Categories, Universal property

In this post I will discuss some basic category theory. I have discussed what a category is, in a previous post. A category basically consists of some objects and maps between these objects are called morphisms. Morphisms are a way to tell which object in our category are related. Now we want the relationship between objects to satisfy some properties. We want to have a morphism from every object to itself. If we have a morphism from some object $A$ to $B$ and a morphism from $B$ to $C$ we want to have another morphism between $A$ to $C$ . So we want a map between $Mor(A,B) \times Mor(B,C)$ and $Mor(A,C)$ Now we have what we call a category. In this category we can define several notions like isomorphism. epimorphism etc.( Look at http://en.wikipedia.org/wiki/Category_theory)

Some examples are category of sets in which the objects are sets and morphisms are map between them, category of vector spaces in which objects are vector spaces and morphism are linear maps between them , category of groups whose morphisms are the homomorphisms between the groups.

We can give categorical definition to the familiar objects. For examples we can call a group as a category with a single object in which every morphism is an isomorphism ( why?) We can call a groupoid to be a category in which every morphism to be isomorphism ( it can have more than one object). We call a morphism between an object $A$ and $B$ a isomorphism if there exist a morphism from $B$ to $A$ such the compositions are identity morphisms on the respective objects.

Informally a category describes some kind of the structure of the objects. The object can have more structure to them but when we look at it as just an object in this category we are just interested in the particular structure of object. So to completely understand the object we may want to understand the relation of this structure to other structures. So we have the notion of functors. For example if for any two objects $A,B$ in the category which are related by a morphism $f:A \to B$ , if there exist another category with objects $F(A)$ and $F(B)$ which are related by the morphism $F(f)$ we call the map $F$ as a functor between these categories. Depending in the direction of $F(f)$ they are called covariant or contravariant functors. That if $F(f)$ is a morphism from $F(A)$ to $F(B)$ it is called covariant functor. If $F(f)$ maps from $F(B)$ to $F(A)$ we call it contravariant. Functors basically say that it two object are related some property of the objects are related.

An example of the a covariant functor is the forgetful functor which maps a vector space to the set. Under this functor a vector space (as a set) is mapped to another. By looking at the linear map between the vector space as just a map between two sets we are forgetting the linear structure of spaces and the map. Hence the name forgetful functor.

An example of contravariant functor is map from category of vector spaces to their dual spaces. If $T: A \to B$ is a linear map then we have a transpose $T^t: B^* \to A^*$ .

For category of objects with a partial order i.e, there is a morphism $<:A \to B$ if $A < B$ , we have monotonic properties associated with it which can be realised as functors.

Universal property:
Given a category we would like to construct new objects from the old ones. We generally do this using what is called universal property. It is one of the most important notion in category theory. What we do is that to construct a new object with some properties we would like it to have we prove that if it exists it is essentially unique ( up to isomorphism) and then exhibit one such object. To illustrate it we use the example of tensor product of two vector spaces.

If $V$ and $W$ are vector spaces then the tensor product is the object such that there is a bilinear map $f: V \times W \to V \otimes W$ and every bilinear map $g: V \times W \to S$ factors through this object. This kind of definition is what we call as universal property. We can check that if there is another object satisfying same property then the two objects are isomorphic. We can infact show that such object exist by explicit construction using the generators and relations.

We can have a notion of product in a category. Given any two objects and maps from some other object $Y$ to these object $X_1$ and $X_2$ then we require that these maps are factorised uniquely through another object called $X_1 \times X_2$ . Tha following is what is called, a commutative diagram, and shows this property. ( Show that product if exists is unique upto isomorphism) Dotted lines refer to unique

1)The category of sets has a product. Our cartesian product is an indeed a product under our definition. Similarly we have products in categories of groups, rings etc.
2)The category of sets ( or sets with extra structure) under inclusion form a category. Under this category we have a product which is nothing is nothing but the intersection of the sets. If the morphism was instead containment the product would have been the union.
3) In a category of objects with complete order we can realise maximum and minimum similarly.

We can actually reverse the arrows in the above diagram and get a co-product. That is every pair of maps from object $X_1$ and $X_2$ to some object $Y$ are factored through $X_1 \coprod X_2$ as follows.

We can similarly define several other notions like fibered products , direct limits , inverse limits etc.

Finally I will end with Yoneda’s lemma which is like a vast generalisation of Cayley’s theorem of groups. It roughly says that to study a category it is enough to category of all functors from it to the category of sets. For example if A is an object in the category then for every $f:B \to C$ we get a corresponding map between the $Mor (C, A)$ and $Mor (B, A)$ by composition of each morphism with $f$ . $Mor (B, A)$ and $Mor (C, A)$ are objects in the category of sets.
So Yoneda’s lemma says that to understand the object $A$ it is sufficient to understand the set of morphism to $A$ . In categorical terms we had a contravariant functor from our category to the category of sets. And this functor determines $A$ upto isomorphism. This lemma infact formalises the general idea of studying things like representation of Groups, modules over rings to understand the structure of the group and rings respectively.