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A Tale of Two Set Theories
We describe the relationship between two versions of Tarski-Grothendieck set
theory: the first-order set theory of Mizar and the higher-order set theory of
Egal. We show how certain higher-order terms and propositions in Egal have
equivalent first-order presentations. We then prove Tarski's Axiom A (an axiom
in Mizar) in Egal and construct a Grothendieck Universe operator (a primitive
with axioms in Egal) in Mizar
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
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