Hom functor

Math helps us see how things link.

We can look at paths between two spots. A hom-functor is a way to study these paths.

It looks at how one object connects to another.

One way moves in one direction. This is called a covariant functor. The other way moves in the opposite direction. This is called a contravariant functor.

These paths can form a special kind of map. Some math worlds even have their own internal paths. This can create a way to build a language. Math is full of these clever links!

In math, we study how things connect. We call these connections morphisms.

A hom-functor is a special way to study these connections. It turns them into sets of things.

There are two main types of hom-functors. The first type is called a covariant functor. It keeps the direction of the connections the same. The second type is a contravariant functor. This one reverses the direction of the arrows.

When we use both sides, we get a bifunctor. This is a tool that works with two objects at once. Some math worlds are very special. They have an internal Hom functor. This means the connections live inside the same world. We call these special worlds closed categories.

These internal connections can even act like a language. This is called an internal language. It helps us build rules for the math world. For example, some systems use a tool called lambda calculus. These tools help math feel like a real way to speak and think.

In math, we often look at how things connect. We call these connections morphisms. A hom-functor is a special tool used in category theory. It takes these connections and turns them into sets. This helps us study the relationships between different objects. It is a way to see the structure of a whole math world.

There are two ways a hom-functor can work. The first way is called a covariant functor. It keeps the direction of the connections the same as they move. This happens when we fix one object and look at the others. The second way is called a contravariant functor. This one actually reverses the direction of the arrows. This happens when we fix the second object instead.

Sometimes, we use both types of connections at once. This creates something called a bifunctor. A bifunctor works with two objects at the same time. It can be contravariant on one side and covariant on the other. This is a very useful way to map connections. It helps mathematicians see how different parts of a category interact.

Some math worlds are very special and organized. These are called closed categories. In these worlds, the connections live inside the category itself. We call this an internal Hom functor. This is different from a normal hom-functor that uses sets. It allows the math to stay within its own rules.

These internal connections can even act like a language. We call this an internal language. It helps us build rules for how the math world works. One famous example is called simply typed lambda calculus. This is used in certain types of closed categories. These tools help us turn math into a way of speaking and thinking.

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.