Timeline for Elementary Equivalence =? Homotopy Equivalence
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| Jun 23, 2012 at 10:58 | comment | added | Mirco A. Mannucci | @Sreven thanks for the tip! Perhaps you can also point this question to him @Francois yes, thanks for the addendum. But I still do not get it: the direction, if I understand you properly, goes from model cats to accessible cats. I would think that for my questions a relevant result should point the other way round: accessible cats are the cats of models of some theory, so if those could be given an additional homotopy structure I would have what I am after. Maybe the second paper you have pointed me is closer to the mark, I will read it. | |
| Jun 20, 2012 at 22:58 | answer | added | o a | timeline score: 3 | |
| Jun 20, 2012 at 18:00 | comment | added | François G. Dorais | I should have been clearer in my earlier comment. In that paper, Rosicky shows that weak equivalences in a combinatorial model category form an accessible category. Some accessible categories are of the form $\mathbf{Elem}(T)$ and some are not. In another recent paper, Beke and Rosicky characterize exactly which accessible categories come from abstract elementary classes: arxiv.org/abs/1005.2910 Maybe the two bridges connect? | |
| Jun 20, 2012 at 17:34 | comment | added | Steven Gubkin | I would see if you can talk to Mike Shulman - he seems to be into this kind of stuff. | |
| Jun 20, 2012 at 15:57 | comment | added | Mirco A. Mannucci | anyway, I will check into rosicky's paper. Looks intriguing.. | |
| Jun 20, 2012 at 15:56 | comment | added | Mirco A. Mannucci | Francois, thanks for your comment. Yes, I am aware of accessible categories (via Makkai's memoir on the AMS). But my question does not hinge on the categorical aspects of Model Theory, I am interested in its homotopical aspects (or, if you prefer, in its higher order categorical aspects). For instance, I would like (if possible) to talk about continuous deformations of structures, where continuous deformation basically means that I can deform two elementary equivalent structures into one another, through some intermediate structures which are also elementary equivalent to the two poles. | |
| Jun 20, 2012 at 15:25 | comment | added | François G. Dorais | The categorical aspects of model theory are fairly well developed though not widely known. They are closely tied to accessible categories, as I explained in this old answer: mathoverflow.net/questions/13155/… Accessible categories are also connected with Quillen model categories, perhaps you can find a partial answer or some hints in this paper by Rosicky: arxiv.org/abs/0708.2185 | |
| Jun 20, 2012 at 14:58 | history | edited | Mirco A. Mannucci | CC BY-SA 3.0 |
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| Jun 20, 2012 at 14:45 | comment | added | Mirco A. Mannucci | continues: in y question I just made an educated guess as to which maps one could choose, namely the elementary embeddings, but on a second thought they may be way too restrictive to get something really cool. Any guess on your part? After one has selected the weak equivalences, one would need at the very least some piece of the homotopy machinery to actually compute anything, for instance some notion like cilinders. | |
| Jun 20, 2012 at 14:40 | comment | added | Mirco A. Mannucci | Andreas, you are a bit too clever for me :). I might as well tell you the real story: my early morning dream would be to have some real homotopy theory for the category of structures of a given first-order language (or perhaps some suitable subcategory thereof). Now, what do I mean by a real homotopy theory? Ideally (but very unlikely) something I mentioned in the question, a full-blown quillen model structure, so that I can do all the homotopy jazz. I would be happy for much less, but how much less? I need first of all to select some maps (the weak equivalences). | |
| Jun 20, 2012 at 14:12 | comment | added | Andreas Blass | Is it intentional that, after defining the notion of elementary equivalence, you never use it in your question but rather talk about elementary embeddings and elementary substructures? | |
| Jun 20, 2012 at 11:16 | history | edited | Mirco A. Mannucci | CC BY-SA 3.0 |
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| Jun 20, 2012 at 10:52 | history | asked | Mirco A. Mannucci | CC BY-SA 3.0 |