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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 k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nforum 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 sheaves 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.
   • CommentAuthorDavid_Corfield
   • CommentTimeFeb 19th 2014
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexThis is another PhD thesis supervised by Steve Awodey. This time by Spencer Breiner ([arxiv](http://arxiv.org/abs/1402.2600)). >Although contemporary model theory has been called "algebraic geometry minus fields", the formal methods of the two fields are radically different. This dissertation aims to shrink that gap by presenting a theory of logical schemes, geometric entities which relate to first-order logical theories in much the same way that algebraic schemes relate to commutative rings. >The construction relies on a Grothendieck-style representation theorem which associates every coherent or classical first-order theory with an affine scheme: a topological groupoid (the spectrum of the theory) together with a sheaf of (local) syntactic categories. The groupoid is constructed from the semantics of the theory (models and isomorphisms) and topologized using a Stone-type construction. The sheaf of categories can be regarded as a logical theory varying over the spectrum, and its global sections recover the theory up to semantic equivalence. These affine pieces can be glued together to give more general logical schemes and these are studied using methods from algebraic geometry. The final chapter also presents some connections between schemes and other areas of logic such as model theory, type theory and topos theory.

   This is another PhD thesis supervised by Steve Awodey. This time by Spencer Breiner (arxiv).

   Although contemporary model theory has been called “algebraic geometry minus fields”, the formal methods of the two fields are radically different. This dissertation aims to shrink that gap by presenting a theory of logical schemes, geometric entities which relate to first-order logical theories in much the same way that algebraic schemes relate to commutative rings.

   The construction relies on a Grothendieck-style representation theorem which associates every coherent or classical first-order theory with an affine scheme: a topological groupoid (the spectrum of the theory) together with a sheaf of (local) syntactic categories. The groupoid is constructed from the semantics of the theory (models and isomorphisms) and topologized using a Stone-type construction. The sheaf of categories can be regarded as a logical theory varying over the spectrum, and its global sections recover the theory up to semantic equivalence. These affine pieces can be glued together to give more general logical schemes and these are studied using methods from algebraic geometry. The final chapter also presents some connections between schemes and other areas of logic such as model theory, type theory and topos theory.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeFeb 19th 2014
   • PermaLink
   Author: Urs
   Format: MarkdownItexCould you add a pointer to that to the entry on model theory?

   Could you add a pointer to that to the entry on model theory?

3. • CommentRowNumber3.
   • CommentAuthorDavid_Corfield
   • CommentTimeFeb 19th 2014
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexI think it needs to go in a few places. I'll take a look and see where when I have a moment.

   I think it needs to go in a few places. I’ll take a look and see where when I have a moment.

4. • CommentRowNumber4.
   • CommentAuthorzskoda
   • CommentTimeFeb 19th 2014
   • (edited Feb 19th 2014)
   • PermaLink
   Author: zskoda
   Format: MarkdownItexFor the generalized idea (in model theory) of the algebraic geometric idea of point = (special: prime, maximal) ideal, one expects to have use of types, e.g. [[Galois type]]s and of classical [[type+(in+model+theory)]] (not to confuse with type in type theory). Somehow this nice thesis does not touch on this idea (that algebraic geometric points are in generalizations related to types of that kind) which is very widely spread in model theory.

   For the generalized idea (in model theory) of the algebraic geometric idea of point = (special: prime, maximal) ideal, one expects to have use of types, e.g. Galois types and of classical type+(in+model+theory) (not to confuse with type in type theory). Somehow this nice thesis does not touch on this idea (that algebraic geometric points are in generalizations related to types of that kind) which is very widely spread in model theory.

5. • CommentRowNumber5.
   • CommentAuthorTodd_Trimble
   • CommentTimeFeb 19th 2014
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexZoran, I didn't know about all these nice articles you've written on model theory! (I guess you saw my threads on [[geometric stability theory]] and [[stability in model theory]] earlier.)

   Zoran, I didn’t know about all these nice articles you’ve written on model theory! (I guess you saw my threads on geometric stability theory and stability in model theory earlier.)

6. • CommentRowNumber6.
   • CommentAuthorDavid_Corfield
   • CommentTimeMar 5th 2014
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexAnd another Awodey et al paper, [Topos Semantics for Higher-Order Modal Logic](http://arxiv.org/abs/1403.0020). Will try to find places for both when I have a moment. I wonder if all these Awodey + supervisee papers can be homotopified.

   And another Awodey et al paper, Topos Semantics for Higher-Order Modal Logic. Will try to find places for both when I have a moment.

   I wonder if all these Awodey + supervisee papers can be homotopified.