Corpus ID: 124634880


  author={Jonathon Funk and Pieter J. W. Hofstra and Benjamin Steinberg},
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
The Isotropy Group for the Topos of Continuous G-Sets
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. WeExpand
Locally anisotropic toposes
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 callExpand
Covariant Isotropy of Grothendieck Toposes
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
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
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 weakerExpand
Isotropy of Algebraic Theories
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
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-commutativeExpand
Aspects of Isotropy in Small Categories
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 toposExpand
Inner automorphisms of presheaves of groups
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 anyExpand
The localic Istropy group of a topos
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


The Universal Covering of an Inverse Semigroup
It is proved that the fundamental group of coincides with the maximum group image of T in terms of the universal locally constant covering of its classifying topos. Expand
Sketches of an Elephant: A Topos Theory Compendium Volume 1
Pseudo-distributive Laws
The definition of a pseudo-distributive law between pseudo-monads is developed, and it is shown how the definition and the main theorems about it may be used to model several such structures simultaneously. Expand
Sheaves in geometry and logic: a first introduction to topos theory
This text presents topos theory as it has developed from the study of sheaves. Sheaves arose in geometry as coefficients for cohomology and as descriptions of the functions appropriate to variousExpand
Inverse Semigroups, the Theory of Partial Symmetries
Introduction to inverse semigroups elementary properties of inverse semigroups Ehresmann's maximum enlargement theorem presentations of inverse monoids inverse semigroups and formal languages theExpand
A 2-Categories Companion
This paper is a rather informal guide to some of the basic theory of 2-categories and bicategories, including notions of limit and colimit, 2-dimensional universal algebra, formal category theory,Expand
Lectures on etale groupoids, inverse semigroups and quantales
complete pseudogroups L∨ bb
Categories for the Working Mathematician
I. Categories, Functors and Natural Transformations.- 1. Axioms for Categories.- 2. Categories.- 3. Functors.- 4. Natural Transformations.- 5. Monics, Epis, and Zeros.- 6. Foundations.- 7. LargeExpand
Categorical Logic and Type Theory
  • B. Jacobs
  • Computer Science, Mathematics
  • Studies in logic and the foundations of mathematics
  • 2001
This chapter discusses fibred category theory, which is concerned with the role of type theory in the development of categorical identity and its role in the construction of types. Expand
Ordered groupoids and etendues
Kock et Moerdijk ont montre que chaque etendue est engendree par un site dont tout morphisme est monic. Dans cet article nous donnons une caracterisation alternative des etendues en termes deExpand