Below is a synthesis of how our “wormhole/spacetime region” metaphor—enriched by products, exponentials, and subobject classifiers—maps onto the **definition of an (elementary) topos**. An **elementary topos** is a category $\mathcal{E}$ that has: 1. **Finite limits** (making it a _cartesian_ category), 2. Is **cartesian closed** (i.e. for any objects $A, B \in \mathcal{E}$, there is an exponential object $B^A$ with the usual “eval” and “currying” universal properties), 3. Possesses a **subobject classifier** $\Omega$, which generalizes the notion of the two-element set $\{\text{true},\text{false}\}$ in $\mathbf{Set}$. A topos may be conceived as a “category that behaves like $\mathbf{Set}$” with respect to finite limits, function spaces, and characteristic functions of sub-objects. Here is how our geometric “wormhole” picture faithfully reflects each piece. --- ## 1. Finite Limits   ⟹ “Cartesian” Structure **Objects** = “Regions of spacetime.” **Morphisms** $f : A \to B$ = “Wormhole networks” from region $A$ to region $B$. - **Terminal Object**: A “trivial region” (like a single point). Any region $A$ has exactly one wormhole map (collapses) into this point. - **Binary Products** $A \times B$: The “joint” or “product” region containing all ordered pairs $(a,b)$. Wormholes out of $A \times B$ factor into wormholes from each component. - **Equalizers**, **Pullbacks**, etc.: We can form sub-regions capturing, e.g., “where two parallel wormhole networks coincide,” or “the inverse image of a sub-region under a wormhole map.” --- ## 2. Cartesian Closed   ⟹  “All Wormhole Configurations from $A$ to $B$” as Regions Objects $B^A$ Being **cartesian closed** adds that for any two objects $A, B$ there is an **exponential** object $B^A$, together with an **evaluation** map $\mathrm{ev} : B^A \times A \;\longrightarrow\; B$ satisfying the usual universal/currying property. **Metaphor**: - $B^A$ is interpreted as the “**space (region) of all wormhole distributions** from $A$ to $B$.” - A _point_ of $B^A$ (in a set-like or topological sense) _is_ a particular wormhole network $A \to B$. - The evaluation map $\mathrm{ev}(G,a) = G(a)$ says: pick a “global configuration of wormholes” $G$ and a point $a\in A$; traveling through $G$ at $a$ lands you in $B$. - **Currying**: any morphism $f : X \times A \to B$ corresponds uniquely to a morphism $\tilde{f} : X \to B^A$ that assigns, to each $x\in X$, an entire wormhole network $\tilde{f}(x) : A \to B$. Thus, the idea that a category is cartesian closed—“has function spaces as legitimate objects”—becomes: **"it has a region that _collects all wormhole distributions_ from $A$ to $B$.”** --- ## 3. Subobject Classifier   ⟹  “Region of Truth Values” Finally, in a topos we have a **subobject classifier** $\Omega$. This means there is an object $\Omega$ (the “object of truth values”) and a distinguished **truth** morphism $\top: 1 \to \Omega$ such that _every_ subobject $U \hookrightarrow A$ (i.e. sub-region of $A$) is given by a **characteristic** morphism $\chi_U : A \to \Omega$. Concretely, $\chi_U(a)$ indicates whether $a$ lies in $U$, in a generalized sense. **Metaphor**: - $\Omega$ is the “**universal region of truth labels**,” or “space of truth values,” potentially more sophisticated than just $\{\text{true}, \text{false}\}$. - The characteristic map $\chi_U : A \to \Omega$ says: each point $a \in A$ is assigned a truth label in $\Omega$ that indicates membership in $U$. - Pullback stability of sub-object classifiers is read as: if $f : B \to A$ is a wormhole map, then the sub-region $f^{-1}(U)\subseteq B$ is characterized by $\chi_{f^{-1}(U)}(b) = \chi_U(f(b))$. Hence, **labeling sub-regions** (sub-objects) and **mapping them to truth values** is an intrinsic part of the geometry, ensuring each sub-region can be “classified.” --- ## 4. Claiming the Metaphor is Adequate to a Topos Putting these three pillars together: 1. **Finite Limits** (all the basic “wormhole geometry” of products, equalizers, sub-regions, etc.), 2. **Cartesian Closure** (the “region of all wormhole configurations” plus evaluation), 3. **Subobject Classifier** (the “universal region of truth values” labeling sub-regions), is precisely the definition of an elementary topos. Our wormhole / spacetime-region metphor attempts to: - **account** for each structural requirement in a topos: we see how objects, morphisms, and subobjects are formed, how exponentials can be viewed as “all wormhole networks,” and how subobject classifiers amount to “universal truth labeling.” - **match** the universal properties: each construction (product, evaluation, characteristic map) is explained by a direct geometric or physical-sounding scenario (joint region, apply a chosen wormhole network, classify membership in a sub-region). - **unify** them in a single coherent picture that is conceptually faithful to the topos axioms. The hope is that we can gain some sort of intuition of toposes using this “wormhole” metaphor: - If $\mathcal{E}$ is a topos, each object in $\mathcal{E}$ can be _imagined_ as a “spacetime region,” each morphism as a “wormhole network,” each exponential as “the region of all such networks,” and $\Omega$ as “the region of truth labels.” The wormhole attempts to provide a faithful metaphor for each of the key ingredients that define a topos. It captures finite-limit structure, cartesian closure, and subobject classification within a single, visually and conceptually unified narrative, and many more constructions, as we'll soon see. --- ## Toposes are Regular and Coherent Categories It is important to note that regularity and coherence follow from this definition. ## 1. Why a topos is a [[A1.3 - Regular Categories |regular category]] A topos, by definition, is: - A category with **all finite limits**. - **Cartesian closed**. - Equipped with a **subobject classifier** $\Omega$. From this it follows that a topos is in fact **exact** (Barr-exact). Being exact immediately implies being regular, because the definition of exactness strengthens the regularity requirement that “every kernel pair has a coequalizer” to “every _equivalence relation_ is the kernel pair of its coequalizer,” i.e. every equivalence relation is _effective_. Concretely: 1. **Existence of coequalizers of kernel pairs**: In a topos, for any morphism $f \colon X \to Y$, the kernel pair of $f$ (an internal equivalence relation on $X$) can be coequalized. This can be seen by combining finite-limit constructions with the fact that every equivalence relation can be turned into a suitable subobject (using the subobject classifier) and then “quotiented” using the logic of the topos. 2. **Pullback-stability of regular epis**: Regular epimorphisms in a topos can be described as the coequalizers of kernel pairs, and these are stable under pullback. In fact, in an elementary topos, _all_ epis are regular epis. Hence a topos is a regular category (indeed, more strongly, an exact category). --- ## 2. Why a topos is a [[A1.4 - Coherent Categories |coherent category]] Informally, a **coherent category** is one that: - Has finite limits (so one can interpret finite conjunctions $\wedge$, and equality). - Admits finite colimits _of monomorphisms_ / sub-objects in a way that internalizes “finite disjunctions” $\vee$. - Is regular, ensuring images behave in a suitably “logical” way. Equivalently, a coherent category is often described as one in which the internal logic can interpret **coherent logic** (finite conjunctions, finite disjunctions, and equality). In a topos, we have even more structure: - We already have **finite limits** (so we can interpret $\wedge$ and “there exists an element of” $\exists$). - Because the topos is **regular**, we have good image factorizations (so we can interpret “there exists” $\exists$ properly). - **Finite joins** (unions of subobjects) can be constructed via finite coproducts and images of monos. Concretely, if $A, B \subseteq X$ are subobjects, one forms their sum $A + B \to X$ (injective in a topos), then takes the (regular epi, mono) factorization (the image) back into $X$. This yields a monomorphism / subobject that behaves as $A \cup B$. Hence the logical connectives for _coherent_ logic are all interpretable inside an elementary topos: - $\wedge$ via pullback (finite limits), - $\vee$ via finite coproduct + image factorization, - “There exists” $\exists$ via regular epis and images, - “Equality”, $=$, via the diagonal and finite limits. Thus every elementary topos is a coherent category. --- ### Summary of the reasoning 1. **Topos   $\implies$ Exact  $\implies$ Regular** Because a topos has all finite limits, plus a subobject classifier, one shows that every internal equivalence relation is the kernel pair of its own coequalizer (i.e., it is _effective_). That makes the topos an _exact_ (Barr-exact) category, hence a fortiori a _regular_ category. 2. **Topos  $\implies$ Coherent** Being regular plus having a suitable way to form finite unions of subobjects (using coproducts and the regular image factorization) ensures that the internal logic can interpret “finite disjunctions.” Combined with finite limits (for “finite conjunction” and equality) and regular epis (for “there exists”), one obtains exactly the fragment known as _coherent logic_. Thus a topos is coherent. In fact, topoi can interpret even richer logics (not just coherent logic but full **intuitionistic** higher-order logic, thanks to cartesian closure, exponentials, and the subobject classifier). But as soon as one can interpret that (strictly weaker) coherent fragment, the category is called _coherent_.