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added in an Examples-section to stable factorization system the statement that in an adhesive category, in particular in a topos, the (epi, mono)-factorization is stable.
ah, do I need to say “co-adhesive”? I should fix that, but my bus is about to arrive…
So I fixed something here and there.
Eventually I really wanted to put more examples there. But not right now.
I missed something — why does an adhesive category even have (epi,mono) factorizations?
I had fixed it by #3.
Here is a question: how known is it that the $n$-connected/$n$-truncated factorization system in an $\infty$-topos is stable?
(For $n = -1$ this is in HTT, of course. I suppose generally it follows directly from the fact that truncation is an idempotent monad on $Type$? )
The examples section at stable factorization system still says “In an a topos, epimorphism are stable under pullback and hence the (epi, mono) factorization system in an adhesive category is stable”.
I think stability of the $n$-connected/$n$-truncated factorization system is HTT 6.5.1.16(6).
still says
Ah, sorry. Fixed now. (I should stop editing entries in a haste at a bus stop…)
I think stability of the n-connected/n-truncated factorization system is HTT 6.5.1.16(6).
Ah, great. I had missed that. Thanks!!
But still my question: in the type theory $n$-truncation is a reflection on $Type$. Does that not also directy imply the statement?
Yes, once you know also that dependent sums preserve n-truncated types.
Mike,
what would be the preferred way to cite your result that generalizes the equivalence “stable” = “comes from lex reflector” for reflective factorization systems as stated at stable factorization system - stable reflective factorization systems, but now generalized from categories to $\infty$-categories?
Your blog notes? Our QFT writeup? Should we have an $n$Lab entry on it?
In either case, I’d like to have a stable (though not necessarily reflective) way to cite this.
I suppose that would be good. Maybe there should be an nLab entry. But I don’t really have time to write out any details right now…
Okay, thanks. I have added a brief pointer to your HoTT posting to the entry.
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