Timeline for Reference request: Models of isomorphic languages result into isomorphic categories
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18 events
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| Jan 30, 2016 at 17:53 | comment | added | Salvo Tringali | @François. I hadn't realized it was a paper (sic!). But then, why don't you post your comment as an answer? It's much more than what I was looking for (which means a plus to me, not a minus). #Andrea. Yes, the claim is precisely the one at the end of your last comment. But nothing similar is discussed in Johnstone's Elephant, AFAICS. | |
| Jan 30, 2016 at 17:40 | comment | added | Andrea Gagna | [...] from the set of formulas of $\sigma$ to the set of formulas of $\sigma'$ (just define it inductively) and in particular to any sequent over the signature $\sigma$, we get a sequent over the signature $\sigma'$; so, it $\mathbb T = (\sigma, \Xi)$ is a theory with signature $\sigma$, we get a corresponding "isomorphic" theory $\mathbb T' = (\sigma', \Xi')$. Following definition D.1.2.12, we get the category $\mathbb T\text{-}\mathbf{Mod}(\mathrm{Set})$ of set-models of $\mathbb T$ (possibly only with elementary morphisms). The claim is that the isomorphism $\phi^*$ restricts to these cats. | |
| Jan 30, 2016 at 17:39 | comment | added | François G. Dorais | Hájek's article is here: digizeitschriften.de/dms/img/?PID=GDZPPN002043920 You asked for a reference for the fact that "isomorphic languages should result into isomorphic categories [of models]"; Hájek does quite a lot more since he considers translations that aren't just symbol substitutions. I doubt the trivial case you're interested in has been published on its own. | |
| Jan 30, 2016 at 17:30 | comment | added | Andrea Gagna | I think Peter's comment solve the question. In section D of the Elephant , the category of $\sigma$-structures of sets for a signature $\sigma$ is denoted $\sigma\text{-}\mathbf{Str}(\mathrm{Set})$ (see Definition D.1.2.1). Since it is functorial on the signatures, if $\phi\colon \sigma \to \sigma'$ is an isomorphism of signatures we have an isomorphism of categories from $\sigma\text{-}\mathbf{Str}(\mathrm{Set})$ to $\sigma'\text{-}\mathbf{Str}(\mathrm{Set})$ (unfortunately, I don't have any reference for this functoriality). Further, notice that $\phi$ induce a morphism $\tilde\phi$ [...] | |
| Jan 30, 2016 at 17:11 | comment | added | Salvo Tringali | Unfortunately, I don't have Hájek's book and can't read German, but isn't the book about categorical logic (or, as some authors call it, cat model theory)? If so, then a better ref would be Sect. D in Johnstone's Elephant (see Peter's comment above). Now, it's a fact that model theory is a "fragment" of categorical logic: In the words of Makkai and Paré (from the introduction of their book), model theory is ``the study of ordinary, "set-valued", or ensembliste models of logical theories.'' So yes, my question could also be framed in the language of cats (and in a much more general form). | |
| Jan 30, 2016 at 16:21 | comment | added | François G. Dorais | OK. I think you're homing in on the idea of 'syntactic model' of Petr Hájek [Logische Kategorien, Arch. Math. Logik Grundlagenforsch. 13 1970 168–193] though Hájek pursues the idea in far too much generality for your purposes. | |
| Jan 30, 2016 at 15:46 | comment | added | Salvo Tringali | @François. The link provides a notion of 'interpretation of a structure within another structure', while I'm using a locution of the form 'a model of a language $L=(\sigma, \Xi)$ of type $\mathscr{L}_\infty$' to mean a $\sigma$-structure that satisfies the axioms in $\Xi$. Here, $\sigma$ is the signature of $L$ and $\Xi$ a (possibly empty) set of wffs derived from $\sigma$ and the (logical and non-logical) symbols of $\mathscr{L}_\infty$ according to the formation rules of the extended first-order logic specified in the OP, and 'to satisfy' just means what you guess. | |
| Jan 30, 2016 at 15:31 | comment | added | François G. Dorais | See also Adámek & Rosicky Locally Presentable and Accessible Categories. | |
| Jan 30, 2016 at 15:25 | comment | added | François G. Dorais | Never mind, I figured out what was confusing me in the wording. Are you looking for something different than a special case of 'interpretation' - wikiwand.com/en/Interpretation_%28model_theory%29 - or are you wondering about the leap to infinitary logic? | |
| Jan 30, 2016 at 15:22 | comment | added | Salvo Tringali | (...) (in the sense of categorical model theory) of an $\mathscr{L}_{\infty}$ language and models of a sketch give rise to equivalent categories (something stronger is true, but never mind). This fact is not related to the OP, yet it answers another question that I would have liked to ask for a while, so I thought to record it here. | |
| Jan 30, 2016 at 15:19 | comment | added | Salvo Tringali | @FrançoisG.Dorais. Not sure I understand your question properly, but I'm used to consider the terms 'model' and 'interpretation' as synonyms, consistently with Chang and Keisler's Model Theory. Do you perhaps mean 'structure' when you write 'model'? #Peter. AFAICS, Makkai and Reyes' book (which, unfortunately, I don't have access to) is mostly focused on languages of type $\mathscr{L}_{\infty,\omega}$. However, it seems the gap is filled in Makkai and Paré's Accessible Categories: The Foundations of Categorical Model Theory, where, among other things, it is proved that models (...) | |
| Jan 30, 2016 at 15:07 | comment | added | François G. Dorais | I'm a little confused by the end of the question. Are you looking for a definition of 'model' or a definition of 'interpretation'? | |
| Jan 30, 2016 at 13:16 | comment | added | Salvo Tringali | (...) languages $f: L_1\to L_2$ (whatever this may be...), for which $L_1$ and $L_2$ are languages of type $\mathscr{L}_\infty$, induces a functor $F: {\bf Str}(L_1)\to{\bf Str}(L_2)$. I mean, the result I'm seeking should take into account the axioms that, along with the information provided by the signature, forge the structures modeling the specific language under consideration, right? Thanks anyway for your comment and the pointers (I was already surfing through Johnstone's Elephant, but hadn't thought of Makkai and Reyes). | |
| Jan 30, 2016 at 13:04 | comment | added | Salvo Tringali | @Peter. I assume that a "map of signatures" is formally a triple $(f,\sigma_1,\sigma_2)$, where $\sigma_i=(\Sigma_{\rm f}^{(i)},\Sigma_{\rm r}^{(i)},\varrho_i)$ is a signature ($i=1,2$) and $f$ is a fnc $\Sigma_{\rm f}^{(1)}\cup\Sigma_{\rm r}^{(1)}\to\Sigma_{\rm f}^{(2)}\cup\Sigma_{\rm r}^{(2)}$ s.t. $f[\Sigma_{\rm f}^{(1)}]\subseteq\Sigma_{\rm f}^{(2)}$, $f[\Sigma_{\rm r}^{(1)}]\subseteq\Sigma_{\rm r}^{(2)}$ and $\varrho_2(f(\zeta))=\varrho_1(\zeta)$ for every $\zeta\in\Sigma_{\rm f}^{(1)}\cup\Sigma_{\rm r}^{(1)}$. If so, I think that I'd rather look for the fact that any morphism of (...) | |
| Jan 30, 2016 at 11:11 | comment | added | Peter LeFanu Lumsdaine | Not time for a full answer now, but a few pointers. Firstly, the specific result I’d look for is the fact that the category of models is functorial in the signature, i.e. any map of signatures $f : \rho \to \sigma$ induces a functor $f^* : \textbf{Str}(\sigma) \to \textbf{Str}(\rho)$, and this action makes $\mathbf{Str}(-)$ a functor from signatures to categories. The isomorphism result then follows immediately. Secondly, two good books to check are Johnstone’s Elephant, and Makkai and Reyes’ First-order Categorical Logic. | |
| S Jan 30, 2016 at 10:28 | history | suggested | Rahman. M |
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| Jan 30, 2016 at 10:08 | review | Suggested edits | |||
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| Jan 29, 2016 at 17:13 | history | asked | Salvo Tringali | CC BY-SA 3.0 |