#### Date of Award

6-2010

#### Document Type

Union College Only

#### Degree Name

Bachelor of Science

#### Department

Mathematics

#### First Advisor

Susan Nieﬁeld

#### Language

English

#### Keywords

topos, theory, category, examples, review

#### Abstract

A topos is a category satisfying certain axioms. By satisfying the topos axioms, a category can be treated as the category of sets for many op-erations and constructions. Some mathematicians view topos theory as a generalized set theory, while others use topos theory for topology, logic, and even physics. We begin with a brief historical sketch of the creation of topos theory, observing both the categorical and geometric interests that eventually com-bined to ﬁnalize today’s topos theory. This is followed by a brief category theory review in order to remind the reader of terms and constructions that will be used in the topos theory section. In chapter 3 we state what a category must satisfy to be considered a topos. We prove that the category of sets is a topos, as well as some other classic topos examples. Some special properties of toposes arise from these proofs. Then, in chapter 4, we deﬁne morphisms between toposes, and give examples of morphisms dealing with the topos examples from chapter 3.

#### Recommended Citation

Zawodniak, Matthew D., "An introduction to topos theory" (2010). *Honors Theses*. 1247.

https://digitalworks.union.edu/theses/1247