisation systems, see ) 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  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 .
(Joint work with Ignacio López-Franco.)
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