Examples of universal properties - Revisited

#category_theory

In light of Yoneda's Lemma, it is useful to revisit Universal Properties I - Inspiring Examples to see how each can be framed in the same language: as a natural isomorphism between a hom-functor and another $Set$ -valued functor.

Examples in $Set$

The Cartesian product of two sets

If $X$ and $Y$ are sets, then there is a natural bijection between the set of maps $Z \to X \times Y$ , and the set of pairs of maps $Z \to X$ , $Z \to Y$ . In other words, there is a natural bijection

ϕ_{Z} : {Hom}_{Set} (Z, X \times Y) \tilde{\to} {(g_{1}, g_{2}) ∣ g_{1} \in {Hom}_{Set} (Z, X), g_{2} \in {Hom}_{Set} (Z, Y)} .

This bijection requires the two projection maps $π_{1} : X \times Y \to X$ and $π_{2} : X \times Y \to Y$ defined by $(x, y) \mapsto x$ and $(x, y) \mapsto y$ , respectively. The bijection $ϕ_{Z}$ above is defined by sending each set map $f : Z \to X \times Y$ to the pair of set maps $π_{1} \circ f : Z \to X$ , $π_{2} \circ f : Z \to Y$ .

Examples in $Ab$

The direct sum of two abelian groups

If $A$ and $B$ are abelian groups, their direct sum is the abelian group $A \oplus B$ . As a set, it consists of all pairs of formal sums $a + b$ with $a \in A$ and $b \in B$ . The operation is defined "component-wise": $(a + b) + (a^{'} + b^{'}) = (a + a^{'}) + (b + b^{'})$ . (Although not common, one could reasonably argue that a different notation should be used for the formal sum symbol, such as $a \oplus b$ .) One can verify that $A \oplus B$ is an abelian group, and that it comes equipped with two injective group morphisms $i_{1} : A \to A \oplus B$ and $i_{2} : B \to A \oplus B$ . Moreover, there is a natural bijection between group morphisms from $A \oplus B$ and pairs of groups morphisms $A \to C$ , $B \to C$ :

ϕ_{C} : {Hom}_{Ab} (A \oplus B, C) \tilde{\to} {(g_{1}, g_{2}) ∣ g_{1} \in {Hom}_{Ab} (A, C), g_{2} \in {Hom}_{Ab} (B, C)}

This bijection is very similar to that for the direct product of sets, above. Each group morphism $f : A \oplus B \to C$ is sent to the pair of group morphisms $f \circ i_{1} : A \to C$ , $f \circ i_{2} : B \to C$ .

Examples in $Grp$

The quotient group construction

If $N$ is a normal subgroup of a group $G$ , then there is a natural bijection between the set of group morphisms $G / N \to H$ , and the of group morphisms $G \to H$ with $N$ contained in the kernel. In other words, there is a natural bijection

ϕ_{H} : {Hom}_{Grp} (G / N, H) \tilde{\to} {g : G \to H ∣ g \in {Hom}_{Grp}, N \leq \ker (g)} .

This bijection requires the "canonical" projection morphism $π : G \to G / N$ . Using that map, the bijection $ϕ_{H}$ sends each group morphism $f : G / N \to H$ to the group morphism $f \circ π : G \to H$ .

Examples in $R -Mod$

The tensor product construction

Suppose $R$ is a commutative ring and $M$ and $N$ are left $R$ -modules. By taking the standard $R$ -module structure (i.e., $(R, R)$ -bimodule structure) on $M$ and the canonical $(R, Z)$ -bimodule structure on $N$ , we can form the tensor product $M \otimes_{R} N$ . The result is an $(R, Z)$ -bimodule, i.e., a left $R$ -module. There is a natural bijection between $R$ -module morphisms $M \otimes_{S} N \to P$ and certain set maps:

ϕ_{P} : {Hom}_{R -Mod} (M \otimes_{R} N, P) \tilde{\to} {g : M \times N \to P ∣ g bilinear, R -balanced (R, Z) -set map}

Examples in Set