🧮Topos Theory Unit 6 Review

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 $m: A \to B$ $m : A \to B$ in $\mathcal{C}$ $C$ , unique characteristic function $\chi_m: B \to \Omega$ $χ_{m} : B \to Ω$ exists making following diagram a pullback:
```
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)

🧮Topos Theory Unit 6 Review