Wreaths in Span(Set)
Steve Lack and Ross Street introduced wreaths as a generalisation of distributive laws between monads in their paper The formal theory of Monads II. This paper presents a nice story about how wreaths arise from considering the free completion of a 2-category under Kleisli objects and provides some interesting examples of wreaths. In particular, it is shown that orthogonal factorisation systems on a category can be expressed as wreaths in \mathsf{Span}(\mathsf{Set}).
In this talk we’ll try to understand general wreaths in \mathsf{Span}(\mathsf{Set}) as expressing a weaker notion of factorisation system on a category. We’ll relate these factorisation structures to both familial functors and crossed double categories, and also present examples of where they naturally occur.