Dear Andre, I believe that you misunderstood the comment made in Bunge-Hermida (2011) concerning model structures. In particular, I do not know what you meant by your saying "I disagree with your conclusion". We say precisely in Section 3 (Remark 3.6) that the two model structures mentioned - namely the Joyal-Tierney and the Lack model structures on Cat(S), S a Grothendieck topos, are different, and why. I think then that we agree. Best,Marta
From: joyal.andre@uqam.ca To: categories@mta.ca Subject: categories: RE: size_question_encore Date: Sun, 10 Jul 2011 09:30:07 -0400
dear Marta,
I apologise, I had forgoten our conversation! My memory was never good, and it is getting worst.
You wrote:
No, I am not thinking of the analogue of Steve Lack's model structure since, strictly speaking, it has nothing to do with stacks. Comments to that effect (with which Steve agrees) are included in the Bunge-Hermida paper. It was actually a surprise to discover that after trying to do what you suggest and failing.
I disagree with your conclusion. I looked at your paper with Hermida. We are not talking about the same model structure. The fibrations in 2Cat(S) defined by Steve Lack (your definition 7.1) are too weak when the topos S does not satify the axiom of choice. Equivalently, his generating set of trivial cofibrations is too small.
Nobody has read my paper with Myles <A.Joyal, M.Tierney: Classifying spaces for sheaves of simplicial groupoids, JPAA, Vol 89, 1993>.
Best, André
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