155 research outputs found
The Galois group of a stable homotopy theory
To a "stable homotopy theory" (a presentable, symmetric monoidal stable
-category), we naturally associate a category of finite \'etale algebra
objects and, using Grothendieck's categorical machine, a profinite group that
we call the Galois group. We then calculate the Galois groups in several
examples. For instance, we show that the Galois group of the periodic
-algebra of topological modular forms is trivial and that
the Galois group of -local stable homotopy theory is an extended version
of the Morava stabilizer group. We also describe the Galois group of the stable
module category of a finite group. A fundamental idea throughout is the purely
categorical notion of a "descendable" algebra object and an associated analog
of faithfully flat descent in this context.Comment: 93 pages. To appear in Advances in Mathematic
Anti-Foundational Categorical Structuralism
The aim of this dissertation is to outline and defend the view here dubbed âanti-foundational categorical structuralismâ (henceforth AFCS). The program put forth is intended to provide an answer the question âwhat is mathematics?â. The answer here on offer adopts the structuralist view of mathematics, in that mathematics is taken to be âthe science of structureâ expressed in the language of category theory, which is argued to accurately capture the notion of a âstructural propertyâ. In characterizing mathematical theorems as both conditional and schematic in form, the program is forced to give up claims to securing the truth of its theorems, as well as give up a semantics which involves reference to special, distinguished âmathematical objectsâ, or which involves quantification over a fixed domain of such objects. One who wishesâcontrary to the AFCS viewâto inject mathematics with a âstandardâ semantics, and to provide a secure epistemic foundation for the theorems of mathematics, in short, one who wishes for a foundation for mathematics, will surely find this view lacking. However, I argue that a satisfactory development of the structuralist view, couched in the language of category theory, accurately represents our best understanding of the content of mathematical theorems and thereby obviates the need for any foundational program
Noncommutative Geometry in the Framework of Differential Graded Categories
In this survey article we discuss a framework of noncommutative geometry with
differential graded categories as models for spaces. We outline a construction
of the category of noncommutative spaces and also include a discussion on
noncommutative motives. We propose a motivic measure with values in a motivic
ring. This enables us to introduce certain zeta functions of noncommutative
spaces.Comment: 19 pages. Minor corrections and one reference added; to appear in the
proceedings volume of AGAQ Istanbul, 200
A brief review of abelian categorifications
This article contains a review of categorifications of semisimple
representations of various rings via abelian categories and exact endofunctors
on them. A simple definition of an abelian categorification is presented and
illustrated with several examples, including categorifications of various
representations of the symmetric group and its Hecke algebra via highest weight
categories of modules over the Lie algebra sl(n). The review is intended to
give non-experts in representation theory who are familiar with the topological
aspects of categorification (lifting quantum link invariants to homology
theories) an idea for the sort of categories that appear when link homology is
extended to tangles.Comment: latex, 35 pages, 4 eps figure
An Abstract Approach to Consequence Relations
We generalise the Blok-J\'onsson account of structural consequence relations,
later developed by Galatos, Tsinakis and other authors, in such a way as to
naturally accommodate multiset consequence. While Blok and J\'onsson admit, in
place of sheer formulas, a wider range of syntactic units to be manipulated in
deductions (including sequents or equations), these objects are invariably
aggregated via set-theoretical union. Our approach is more general in that
non-idempotent forms of premiss and conclusion aggregation, including multiset
sum and fuzzy set union, are considered. In their abstract form, thus,
deductive relations are defined as additional compatible preorderings over
certain partially ordered monoids. We investigate these relations using
categorical methods, and provide analogues of the main results obtained in the
general theory of consequence relations. Then we focus on the driving example
of multiset deductive relations, providing variations of the methods of matrix
semantics and Hilbert systems in Abstract Algebraic Logic
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