In order to give the reader an *intuition* of categorical or topos theoretical concepts, we adopt a long running metaphor, wherein the reader is invited to visualize objects as "(spacetime) regions", and morphisms as "wormholes", connecting points in one region to points in another. The metaphor is not meant to be taken literally; it is meant rather to evoke a deep seated intuition, and to enable _visualizing_ the concepts under discussion. Below is an attempt to make precise what we mean by “spacetime regions” and “wormholes” in the metaphor—and how these notions map exactly onto the usual categorical / topos-theoretic definitions. --- ## 1. Spacetime Regions = Objects in a Category - In our metaphor, a **“spacetime region”** is simply an **object** $X$ in a category $\mathcal{C}$. - If $\mathcal{C}$ is a **topos**, this means $X$ might be, for example, a set, a sheaf, or something more sophisticated (like a manifold, a ringed space, etc.) internal to that topos, depending on the context. The reason we will regularly invoke the metaphorical notion of a “region” is to invoke a sort of visual cue, one which (we hope) will enable the reader to keep in mind that: - an **object** in a topos can often be viewed as a “space-like” or “region-like” structure (e.g. a topological space, a sheaf of sets, etc.), and - even when $\mathcal{C}$ is more abstract, we may still (if we so choose) _pretend_ that each object is “some chunk of geometry” or “some cosmic zone", in order to enable grounding our imagination in some form over which we have some intuitive control (or grasp). $\boxed{ \text{“Spacetime region”} \;\;\longleftrightarrow\;\; \text{Object in } \mathcal{C}\text{ (or in a topos)} }$ We use the label **“spacetime”** loosely to evoke a geometric or physical intuition. However: - If $\mathcal{C}$ is $\mathbf{Set}$, these “regions” are literally just **sets**. - If $\mathcal{C}$ is $\mathbf{Top}$, these “regions” are **topological spaces**. - If $\mathcal{C}$ is a **Grothendieck topos**, these “regions” are **sheaves** on some site, or **objects** in an abstract sense, etc. In each case, the idea is that an **object** is some “realm” of points or data. We simply call it a “region” to keep a geometric / physical flavor. --- ## 2. Wormholes = Morphisms Between Objects - A **“wormhole network / distribution”** in our metaphor is just a **morphism** $f : X \to Y$ in a category $\mathcal{C}$. - In a topos (or geometric context), such a morphism is typically a **continuous/smooth map**, or a **sheaf morphism**, or an **internal function**—depending on the exact nature of $\mathcal{C}$. $\boxed{ \text{“Wormhole”} \;:\; X \longrightarrow Y \;\;\longleftrightarrow\;\; \text{Morphism in } \mathcal{C}\text{ (or in a topos)} }$ We borrow the term **“wormhole”** to evoke an image of a “direct path” that takes every point of $X$ to a point in $Y$ _in a single jump_. In ordinary category language: - A morphism $f : X \to Y$ “assigns to each point $x \in X$ a point $f(x) \in Y$.” - This is reminiscent of a “shortcut” or “tunnel” that instantaneously lands you from any point of $X$ to some single point in $Y$. So the wormhole metaphor is a _fanciful re-interpretation_ of “for each $x\in X$, there’s exactly one image $f(x)\in Y$.” --- ## 3. Higher Notions: Subobjects, Limits, Sheaves Once we identify: - **Objects** $\leftrightarrow$ “spacetime regions,” - **Morphisms** $\leftrightarrow$ “wormholes,” we can reinterpret more advanced categorical constructions: 1. **Subobject** $U\subseteq X$: “A sub-region $U$ inside the region $X$.” 2. **Limit** or **Colimit** of a diagram: “A universal merging or splitting of regions via wormholes.” 3. **Sheaves**: “Local data assigned to each sub-region that glue together on overlaps,” re-labeled as “fields or distributions on the entire cosmos.” --- ## 4. Physical vs. Categorical Precision In actual general relativity or quantum field theory, a “wormhole” is a _possible_ topological feature bridging distant regions of spacetime. Our usage here is: - **Purely categorical**: any morphism is called a “wormhole.” - The “spacetime region” might not be an actual Minkowski manifold or a region in a physically realistic sense. We do **not** claim to model real gravitational wormholes. Instead, we adopt “wormhole” as a **visual mnemonic** signifying: "there is a unique path from any point of $X$ to a single point in $Y$.” This <spacetime regions, wormhole> language helps induce a **visual intuition** of the structures we will describe, and the metaphor will help continually suggest a certain perspective (an internal perspective?) that is often worth keeping top of mind, namely: - that each object can plausibly be “entered” to reveal rich structure within, - that each morphism involves “travel routes” from every point of the source to every (in some sense) unique point in the target, - that composition of morphisms somehow involves chaining these routes. This is our physically inspired mental model for category theory’s fundamental data: objects and morphisms. --- ## 5. Summary of the Formal Connection $\begin{aligned} &\textbf{Category } \mathcal{C} \\ &\quad\bullet \;\text{Objects } X,Y \quad \longleftrightarrow \quad\text{“Spacetime regions”} \\ &\quad\bullet \;\text{Morphisms } (X\to Y) \quad \longleftrightarrow \quad \text{“Wormholes from region }X\text{ to region }Y\text{”} \end{aligned}$ If $\mathcal{C}$ is a **topos**, each “region” is a **(sheaf-like) object** in that topos, and each “wormhole” is an **(internal) geometric morphism** or **map**. We do **not** require real spacetime structures, nor do we require exotic physics. We merely **reuse** the _words_ “spacetime region” and “wormhole” to evoke a geometry-laden, travel-laden image that stands in for **objects** and **morphisms**. **Physically**: You might imagine “a region” as a chunk of universe, and “a wormhole” as a direct route from one chunk to another. **Categorically**: An **object** in a category is some set/space/sheaf/structure, and a **morphism** is a function/ map that picks out exactly one image in the target for each element in the source. --- **6. Caveat: Metaphor, not physical reality** While it’s a powerful mental model for category theory, there’s no reason to believe **real** Lorentzian geometry allows infinite families of stable wormholes that replicate every categorical morphism. From a **physics** standpoint, we do **not** have evidence that the full “wormhole-laced” topos structure can be literally realized. We only have a **useful** analogy—**not** a physically validated blueprint for infinite cosmic highways. 1. **Lorentzian wormholes** are specific solutions in general relativity requiring exotic conditions to connect two distinct regions with a single tunnel. 2. **Categorical wormholes** are **all** morphisms $X\to Y$. From the vantage of physics, that would require an unbelievably large or infinite network of “bridges,” one for each point in $X$, to distinct points in $Y$. That is not physically plausible as we know it. 3. The metaphor is, by and large, **only** that—**a metaphor**—though it’s “more than a pun” in the sense that it does evoke the idea of “direct bridging." --- ## 7. Wormholes in Lorentzian Geometry In **classical general relativity**, a “wormhole” (often called an **Einstein–Rosen bridge**) is a (hypothesized) topological feature of a Lorentzian manifold $(M,g)$ that: 1. **Connects distinct regions** of spacetime $M$ (or possibly connects two separate spacetimes) by a “bridge” or “throat.” 2. Potentially allows causal or at least topological travel from region $A \subseteq M$ to region $B \subseteq M$ more quickly than going through normal “outside” space. Mathematically, one usually says that the **spacetime manifold** has a nontrivial topology that includes at least one “handle” or “tunnel” linking these two (otherwise distant) regions. Solutions of the Einstein Field Equations that exhibit such handles typically require: - **Exotic matter** (negative energy density, or violations of energy conditions) to keep the wormhole open, - Or they remain ephemeral / collapse too quickly to be traversed. Many physicists consider stable, traversable wormholes **speculative**. They appear in advanced solutions of the Einstein equations (especially if one allows negative energy or certain “exotic” fields), but we don’t know if they exist physically. Even if real wormholes exist, they’d be far from the “one wormhole per point” scenario we’ve described categorically. Instead, a Lorentzian “wormhole” is usually a single “bridge” that, at best, connects each point of region $A$ to some set of points in region $B$ _via the same tunnel geometry_—and that’s if it’s traversable. Hence physically, wormholes are complicated, exotic solutions. They are not simply “for each point in $A$, there’s a unique exit in $B$.” The actual geometry is far more constrained. Physically, if you imagine building an **Einstein–Rosen** style wormhole for each point $x \in X$, you’d have an **infinite** (likely uncountable) family of separate throats, each “exiting” into a distinct point of $Y$. This is beyond the usual “two mouths” wormhole concept. It’d be an unbelievably complicated geometry: essentially “a net of wormholes,” one for each $x\in X$. That’s not usually how classical wormholes are described in relativity. Having a separate wormhole for each point in $X$ would require some unbelievably infinite “wormhole factory,” presumably with total negative energy of enormous magnitude. Nothing in standard physics suggests such solutions exist or are even consistent with quantum gravity constraints. If some **extremely** advanced theory (like a wilder form of quantum gravity or baby universes) allowed you to create “private wormholes” for each point in $X$, that’d be a scenario so beyond mainstream physics that it’s purely speculative. Even then, the stability issues, energy conditions, and topological constraints would be enormous. Hence in **practical** physical terms, the “categorical wormhole” concept is best viewed as **an analogy** or **imaginative device** for internalizing how a function/morphism “collapses” an entire domain to single points in the codomain. We’re **not** claiming that physically, such a structure is literally realized.