* The stabilisation theorem [not found] <B3BDD01E-418D-4AF6-A573-D8BD2C70FB2C@wanadoo.fr> @ 2010-05-15 14:06 ` Joyal, André 2010-05-16 18:44 ` John Baez 0 siblings, 1 reply; 3+ messages in thread From: Joyal, André @ 2010-05-15 14:06 UTC (permalink / raw) To: Bob Rosebrugh Dear John, I wrote: >> The Breen-Baez-Dolan Stabilisation Hypothesis is a theorem. You wrote: >It seems I understand everything except this sentence. I have a pretty good idea of how it can be proved. It is like a road map. The steps are not difficult to understand. Here is a sketch. 1) Let me denote by E(n) the theory of n-fold monoids and by E(infty) the theory of symmetric monoids. If U[n] dnotes the quasi-category of n-types, then the map Model(E(infty),U[n]) --->Model(E(n+2), U[n]) induced by the canonical map E(n+2)-->E(infty) is an equivalence of quasi-categories. This follows from the fact that the map E(n+2)-->E(infty) is a n-equivalence. 2) If T is any finite limit sketch, then the equivalence above induces an equivalence of quasi-categories Model(E(infty),Model(T,U[n])) --->Model(E(n+2), Model(T,U[n])) In particular, if T is the theory of n-categories T_n, we obtain an equivalence of quasi-categories Model(E(infty),Cat(n)) --->Model(E(n+2), Cat(n)) where Cat(n) is the quasi-category of (weak)-n-category. QED Too simple to be true? I am ready to give more details if you want. Best regards, André [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 3+ messages in thread
* Re: The stabilisation theorem 2010-05-15 14:06 ` The stabilisation theorem Joyal, André @ 2010-05-16 18:44 ` John Baez 2010-05-18 3:27 ` joyal.andre 0 siblings, 1 reply; 3+ messages in thread From: John Baez @ 2010-05-16 18:44 UTC (permalink / raw) To: categories André wrote: Too simple to be true? > No, I always hoped for a simple proof! And this proof is not "too simple". It seems all the hard work is packed into the formalism that underlies the proof. And that's how it should be, I think. So, I'm happy. Best, jb [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 3+ messages in thread
* Re: The stabilisation theorem 2010-05-16 18:44 ` John Baez @ 2010-05-18 3:27 ` joyal.andre 0 siblings, 0 replies; 3+ messages in thread From: joyal.andre @ 2010-05-18 3:27 UTC (permalink / raw) To: John Baez, categories Dear John, > It seems all the hard work is packed into the formalism that > underlies the proof. I cannot resist describing a key idea of my proof. It is to construct categories from homotopy types rather than from sets. The notion of category is homotopy essentially algebraic: "A category is essentially the same thing as a complete Segal object X satisfying an extra condition: the (source, target) map X_1-->X_0 times X_0 is a 0-cover (a map is a 0-cover if its homotopy fibers are discrete). This is like constructing the "natural" (or folk") model structure on Cat from a model structure on simplicial diagrams of spaces (spaces = simplical sets). A similar description applies to (weak) n-categories. The Stabilisation Hypothesis was a great conjecture. Let me congratulate you and Jim Dolan for formulating it. Best, André -------- Message d'origine-------- De: categories@mta.ca de la part de John Baez Date: dim. 16/05/2010 14:44 À: categories Objet : categories: Re: The stabilisation theorem André wrote: Too simple to be true? > No, I always hoped for a simple proof! And this proof is not "too simple". It seems all the hard work is packed into the formalism that underlies the proof. And that's how it should be, I think. So, I'm happy. Best, jb [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 3+ messages in thread
end of thread, other threads:[~2010-05-18 3:27 UTC | newest] Thread overview: 3+ messages (download: mbox.gz / follow: Atom feed) -- links below jump to the message on this page -- [not found] <B3BDD01E-418D-4AF6-A573-D8BD2C70FB2C@wanadoo.fr> 2010-05-15 14:06 ` The stabilisation theorem Joyal, André 2010-05-16 18:44 ` John Baez 2010-05-18 3:27 ` joyal.andre
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