Full Record

Author | McBride, Aaron |

Title | Grothendieck Group Decategorifications and Derived Abelian Categories |

URL | http://hdl.handle.net/10393/33000 |

Publication Date | 2015 |

Date Accessioned | 2015-10-08 19:56:18 |

University/Publisher | University of Ottawa |

Abstract | The Grothendieck group is an interesting invariant of an exact category. It induces a decategorication from the category of essentially small exact categories (whose morphisms are exact functors) to the category of abelian groups. Similarly, the triangulated Grothendieck group induces a decategorication from the category of essentially small triangulated categories (whose morphisms are triangulated functors) to the category of abelian groups. In the case of an essentially small abelian category, its Grothendieck group and the triangulated Grothendieck group of its bounded derived category are isomorphic as groups via a natural map. Because of this, homological algebra and derived functors become useful in surprising ways. This thesis is an expository work that provides an overview of the theory of Grothendieck groups with respect to these decategorications. |

Subjects/Keywords | additive categories; abelian categories; exact categories; triangulated categories; Grothendieck groups; category theory; cochain complexes; homotopy category; cohomology; bounded derived category; derived functors |

Language | en |

Country of Publication | ca |

Record ID | handle:10393/33000 |

Repository | ottawa |

Date Indexed | 2018-01-03 |

Issued Date | 2015-01-01 00:00:00 |

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…and triangulated
categories. Its purpose is to make the work as accessible as possible, and only some
basic *category* *theory* and some undergraduate abstract algebra is assumed. Since
references will be made to this chapter in the chapters that follow…

…readers with a
background in *category* *theory* may choose to go directly to Chapter 2.
We will use the convention of writing X ∈ C in place of X ∈ Ob C. A *category*
C is essentially small if there exists a small *category* D and an equivalence C → D.
To avoid…

…List of Symbols
viii
Ob C The class of objects of a *category* C.
Prj A The collection of projective objects in an abelian *category* A; see Definition 1.2.21.
Quis A The collection of quasi-isomorphisms in Mor A where A is an abelian *category*.
RF The…

…total right derived functor of a left exact functor; see Definition 4.2.2.
Rn F The n-th right derived functor of a left exact functor; see Definition 4.2.3.
R-mod The *category* of left R-modules for some ring with unity R; see Example 1.1.2.
R-modfg The…

…*category* of finitely generated left R-modules for some ring with unity R;
see Example 1.3.12.
X mono The collection of all monomorphism with codomain X in some *category* that is
clear from the context; see Definition 1.2.2 (∼ on X mono is defined there…

…x29;.
X k Tthe k-shift functor on an object X in the graded *category* of an abelian *category*; see Definition 3.1.5.
(X • , dX ) k The k-shift on an object (X • , dX ) in Kom∗ A; see Definition 3.1.11.
Introduction
Due to the all…

…encompassing nature of categories in modern mathematics, most categories are simply unwieldy beasts. For example, the task of classifying the indecomposable objects of an additive *category* up to isomorphism is a natural problem but
typically a daunting feat…

…recently gained a wide interest is decategorification. In a nutshell, decategorification is the study of set-theoretic invariants
for *category*-theoretic notions; its philosophy is that an invariant for an object, a
morphism, a *category*, a functor, or a…