3.1 Isomorphisms and their properties

Isomorphisms in category theory show when objects are essentially the same. They have inverse morphisms that compose to identity. This concept helps us understand equivalence and structural similarities between different mathematical objects.

Isomorphisms preserve important properties, allowing us to reason about objects more abstractly. They have unique inverses and follow the two-out-of-three property. This powerful tool simplifies our study of categories and their relationships.

Isomorphisms in Category Theory

Isomorphism in category theory

• Morphism $f: A \to B$ in category $\mathcal{C}$ has an inverse morphism $g: B \to A$ satisfying $g \circ f = id_A$ and $f \circ g = id_B$
• Captures notion of two objects being "essentially the same" within a category
  • Isomorphic objects ($A$ and $B$) have the same categorical properties
• Strong equivalence relation between objects
  • Isomorphic objects represent different representations of the same abstract concept (groups $\mathbb{Z}$ and $\mathbb{Z} \times \{1\}$, vector spaces $\mathbb{R}^2$ and $\mathbb{R}^3$)

Invertibility of isomorphisms

• Isomorphism $f: A \to B$ with inverse $g: B \to A$ is invertible
  • By definition, $g \circ f = id_A$ and $f \circ g = id_B$
  • Shows $f$ has a two-sided inverse, making it invertible
• Inverses of isomorphisms are unique
  • Suppose $f: A \to B$ has two inverses $g_1, g_2: B \to A$
  • Then $g_1 = g_1 \circ id_B = g_1 \circ (f \circ g_2) = (g_1 \circ f) \circ g_2 = id_A \circ g_2 = g_2$
  • Demonstrates $g_1 = g_2$, proving uniqueness of the inverse (identity morphism, composition)

Two-out-of-three property

• For morphisms $f: A \to B$, $g: B \to C$, and $h: A \to C$, if any two are isomorphisms, the third is also an isomorphism
• Case 1: $f$ and $g$ are isomorphisms, then $h = g \circ f$ is an isomorphism
  • Inverse of $h$ is $f^{-1} \circ g^{-1}$
• Case 2: $f$ and $h$ are isomorphisms, then $g = h \circ f^{-1}$ is an isomorphism
  • Inverse of $g$ is $f \circ h^{-1}$
• Case 3: $g$ and $h$ are isomorphisms, then $f = g^{-1} \circ h$ is an isomorphism
  • Inverse of $f$ is $h^{-1} \circ g$
• Useful for proving isomorphisms in a compositional manner (functors, natural transformations)

Isomorphism invariance

• Categorical properties preserved by isomorphisms
  • Property holding for object $A$ also holds for any isomorphic object $B$
• Examples of isomorphism-invariant properties
  • Initial and terminal objects
  • Monomorphisms and epimorphisms
  • Products and coproducts
  • Limits and colimits
• Allows reasoning about objects and properties up to isomorphism
  • Focus on essential characteristics rather than specific representations
• Simplifies study of categories by identifying "essentially the same" objects
  • Classify objects based on categorical properties and behavior (universal properties, adjunctions)