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Linked to from https://ncatlab.org/nlab/show/orthogonal+factorization+system and https://ncatlab.org/nlab/show/final+functor
Todo: add more proofs of this result.
For some reason the xymatrixes were causing errors so I had to comment them out to submit. Here is an example error:
An error occurred when running pdflatex on the following diagram. \xymatrix@=5em{e \ar[r]^\gamma \ar[dr]_{\gamma’} & GFc \ar[d]^{Gf} \ & GFc’} The error was: Timed out
How can I fix this?
For the xymatrix
to display, they must not be enclosed by any other environment and the delimiters must be aligned to the far left.
(I was going to fix it, but, since last night, all attempts to submit tikzcd
or xymatrix
to the $n$Lab time out.)
According to Richard here, the server has apparently reached the point where it can’t handle the size of the $n$Lab anymore, and we need to wait for Richard to, thankfully, migrate us to a better server.
I think editing this page may be triggering the problems, as it is deeply nested in the rest of the nLab, so it may be best to hold off for a while.
where do the sources for the tikzcd diagrams on this page live?
The page source only has stuff like
<img src="/nlab/diagrams/show/20220421002920849955">
Answered here :-).
I corrected the title of the first referenced paper. Thanks RIchard for all the work.
Great! So far the edits seem to have gone through fine. There is no diff functionality yet, but if people wish to see what has changed they can of course do so locally, e.g.:
curl https://ncatlab.org/nlab/history/source/comprehensive%20factorization%20system/5 > /tmp/revision_5
curl https://ncatlab.org/nlab/history/source/comprehensive%20factorization%20system/6 > /tmp/revision_6
vimdiff /tmp/revision_5 /tmp/revision_6
Here, for those who are not familiar with it, curl is just a command-line way to fetch the content of a webpage (or execute other HTTP methods), one can equally well visit the page and download the content.
I think an object of $E$ should be an ordered pair $(d,[\alpha:d\to Fc])$, not the other way around (since final functors talk about the comma categories $d/F$). Could someone double-check?
It’s also unclear why “It suffices to consider the case of a zig-zag of length one”: the argument works by induction if every morphism in the zig-zag is between objects of the form $Fx$, but that need not be the case.
I guess the idea is that you can modify a zig-zag $GFc \leftrightarrow Gx \leftrightarrow Gy \leftrightarrow GFc'$ to $GFc \leftrightarrow GFx' \leftrightarrow GFy' \leftrightarrow GFc'$ using that each $x/F$ is inhabited, and then the argument applies to this modified zig-zag.
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