Let be the class of final functors and be the class of discrete fibrations. Then is an orthogonal factorization system of Cat, called the comprehensive factorization system.
Let be a functor. Define as the left Kan extension of the constant presheaf at the singleton along . Explicitly, is the set of connected components of . Let , so an object of is an ordered pair where denotes the connected component of . Then it is not hard to verify that mapping is final, the canonical is a discrete fibration, and .
Now we show that and are replete subcategories of . Clearly they include all isomorphisms.
If functors and are final, then we show that is final. For , there is a element of , and thence an element of , so we obtain an element of . Now we must show that any two elements are connected. Since is final, elements and of are connected. It suffices to consider the case of a zig-zag of length one: a morphism such that
By finality of , the elements and of are connected. A zig-zag path between them, by precomposition with , becomes a zig-zag path between and . So is final.
The proof that discrete fibrations form a subcategory is omitted.
Now we must show that the lifting problem
has a unique solution when and .
We prove uniqueness first. For , let . Then must be the unique lifting of , and the domain of this lifting, proving uniqueness of on objects. For in , must be the unique lifting of , so is unique (if it exists).
Now we must show that this is well-defined, functorial, and a solution to the lifting problem. If is another element of , then WLOG let such that
Lifting this diagram, we see that and must lift to morphisms with identical domain, so is well-defined on objects.
For in , let , and by the diagram
we see that and must lift to morphisms with identical domains, so has domain .
Functoriality now follows easily from uniqueness of lifting for a discrete fibration, and it is not hard to show that is a solution to the lifting problem.
Ross Street, R. F. C. Walters, The Comprehensive Factorization of Functor, Bull. Amer. Math. Soc. 79(5): 936-941 (bams:1183534973)
Fosco Loregian, Emily Riehl, Categorical Notions of Fibration (arXiv:1806.06129)
Clemens Berger, Ralph M. Kaufmann, Comprehensive Factorization Systems, Tbilisi Math. J. 10 (2017), 255-277 (doi:10.1515/tmj-2017-0112)
Paolo Perrone, Walter Tholen, Kan extensions are partial colimits, Applied Categorical Structures, 2022. (arXiv:2101.04531)
Internal comprehensive factorisations (and torsors) are considered in:
Last revised on March 16, 2023 at 18:17:28. See the history of this page for a list of all contributions to it.