6.2 Definition and properties of subobject classifiers

Subobject classifiers are powerful tools in category theory that generalize truth values. They're objects with special properties that allow us to classify subobjects in a category, providing a way to think about "truth" in different mathematical contexts.

In Set, the subobject classifier is the two-element set {0,1}, while in Top, it's the Sierpinski space. These examples show how subobject classifiers adapt to different mathematical structures, enabling us to reason about truth and logic in various settings.

Subobject Classifiers in Category Theory

Definition of subobject classifiers

• Subobject classifier denotes object $\Omega$ in category $\mathcal{C}$ with terminal object $1$ and morphism $true: 1 \to \Omega$ (truth arrow) generalizing concept of truth values
• Universal property characterizes subobject classifiers ensuring for every monomorphism $m: A \to B$ in $\mathcal{C}$, unique characteristic function $\chi_m: B \to \Omega$ exists making following diagram a pullback:

  </>Code
  A ---m---> B
  |         |
  |         | χ_m
  v         v
  1 --true-> Ω

• Pullback property establishes $A$ as pullback of $\chi_m$ along $true$, capturing notion of subobject
• Bijective correspondence links subobjects of $B$ with morphisms $B \to \Omega$, providing classification of subobjects

Uniqueness of subobject classifiers

• Assume two subobject classifiers $(\Omega, true)$ and $(\Omega', true')$ in category
• Construct isomorphism $f: \Omega \to \Omega'$ using universal property of $\Omega'$, defining $f$ as characteristic function of $true: 1 \to \Omega$
• Construct inverse $g: \Omega' \to \Omega$ using universal property of $\Omega$, defining $g$ as characteristic function of $true': 1 \to \Omega'$
• Prove $f \circ g = id_{\Omega'}$ and $g \circ f = id_\Omega$ using uniqueness of characteristic functions
• Conclude $\Omega$ and $\Omega'$ are isomorphic, establishing uniqueness up to isomorphism

Existence in Set and Top

• Subobject classifier in Set:
  1. Define $\Omega = \{0, 1\}$ (two-element set)
  2. Establish $true: 1 \to \{0, 1\}$ mapping singleton to 1
  3. Characteristic functions correspond to indicator functions of subsets
• Subobject classifier in Top (category of topological spaces):
  1. Define $\Omega = \{0, 1\}$ with Sierpinski topology (open sets: $\{\emptyset, \{1\}, \{0,1\}\}$)
  2. Establish $true: 1 \to \Omega$ mapping point to 1
  3. Characteristic functions correspond to open subsets of spaces
• Non-existence in some categories demonstrated by category of groups lacking subobject classifier

Relationship to truth values

• Topos represents category with additional structure: cartesian closed, possesses all finite limits and colimits, has subobject classifier
• Subobject classifier generalizes truth values with elements of $\Omega$ representing truth values and morphisms $1 \to \Omega$ corresponding to propositions
• Internal logic of topos enabled by subobject classifier allowing logical operations (conjunction, disjunction, negation, implication) with Heyting algebra structure on $\Omega$
• Relationship to intuitionistic logic highlighted by law of excluded middle potentially not holding, different toposes yielding different logical systems
• Non-classical truth values exemplified by sheaf toposes (truth values as open sets) and effective topos (realizability interpretation of logic)