Summary Categorifying Group Theory Hoang Xuan Sinhs Thesis arxiv.org
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Hoang Xuan Sinh's thesis explores Gr-categories, which are monoidal categories with inverses for all objects and morphisms.
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Key Points
- Hoang Xuan Sinh's thesis explores the concept of 'Gr-categories', also known as '2-groups'
- Gr-categories are monoidal categories where all objects and morphisms have inverses
- Monoidal categories were first discussed by Benabou and Mac Lane in 1963
- Hoang Xuan Sinh proved that every Gr-category is equivalent to a strict one
- The concept of categorifying group theory is discussed in the thesis
- The thesis explores the geometry of 2-categories and their classifying spaces
- Homotopy equivalence and the classification of n-types are also mentioned
- Monoidal natural transformations and equivalence for Gr-categories and Pic-categories are defined
Summaries
18 word summary
Hoang Xuan Sinh's thesis focuses on Gr-categories, or 2-groups, monoidal categories with inverses for all objects and morphisms.
36 word summary
Hoang Xuan Sinh's thesis explores the concept of Gr-categories, also known as 2-groups, which are monoidal categories where all objects and morphisms have inverses. She studied under Alexander Gro during the Vietnam War, writing her thesis
482 word summary
Hoang Xuan Sinh's thesis, written during the Vietnam War, explores the concept of 'Gr-categories', also known as '2-groups', which are monoidal categories where all objects and morphisms have inverses. She studied under Alexander Gro
Hoang Xuan Sinh worked on her thesis while teaching during the day and wrote in French with her distant teacher's guidance. There were concerns about her ability to return after going to France to defend her thesis, but Lady Ha Th? Qu?
The concept of Gr-categories, which are a generalization of monoidal categories, was introduced by Hoang Xuan Sinh in her thesis. Monoidal categories were first discussed by Benabou and Mac Lane in 1963, but they were
Monoidal bicategories, monoidal tricategories, and other similar concepts are important in group theory. The unitors and associators, along with the associahedra, are crucial to Mac Lane's coherence theorem. Hoang Xuan Sin
In this excerpt, the author discusses the concept of categorifying group theory. They explain that in a Gr-category, the associator must obey the pentagon identity and can be represented as a map from G3 to A. They also mention that not
In 1978, Hoang Xuan Sinh proved that every Gr-category is equivalent to a strict one. However, this seems contradictory because every Gr-category is also equivalent to a skeletal one. The key difference is that a strict Gr-category is
The Gr-category discussed in this document is characterized by the fact that all objects are isomorphic. A strict Gr-category may have a nontrivial Sinh invariant if it is not skeletal. Hoang Xuan Sinh's thesis builds on the discovery
The Picard group of a topological space X depends on both its topology and its holomorphic structure. It is best understood as a topological group with connected components that are projective algebraic varieties. Gr-categories, which are a categorification of
Homotopy equivalence is defined as the existence of maps between two spaces that can be continuously deformed into each other. Classifying spaces up to homotopy equivalence is complicated, but it becomes more manageable when focusing on connected CW complexes, which are
In his work, Hoang Xuan Sinh proposed the idea that n-types can be classified up to homotopy equivalence by algebraic structures called n-groupoids. An n-groupoid consists of objects, morphisms, 2-morphisms
A monoidal natural transformation is defined as a monoidal natural isomorphism. Equivalence for Gr-categories and Pic-categories is defined based on equivalence as monoidal categories and symmetric monoidal categories, respectively. Theorems 13 and 14 state
In this document, Hoang Xuan Sinh's thesis on categorifying group theory is referenced. The thesis explores the geometry of 2-categories and their classifying spaces, and it cites relevant literature on the topic. Some of the key works mentioned
The summary of the text excerpt includes various citations and links to academic papers and resources related to group theory and category theory. These include papers by Andre Joyal, Saunders Mac Lane, Fernando Muro and Andrew Tonks, Thomas Nikolaus, Urs Sch