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At long last I created ternary factorization system. Also stubs for k-ary factorization system, strict factorization system, and surjective geometric morphism.
I put in my definitions at k-ary factorisation system.
I’ve added to ternary factorization system the example of a span of categories that is both a left and a right fibration, and the condition that makes it a two-sided fibration. There should be a simple common generalization of these and the ambifibrations of the second-last example.
I think the middle arrow in the three-way factorization has something to do with the unique transformation that makes the right-fibration structure a colax morphism of $L$-algebras, where $L$ is the monad for left fibrations, so that requiring the arrow to be invertible would (via the usual distributive-law business) be the same as requiring the left- and right-fibration structures to underlie an $M$-algebra, $M$ being the two-sided-fibration monad. But it’s fiddly – has anyone seen this explicitly worked out anywhere?
Nice, thanks! Actually, I think that can be regarded as a special case of the ambifibrations example: regard E as over $A\times B$, where $A\times B$ has the factorization system (E,M) with E being morphisms of the form (1,g) and M being morphisms of the form (f,1). (That factorization system has the amusing property that (M,E) is also a factorization system!)
What you say about distributive laws sounds reasonable, but I haven’t seen it worked out either. If you work it out, I think it would be worth adding to two-sided fibration.
D’oh, the product category – I should have thought of that!
I think two-sided fibration could use some expanding and rearranging anyway, so I’ll commit myself in public to doing that just so I can’t put it off for too long.
I have added to k-ary factorization system Example-pointers to, first of all, ternary factorization system (that link was missing) and then to
which I’d think is the archetypical example.
I made more explicit the “straightforward exercise in orthogonality”. Hope it helps to visualize the underlying argument! (in a second moment I’ll turn the .png images into real-code diagrams)
Nice!
I have believed for 13 years that strict ternary factorization systems were equivalent to triple distributive laws, but today I realized that’s false, because $R_2L_1$ may not be closed under composition. So I removed this claim from the page and added some facts and an example from the Pultr-Tholen paper showing why it is false, and conjectured that this is the only obstacle.
Now I’m curious what the abstract distributive-law-like structure corresponding to a ternary factorization system is. It looks like we should have
Given these data, we can define a multiplication for the composite $A B C$ as
$A B C A B C \xrightarrow{A B \beta B C} A B A B C B C \xrightarrow{A \alpha B \gamma C} A A B B B C C \xrightarrow{\mu_A \mu_B^2 \mu_C} A B C$and I hope it should be a monad. Has anyone seen anything like this before?
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