This post is a continuation of the previous post – A mild introduction to Categories. Before I jump into the technical details of Categorical stuff like Functors, I would like to introduce the notion of a universal mapping property (henceforth referred to as UMP).

I won’t bog you down with the details of jargon that one has to absorb before getting the definition. Those interested in the definition of a UMP may refer the Wikipedia article here. Loosely speaking, a UMP of an object in a category is a property that characterizes it completely. In this post, I shall only present the universal properties of some familiar objects. We start with the property of a kernel of a group homomorphism.

**Kernel of a group homomorphism: **Let **Grp **be the category of groups. (One may similarly define the UMP of the kernel of a ring, module, field homomorphism). Let and be groups and be a group homomorphism. To familiarize ourselves with the categorical notation, we continue this example in categorical language- Let (**Grp**) and . There exists an object (**Grp**) and a morphism such that the pair satisfies the following properties:

- as a morphism (group homomorphism) from to .
- Whenever $latex (K’, \iota’ \in \text{Mor}(K’,
~~A))$ is such that , then there is a unique morphism such that .~~

In group-theoretic parlance, kernel (of a homomorphism) is a group which has an embedding into the preimage and whose image under the homomorphism is trivial. The group is unique in that any other group that has these properties must factor uniquely via the kernel. This property of the kernel, in fact, characterizes it.

We begin with examples that create new rings from old ones:

**Direct sum of -Modules:** Given -modules , the direct product of and is an -module along with (inclusion) maps such that given any -module and any -module homomorphisms , there is a unique homomorphism such that .

Note: This can be generalized to an arbitrary family of submodules of an -module. The existence follows from by taking all but finitely many nonzero elements of each submodule.

**Polynomial ring in one variable over a ring: **Given rings , a ring-homomorphism and an element , there exists a unique ring-homomorphism extending such that .

**Quotient Ring: **Given a ring and its ideal (possibly trivial or the whole ring), the quotient ring is a ring along with ring homomorphism such that

- ,
- For every ring and homomorphism with , there is a unique ring homomorphism such that .

The UMP of a free module (or free group) is in some sense, more fundamental.

**Free -module: **Let be a set and be a ring. A free -module over is an -module along with an (injective) set map such that given any -module and a set map , there is a unique -module homomorphism such that .

The UMP of a ring of fractions of a ring is often useful.

**Ring of fractions of a ring: **Given a domain and a multiplicatively closed subset of , (that contains 1 but no zerodivisors of ), the ring of fractions of is a ring (denoted by) along with a map such that given a ring and ring homomorphism such that consists of units (of ), then there is a unique homomorphism such that .

Note: If we take to be a domain and then we get the field of fractions of .

We give two examples to show that UMPs exist for non-*algebraic *objects too. (whatever that means!)

**Quotient of a set:** Let be a set with an equivalence relation on it. The quotient of with respect to is a set with a map such that given any set and a map with whenever , then there is a unique map such that .

**Product of sets: **Let be an arbitrary family of sets. We define the product set of ‘sto be a set alongwith (projection) maps such that for any __ (complete this!)

The above two properties – quotient and product of sets generalize immediately and can be used to define the UMP for Quotient topology and Box topology in the category **Top** of topological spaces. To state the UMP of the Product topology, one uses an analog of Direct Sum of Modules stated above.

**Exercise:** Deduce the UMP for cokernel of a module homomorphism.

**Uniqueness of a UMP: **The universal property of an object need not be unique. A trivial example to see this is as follows: The identity element of a group (denote it by ) satisfies the two following properties:

- Given any group , there is a unique homomorphism .
- Given any group , there is a unique homomorphism .

**Future post:**

- A small application of UMP.

## 7 comments

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October 30, 2010 at 15:05

AnonymousAs an application i wish we could solve a problem using categories.

It is a fact that the fundamental group of a topological group is abelian. Consider the functor Top–>Set and to establish the result we must prove that the Image of the class of topological groups is a subclass of Ab. Any insights for a further step?

November 1, 2010 at 11:51

abhishekparabProving it directly is very easy..

Usually proving the functoriality of something requires the same efforts as proving without using categories. For example, to prove that is a functor from Top to Gp requires the same efforts as proving that the equivalence class of loops forms a group. The importance of categories lies more in what follows from the fact that is a functor.

November 1, 2010 at 23:26

AnonymousPerhaps you are true but can you say that these two methods are essentially the same?

November 22, 2011 at 19:58

AnonymousIn the definition of kernel of a grp homomorphism, is there any problem with function composition in the following statement when you are considering f: A to B, i’ in Mor(K’, B)) and f composed with i’? Whenever (K’, \iota’ \in \text{Mor}(K’, B)) is such that f \circ \iota’ = 0, then there is a unique morphism \Phi \in \text{Mor}(K’, K) such that \iota \circ \Phi = \iota’.

November 26, 2011 at 05:39

abhishekparabThanks for pointing out the typo:

is a map into and not . I have corrected it now. Thanks.

Actually, I wrote this when I was getting familiar with category theory. Hence I didn’t quite notice mistakes like the collection of objects in a category don’t form a set. These things run deeper. There are categories of categories, n-category, infinity category etc. Its dizzied abstraction gets one giddy!

January 12, 2012 at 17:26

kamnican u show me ump in case of poset?

January 12, 2012 at 19:42

abhishekparabThere may be many ways in which a poset can be thought of as a category. Here is one –

Let (X, b) or has one element (if a <= b). It doesn't matter what the one element is.

Now verify that this categorical definition and the old definition agree.

Exercise : One can also define a group as a category with one object.