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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: it’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)
(double post)
Reverted the previous edit. I think there’s an error in Hirschhorn, Model Categories and Their Localizations.
In definition 18.1.8, the hypotheses are:
Let M be a simplicial model category and let C be a small category. If X is a C-diagram in M
and then proceeds to define holim(X) via the equalizer defining the end
$holim(X) = \int_{c \in C} X_c^{B(C \downarrow c)}$I’m not sure I believe that statement. E.g. if we take M=sSet, let $A = \Delta^0$ and let $B$ be the graph $0 \to 1 \to 2 \to 3 \to 0$. Let $X$ be the diagram of two parallel arrows $A \to B$, both sending the point to $0$.
$B$ is the 1-sphere and this equalizer should compute the loop space at $0$, so the homotopy limit should be homotopy equivalent to $\mathbb{Z}$.
However, this end reduces to computing the pullback of $A \to B \times B \leftarrow B^{\Lambda^2_2}$. In particular, it is a finite simplicial set, so it can’t have the right homotopy type.
Either I’ve made an error along the way, or it’s missing a fibrancy assumption on $X$.
As an aside, the homotopy limit page has a lot more to say about colimits than limits
Either I’ve made an error along the way, or it’s missing a fibrancy assumption on X.
The fibrancy assumption on X is present in the correct place, see, for example, Theorem 18.5.3 in Hirschhorn.
Hirschhorn’s homotopy limit functor must be derived in order to compute the correct answer.
This is an unfortunate mismatch with older terminology (including Bousfield–Kan), who use “homotopy (co)limit” to refer to what we would call “weighted (co)limit” (with respect to a certain choice of weight).
These weighted (co)limits do compute homotopy (co)limits, but only under additional (co)fibrancy conditions.
added pointer to:
added pointer to:
added pointer to
(though have any of Bruckner’s articles been published?)
have fixed a few oddities in the formulation of the definition (here)
also deleted this line from the References-section:
[In Cisinski 1999] it was clarified that every cofibrant object in the Thomason model structure is a poset (although this is already explicitly mentioned in Thomason’s paper – see the beginning of section 5).
and instead added explicit statement of and reference to Thomason’s proposition 5.7 here
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