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Idea-section and one further reference at Thomason model structure.
I remember Mike once said on the blog somewhere that there might be some problem with Thomason's original claim that cofibrant objects in this structure are posets. I made a brief remark on this, but I can't find Mike's original comment.
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created cohomology of a category just for completeness in the course of Mike and my discussion on the blog
I can't tell you much more... all I remember is what you said (someone said there might be a problem with it). I don't even remember who it was. (-: So it could just be a malicious rumor. I certainly haven't read the proof myself.
I added the reference
The result on calculi of left fractions is especially interesting: itâ€™s not often that we get a privileged direction in category theory!
Of course, the asymmetry comes from $Ex$, not from category theory.
gave these two reference more proper publication data:
Lennart Meier, Viktoriya Ozornova, Fibrancy of partial model categories, Homology, Homotopy and Applications, Volume 17 (2015) Number 2 (arXiv:1408.2743, doi:10.4310/HHA.2015.v17.n2.a5)
Lennart Meier, Fibrancy of (Relative) Categories, talk at Young Topologists Meeting 2014 (slides pdf)
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