3.4 Examples in concrete categories

Isomorphisms in set theory reveal deep connections between mathematical structures. They show when objects are essentially the same, despite surface differences. This concept extends to various mathematical categories, like groups and topological spaces.

Initial and terminal objects are special elements in categories with universal properties. The empty set and singleton sets play these roles in Set. Understanding these concepts helps simplify proofs and identify key structures across different mathematical domains.

Isomorphisms in set category

Group isomorphisms and category theory

• In the category of groups (Grp), an isomorphism is a group isomorphism which is a bijective group homomorphism
  • A group homomorphism $f: G \to H$ preserves the group operation $f(g_1 \cdot g_2) = f(g_1) \cdot f(g_2)$ for all $g_1, g_2 \in G$
• To show two groups $G$ and $H$ are isomorphic in Grp, find a bijective group homomorphism $f: G \to H$
  • The inverse function $f^{-1}: H \to G$ will also be a group homomorphism since $f$ is bijective
• Isomorphic groups have the same group structure and properties (order, subgroups, etc.)

Initial and terminal objects in sets

• In the category of sets (Set):
  • The empty set $\emptyset$ is an initial object since there exists a unique function $f: \emptyset \to A$ for any set $A$
  • Any singleton set $\{x\}$ is a terminal object because for any set $A$, there exists a unique function $g: A \to \{x\}$ mapping every element of $A$ to $x$
• Initial and terminal objects are unique up to isomorphism in a category

Examples of initial/terminal objects

• In the category of groups (Grp):
  • The trivial group $\{e\}$ is both an initial and terminal object
    • For any group $G$, there exists a unique group homomorphism $f: \{e\} \to G$ mapping $e$ to the identity element of $G$
    • There also exists a unique group homomorphism $g: G \to \{e\}$ mapping every element of $G$ to $e$
• In the category of topological spaces (Top):
  • The one-point space $\{*\}$ is a terminal object since for any topological space $X$, there exists a unique continuous function $f: X \to \{*\}$
• In the category of rings (Ring):
  • The zero ring $\{0\}$ is an initial object
  • The ring of integers $\mathbb{Z}$ is an initial object in the category of rings with unity

Applications of categorical concepts

• Use isomorphisms and initial/terminal objects to:
  • Prove two objects in a concrete category are isomorphic by finding a bijective morphism between them
  • Identify the initial and terminal objects in a given concrete category based on their universal properties
  • Simplify compositions involving initial or terminal objects
• Example: In Set, prove that for any set $A$, the composition $f \circ g: \emptyset \to A$ is the unique function from $\emptyset$ to $A$, where $g: \emptyset \to \{x\}$ and $f: \{x\} \to A$
  1. $g$ is the unique function from $\emptyset$ to $\{x\}$ (initial object property)
  2. $f$ is an arbitrary function from $\{x\}$ to $A$
  3. The composition $f \circ g$ is a function from $\emptyset$ to $A$
  4. By the initial object property, $f \circ g$ must be the unique function from $\emptyset$ to $A$
• Example: Prove that the fundamental group of a topological space is a group up to isomorphism in Grp