Distributive laws and factorization

Factorization Systems Induced by Weak Distributive Laws

“…This construction extends that in [10]. If E → C ← M is a bilinear factorization system, then applying the notations in Section 3.1, for any morphism in C 2 as in the first diagram in Eq.…”

“…These isomorphisms amount to a biequivalence between the bicategory of spans and the bicategory of profunctors between discrete categories-the latter called SetMat e.g. in [3] and [10].…”

“…2.7. By the axioms of a demimonad in Section 2.2, the identity morphism of any object a is equal to η(a) = (1 a 1 a , 1 a 1 a ) and the composition law is Since we are dealing simultaneously with three categories E, M and M E, the following arrow notation (which we learnt from [10]) will turn out to be handy. For morphisms in E, we use arrows of the type , for morphisms in M we draw .…”

“…It was proved by Rosebrugh and Wood in [10] that orthogonal factorization systems arising from strict ones are precisely those corresponding to strict algebras for the 2-monad (−) 2 on Cat via the correspondence in [8]. Furthermore, in [10] also an equivalence between strict factorization systems and distributive laws (in the sense of [1] or rather of [11]) in the bicategory of spans (possessing an equivalent realization SetMat; cf. Section 2.3) was established.…”

Categorical Quantum Models and Logics

“…11 "Factorisation" can be taken more literally by viewing M and E as subcategories of C and saying C = M • E [187]. In any factorisation system, each of the classes E and M determines the other via 'orthogonality' [33, Proposition 5.5.3].…”