# ISOTROPY AND CROSSED TOPOSES

@inproceedings{Funk2012ISOTROPYAC, title={ISOTROPY AND CROSSED TOPOSES}, author={Jonathon Funk and Pieter J. W. Hofstra and Benjamin Steinberg}, year={2012} }

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- spective. Our main conceptual tool is a monad on the category of grouped toposes. Its algebras correspond to a generalized notion of crossed module, which we call a crossed topos. As an application, we present a topos-theoretic characterization and generaliza- tion of the 'Cliord, fundamental… Expand

#### 18 Citations

The Isotropy Group for the Topos of Continuous G-Sets

- Mathematics
- 2017

The objective of this thesis is to provide a detailed analysis of a new invariant for Grothendieck topoi in the special case of the topos of continuous G-sets and continuous G-equivariant maps. We… Expand

Locally anisotropic toposes

- Mathematics
- 2017

This paper continues the investigation of isotropy theory for toposes. We develop the theory of isotropy quotients of toposes, culminating in a structure theorem for a class of toposes we call… Expand

Covariant Isotropy of Grothendieck Toposes

- Mathematics
- 2021

We provide an explicit characterization of the covariant isotropy group of any Grothendieck topos, i.e. the group of (extended) inner automorphisms of any sheaf over a small site. As a consequence,… Expand

Polymorphic Automorphisms and the Picard Group

- Computer Science, Mathematics
- FSCD
- 2021

This work applies a syntactical characterization of the group of such automorphisms associated with an algebraic theory to the wider class of quasi-equational theories and proves that the isotropy group of a strict monoidal category is precisely its Picard group of invertible objects. Expand

Quotient Categories and Phases

- Mathematics, Physics
- 2018

We study properties of a category after quotienting out a suitable chosen group of isomorphisms on each object. Coproducts in the original category are described in its quotient by our new weaker… Expand

Isotropy of Algebraic Theories

- Mathematics, Computer Science
- MFPS
- 2018

The main technical result is a syntactic characterization of the isotropy group of an algebraic theory, and the usefulness of this characterization is illustrated by applying it to various concrete examples of algebraic theories. Expand

Recent developments in inverse semigroup theory

- Mathematics
- Semigroup Forum
- 2019

After reviewing aspects of the development of inverse semigroup theory, we describe an approach to studying them which views them as ‘non-commutative meet semilattices’. This leads to non-commutative… Expand

Aspects of Isotropy in Small Categories

- Mathematics
- 2017

In the paper [FHS12], the authors announce the discovery of an invariant for Grothendieck toposes which they call the isotropy group of a topos. Roughly speaking, the isotropy group of a topos… Expand

Inner automorphisms of presheaves of groups

- Mathematics
- 2021

It has been proven by Schupp and Bergman that the inner automorphisms of groups can be characterized purely categorically as those group automorphisms that can be coherently extended along any… Expand

The localic Istropy group of a topos

- Mathematics
- 2017

It has been shown by J.Funk, P.Hofstra and B.Steinberg that any Grothendieck topos T is endowed with a canonical group object, called its isotropy group, which acts functorially on every object of T.… Expand

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