Hom-functor

Definition

The hom-functor is a functor that assigns to each object in a category the set of morphisms (arrows) from a fixed object to that object. It captures the idea of relationships between objects through their morphisms, making it a crucial tool for understanding how different objects interact within a category. This concept connects deeply with faithful, full, and essentially surjective functors, as these properties can describe how well hom-functors reflect the structure of morphisms in categories.

5 Must Know Facts For Your Next Test

1. The hom-functor is typically denoted as $$ ext{Hom}(A, -)$$, where $$A$$ is the fixed object and $$-$$ represents any object in the category.
2. When dealing with sets, the hom-functor maps each object to the set of morphisms from $$A$$ to that object, illustrating how many ways there are to map from one object to another.
3. If a functor is full, it means that the hom-functor reflects all morphisms between objects, capturing complete information about relationships in the category.
4. A hom-functor can be seen as representing the 'structure' of a category by focusing on the relationships defined by morphisms instead of just the objects themselves.
5. In categories like $$ extbf{Set}$$, where objects are sets and morphisms are functions, the hom-functor becomes particularly useful in understanding function spaces.

Review Questions

• How does the hom-functor illustrate the relationships between objects in a category?
  • The hom-functor illustrates relationships by mapping each object to the set of morphisms from a fixed object to that object. This mapping enables us to see not only what objects exist but also how they are interconnected through morphisms. By examining these relationships, we gain insights into the structure of the category and how different objects interact with one another.
• Discuss how properties like being faithful or full relate to the hom-functor.
  • Being faithful means that the hom-functor injectively maps morphisms from one object to another, ensuring that distinct morphisms are preserved. If a functor is full, it means that every morphism in the target category can be realized by a morphism coming from the source category via the hom-functor. These properties highlight how well the hom-functor captures the essence of morphisms within the categories involved, reflecting their relationships accurately.
• Evaluate the importance of hom-functors in connecting category theory concepts and their applications in other mathematical areas.
  • Hom-functors play a critical role in bridging various concepts in category theory with other mathematical disciplines by providing a framework for understanding how objects relate through morphisms. This connection allows mathematicians to apply categorical thinking to areas such as topology, algebra, and functional analysis. By analyzing morphism sets, researchers can uncover deeper insights about continuity, transformation properties, and structural relations across different branches of mathematics.

Related terms

Functor: A structure-preserving map between categories that associates objects and morphisms in one category to objects and morphisms in another.

Natural Transformation: A way of transforming one functor into another while preserving the categorical structure, involving a collection of morphisms that fit together nicely.

Set: A fundamental concept in mathematics, often used to define collections of distinct objects and serves as the primary type for sets of morphisms in category theory.