Hom functor

In category theory, mathematicians study the relationships between objects. These relationships are called morphisms, which we often visualize as arrows. A hom-functor is a mathematical tool that transforms these morphisms into sets. Specifically, it maps objects and morphisms from a category into the category of sets, known as Set. This process allows mathematicians to use the properties of sets to understand the structure of more complex categories.

To understand how a hom-functor works, we must look at how it handles morphisms. Consider a locally small category, which is a category where the collections of morphisms between any two objects are actual sets. If we fix one object, say object A, we can create a covariant functor. This functor, denoted as Hom(A, –), maps every object X in the category to the set of all morphisms from A to X. When we apply this functor to a morphism between X and Y, it maps the set of connections to a new set of connections in the same direction. This preservation of direction is why we call it covariant.

We can also create a different type of functor by fixing the second object instead. If we fix object B, we create a contravariant functor denoted as Hom(–, B). This functor maps an object X to the set of morphisms from X to B. However, it behaves differently when it encounters a morphism. If there is a morphism from X to Y, the functor maps it in the opposite direction, from the set of morphisms from Y to B back to the set of morphisms from X to B. This arrow-reversing behavior is the defining trait of contravariance. This specific functor is sometimes called the functor of points for object B.

When we do not fix either object, we can look at both sides at once. This creates a bifunctor, which is a functor that takes two arguments. The notation Hom(–, –) represents this bifunctor mapping from the product of the opposite category and the original category into Set. It is contravariant in its first argument and covariant in its second. This means it can handle changes to both the starting and ending objects simultaneously. This complex structure is essential for understanding how different parts of a category interact through composition.

A very important result in this field is Yoneda's lemma. This lemma describes how natural transformations work between hom-functors. It states that every morphism between objects gives rise to a natural transformation between their corresponding functors. Specifically, a morphism from A' to A creates a transformation between their covariant functors. Similarly, a morphism from B to B' creates a transformation between their contravariant functors. This implies that hom-functors provide a way to embed a category into a category of functors, maintaining its essential structure.

Some advanced mathematical systems possess a special type of connection called an internal Hom functor. In most cases, hom-functors result in sets that live outside the original category. However, in a closed category, the internal Hom functor takes values within the category itself. This is written as [X, Z] or similar notation to show it is like a product. In a closed monoidal category, this internal Hom is an adjoint functor to the internal product functor. When the product is a Cartesian product, the internal Hom is called an exponential object.

These internal Hom functors can even form a formal language, known as the internal language of the category. For example, Cartesian closed categories have an internal language called simply typed lambda calculus. Other systems, like closed symmetric monoidal categories, use a linear type system. These languages allow mathematicians to treat the objects and morphisms of a category as if they were part of a logical or computational system. This connects abstract category theory to the foundations of computer science and logic.

Finally, hom-functors have specific properties depending on the type of category being studied. In an abelian category, the covariant Hom functor is left-exact. If the object being used is projective, the functor becomes exact. In the study of modules, the Hom functor is adjoint to the tensor product functor. Furthermore, the contravariant version of the hom-functor is known as a presheaf, while the covariant version is a copresheaf. These classifications help mathematicians place hom-functors into a much larger map of mathematical ideas.