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2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory internal-categories k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthorjesse
   • CommentTimeMay 8th 2017
   • PermaLink
   Author: jesse
   Format: MarkdownItexFor the sake of having a reference to link to later, I've written [[diagram of a first-order structure]]. This is just a construction where you take a theory $T$ and expand it to a new theory $T'$ by naming one of its models $M$ with constant symbols for each element of $M$ while additionally stipulating those constant symbols have to behave like they came from $M$. If those additional stipulations were only quantifier-free, the models of $T'$ are those models of $T$ containing $M$ as an induced substructure. If those additional stipulations were all the first-order sentences satisfied by the elements of $M$, then the models of $T'$ are those models of $T$ containing $M$ as an [[elementary embedding|elementary substructure]].

   For the sake of having a reference to link to later, I’ve written diagram of a first-order structure. This is just a construction where you take a theory $T$ and expand it to a new theory $T'$ by naming one of its models $M$ with constant symbols for each element of $M$ while additionally stipulating those constant symbols have to behave like they came from $M$.

   If those additional stipulations were only quantifier-free, the models of $T'$ are those models of $T$ containing $M$ as an induced substructure.

   If those additional stipulations were all the first-order sentences satisfied by the elements of $M$, then the models of $T'$ are those models of $T$ containing $M$ as an elementary substructure.

2. • CommentRowNumber2.
   • CommentAuthorjesse
   • CommentTimeMay 19th 2017
   • (edited May 19th 2017)
   • PermaLink
   Author: jesse
   Format: MarkdownItexI added a remark (currently Remark 4.1) to [[diagram of a first-order structure]] that passing from $T$ to $T_{\mathsf{Diag(M)}}$ (resp. $\mathsf{EDiag}$) is functorial on interpretations the way that one would expect.

   I added a remark (currently Remark 4.1) to diagram of a first-order structure that passing from $T$ to $T_{\mathsf{Diag(M)}}$ (resp. $\mathsf{EDiag}$) is functorial on interpretations the way that one would expect.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeMay 19th 2017
   • (edited May 19th 2017)
   • PermaLink
   Author: Urs
   Format: MarkdownItexJesse, with all the nice material that you are adding on model theory. Might you have the energy to expand the (presently puny) table of contents at * _[[model theory - contents]]_ ? (This entry is the one that is included as the "floating table of contents" in the top right of all the model theory entries.) As an example for what I have in mind, compare to * _[[topology - contents]]_

   Jesse,

   with all the nice material that you are adding on model theory. Might you have the energy to expand the (presently puny) table of contents at

   • model theory - contents

   ?

   (This entry is the one that is included as the “floating table of contents” in the top right of all the model theory entries.)

   As an example for what I have in mind, compare to

   • topology - contents
4. • CommentRowNumber4.
   • CommentAuthorjesse
   • CommentTimeMay 19th 2017
   • PermaLink
   Author: jesse
   Format: MarkdownItexThanks for the reminder! It's now on my to-do list.

   Thanks for the reminder! It’s now on my to-do list.