Equalizers

Definition

Equalizers are morphisms in category theory that capture the notion of the uniqueness of an object relative to two parallel morphisms. In a category, an equalizer of two morphisms $f: A \to B$ and $g: A \to B$ is an object $E$ together with a morphism $e: E \to A$ such that $f \circ e = g \circ e$, and this property uniquely characterizes the object up to isomorphism. They play a crucial role in defining limits and understanding completeness and cocompleteness within categories, as well as providing a foundation for exploring the relationships between objects in elementary topoi.

5 Must Know Facts For Your Next Test

1. Equalizers can be thought of as a way to recover 'common behavior' between two morphisms by identifying elements that map to the same target.
2. In the category of sets, the equalizer corresponds to the set of elements in A that are sent to the same element in B by f and g.
3. Equalizers are important for defining the structure of limits in categories, particularly because they help establish how objects relate through morphisms.
4. The concept of equalizers extends to other structures in category theory, allowing for a broader understanding of how objects can interact via morphisms.
5. In elementary topoi, equalizers contribute to understanding logical relationships and implications among different types of objects and morphisms.

Review Questions

• How do equalizers help in understanding the structure of limits within a category?
  • Equalizers are crucial for defining limits because they identify common structures among morphisms. By capturing the elements that satisfy multiple conditions simultaneously, equalizers help construct limit objects, which represent the most general way these morphisms interact. This uniqueness property provided by equalizers ensures that we can characterize limits consistently across different categories.
• Discuss the role of equalizers in establishing relationships between objects in elementary topoi.
  • In elementary topoi, equalizers provide a framework for understanding logical implications and relationships between various types of objects. They allow us to define equivalences among morphisms based on shared targets, thus enabling a rich interplay between different structures. This capability is essential for exploring concepts like subobjects and their interactions within the categorical context, fostering deeper insights into logical frameworks.
• Evaluate how equalizers relate to other categorical constructs like coproducts and universal properties within both complete and cocomplete categories.
  • Equalizers serve as fundamental building blocks within both complete and cocomplete categories by providing insights into how objects interact through morphisms. While coproducts focus on combining objects, equalizers focus on commonality among them. Both constructs highlight universal properties—equalizers showcase uniqueness while coproducts reflect a form of aggregation. Together, they deepen our understanding of categorical structures by illustrating diverse ways objects can be related and interact through morphisms.

Related terms

Limits:

Limits are a way to capture the idea of 'universal properties' in categories, such as cones and pullbacks, that generalize concepts from set theory.

Coproducts:

Coproducts are the categorical dual of products and represent the 'sum' or 'disjoint union' of objects within a category.

Universal Property:

A universal property characterizes an object in terms of its relationships with other objects and morphisms, often encapsulating the essence of limits or colimits.