Topos theory

1. What is Topos Theory?

A topos is a category that behaves like a generalized universe of sets, equipped with enough structure to support mathematical reasoning, including logic, geometry, and set-like operations. Formally, a topos is a category with:

Topoi generalize $\mathbf{Sets}$, the category of sets and functions, which is the standard universe for classical mathematics. They provide a framework where "sets" can have additional structure, and logic can be non-classical, such as intuitionistic or spatial logic.

2. $\mathbf{Sets}$ as a Degenerate Sheaves Topos

The category $\mathbf{Sets}$ is a topos, and it can be viewed as a sheaves topos over a trivial topological space, specifically a one-point space. Let’s unpack this:

• Sheaves and Sites: A sheaf is a mathematical object that assigns data (e.g., sets, groups) to open sets of a topological space in a way that is "locally consistent" (data on smaller open sets glue together to define data on larger ones). A sheaves topos is the category $\mathbf{Sh}(X)$ of sheaves on a topological space $X$, or more generally, sheaves on a site (a category with a Grothendieck topology).

• One-Point Space: Consider a topological space $X=\{\ast \}$ with one point. The only open set is $\{\ast \}$, and the category of sheaves on $X$, denoted $\mathbf{Sh}(\{\ast \})$, is equivalent to $\mathbf{Sets}$. This is because a sheaf on a one-point space assigns a single set to $\{\ast \}$, with no gluing conditions (since there are no smaller open sets). Thus, a sheaf is just a set, and morphisms are functions between sets.

3. How Does a Sheaf Topos Generalize Classical Logic?

In a topos, logic is internalized via the subobject classifier, denoted $\mathrm{\Omega }$.

In $\mathbf{Sets}$, $\mathrm{\Omega }=\{\text{true},\text{false}\}$, and subobjects (subsets) of a set $A$ correspond to characteristic functions (indicator function) $A\to \{\text{true},\text{false}\}$. This gives classical (Boolean) logic, where propositions are either true or false.

In a general topos, $\mathrm{\Omega }$ is more complex. For a sheaves topos $\mathbf{Sh}(X)$ over a topological space $X$, the subobject classifier $\mathrm{\Omega }$ is the sheaf of "open sets":

• For an open set $U\subset X$, the value $\mathrm{\Omega }(U)$ is the set of open subsets of $U$.
• Restriction maps $\mathrm{\Omega }(U)\to \mathrm{\Omega }(V)$ for $V\subset U$ send an open subset $W\subset U$ to $W\cap V$.

This $\mathrm{\Omega }$ classifies subobjects (subsheaves) via characteristic morphisms, but the truth values in $\mathrm{\Omega }(U)$ are "local" to each open set $U$. This leads to a logic where truth varies across the space $X$, generalizing classical logic by allowing truth values to be spatially or contextually dependent.

4. Sheaf Topos Over a Two-Element Set vs. One-Element Set

Let’s compare the sheaf topos over a two-element set with the one-element set case ($\mathbf{Sets}$).

One-Element Set ($\mathbf{Sets}$):

• Space: $X=\{\ast \}$, with topology $\{\mathrm{\varnothing },\{\ast \}\}$.
• Sheaves: A sheaf assigns a set to $\{\ast \}$, so $\mathbf{Sh}(\{\ast \})\cong \mathbf{Sets}$.
• Subobject Classifier: $\mathrm{\Omega }=\{\text{true},\text{false}\}$.
• Logic: Classical (Boolean). Propositions are globally true or false.

Two-Element Set:

Consider a two-element set $X=\{a,b\}$ with the discrete topology, where every subset is open: $\{\mathrm{\varnothing },\{a\},\{b\},\{a,b\}\}$.

• Sheaves: A sheaf $F$ on $X$ assigns:

  • Sets $F(\{a\})$, $F(\{b\})$, $F(\{a,b\})$ to the open sets $\{a\},\{b\},\{a,b\}$.
  • Restriction maps, e.g., $F(\{a,b\})\to F(\{a\})$, satisfying identity and composition laws.
    Since the topology is discrete, a sheaf is equivalent to a pair of sets $(A,B)$, where $F(\{a\})=A$, $F(\{b\})=B$, and $F(\{a,b\})=A\times B$.
    Thus, $\mathbf{Sh}(X)$ is equivalent to $\mathbf{Sets}\times \mathbf{Sets}$, the category of pairs of sets $(A,B)$ with morphisms $(f_a,f_b):(A,B)\to (A^{\prime },B^{\prime })$.

• Subobject Classifier: In $\mathbf{Sh}(X)$, the subobject classifier $\mathrm{\Omega }$ is the sheaf where:

  • $\mathrm{\Omega }(\{a\})=\{\mathrm{\varnothing },\{a\}\}\cong \{0,1\}$.
  • $\mathrm{\Omega }(\{b\})=\{\mathrm{\varnothing },\{b\}\}\cong \{0,1\}$.
  • $\mathrm{\Omega }(\{a,b\})=\{\mathrm{\varnothing },\{a\},\{b\},\{a,b\}\}\cong \{0,1{\}}^2$.
    Restriction maps are induced by intersections.

The subobject classifier let us define subobjects in the same way that the characteristic map ${\mathcal{X}}_A:U\to \{0,1\}$ let us define a subset $A\subseteq U$.

5. Propositions in These Logics

In $\mathbf{Sets}$ (One-Element Set):

• Universe: An object $M$ in $\mathbf{Sets}$, representing the set of "men."
• Predicate: "Mortal" is a subobject $\text{Mortal}\hookrightarrow M$, represented by a characteristic map $\varphi :M\to \{\text{true},\text{false}\}$.
• Proposition: "All men are mortal" is the statement that $\text{Mortal}=M$, i.e., $\mathrm{\forall }m\in M,\varphi (m)=\text{true}$.
• Example:
  • If $M=\{\text{John},\text{Paul}\}$ and $\text{Mortal}=\{\text{John},\text{Paul}\}$, the proposition is true.
  • If $\text{Mortal}=\{\text{John}\}$, the proposition is false.

In $\mathbf{Sh}(\{a,b\})$ (Two-Element Set):

• Universe: An object $M$ in $\mathbf{Sh}(X)$, e.g., $M(\{a\})=M_a$, $M(\{b\})=M_b$, $M(\{a,b\})=M_a\times M_b$.
• Predicate: "Mortal" is a subsheaf $\text{Mortal}\hookrightarrow M$, with $\text{Mortal}(\{a\})\subseteq M_a$, $\text{Mortal}(\{b\})\subseteq M_b$, etc.
• Proposition: "All men are mortal" means $\text{Mortal}=M$ locally. Its truth value is $(t_a,t_b)$, where $t_a$ is true if all men in $M_a$ are mortal, and similarly for $t_b$.
• Example:
  • If $\text{Mortal}(\{a\})=M_a$ and $\text{Mortal}(\{b\})⊊M_b$, the proposition has truth value $(\text{true},\text{false})$.

6. Interpretation

We can interpret:

• Point $a$ as Universe A, with its own set of objects, properties, and logic (modeled by a copy of $\text{Sets}$).
• Point $b$ as Universe B, similarly with its own independent set of objects, properties, and logic.
• The topos $\text{Sh}(X)$ as a mathematical framework that describes both universes simultaneously, allowing us to reason about propositions across these universes.

The discrete topology ensures that $\{a\}$ and $\{b\}$ are completely independent (no non-trivial gluing conditions), so the universes do not interact or share information unless explicitly combined at the "global" level $\{a,b\}$. This independence makes the parallel universes analogy particularly apt: each universe has its own "reality" (sets and functions), and propositions can have different truth values in each.

The subobject classifier $\mathrm{\Omega }$ in $\text{Sh}(\{a,b\})$ reinforces this interpretation:

• $\mathrm{\Omega }(\{a\})=\{\mathrm{\varnothing },\{a\}\}\cong \{0,1\}$: Truth values in Universe A (false, true).
• $\mathrm{\Omega }(\{b\})=\{\mathrm{\varnothing },\{b\}\}\cong \{0,1\}$: Truth values in Universe B.
• $\mathrm{\Omega }(\{a,b\})=\{\mathrm{\varnothing },\{a\},\{b\},\{a,b\}\}\cong \{0,1{\}}^2$: Global truth values, which are pairs $(t_a,t_b)$, where $t_a$ is the truth value in Universe A, and $t_b$ is the truth value in Universe B.

Propositions in this topos are evaluated locally in each universe:

• A proposition is true in Universe A if its characteristic map at $\{a\}$ assigns the value $1$ (true) to all relevant elements.
• Similarly, it is true in Universe B if true at $\{b\}$.
• Globally, the proposition’s truth is the pair $(t_a,t_b)$, which might be $(\text{true},\text{true})$, $(\text{true},\text{false})$, $(\text{false},\text{true})$, or $(\text{false},\text{false})$.

Let’s reinterpret the proposition "all men are mortal" from the previous example, framing $a$ and $b$ as parallel universes.

• Universe A (point $a$):

  • The set of men is $M(\{a\})=M_a=\{{\text{John}}_a,{\text{Paul}}_a\}$.
  • The predicate "mortal" is a subsheaf $\text{Mortal}(\{a\})\subseteq M_a$.
  • Suppose $\text{Mortal}(\{a\})=\{{\text{John}}_a,{\text{Paul}}_a\}$, so all men in Universe A are mortal. The proposition "all men are mortal" is true in Universe A ($t_a=\text{true}$).

• Universe B (point $b$):

  • The set of men is $M(\{b\})=M_b=\{{\text{John}}_b,{\text{Paul}}_b\}$.
  • Suppose $\text{Mortal}(\{b\})=\{{\text{John}}_b\}$, so not all men are mortal (e.g., Paul is immortal in Universe B). The proposition is false in Universe B ($t_b=\text{false}$).

• Global Level ($\{a,b\}$):

  • The set of men is $M(\{a,b\})=M_a\times M_b$.
  • The subsheaf $\text{Mortal}(\{a,b\})=\text{Mortal}(\{a\})\times \text{Mortal}(\{b\})=\{{\text{John}}_a,{\text{Paul}}_a\}\times \{{\text{John}}_b\}$.
  • The proposition’s truth value is $(t_a,t_b)=(\text{true},\text{false})$, meaning it holds in Universe A but not in Universe B.

This is exactly like saying:

• In Universe A, the laws of nature dictate that all men are mortal.
• In Universe B, some men (like Paul) are immortal, perhaps due to different physical or metaphysical rules.
• Globally, the proposition is not universally true across both universes, reflecting the independent realities.