Full Functor

Definition

A full functor is a type of functor between two categories that maps morphisms (arrows) in such a way that every morphism in the target category is the image of a morphism in the source category. This means that the functor not only preserves the structure of the categories but also ensures that it captures all possible morphisms, making it 'full' in the sense of being complete with respect to the arrows between objects.

5 Must Know Facts For Your Next Test

1. A full functor ensures that for every morphism in the target category, there exists at least one corresponding morphism in the source category.
2. Full functors are essential in demonstrating certain properties of categorical structures, like adjunctions and equivalences.
3. If a functor is both full and faithful, it preserves not only the structure of morphisms but also the distinctness of those morphisms.
4. Full functors can be used to show that certain constructions within category theory, such as limits and colimits, behave well under mappings between categories.
5. Understanding full functors helps to clarify how various mathematical concepts can be related through their categorical representations.

Review Questions

• How does a full functor relate to the concepts of morphisms and their preservation in category theory?
  • A full functor is crucial because it ensures that every morphism in the target category corresponds to at least one morphism in the source category. This relationship emphasizes how full functors preserve not just objects but also all possible arrows between them. By capturing all the morphisms, full functors help maintain the integrity and structure within categorical mappings, allowing for more profound insights into mathematical relationships.
• Discuss the implications of a functor being both full and faithful. How does this enhance our understanding of categorical structures?
  • When a functor is both full and faithful, it means it preserves both the structure of morphisms and their distinctness. This duality provides a more robust framework for understanding categorical relationships because it guarantees that we can fully translate information between categories without loss. This property enhances our understanding by allowing us to treat one category as an accurate representation of another, leading to better insights into their interrelations.
• Evaluate how full functors can affect the construction and properties of limits and colimits in category theory.
  • Full functors significantly influence how limits and colimits are constructed within category theory because they ensure that these constructs maintain their properties when moving between categories. Since full functors capture all necessary morphisms, they enable us to analyze how limits and colimits behave under these mappings, which is essential for understanding more complex categorical phenomena. As such, they play a vital role in ensuring that foundational concepts like universal properties hold true across different settings.

Related terms

Functor:

A functor is a mapping between two categories that preserves the structure of those categories, specifically objects and morphisms.

Natural Transformation: A natural transformation is a way of transforming one functor into another while preserving the categorical structure, allowing for the comparison of different functors.

Injective Functor: An injective functor is a type of functor where distinct morphisms in the source category map to distinct morphisms in the target category, which may or may not be full.