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One of the most interesting novelties in recent foundational studies is Voevodsky's Homotopical Type Theory project (see here).

Finally homotopy theory ideas have entered in a royal fashion the foundational arena!

I wonder if other areas of logic and foundational studies can be tackled from an homotopical standpoint. For instance, in model theory, one encounters the central notion of elementary equivalence:

two structures M and N of the same signature $\sigma$ are called elementarily equivalent if they satisfy the same first-order σ-sentences.σ-sentences.

The question:

take elementary embedding as a notion of weak equivalence, what kind of structure has the associated homotopy category? Perhaps dreaming a little, can one even manage to identify a Quillen model structure on the category of $\sigma$ -structures?

NOTE: Andreas Blass has (rightly) asked why I mention elementary embedding in my question, whereas the title talks about homotopy equivalences. Point well taken: I should reformulate the question in a broader form, as: can you choose some maps in the category as weak equivalences, so that we can have a homotopy category, possibly with a good amount of homotopy limits and colimits to do some real computations?

PS: The dream behind this question is that perhaps one could think of an elementary substructure as a homotopical retract of the ambient structure , and more generally one could come up with a notion of "continuous deformation" of structures, just like in the topological category

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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? –  Andreas Blass Jun 20 '12 at 14:12
    
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). –  Mirco Mannucci Jun 20 '12 at 14:40
    
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. –  Mirco Mannucci Jun 20 '12 at 14:45
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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 –  François G. Dorais Jun 20 '12 at 15:25
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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? –  François G. Dorais Jun 20 '12 at 18:00
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1 Answer

The dream behind this question is that perhaps one could think of an elementary substructure as a homotopical retract of the ambient structure

I have tried and failed to do something similar for models of a sufficiently nice theory, say a first order categorical theory such as ACF; here is what my thoughts were. My desired weak equivalences were: add a finite tuple to a model M and get a model Ma prime(primary) and minimal over $M\cup a$ (acyclic cofibration); represent a model $N$ as the union $\cup M_i$ of an increasing chain of elementary submodels of strictly smaller cardinality (acyclic fibration). The first one is an elementary equivalence between a model and its substructure, and thus fits your suggested definition. However, the second one requires one to extend your category of models and consider the category of families of models; from the categorical point of view, you formally add new limits ignoring those limits you already have in your category. It is then easy to define a model "pre"structure that satisfies some of the axioms of a model category; Lowenheim-Skolem theorem then means that every morphism can be decomposed as a cofibration and an acyclic fibration, and existence of prime and minimal models (that holds for sufficiently nice theories) means that every morphism decomposes as an anyclic cofibration and a fibration (axiom M2 of Quillen). But that's all: I was not able to construct a model category for an interesting theory, say non-locally modular or even ACF itself.

We were only able to construct a model category for a trivial theory T with the empty language; then this all becomes set theory and is described in here and here

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