Direct image functor

Direct Image Functor

The Direct Image Functor is a concept in category theory that plays a fundamental role in understanding the relationship between different categories and their associated morphisms. It is also known as the Pushforward Functor or the Forward Image Functor.

Definition

Let C and D be categories, and let F: C → D be a functor between them. Given an object X in C, the direct image functor F_*: C → D is defined as follows:

For any object X in C, the direct image functor F_* maps X to the object F_*(X) in D.

For any morphism f: X → Y in C, the direct image functor F_* maps f to the morphism F_*(f): F_*(X) → F_*(Y) in D.

The direct image functor preserves the composition of morphisms, i.e., for any morphisms f: X → Y and g: Y → Z in C, we have F_*(g ∘ f) = F_*(g) ∘ F_*(f) in D.

Properties

The direct image functor has several important properties:

1. Functoriality: The direct image functor is a functor, meaning it preserves the structure and composition of morphisms between categories.

2. Image of Objects: The direct image functor maps objects in C to objects in D. This allows us to study the behavior of objects under the functor.

3. Image of Morphisms: The direct image functor maps morphisms in C to morphisms in D. This allows us to study the behavior of morphisms under the functor.

4. Composition Preservation: The direct image functor preserves the composition of morphisms. This property ensures that the functor respects the composition structure of the category.

Applications

The direct image functor has various applications in mathematics and theoretical physics. Some notable applications include:

1. Algebraic Geometry: In algebraic geometry, the direct image functor is used to study the behavior of sheaves under morphisms between algebraic varieties. It allows us to understand how geometric objects are transformed under different mappings.

2. Differential Geometry: In differential geometry, the direct image functor is used to study the behavior of vector fields and differential forms under smooth maps between manifolds. It provides a way to transport geometric structures from one manifold to another.

3. Topology: In topology, the direct image functor is used to study the behavior of continuous maps between topological spaces. It allows us to understand how topological properties are preserved or transformed under different mappings.

See Also

• Functor
• Category Theory
• Algebraic Geometry
• Differential Geometry
• Topology

References

1. Saunders Mac Lane, "Categories for the Working Mathematician", Springer, 1998. 2. Emily Riehl, "Category Theory in Context", Dover Publications, 2016.