Higher Topos Theory (AM-170)

2009 ◽  
Author(s):  
Jacob Lurie
Keyword(s):  
Topos Theory

Model Theory: Geometrical and Set-Theoretic Aspects and Prospects

2003 ◽  
Vol 9 (2) ◽  
pp. 197-212 ◽  
Author(s):  
Angus Macintyre
Keyword(s):  
Model Theory ◽  
Category Theory ◽  
Theoretic Model ◽  
Topos Theory ◽  
Sheaf Theory ◽  
Special Cases ◽  
Algebraic Spaces ◽  
The Subject ◽  
Near Future ◽  
Modern Mathematics

I see model theory as becoming increasingly detached from set theory, and the Tarskian notion of set-theoretic model being no longer central to model theory. In much of modern mathematics, the set-theoretic component is of minor interest, and basic notions are geometric or category-theoretic. In algebraic geometry, schemes or algebraic spaces are the basic notions, with the older “sets of points in affine or projective space” no more than restrictive special cases. The basic notions may be given sheaf-theoretically, or functorially. To understand in depth the historically important affine cases, one does best to work with more general schemes. The resulting relativization and “transfer of structure” is incomparably more flexible and powerful than anything yet known in “set-theoretic model theory”.It seems to me now uncontroversial to see the fine structure of definitions as becoming the central concern of model theory, to the extent that one can easily imagine the subject being called “Definability Theory” in the near future.Tarski's set-theoretic foundational formulations are still favoured by the majority of model-theorists, and evolution towards a more suggestive language has been perplexingly slow. None of the main texts uses in any nontrivial way the language of category theory, far less sheaf theory or topos theory. Given that the most notable interactions of model theory with geometry are in areas of geometry where the language of sheaves is almost indispensable (to the geometers), this is a curious situation, and I find it hard to imagine that it will not change soon, and rapidly.

scholarly journals Goodwillie's calculus of functors and higher topos theory

2018 ◽  
Vol 11 (4) ◽  
pp. 1100-1132 ◽  
Author(s):  
M. Anel ◽  
G. Biedermann ◽  
E. Finster ◽  
A. Joyal
Keyword(s):  
Topos Theory ◽  
Calculus Of Functors

Topos Theory

1996 ◽  
pp. 501-528 ◽  
Author(s):  
S. MacLane ◽  
I. Moerdijk
Keyword(s):  
Topos Theory

scholarly journals Measure theory over boolean toposes

2016 ◽  
Vol 163 (1) ◽  
pp. 1-21
Author(s):  
SIMON HENRY
Keyword(s):  
Time Evolution ◽  
Noncommutative Geometry ◽  
Principal Bundle ◽  
Von Neumann Algebras ◽  
Measure Theory ◽  
Topos Theory ◽  
Von Neumann ◽  
Theoretic Version

AbstractIn this paper we develop a notion of measure theory over boolean toposes reminiscent of the theory of von Neumann algebras. This is part of a larger project to study relations between topos theory and noncommutative geometry. The main result is a topos theoretic version of the modular time evolution of von Neumann algebras which take the form of a canonical $\mathbb{R}^{>0}$-principal bundle over any integrable locally separated boolean topos.

Simply Connected Limits

1990 ◽  
Vol 42 (4) ◽  
pp. 731-746 ◽  
Author(s):  
Robert Paré
Keyword(s):  
Left Adjoint ◽  
Topos Theory ◽  
Essential Ingredient ◽  
Original Definition ◽  
The Subject ◽  
Definition Of ◽  
Simply Connected ◽  
Geometric Morphisms ◽  
Exact Categories

The importance of finite limits in completeness conditions has been long recognized. One has only to consider elementary toposes, pretoposes, exact categories, etc., to realize their ubiquity. However, often pullbacks suffice and in a sense are more natural. For example it is pullbacks that are the essential ingredient in composition of spans, partial morphisms and relations. In fact the original definition of elementary topos was based on the notion of partial morphism classifier which involved only pullbacks (see [6]). Many constructions in topos theory, involving left exact functors, such as coalgebras on a cotriple and the gluing construction, also work for pullback preserving functors. And pullback preserving functors occur naturally in the subject, e.g. constant functors and the Σα. These observations led Rosebrugh and Wood to introduce partial geometric morphisms; functors with a pullback preserving left adjoint [9]. Other reasons led Kennison independently to introduce the same concept under the name semi-geometric functors [5].

scholarly journals Fuzzy Set Theory and Topos Theory

1986 ◽  
Vol 29 (4) ◽  
pp. 501-508 ◽  
Author(s):  
Michael Barr
Keyword(s):  
Fuzzy Sets ◽  
Set Theory ◽  
Fuzzy Set ◽  
Fuzzy Set Theory ◽  
Topos Theory

AbstractThe relation between the categories of Fuzzy Sets and that of Sheaves is explored and the precise connection between them is explicated. In particular, it is shown that if the notion of fuzzy sets is further fuzzified by making equality (as well as membership) fuzzy, the resultant categories are indeed toposes.

scholarly journals A Brief Introduction to Algebraic Set Theory

2008 ◽  
Vol 14 (3) ◽  
pp. 281-298 ◽  
Author(s):  
Steve Awodey
Keyword(s):  
Set Theory ◽  
Algebraic Models ◽  
Topos Theory ◽  
Algebraic Set ◽  
Logical Constraints ◽  
Models Of Set Theory

AbstractThis brief article is intended to introduce the reader to the field of algebraic set theory, in which models of set theory of a new and fascinating kind are determined algebraically. The method is quite robust, applying to various classical, intuitionistic, and constructive set theories. Under this scheme some familiar set theoretic properties are related to algebraic ones, while others result from logical constraints. Conventional elementary set theories are complete with respect to algebraic models, which arise in a variety of ways, such as topologically, type-theoretically, and through variation. Many previous results from topos theory involving realizability, permutation, and sheaf models of set theory are subsumed, and the prospects for further such unification seem bright.

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