Timeline for Category of groups = Category of models of group theory?
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9 events
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| Jan 22, 2010 at 18:22 | comment | added | Pete L. Clark | I don't think this is chimerical at all. Given a language L, one defines L-structures [or slightly more precisely, relational structures] and morphisms between them [often required to be injections]. This is certainly a [concrete] category, even though for some reason it is not standard to say so explicitly in Chapter 1 of model theory books. If you have a theory T of that language, then it is natural to consider the full subcategory of models of T. If you do this with the theory of groups (in the language of monoids), you get back the category of groups. | |
| Jan 22, 2010 at 17:56 | comment | added | Hans-Peter Stricker | Maybe it's some kind of chimera I am chasing: indeed there's only the notion of "category of models of a (general) theory T". What I was wondering about is, why this general approach is not taken by default. | |
| Jan 22, 2010 at 16:26 | comment | added | Reid Barton | @Hans: Google says you are the only one to use the phrase "category of models of group theory", so... can you give an example of what you are talking about? | |
| Jan 22, 2010 at 15:09 | comment | added | Hans-Peter Stricker | @Pete: Indeed I wasn't aware of the importance of "elementary". The answer demonstrates it. Without "elementary" the question sums up to the question whether a group is nothing but a model of group theory, even in the categorical context. Why then are there categories called "category of models of X theory" and not just "category of X's"? | |
| Jan 22, 2010 at 14:48 | comment | added | Qiaochu Yuan | My guess is that Hans wants to know about something like the category of models of the Lawvere theory of groups. (If I understand correctly, this is not what the word "model" means in model theory...?) | |
| Jan 22, 2010 at 14:42 | comment | added | Joel David Hamkins | But he used the tag "model-theory", which suggests that he did intend to consider elementary maps. But if not, then I'm not sure which category you are speaking of. | |
| Jan 22, 2010 at 14:23 | comment | added | Pete L. Clark | Right, but this makes me think that maybe the questioner did not really mean to include the word "elementary" in the question. In that case, the categories really are "the same", i.e., canonically isomorphic. | |
| Jan 22, 2010 at 14:01 | vote | accept | Hans-Peter Stricker | ||
| Jan 22, 2010 at 13:44 | history | answered | Joel David Hamkins | CC BY-SA 2.5 |