Hom

Definition

The term 'Hom' refers to a functor that represents the set of morphisms between two objects in a category. It serves as a fundamental building block in category theory, providing insights into the relationships between objects, particularly when studying their structure and properties. This concept is essential in understanding how different algebraic structures can interact with one another through mappings or functions.

5 Must Know Facts For Your Next Test

1. 'Hom' is denoted as Hom(A, B) for two objects A and B, representing the set of morphisms from A to B.
2. In many algebraic contexts, Hom can be used to define important properties like injectivity and surjectivity of morphisms.
3. 'Hom' is contravariant when applied to dual objects, reflecting how morphisms interact with structures in reverse.
4. The Hom functor is a key player in defining and calculating derived functors like Ext, linking them to more complex homological concepts.
5. In addition to sets of morphisms, Hom can also have additional structure such as topological spaces or modules, enriching its role in various mathematical theories.

Review Questions

• How does the 'Hom' functor illustrate the relationships between objects in a category?
  • 'Hom' serves as a bridge between two objects by capturing all the morphisms that connect them. This showcases not only direct mappings but also reflects on how these mappings interact within the broader structure of the category. By analyzing 'Hom' sets, one can discern properties about the objects themselves, such as whether they can be transformed into each other through various morphisms.
• Discuss the significance of 'Hom' in the context of derived functors such as Ext.
  • 'Hom' is crucial for defining derived functors like Ext because it lays the foundation for understanding how complex relationships arise from simpler ones. The Ext functor, which measures extensions of modules, essentially extends the concept of 'Hom' into a higher dimension, allowing mathematicians to investigate how far apart two objects are and how one can be constructed from another. This extension enables deeper insights into homological algebra and its applications.
• Evaluate how 'Hom' interacts with different structures in category theory and provide an example illustrating this interaction.
  • 'Hom' interacts with various structures in category theory by accommodating additional properties such as commutativity in abelian categories or continuity in topological spaces. For instance, consider the category of abelian groups; here 'Hom' not only provides morphisms but also respects group operations. An example would be examining Hom(Z, G) for an abelian group G, which captures all group homomorphisms from integers Z to G, showcasing how underlying algebraic structures affect mappings and properties.

Related terms

Functor: A mapping between categories that preserves the structure of categories, sending objects to objects and morphisms to morphisms.

Morphism: An abstraction representing a structure-preserving map between two mathematical objects in category theory.

Natural Transformation: A way of transforming one functor into another while preserving the structure of the categories involved.