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Factorization systems and double categories
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We show that factorization systems, both strict and orthogonal, can be equivalently described as double categories satisfying certain properties. This provides conceptual reasons for why the category of sets and partial maps or the category of small categories and cofunctors admit orthogonal factorization systems. The theory also gives an explicit description of various lax morphism classifiers and explains why they admit strict factorization systems.
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On orthogonal factorization systems and double categories
In infinity-category theory, factorization systems embed fully faithfully into double infinity-categories, with an unstraightening theorem and a complete Z/2Z automorphism group for adequate systems.
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