Topoi are special categories that generalize the category of sets. They have finite limits and colimits, are cartesian closed, and contain a subobject classifier. These properties make topoi powerful for studying logic and geometry in category theory.
Topoi differ from cartesian closed categories by having a subobject classifier. Examples include the category of sets, groups, and sheaf topoi. The subobject classifier allows for internalizing logic and reasoning within the topos framework.
Topoi: Definition and Properties
Properties of topoi
- Has all finite limits and colimits enables the construction of complex objects and morphisms from simpler ones
- Cartesian closed allows for the existence of function spaces and higher-order functions within the category
- Contains a subobject classifier that generalizes the concept of characteristic functions and allows for the internalization of logic
- Generalizes the category of sets (Set) and provides a framework for studying logic and geometry within category theory
Topoi vs cartesian closed categories
- Every topos is a cartesian closed category satisfies the properties of having a terminal object, products, and exponential objects
- Not every cartesian closed category is a topos lacks the additional requirement of a subobject classifier
- In a cartesian closed category:
- Terminal object serves as a "single-point space" and is a target for unique morphisms from any object
- Product of two objects represents their Cartesian product and allows for the construction of pairs
- Exponential object $B^A$ represents the set of morphisms from $A$ to $B$ and allows for the construction of function spaces
Examples of topoi
- Category of sets (Set):
- Objects are sets and morphisms are functions between sets
- Serves as the prototypical example of a topos and provides a foundation for understanding the general properties of topoi
- Category of groups (Grp):
- Objects are groups and morphisms are group homomorphisms
- Demonstrates that algebraic structures can form a topos when they satisfy the required properties
- Sheaf topoi:
- Constructed from sheaves on a topological space or a site (a category with a Grothendieck topology)
- Allows for the study of local properties of spaces and is important in algebraic geometry and the theory of schemes
Subobject classifier in topoi
- Special object denoted as $\Omega$ that generalizes the two-element set ${0, 1}$ in Set
- One-to-one correspondence between subobjects of an object $A$ and morphisms from $A$ to $\Omega$ allows for the classification of subobjects
- Morphisms to $\Omega$ can be thought of as "truth values" or "characteristic functions" for subobjects enables the internalization of logic within the topos
- Allows for the study of logic and reasoning within the framework of category theory by providing a way to represent propositions and their truth values