1.3 Duality and opposite categories

Opposite categories flip the direction of morphisms, creating a mirror image of the original category. This concept introduces duality, allowing us to explore relationships between seemingly different structures and simplify proofs.

Dual constructions like products and coproducts, equalizers and coequalizers, and limits and colimits showcase the power of duality. By reversing arrows, we gain new insights into categorical structures and their connections.

Opposite Categories and Duality

Definition of opposite categories

• Opposite category (dual category) construction reverses morphism directions
  • Objects remain identical to original category
  • Morphisms $f: A \to B$ become $f^{op}: B \to A$ in opposite category
• Composition in opposite category reverses order
  • Original: $f: A \to B$ and $g: B \to C$
  • Opposite: $g^{op} \circ f^{op} = (f \circ g)^{op}$
• Notation $C^{op}$ represents opposite category of $C$

Concept of duality in categories

• Duality principle generates dual counterparts by reversing arrows and compositions
• Enhances understanding of categorical structures through symmetry
• Dual notions illuminate relationships (initial object $\leftrightarrow$ terminal object, product $\leftrightarrow$ coproduct)
• Applications include simplifying proofs and connecting concepts (monomorphism $\leftrightarrow$ epimorphism)

Examples of dual constructions

• Product vs Coproduct:
  • Product: $P$ with projections $p_1: P \to A$, $p_2: P \to B$
  • Coproduct: $Q$ with injections $i_1: A \to Q$, $i_2: B \to Q$
• Equalizer vs Coequalizer:
  • Equalizer: $E \to A$ satisfies $f \circ e = g \circ e$
  • Coequalizer: $A \to C$ satisfies $c \circ f = c \circ g$
• Monomorphism vs Epimorphism:
  • Monomorphism: $f: A \to B$ monic if $f \circ g = f \circ h$ implies $g = h$
  • Epimorphism: $f: A \to B$ epic if $g \circ f = h \circ f$ implies $g = h$
• Limit vs Colimit:
  • Limit: Universal cone to diagram
  • Colimit: Universal cocone from diagram

Isomorphism of double opposites

• Statement: $(C^{op})^{op} \cong C$
• Proof outline constructs functor $F: C \to (C^{op})^{op}$
  1. Demonstrate $F$ preserves morphism distinctness (faithful)
  2. Show every $(C^{op})^{op}$ morphism originates from $C$ (full)
  3. Prove all $(C^{op})^{op}$ objects isomorphic to $F$ image (essentially surjective)
• Conclusion establishes $F$ as category equivalence, confirming isomorphism