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