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 $A$ and $B$ be groups and $f: A \to B$ be a group homomorphism. To familiarize ourselves with the categorical notation, we continue this example in categorical language- Let $A, B \in \text{Obj}$ (Grp) and $f \in \text{Mor} (A, B)$. There exists an object $K\in \text{Obj}$ (Grp) and a morphism $\iota \in \text{Mor} (K, A)$ such that the pair $(K, \iota)$ satisfies the following properties:

• $f \circ \iota = 0$ as a morphism (group homomorphism) from $K$ to $B$.
• Whenever $latex (K’, \iota’ \in \text{Mor}(K’, A))$ is such that $f \circ \iota' = 0$, then there is a unique morphism $\Phi \in \text{Mor}(K', K)$ such that $\iota \circ \Phi = \iota'$.

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 $R$-Modules: Given $R$-modules $M_1, M_2$, the direct product of $M_1$ and $M_2$ is an $R$-module $M_1\oplus M_2$ along with (inclusion) maps $\iota_i: M_i \to M_1\oplus M_2\; (i=1,2)$ such that given any $R$-module $N$ and any $R$-module homomorphisms $\phi_i : M_i \to N$, there is a unique homomorphism $\Phi: M_1 \oplus M_2 \to N$ such that $\Phi \circ \iota_i = \phi_i$.

Note: This can be generalized to an arbitrary family of submodules of an $R$-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 $R, S$, a ring-homomorphism $\phi: R \to S$ and an element $a\in S$, there exists a unique ring-homomorphism $\Phi: R[X] \to S$ extending $\phi$ such that $\Phi(X)=a$.

Quotient Ring: Given a ring $R$ and its ideal $I$ (possibly trivial or the whole ring), the quotient ring is a ring $Q$ along with ring homomorphism $\pi: R \to Q$ such that

• $\pi(I)=0$,
• For every ring $S$ and homomorphism $\phi:R\to S$ with $\phi(I)=0$, there is a unique ring homomorphism $\Phi: Q\to S$ such that $\phi = \pi \circ \Phi$.

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

Free $R$-module: Let $S$ be a set and $R$ be a ring. A free $R$-module over $S$ is an $R$-module $F(S)$ along with an (injective) set map $\iota: S \to F(S)$ such that given any $R$-module $M$ and a set map $\phi: S \to M$, there is a unique $R$-module homomorphism $\Phi: F(S) \to M$ such that $\Phi \circ \iota = \phi$.

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

Ring of fractions of a ring: Given a domain $R$ and a multiplicatively closed subset $S$ of $R$, (that contains 1 but no zerodivisors of $R$), the ring of fractions of $R$ is a ring (denoted by) $S^{-1}R$ along with a map $\iota: R \to S^{-1}R$ such that given a ring $Q$ and ring homomorphism $\phi: R \to Q$ such that $\phi(S)$ consists of units (of $Q$), then there is a unique homomorphism $\Phi: S^{-1}R \to Q$ such that $\Phi\circ\iota=\phi$.

Note: If we take $R$ to be a domain and $S=R\backslash \{0\}$ then we get the field of fractions of $R$.

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

Quotient of a set: Let $X$ be a set with an equivalence relation $\tilde{}$ on it. The quotient of $X$ with respect to $\tilde{}$ is a set $\overline{X}$ with a map $\pi: X \to \overline{X}$ such that given any set $Z$ and a map $\phi: X\to Z$ with $\phi(x) = \phi(x')$ whenever $x$ $\tilde{}$ $x'$, then there is a unique map $\Phi: \overline{X}\to Z$ such that $\Phi\circ\pi=\phi$.

Product of sets: Let $X_i\; (i\in I)$ be an arbitrary family of sets. We define the product set of $X_i$‘sto be a set $\Pi_{i\in I} X_i$ alongwith (projection) maps $\displaystyle\pi_i : \Pi_{i\in I} X_i \to X_i$ 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 $e$) satisfies the two following properties:

1. Given any group $G$, there is a unique homomorphism $f:\{e\} \to G$.
2. Given any group $G$, there is a unique homomorphism $f: G \to \{e\}$.

Future post:

• A small application of UMP.