by Lutz Schröder and Horst Herrlich
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.
Reference:
Lutz Schröder and Horst Herrlich: Free factorizations, In Applied Categorical Structures, 9, pp. 571–593, 2001. [preprint]
Bibtex Entry:
@Article{SchroderHerrlich01b,
author = {Lutz Schr{\"o}der and Horst Herrlich},
title = {Free factorizations},
year = {2001},
journal = {Applied Categorical Structures},
volume = {9},
pages = {571--593},
keywords = {factorization retraction graph category word problem},
comment = {<a href="http://www8.informatik.uni-erlangen.de/~schroeder/papers/APCS3.ps">[preprint]</a>},
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.
},
}