A full functor is a type of functor between categories that maps morphisms in a way that preserves the entire structure of the category, meaning it takes every morphism from the source category to a morphism in the target category. If for any two objects in the source category, every morphism between them has a corresponding morphism in the target category, then the functor is considered full. This concept connects closely to the nature of relationships between categories and provides insights into how they relate to representable functors and embeddings.
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A functor is called full if for any two objects A and B in the source category, every morphism from A to B is mapped to a morphism in the target category.
Full functors can be useful for establishing connections between different categories by ensuring that all relationships (morphisms) are preserved.
If a functor is both full and faithful, it not only preserves all morphisms but also ensures that distinct morphisms remain distinct when mapped.
Full functors can help identify whether certain properties or structures are maintained across different categories, making them valuable in categorical analysis.
In contexts involving representable functors, full functors can provide deeper insights into how objects and morphisms interact under different categorical frameworks.
Review Questions
How does a full functor ensure the preservation of morphisms between categories, and what implications does this have for understanding categorical relationships?
A full functor ensures the preservation of morphisms by mapping every morphism between objects in the source category to a corresponding morphism in the target category. This means that all structural relationships within the source are maintained, allowing for a more profound understanding of how different categories relate to each other. By retaining these relationships, full functors enable mathematicians to analyze the properties of categories more effectively, as they can see which morphisms are preserved across categorical mappings.
Discuss the significance of a full functor being both full and faithful, and how this dual property enhances its role in categorical analysis.
When a functor is both full and faithful, it not only ensures that all morphisms from one category are mapped accurately to another but also that distinct morphisms remain distinct. This dual property enhances its role in categorical analysis because it allows for a more precise understanding of how categories are related. A full and faithful functor can be seen as an embedding of one category into another, revealing deeper insights into their structures and providing powerful tools for examining equivalences and isomorphisms between categories.
Evaluate how the concept of full functors relates to representable functors and their applications within the framework of the Yoneda embedding.
The concept of full functors is closely tied to representable functors, especially within the framework established by the Yoneda embedding. Full functors facilitate an understanding of how objects can be represented through natural transformations, which are central to representable functors. By utilizing full functors, one can explore how various objects in one category can be mapped and represented in another, allowing for a richer exploration of structures and relationships. This connection underscores the importance of categorical approaches in abstract algebra and topology, showcasing how full functors contribute significantly to deeper theoretical insights.
A structure-preserving mapping between two categories that assigns to each object in one category an object in another category and to each morphism a morphism, maintaining identity and composition.
Essentially Surjective Functor: A functor is essentially surjective if every object in the target category is isomorphic to an object in the image of the functor, meaning it covers all relevant structures in the target.
A fundamental result in category theory that states that natural transformations correspond to morphisms in a specific way, highlighting the importance of representable functors and their relationship to full functors.