Abstract
Given any morphism, we construct extensions of the original category in which this morphism admits certain factorizations, in particular a (retraction, section)-factorization. To this end, we solve the word problem for a certain type of systems of generators and relations for categories. This also enables us to prove preservation properties for the said extensions, e.g. preservation of a pair of diagonalizing classes of epimorphisms and monomorphisms.
Iterating such extension processes, we obtain factorizable extensions of categories; in particular, we construct a free proper factorization structure on a given category, which leads to a characterization of preimages of proper factorization structures under full embeddings. As a further application, we characterize an absoluteness property regarding factorizations of functorial images of a morphism.
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Schröder, L., Herrlich, H. Free Factorizations. Applied Categorical Structures 9, 571–593 (2001). https://doi.org/10.1023/A:1012586819160
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DOI: https://doi.org/10.1023/A:1012586819160