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1. • CommentRowNumber1.
   • CommentAuthorzskoda
   • CommentTimeMay 25th 2012
   • (edited May 25th 2012)
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   Author: zskoda
   Format: MarkdownItexNew entry [[structure]] but the $n$Lab is down so I save here the final version of editing, which is probably lost in $n$Lab: The concept of a _structure_ is formulated as the basic object of mathematics in the work of [[Bourbaki]]. In [[model theory]], a structure of a language $L$ is the same as [[model]] of $L$ with empty set of extra axioms. Given a first-order language $L$, which consists of symbols (variable symbols, constant symbols, function symbols and relation symbols including $\epsilon$) and quantifiers; a __structure__ for $L$, or $L$-structure is a set $M$ with an __interpretation__ for symbols: * if $R\in L$ is an $n$-ary relation symbol, then its interpretation $R^M\subset M^n$ * if $f\in L$ is an $n$-ary function symbol, then $f^M:M^n\to M$ is a function * if $c\in L$ is a constant symbol, then $c^M\in M$ Interpretation for an $L$-structure inductively defines an interpretation for well-formed formulas in $L$. We say that a sentence $\phi\in L$ is true in $M$ if $\phi^M$ is true. Given a [[theory]] $(L,T)$, which is a language $L$ together with a given set $T$ of sentences in $L$, the interpretation in a structure $M$ makes those sentences true or false; if all the sentences in $T$ are true in $M$ we say that $M$ is a [[model]] of $(L,T)$. Some special cases include __algebraic structures__, which is usually defined as a structure for a first order language with equality and $\epsilon$-relation both with the standard interpretation, no other relation symbols and whose function symbols are interpreted as operations of various arity. This is a bit more general than an __algebraic theory__ as in the latter, one needs to have free algebras so for example fields do not form an algebraic theory but are the algebraic structures for the theory of fields. In [[category theory]] we may talk about functor forgetting structure (formalizing an intuitive, related and in a way more general sense), see * [[stuff, structure, property]] [[!redirects structures]]

   New entry structure but the $n$Lab is down so I save here the final version of editing, which is probably lost in $n$Lab:

   The concept of a structure is formulated as the basic object of mathematics in the work of Bourbaki.

   In model theory, a structure of a language $L$ is the same as model of $L$ with empty set of extra axioms. Given a first-order language $L$, which consists of symbols (variable symbols, constant symbols, function symbols and relation symbols including $\epsilon$) and quantifiers; a structure for $L$, or $L$-structure is a set $M$ with an interpretation for symbols:

   • if $R\in L$ is an $n$-ary relation symbol, then its interpretation $R^M\subset M^n$

   • if $f\in L$ is an $n$-ary function symbol, then $f^M:M^n\to M$ is a function

   • if $c\in L$ is a constant symbol, then $c^M\in M$

   Interpretation for an $L$-structure inductively defines an interpretation for well-formed formulas in $L$. We say that a sentence $\phi\in L$ is true in $M$ if $\phi^M$ is true. Given a theory $(L,T)$, which is a language $L$ together with a given set $T$ of sentences in $L$, the interpretation in a structure $M$ makes those sentences true or false; if all the sentences in $T$ are true in $M$ we say that $M$ is a model of $(L,T)$.

   Some special cases include algebraic structures, which is usually defined as a structure for a first order language with equality and $\epsilon$-relation both with the standard interpretation, no other relation symbols and whose function symbols are interpreted as operations of various arity. This is a bit more general than an algebraic theory as in the latter, one needs to have free algebras so for example fields do not form an algebraic theory but are the algebraic structures for the theory of fields.

   In category theory we may talk about functor forgetting structure (formalizing an intuitive, related and in a way more general sense), see

   • stuff, structure, property

   !redirects structures

2. • CommentRowNumber2.
   • CommentAuthorzskoda
   • CommentTimeMay 25th 2012
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   Author: zskoda
   Format: MarkdownItexThe $n$Lab is back and I did some more editing.

   The $n$Lab is back and I did some more editing.

3. • CommentRowNumber3.
   • CommentAuthorMike Shulman
   • CommentTimeMay 25th 2012
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexThis is good to have, but maybe we could name the page more specifically? Maybe something like [[structure in model theory]]? I am sad to see that you removed the redirect of "structure" to [[stuff, structure, property]], because I think that is the most common use of "structure" on the nLab.

   This is good to have, but maybe we could name the page more specifically? Maybe something like structure in model theory? I am sad to see that you removed the redirect of “structure” to stuff, structure, property, because I think that is the most common use of “structure” on the nLab.

4. • CommentRowNumber4.
   • CommentAuthorzskoda
   • CommentTimeMay 25th 2012
   • (edited May 25th 2012)
   • PermaLink
   Author: zskoda
   Format: MarkdownItexBut this entry [[structure]] has a bottom section pointing to [[stuff, structure, property]]. I will now (with expansion on model side) need to refer many times to [[structure]] (some already did) and will refer to one of the usual senses in mathematics (either Bourbaki version or "algebraic structure" or general model theoretical). The entry is something like expanded _disambiguation entry_, so nobody will be lost by being sent there. My understanding is, besides, that [[stuff, structure, property]] does not define structure itself, but rather _relative_ terms like __forgetting structure__ and alike; and the entry name, I think, would be better to adhere to the named/defined concepts. Isn't it more logical that [[structure]] refers to the entry which has a definition of a structure and redirect [[forgetting structure]] referring to the entry [[stuff, structure, property]] containing the definition of forgetting structure rather than of structure ? > the most common use of "structure" My feeling is that the development of the $n$Lab is gradually to include wider perspective in math, than only category theory and as it does it is likely that the uses will go statistically a bit more mainstream.

   But this entry structure has a bottom section pointing to stuff, structure, property. I will now (with expansion on model side) need to refer many times to structure (some already did) and will refer to one of the usual senses in mathematics (either Bourbaki version or “algebraic structure” or general model theoretical). The entry is something like expanded disambiguation entry, so nobody will be lost by being sent there.

   My understanding is, besides, that stuff, structure, property does not define structure itself, but rather relative terms like forgetting structure and alike; and the entry name, I think, would be better to adhere to the named/defined concepts. Isn’t it more logical that structure refers to the entry which has a definition of a structure and redirect forgetting structure referring to the entry stuff, structure, property containing the definition of forgetting structure rather than of structure ?

   the most common use of “structure”

   My feeling is that the development of the $n$Lab is gradually to include wider perspective in math, than only category theory and as it does it is likely that the uses will go statistically a bit more mainstream.

5. • CommentRowNumber5.
   • CommentAuthorzskoda
   • CommentTimeMay 25th 2012
   • (edited May 25th 2012)
   • PermaLink
   Author: zskoda
   Format: MarkdownItexEntry [[structured set ]]had the first sentence of idea section as follows: > A structured set is, of course, a set equipped with extra [[structure]]. It is even written __extra structure__, what is more specific, but it was referred by [[structure]]. And [[extra structure]] is a redirect which already existed in [[stuff, structure, property]] and _is_ more specific. So I changed just the bracketing to unambiguous: > A structured set is, of course, a set equipped with [[extra structure]]. So, whatever we decide about the final titles of two entries, this will always point to the correct one. On the other hand, [[lax morphism]] has "algebraic [[structure]]" on a category, where it is meant (higher) [[algebraic structure]] in monadic sense. So I added new entry [[algebraic structure]].

   Entry structured sethad the first sentence of idea section as follows:

   A structured set is, of course, a set equipped with extra structure.

   It is even written extra structure, what is more specific, but it was referred by structure. And extra structure is a redirect which already existed in stuff, structure, property and is more specific. So I changed just the bracketing to unambiguous:

   A structured set is, of course, a set equipped with extra structure.

   So, whatever we decide about the final titles of two entries, this will always point to the correct one.

   On the other hand, lax morphism has “algebraic structure” on a category, where it is meant (higher) algebraic structure in monadic sense. So I added new entry algebraic structure.

6. • CommentRowNumber6.
   • CommentAuthorMike Shulman
   • CommentTimeMay 26th 2012
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   Author: Mike Shulman
   Format: MarkdownItexI feel that the categorical point of view is that "structure" is not a meaningful thing in itself, that it is always something you *add* to some other thing you started with. The model-theoretic kind of "structure" is just the particular case of putting structure on a set (or a family of sets). I'm not against having [[structure]] be a disambiguation page, but I think in that case it should just disambiguate, and the model-theoretic point of view should be on a separate page.

   I feel that the categorical point of view is that “structure” is not a meaningful thing in itself, that it is always something you add to some other thing you started with. The model-theoretic kind of “structure” is just the particular case of putting structure on a set (or a family of sets).

   I’m not against having structure be a disambiguation page, but I think in that case it should just disambiguate, and the model-theoretic point of view should be on a separate page.

7. • CommentRowNumber7.
   • CommentAuthorzskoda
   • CommentTimeMay 26th 2012
   • PermaLink
   Author: zskoda
   Format: MarkdownItexOK, I agree it could be also yet additional entry, I will reform that carefully in near future (now first adding more material to related entries). On the other hand the point of view, > The model-theoretic kind of "structure" is just the particular case of putting structure on a set (or a family of sets). requires careful treatment which category we have in mind. Model theorist like the category of models and _elementary_ maps which disagrees with the category of models in the sense of [[model]] or which, in the case of an algebraic theory, for example, disagree with the category of algebras and homomorphisms. Cf. the discussion here: [MathOverflow](http://mathoverflow.net/questions/12629/category-of-groups-category-of-models-of-group-theory), the Hamkins' and Clarks' answers together.

   OK, I agree it could be also yet additional entry, I will reform that carefully in near future (now first adding more material to related entries).

   On the other hand the point of view,

   The model-theoretic kind of “structure” is just the particular case of putting structure on a set (or a family of sets).

   requires careful treatment which category we have in mind. Model theorist like the category of models and elementary maps which disagrees with the category of models in the sense of model or which, in the case of an algebraic theory, for example, disagree with the category of algebras and homomorphisms. Cf. the discussion here: MathOverflow, the Hamkins’ and Clarks’ answers together.

8. • CommentRowNumber8.
   • CommentAuthorzskoda
   • CommentTimeMay 26th 2012
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   Author: zskoda
   Format: MarkdownItexAnybody has access to this ?: * [[Steve Awodey]], _Structure in mathematics and logic: a categorical perspective_, Philosophia Mathematica (3), vol. 4 (1996), pp. 209--237, [doi](http://dx.doi.org/10.1093/philmat/4.3.209) Abstract: > A precise notion of ‘mathematical structure’ other than that given by model theory may prove fruitful in the philosophy of mathematics. It is shown how the language and methods of category theory provide such a notion, having developed out of a structural approach in modern mathematical practice. As an example, it is then shown how the categorical notion of a topos provides a characterization of ‘logical structure’, and an alternative to the Pregean approach to logic which is continuous with the modern structural approach in mathematics. Looks like yet another point of view. Also note that Bourbaki's notion of "structure" is more like what we call "model", while I think (should double check) he allows the infinitary relations unlike model theorists.

   Anybody has access to this ?:

   • Steve Awodey, Structure in mathematics and logic: a categorical perspective, Philosophia Mathematica (3), vol. 4 (1996), pp. 209–237, doi

   Abstract:

   A precise notion of ‘mathematical structure’ other than that given by model theory may prove fruitful in the philosophy of mathematics. It is shown how the language and methods of category theory provide such a notion, having developed out of a structural approach in modern mathematical practice. As an example, it is then shown how the categorical notion of a topos provides a characterization of ‘logical structure’, and an alternative to the Pregean approach to logic which is continuous with the modern structural approach in mathematics.

   Looks like yet another point of view. Also note that Bourbaki’s notion of “structure” is more like what we call “model”, while I think (should double check) he allows the infinitary relations unlike model theorists.

9. • CommentRowNumber9.
   • CommentAuthorMike Shulman
   • CommentTimeMay 26th 2012
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexYes, I'll send it to you.

   Yes, I’ll send it to you.

10. • CommentRowNumber10.
    • CommentAuthorRodMcGuire
    • CommentTimeMay 26th 2012
    • (edited May 26th 2012)
    • PermaLink
    Author: RodMcGuire
    Format: MarkdownItexUmm, "structure" is a "technical" word used in almost any sort of (seemingly or quasi) scientific analysis. People who analyzed things in terms of Earth, Air, Fire, and Water had a structure of oppositions and adjacency that they used. Music, kinship, etc. can be analyzed with structures. In maths various subfields use the word "structure", [(wikipedia)](http://en.wikipedia.org/wiki/Mathematical_structure), in quite specific senses, most of which are not compatible with each other. I feel that any use of "structure" in the nLab should be linkable to a page that explains in which particular math sense the word is to be interpreted. On a more meta level I see 3 different usages of "structure" in maths: 1. "Pure structure" which seems to be type theory. 2. "Implicit structure" where say a category defined by objects and arrows automatically has Homs and a endofunctor category. 3. "Added structure" where say one takes a set $S$ (regarded as a category) and "enhances" it with a function $+: S\times S \to S$ to maybe eventually wind up with a semi-group. Much of the math specific definitions of "structure"; seem to fall within the "added" rather than "implicit" notions, though if one starts with say a monoid structure there are several way to see its implicit structure in terms of simpler things, some that may involve structures that have been enhanced. Anyway, any use of "structure" should be linkable to a page that explains the intended restricted meaning in a particular subfield, and that page should note (maybe in its title) whether it is talking about pure, implicit, or additional structure. Thus I advocate for a page called "structure" to mainly serve as a disambiguation page for the various notions of structure. However no other pages should link to it unless maybe as a stop-gap for a particular definition that doesn't yet exist.

    Umm, “structure” is a “technical” word used in almost any sort of (seemingly or quasi) scientific analysis. People who analyzed things in terms of Earth, Air, Fire, and Water had a structure of oppositions and adjacency that they used. Music, kinship, etc. can be analyzed with structures.

    In maths various subfields use the word “structure”, (wikipedia), in quite specific senses, most of which are not compatible with each other.

    I feel that any use of “structure” in the nLab should be linkable to a page that explains in which particular math sense the word is to be interpreted.

    On a more meta level I see 3 different usages of “structure” in maths:

    1. “Pure structure” which seems to be type theory.
    2. “Implicit structure” where say a category defined by objects and arrows automatically has Homs and a endofunctor category.
    3. “Added structure” where say one takes a set $S$ (regarded as a category) and “enhances” it with a function $+: S\times S \to S$ to maybe eventually wind up with a semi-group.

    Much of the math specific definitions of “structure”; seem to fall within the “added” rather than “implicit” notions, though if one starts with say a monoid structure there are several way to see its implicit structure in terms of simpler things, some that may involve structures that have been enhanced.

    Anyway, any use of “structure” should be linkable to a page that explains the intended restricted meaning in a particular subfield, and that page should note (maybe in its title) whether it is talking about pure, implicit, or additional structure.

    Thus I advocate for a page called “structure” to mainly serve as a disambiguation page for the various notions of structure. However no other pages should link to it unless maybe as a stop-gap for a particular definition that doesn’t yet exist.

11. • CommentRowNumber11.
    • CommentAuthorzskoda
    • CommentTimeMay 28th 2012
    • (edited May 28th 2012)
    • PermaLink
    Author: zskoda
    Format: MarkdownItexOK, I have done some major _restructuring_, following the advice of the people in this discussion. Now we have [[structure]] (disambiguation entry), [[structure in model theory]], [[algebraic structure]], [[stuff, structure, property]], [[structuralism]] (tagged under category: philosophy); while awaited and cited [[structure after Bourbaki]] still waits for creation. Or should it be "structure in Bourbaki"; well maybe not as it could suggest the structure in the work of Bourbaki, how his works are structured. After is unambiguous.

    OK, I have done some major restructuring, following the advice of the people in this discussion. Now we have structure (disambiguation entry), structure in model theory, algebraic structure, stuff, structure, property, structuralism (tagged under category: philosophy); while awaited and cited structure after Bourbaki still waits for creation. Or should it be “structure in Bourbaki”; well maybe not as it could suggest the structure in the work of Bourbaki, how his works are structured. After is unambiguous.

12. • CommentRowNumber12.
    • CommentAuthorDavidRoberts
    • CommentTimeMay 28th 2012
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    Author: DavidRoberts
    Format: MarkdownItexBourbaki structure?

    Bourbaki structure?

13. • CommentRowNumber13.
    • CommentAuthorzskoda
    • CommentTimeMay 28th 2012
    • (edited May 28th 2012)
    • PermaLink
    Author: zskoda
    Format: MarkdownItexIt came to my mind, but it is strange as Bourbaki would never call it that way and the books about it say things like "Bourbaki's notion of structure". I don't know.

    It came to my mind, but it is strange as Bourbaki would never call it that way and the books about it say things like “Bourbaki’s notion of structure”. I don’t know.

14. • CommentRowNumber14.
    • CommentAuthorDavidRoberts
    • CommentTimeMay 28th 2012
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    Author: DavidRoberts
    Format: MarkdownItexIt is unambiguous though, and there are lots of concepts that get named for a person (or in this case a 'person') and they never called it that. (although cf Coxeter's 'My group'). Why not call it 'Bourbaki's notion of structure'?

    It is unambiguous though, and there are lots of concepts that get named for a person (or in this case a ’person’) and they never called it that. (although cf Coxeter’s ’My group’). Why not call it ’Bourbaki’s notion of structure’?

15. • CommentRowNumber15.
    • CommentAuthorzskoda
    • CommentTimeMay 28th 2012
    • (edited May 28th 2012)
    • PermaLink
    Author: zskoda
    Format: MarkdownItexIn principle both are OK, the long, shy and somewhat received version, as your shorter one. If we decide for the shorter one we play naming (we play Bourbaki). To add to historical remarks, Borel called [[Borel subgroup]] "maximal connected solvable subgroup" (and understood also closed), including in his book _Linear algebraic groups_, but yield to "Borel subgroup" in the second edition. When I was introduced to Borel in 2002 the person who introduced me said this is Zoran Škoda who wrote his thesis on _quantum Borel subgroups_ (well it was about an approach to quantum $G/B$, but OK), and Borel asked me, why do you need it, and I said "for representation theory" (well I meant coherent states for quantum groups and alike aspects), and he asked with a smile "Borel-Weil?", I said "Yes!" It was funny.

    In principle both are OK, the long, shy and somewhat received version, as your shorter one. If we decide for the shorter one we play naming (we play Bourbaki).

    To add to historical remarks, Borel called Borel subgroup “maximal connected solvable subgroup” (and understood also closed), including in his book Linear algebraic groups, but yield to “Borel subgroup” in the second edition. When I was introduced to Borel in 2002 the person who introduced me said this is Zoran Škoda who wrote his thesis on quantum Borel subgroups (well it was about an approach to quantum $G/B$, but OK), and Borel asked me, why do you need it, and I said “for representation theory” (well I meant coherent states for quantum groups and alike aspects), and he asked with a smile “Borel-Weil?”, I said “Yes!” It was funny.