7.3 Products, equalizers, and pullbacks

Products, equalizers, and pullbacks are key constructions in category theory. They represent different ways of combining or comparing objects and morphisms, each with its own universal property that defines its behavior across categories.

These constructions appear in various mathematical contexts, from set theory to group theory and topology. Understanding their universal properties helps us see common patterns across different mathematical structures and solve complex problems more efficiently.

Products, Equalizers, and Pullbacks

Products, equalizers, and pullbacks

• Product consists of an object $A \times B$ and morphisms $\pi_1: A \times B \to A$, $\pi_2: A \times B \to B$ satisfying a universal property
  • For any object $C$ and morphisms $f: C \to A$, $g: C \to B$, there exists a unique morphism $\langle f, g \rangle: C \to A \times B$ such that $\pi_1 \circ \langle f, g \rangle = f$ and $\pi_2 \circ \langle f, g \rangle = g$
• Equalizer consists of an object $E$ and a morphism $e: E \to A$ satisfying $f \circ e = g \circ e$ for given morphisms $f, g: A \to B$
  • For any object $C$ and morphism $h: C \to A$ such that $f \circ h = g \circ h$, there exists a unique morphism $u: C \to E$ such that $e \circ u = h$
• Pullback consists of an object $P$ and morphisms $p_1: P \to A$, $p_2: P \to B$ satisfying $f \circ p_1 = g \circ p_2$ for given morphisms $f: A \to C$, $g: B \to C$
  • For any object $Q$ and morphisms $q_1: Q \to A$, $q_2: Q \to B$ such that $f \circ q_1 = g \circ q_2$, there exists a unique morphism $u: Q \to P$ such that $p_1 \circ u = q_1$ and $p_2 \circ u = q_2$

Construction in various categories

• Products
  • In $\mathbf{Set}$, the product of sets $A$ and $B$ is the Cartesian product $A \times B = \{(a, b) \mid a \in A, b \in B\}$ with projection functions ($\pi_1$, $\pi_2$)
  • In $\mathbf{Grp}$, the product of groups $G$ and $H$ is the direct product $G \times H = \{(g, h) \mid g \in G, h \in H\}$ with componentwise group operation and projection homomorphisms ($\pi_1$, $\pi_2$)
• Equalizers
  • In $\mathbf{Set}$, the equalizer of functions $f, g: A \to B$ is the subset $E = \{a \in A \mid f(a) = g(a)\}$ with the inclusion map $e: E \to A$
  • In $\mathbf{Grp}$, the equalizer of group homomorphisms $f, g: G \to H$ is the subgroup $E = \{g \in G \mid f(g) = g(g)\}$ with the inclusion homomorphism $e: E \to G$
• Pullbacks
  • In $\mathbf{Set}$, the pullback of functions $f: A \to C$ and $g: B \to C$ is the subset $P = \{(a, b) \in A \times B \mid f(a) = g(b)\}$ with projection maps $p_1: P \to A$ and $p_2: P \to B$
  • In $\mathbf{Grp}$, the pullback of group homomorphisms $f: G \to K$ and $g: H \to K$ is the subgroup $P = \{(g, h) \in G \times H \mid f(g) = g(h)\}$ with projection homomorphisms $p_1: P \to G$ and $p_2: P \to H$

Universal properties

• Products
  1. Define the unique morphism $\langle f, g \rangle: C \to A \times B$ by $\langle f, g \rangle(c) = (f(c), g(c))$ for any $c \in C$
  2. Prove uniqueness: If $h: C \to A \times B$ satisfies $\pi_1 \circ h = f$ and $\pi_2 \circ h = g$, then $h(c) = (\pi_1(h(c)), \pi_2(h(c))) = (f(c), g(c)) = \langle f, g \rangle(c)$ for any $c \in C$
• Equalizers
  1. Define the unique morphism $u: C \to E$ by $u(c) = h(c)$ for any $c \in C$
  2. Prove uniqueness: If $v: C \to E$ satisfies $e \circ v = h$, then $v(c) = e(v(c)) = h(c) = u(c)$ for any $c \in C$
• Pullbacks
  1. Define the unique morphism $u: Q \to P$ by $u(q) = (q_1(q), q_2(q))$ for any $q \in Q$
  2. Prove uniqueness: If $v: Q \to P$ satisfies $p_1 \circ v = q_1$ and $p_2 \circ v = q_2$, then $v(q) = (p_1(v(q)), p_2(v(q))) = (q_1(q), q_2(q)) = u(q)$ for any $q \in Q$

Existence and uniqueness problems

• Existence
  • Products, equalizers, and pullbacks may not exist in every category ($\mathbf{Pos}$, $\mathbf{Mon}$)
  • A category is complete if it has all small limits, including products, equalizers, and pullbacks ($\mathbf{Set}$, $\mathbf{Grp}$, $\mathbf{Ab}$, $\mathbf{Top}$)
• Uniqueness up to unique isomorphism
  • Products: If $A \times B$ and $A \times' B$ are both products, there exists a unique isomorphism $\varphi: A \times B \to A \times' B$ such that $\pi'_1 \circ \varphi = \pi_1$ and $\pi'_2 \circ \varphi = \pi_2$
  • Equalizers: If $e: E \to A$ and $e': E' \to A$ are both equalizers of $f, g: A \to B$, there exists a unique isomorphism $\varphi: E \to E'$ such that $e' \circ \varphi = e$
  • Pullbacks: If $P$ and $P'$ are both pullbacks of $f: A \to C$ and $g: B \to C$, there exists a unique isomorphism $\varphi: P \to P'$ such that $p'_1 \circ \varphi = p_1$ and $p'_2 \circ \varphi = p_2$