scholarly journals Connected Lie Groupoids are Internally Connected and Integral Complete in Synthetic Differential Geometry

2017 ◽  
Author(s):  
Matthew Burke
Keyword(s):  
Differential Geometry ◽  
Lie Groupoids ◽  
Synthetic Differential Geometry

scholarly journals THE UNITY AND IDENTITY OF DECIDABLE OBJECTS AND DOUBLE-NEGATION SHEAVES

2018 ◽  
Vol 83 (04) ◽  
pp. 1667-1679
Author(s):  
MATÍAS MENNI
Keyword(s):  
Differential Geometry ◽  
Algebraic Geometry ◽  
Algebraic Topology ◽  
Full Subcategory ◽  
Sufficient Conditions ◽  
Double Negation ◽  
Synthetic Differential Geometry ◽  
Simplicial Sets ◽  
Image Position

AbstractLet ${\cal E}$ be a topos, ${\rm{Dec}}\left( {\cal E} \right) \to {\cal E}$ be the full subcategory of decidable objects, and ${{\cal E}_{\neg \,\,\neg }} \to {\cal E}$ be the full subcategory of double-negation sheaves. We give sufficient conditions for the existence of a Unity and Identity ${\cal E} \to {\cal S}$ for the two subcategories of ${\cal E}$ above, making them Adjointly Opposite. Typical examples of such ${\cal E}$ include many ‘gros’ toposes in Algebraic Geometry, simplicial sets and other toposes of ‘combinatorial’ spaces in Algebraic Topology, and certain models of Synthetic Differential Geometry.

scholarly journals Differential forms with values in groups

1982 ◽  
Vol 25 (3) ◽  
pp. 357-386 ◽  
Author(s):  
Anders Kock
Keyword(s):  
Differential Geometry ◽  
Lie Algebra ◽  
Classical Theory ◽  
Lie Group ◽  
Differential Form ◽  
Differential Forms ◽  
Synthetic Differential Geometry ◽  
Exterior Differentiation ◽  
Group Object

In the context of synthetic differential geometry, we present a notion of differential form with values in a group object, typically a Lie group or the group of all diffeomorphisms of a manifold. Natural geometric examples of such forms and the role of their exterior differentiation is given. The main result is a comparison with the classical theory of Lie algebra valued forms.

scholarly journals Material groupoids and algebroids

2018 ◽  
Vol 24 (3) ◽  
pp. 796-806 ◽  
Author(s):  
Marcelo Epstein ◽  
Manuel de León
Keyword(s):  
Differential Geometry ◽  
Continuum Mechanics ◽  
Index Of Refraction ◽  
Lie Algebroid ◽  
Lie Groupoids ◽  
Material Body ◽  
Material Defects ◽  
Lie Groupoid ◽  
Constitutive Properties

Lie groupoids and their associated algebroids arise naturally in the study of the constitutive properties of continuous media. Thus, continuum mechanics and differential geometry illuminate each other in a mutual entanglement of theory and applications. Given any material property, such as the elastic energy or an index of refraction, affected by the state of deformation of the material body, one can automatically associate to it a groupoid. Under conditions of differentiability, this material groupoid is a Lie groupoid. Its associated Lie algebroid plays an important role in the determination of the existence of material defects, such as dislocations. This paper presents a rather intuitive treatment of these ideas.

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