6.1 Subobjects and characteristic functions

Subobjects in categories extend the idea of "parts" or "subsets" to abstract mathematical structures. They're represented by equivalence classes of monomorphisms, allowing us to compare and analyze substructures across different mathematical domains.

Characteristic functions of subobjects generalize indicator functions, connecting subobjects to logic and type theory. This concept finds applications in various fields, from set theory to algebraic geometry and computer science, highlighting its versatility and importance.

Subobjects in Categories

Definition of subobjects

• Subobject represents equivalence class of monomorphisms capturing notion of "part" or "subset" in abstract categories
• Two monomorphisms deemed equivalent if they factor through each other allowing different representations of same substructure
• Denoted $m: A \rightarrow B$, where $m$ is a monomorphism embedding substructure $A$ into larger structure $B$
• Generalizes familiar concepts like subsets (Set), subgroups (Group), and subspaces (Top)

Subobjects vs monomorphisms

• Monomorphisms serve as representatives for subobjects but not in one-to-one correspondence
• Factorization property defines equivalence: $f: A \rightarrow C$ and $g: B \rightarrow C$ represent same subobject if isomorphisms $h: A \rightarrow B$ and $k: B \rightarrow A$ exist with $f = g \circ h$ and $g = f \circ k$
• Subobjects form partially ordered set based on factorization relation enabling comparison of "size" or "inclusion"
• Distinction crucial for understanding abstract structure preservation in category theory

Characteristic functions of subobjects

• Morphism $\chi_m: B \rightarrow \Omega$ where $\Omega$ is subobject classifier generalizing indicator functions
• Constructed using universal property of subobject classifier ensuring unique factorization through true morphism $true: 1 \rightarrow \Omega$
• Exhibits uniqueness for each subobject and pullback property allowing recovery of subobject
• Connects to internal logic and type theory in elementary toposes

Applications in various categories

• Set: Subobjects as subsets with characteristic functions acting as indicator functions (0 and 1)
• Top: Open subsets as subobjects utilizing local homeomorphisms for characteristic functions
• Algebraic geometry: Closed subschemes as subobjects related to ideal sheaves
• Computer science: Modeling data structures (trees, graphs) and type systems in programming languages
• Logic: Internal logic of toposes leveraging subobject classifiers for intuitionistic reasoning