Isotropy of Algebraic Theories

@article{Hofstra2018IsotropyOA,
  title={Isotropy of Algebraic Theories},
  author={Pieter J. W. Hofstra and Jason Parker and P. Scott},
  journal={Electr. Notes Theor. Comput. Sci.},
  year={2018},
  volume={341},
  pages={201-217}
}
Abstract To every small category or topos one may associate its isotropy group, which is an algebraic invariant capturing information about the behaviour of automorphisms. We investigate this invariant in the particular situation of algebraic theories, thus obtaining a group-theoretic invariant of algebraic theories. This invariant encodes a notion of inner automorphism relative to the theory. Our main technical result is a syntactic characterization of the isotropy group of an algebraic theory… Expand

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References

SHOWING 1-10 OF 17 REFERENCES
ISOTROPY AND CROSSED TOPOSES
In memory of Hugh Millington Abstract. Motivated by constructions in the theory of inverse semigroups and etale groupoids, we dene and investigate the concept of isotropy from a topos-theoretic per-Expand
An inner automorphism is only an inner automorphism, but an inner endomorphism can be something strange
The inner automorphisms of a group G can be characterized within the category of groups without reference to group elements: they are precisely those au- tomorphisms of G that can be extended, in aExpand
Scheme representation for first-order logic
TLDR
This dissertation aims to shrink the 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. Expand
Core algebra revisited
  • P. Freyd
  • Computer Science, Mathematics
  • Theor. Comput. Sci.
  • 2007
Reynolds's work in parametric polymorphism when specialized to a particular example gives rise to the notion of the core of a category and its associated equational theory of core algebras.
Some aspects of the SD-world
We survey a few of the many results now known about the self-distributivity law and selfdistributive structures, with a special emphasis on the associated word problems and the algorithms solvingExpand
Phd by thesis
Deposits of clastic carbonate-dominated (calciclastic) sedimentary slope systems in the rock record have been identified mostly as linearly-consistent carbonate apron deposits, even though mostExpand
Higher Isotropy
  • Accepted for publication in Theory and Applications of Categories,
  • 2018
Theories, Sites, Toposes
Toposes
  • Oxford University Press,
  • 2017
Theory and Applications of Categories 26
  • pp. 660–709,
  • 2012
...
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