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1. • CommentRowNumber1.
   • CommentAuthorMike Shulman
   • CommentTimeOct 28th 2010
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexAt long last I created [[ternary factorization system]]. Also stubs for [[k-ary factorization system]], [[strict factorization system]], and [[surjective geometric morphism]].

   At long last I created ternary factorization system. Also stubs for k-ary factorization system, strict factorization system, and surjective geometric morphism.

2. • CommentRowNumber2.
   • CommentAuthorTobyBartels
   • CommentTimeNov 1st 2010
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   Author: TobyBartels
   Format: MarkdownItexI put in my definitions at [[k-ary factorisation system]].

   I put in my definitions at k-ary factorisation system.

3. • CommentRowNumber3.
   • CommentAuthorFinnLawler
   • CommentTimeMar 23rd 2011
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   Author: FinnLawler
   Format: MarkdownItexI'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?

   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?

4. • CommentRowNumber4.
   • CommentAuthorMike Shulman
   • CommentTimeMar 23rd 2011
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   Author: Mike Shulman
   Format: MarkdownItexNice, 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]].

   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.

5. • CommentRowNumber5.
   • CommentAuthorFinnLawler
   • CommentTimeMar 24th 2011
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   Author: FinnLawler
   Format: MarkdownItexD'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.

   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.

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeNov 29th 2012
   • (edited Nov 29th 2012)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have added to _[[k-ary factorization system]]_ Example-pointers to, first of all, _[[ternary factorization system]]_ (that link was missing) and then to * [[Postnikov system]], [[n-image]]-factorization which I'd think is the archetypical example.

   I have added to k-ary factorization system Example-pointers to, first of all, ternary factorization system (that link was missing) and then to

   • Postnikov system, n-image-factorization

   which I’d think is the archetypical example.

7. • CommentRowNumber7.
   • CommentAuthorFosco
   • CommentTimeJan 19th 2014
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   Author: Fosco
   Format: MarkdownItexI 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)

   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)

8. • CommentRowNumber8.
   • CommentAuthorMike Shulman
   • CommentTimeFeb 22nd 2019
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexUpdated the broken codecogs pictures to tikzcd. <a href="https://ncatlab.org/nlab/revision/diff/ternary+factorization+system/7">diff</a>, <a href="https://ncatlab.org/nlab/revision/ternary+factorization+system/7">v7</a>, <a href="https://ncatlab.org/nlab/show/ternary+factorization+system">current</a>

   Updated the broken codecogs pictures to tikzcd.

   diff, v7, current

9. • CommentRowNumber9.
   • CommentAuthorRichard Williamson
   • CommentTimeFeb 22nd 2019
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   Author: Richard Williamson
   Format: MarkdownItexNice!

   Nice!

10. • CommentRowNumber10.
    • CommentAuthorMike Shulman
    • CommentTimeJul 6th 2023
    • PermaLink
    Author: Mike Shulman
    Format: MarkdownItexI 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. <a href="https://ncatlab.org/nlab/revision/diff/ternary+factorization+system/8">diff</a>, <a href="https://ncatlab.org/nlab/revision/ternary+factorization+system/8">v8</a>, <a href="https://ncatlab.org/nlab/show/ternary+factorization+system">current</a>

    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.

    diff, v8, current

11. • CommentRowNumber11.
    • CommentAuthorMike Shulman
    • CommentTimeJul 6th 2023
    • PermaLink
    Author: Mike Shulman
    Format: MarkdownItexNow I'm curious what the abstract distributive-law-like structure corresponding to a ternary factorization system is. It looks like we should have * Three monads $A,B,C$ * Distributive laws $\alpha : B A \to A B$ and $\gamma : C B \to B C$ * A morphism $\beta : C A \to A B C$ * Two unit laws for $\beta$, e.g. the composite $C \xrightarrow{C \eta_A} C A \xrightarrow{\beta} A B C$ is equal to $C \xrightarrow{\eta_A \eta_B C} A B C$. * Two composition laws relating $\beta$ to $\alpha$ and $\gamma$, e.g. the composite $C A A \xrightarrow{\beta A} A B C A \xrightarrow{A B \beta} A B A B C \xrightarrow{A \alpha B C} A A B B C \xrightarrow{\mu_A \mu_B C} A B C$ is equal to the composite $C A A \xrightarrow{C \mu_A} C A \xrightarrow{\beta} A B C$ using the multiplication of $A$. 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?

    Now I’m curious what the abstract distributive-law-like structure corresponding to a ternary factorization system is. It looks like we should have

    • Three monads $A,B,C$
    • Distributive laws $\alpha : B A \to A B$ and $\gamma : C B \to B C$
    • A morphism $\beta : C A \to A B C$
    • Two unit laws for $\beta$, e.g. the composite $C \xrightarrow{C \eta_A} C A \xrightarrow{\beta} A B C$ is equal to $C \xrightarrow{\eta_A \eta_B C} A B C$.
    • Two composition laws relating $\beta$ to $\alpha$ and $\gamma$, e.g. the composite $C A A \xrightarrow{\beta A} A B C A \xrightarrow{A B \beta} A B A B C \xrightarrow{A \alpha B C} A A B B C \xrightarrow{\mu_A \mu_B C} A B C$ is equal to the composite $C A A \xrightarrow{C \mu_A} C A \xrightarrow{\beta} A B C$ using the multiplication of $A$.

    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?