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 an 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.
Dually, there is an orthogonal factorisation system on for which is the class of initial functors and is the class of discrete opfibrations.
Note there is a mistake in the proof of the main theorem of the paper above, as noted on page 74 of:
Max Kelly, Basic concepts of enriched category theory, London Math. Soc. Lec. Note Series 64, Cambridge Univ. Press (1982), Reprints in Theory and Applications of Categories 10 (2005) 1-136 [ISBN:9780521287029, tac:tr10, pdf]
Fosco Loregian, Emily Riehl, Categorical Notions of Fibration, Expositiones Mathematicae 38, 2020. (arXiv:1806.06129, doi:10.1016/j.exmath.2019.02.004)
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, Kan Extensions are Partial Colimits. Applied Categorical Structures 30, 685–753 (2022). (arXiv:2101.04531. doi:10.1007/s10485-021-09671-9)
Internal comprehensive factorisations (and torsors) are considered in:
Last revised on January 10, 2024 at 16:27:39. See the history of this page for a list of all contributions to it.