The Yoneda Constraint

Let $\mathscr{C}$ be a category whose objects are observational contexts---complete specifications of experimental setups, reference frames, or measurement configurations---and whose morphisms $f: C \to C'$ represent refinements or transformations between contexts.

::: {#ax:yoneda .axiom}

From Presheaves to Quantum Mechanics

The main results of [@long2026yoneda] establish that:

1. The monoidal structure on $\mathscr{C}$ forces fibers $S(C)$ to carry vector space structure over $\mathbb{C}$.

1. Perspectival consistency---coherent pairings across common refinements---yields Hermitian inner products, making each fiber a Hilbert space $\mathcal{H}_C$.

1. Observables arise as self-adjoint natural transformations $\alpha : S \Rightarrow S$.

1. The Born rule $p(\lambda) = |\langle e_\lambda, \psi \rangle|^2$ follows from the Yoneda isomorphism combined with Gleason's theorem.

1. Unitary dynamics are natural automorphisms $U_t : S \Rightarrow S$, generated by a Hamiltonian via Stone's theorem.

The Presheaf Topos

The category $\widehat{\mathscr{C}} = [\mathscr{C}^{\mathrm{op}}, \mathbf{Set}]$ of all presheaves on $\mathscr{C}$ is a topos: it has a subobject classifier $\Omega$ (whose values are sieves, providing multi-valued quantum logic), all finite limits and colimits, and internal function objects. The presheaf topos is the "universe of discourse" for Quantum Perspectivism.

A key observation for the present paper: when we equip $\mathscr{C}$ with a Grothendieck topology $J$, we can pass from the presheaf topos to the sheaf topos $\mathrm{Sh}(\mathscr{C}, J)$, a subtopos of $\widehat{\mathscr{C}}$ in which presheaves satisfy a gluing condition. This passage from presheaves to sheaves is the mathematical mechanism by which spacetime geometry emerges from abstract perspectival structure.

AQFT from the Yoneda Constraint: Detailed Derivation {#sec:aqft}

Algebraic quantum field theory (AQFT), formulated by Haag and Kastler [@haag1964], provides a mathematically rigorous framework for quantum field theory based on assigning algebras of observables to spacetime regions. We now show that the Haag--Kastler axioms are not independent postulates but consequences of the Yoneda Constraint when the category of contexts is specialized to spacetime regions.

The Category of Spacetime Regions

Let $(M, g)$ be a globally hyperbolic Lorentzian manifold. Define the category $\mathbf{Loc}(M)$ whose objects are causally convex open subsets $\mathcal{O}\subseteq M$ and whose morphisms are inclusions $\iota : \mathcal{O}_1 \hookrightarrow \mathcal{O}_2$ whenever $\mathcal{O}_1 \subseteq \mathcal{O}_2$. This is a poset category: there is at most one morphism between any two objects.

::: definition

The reversal to the opposite category is natural from the perspectival viewpoint: a larger region provides a coarser perspective (less localized information), and the morphism points from coarse to fine.

The Net of Observables as a Presheaf

By the Yoneda Constraint, a physical system on $M$ is a presheaf $$\begin{equation}

\mathcal{A}: \mathscr{C}_{\mathrm{st}}^{\mathrm{op}}\to \mathbf{Set}.

\end{equation}$$ Since $\mathscr{C}_{\mathrm{st}} = \mathbf{Loc}(M)^{\mathrm{op}}$, this is equivalently a covariant functor $$\begin{equation}

\mathcal{A}: \mathbf{Loc}(M) \to \mathbf{Set}.

\end{equation}$$ The Yoneda Constraint, combined with the linearization argument (monoidal structure on contexts forces vector space structure) and the algebraic structure induced by composition of observables, promotes $\mathbf{Set}$ to the category of $C^*$-algebras:

::: {#thm:aqft-net .theorem}

1. monoidal structure on $\mathscr{C}_{\mathrm{st}}$ arising from causal disjointness,

1. perspectival consistency (Hermitian inner product on fibers),

1. the algebraic closure requirement that observables form a $C^$-algebra,*

then $S$ determines a covariant functor $\mathcal{A}: \mathbf{Loc}(M) \to C^\text{-}\mathbf{Alg}$, i.e., a net of $C^$-algebras over spacetime.

:::

::: proof

Proof. The Yoneda Constraint gives a presheaf $S : \mathscr{C}_{\mathrm{st}}^{\mathrm{op}}\to \mathbf{Set}$, equivalently a functor $S : \mathbf{Loc}(M) \to \mathbf{Set}$. For each open $\mathcal{O}$, the fiber $S(\mathcal{O})$ carries the structure of a complex vector space (by the linearization proposition of [@long2026yoneda]) and a Hermitian inner product (by perspectival consistency).

The observables on $\mathcal{O}$ are self-adjoint natural endomorphisms of the restriction of $S$ to $\mathcal{O}$, which form a real vector space closed under the Jordan product $A \circ B = \frac{1}{2}(AB + BA)$. The algebraic closure requirement promotes this to a $C^*$-algebra $\mathcal{A}(\mathcal{O})$.

For an inclusion $\iota : \mathcal{O}_1 \hookrightarrow \mathcal{O}_2$, the functoriality of $S$ gives a map $S(\iota) : S(\mathcal{O}_1) \to S(\mathcal{O}_2)$, which induces an injective $*$-homomorphism $\mathcal{A}(\iota) : \mathcal{A}(\mathcal{O}_1) \hookrightarrow \mathcal{A}(\mathcal{O}_2)$ by naturality. Functoriality ($S(\iota_2 \circ \iota_1) = S(\iota_2) \circ S(\iota_1)$) ensures that $\mathcal{A}$ is a covariant functor. ◻

:::

Derivation of the Haag--Kastler Axioms

We now derive each of the standard Haag--Kastler axioms as consequences of the Yoneda-perspectival structure.

Isotony

::: {#prop:isotony .proposition}

::: proof

Proof. This is immediate from the functoriality of $\mathcal{A}$. The inclusion $\iota : \mathcal{O}_1 \hookrightarrow \mathcal{O}_2$ induces $\mathcal{A}(\iota) : \mathcal{A}(\mathcal{O}_1) \to \mathcal{A}(\mathcal{O}_2)$, and the faithfulness of the Yoneda embedding guarantees that $\mathcal{A}(\iota)$ is injective. Indeed, if $\mathcal{A}(\iota)(A) = 0$ for some $A \in \mathcal{A}(\mathcal{O}_1)$, then $A$ would be relationally invisible from the perspective of $\mathcal{O}_2$, contradicting the Yoneda Constraint's assertion that all relational data is physically meaningful. ◻

:::

Locality (Einstein Causality)

::: {#prop:locality .proposition}

::: proof

Proof. Two spacelike-separated regions represent causally independent perspectives: no morphism in $\mathscr{C}_{\mathrm{st}}$ connects them directly (neither is contained in the other, and no causal signal can travel between them). In the Yoneda framework, the observational data obtained from causally independent perspectives must be jointly consistent but mutually non-interfering.

Formally, the monoidal structure on $\mathscr{C}_{\mathrm{st}}$ arising from causal disjointness gives $\mathcal{O}_1 \otimes \mathcal{O}_2$ (the "parallel combination" of the two perspectives). The presheaf condition requires $$\begin{equation}

S(\mathcal{O}_1 \otimes \mathcal{O}_2) \cong S(\mathcal{O}_1) \otimes S(\mathcal{O}_2),

\end{equation}$$ which at the algebraic level translates to $$\begin{equation}

\mathcal{A}(\mathcal{O}_1 \cup \mathcal{O}_2) \cong \mathcal{A}(\mathcal{O}_1) \otimes \mathcal{A}(\mathcal{O}_2)

\end{equation}$$ (where $\otimes$ is the $C^*$-algebraic tensor product). In the tensor product algebra, elements of the two tensor factors commute. Since the algebras $\mathcal{A}(\mathcal{O}_1)$ and $\mathcal{A}(\mathcal{O}_2)$ embed into $\mathcal{A}(\mathcal{O})$ through the respective inclusions, they commute within $\mathcal{A}(\mathcal{O})$. ◻

:::

Covariance

::: {#prop:covariance .proposition}

::: proof

Proof. An isometry $\alpha$ of $(M, g)$ induces a functor $\alpha_ : \mathbf{Loc}(M) \to \mathbf{Loc}(M)$ given by $\mathcal{O}\mapsto \alpha(\mathcal{O})$. By the functoriality of the presheaf $S$ and the Yoneda Constraint (which requires that relational structure be preserved under symmetries of the context category), $\alpha$ lifts to a natural isomorphism $\beta_\alpha : \mathcal{A}\circ \alpha_^{-1} \Rightarrow \mathcal{A}$. The naturality condition is precisely the compatibility with inclusions stated above.

In representation-theoretic terms, if $\mathcal{A}$ is represented on a Hilbert space $\mathcal{H}$ via a state $\omega$ (the GNS construction), then $\beta_\alpha$ lifts to a unitary operator $U(\alpha)$ on $\mathcal{H}$ implementing the symmetry: $$\begin{equation}

U(\alpha) \pi_\omega(A) U(\alpha)^{-1} = \pi_\omega(\beta_\alpha(A)).

\end{equation}$$ The group property $U(\alpha \circ \alpha') = U(\alpha) U(\alpha')$ follows from the functoriality of $\alpha \mapsto \alpha_*$. ◻

:::

Spectrum Condition

::: {#prop:spectrum .proposition}

::: proof

Proof sketch. The causal structure of $(M, g)$ is encoded in the morphism structure of $\mathscr{C}_{\mathrm{st}}$: a morphism $\mathcal{O}_2 \to \mathcal{O}_1$ exists only when $\mathcal{O}_1 \subseteq \mathcal{O}_2$, and the causal convexity requirement ensures that the ordering respects the causal structure. The Yoneda Constraint, applied to the translation subgroup of isometries, requires that the presheaf data transform covariantly under translations.

The vacuum state, defined as the unique translation-invariant state (the global section of the presheaf invariant under the translation automorphisms), determines a representation via GNS construction. The covariance condition, combined with the positivity of the inner product from perspectival consistency, forces the spectral measure of $P^\mu$ to be supported in $\overline{V^+}$. The detailed argument uses the Borchers--Buchholz theorem relating positivity of energy to the causal structure of the net. ◻

:::

The Vacuum

::: {#prop:vacuum .proposition}

::: proof

Proof sketch. In the Yoneda framework, the vacuum is the "perspective from everywhere"---the unique global section of the state presheaf that is invariant under all automorphisms of $\mathscr{C}_{\mathrm{st}}$ induced by isometries. Its existence follows from the amenability of the Poincaré group (or the isometry group of $(M,g)$) and the Ryll-Nardzewski fixed-point theorem applied to the weak-$*$ compact convex set of states. Cyclicity follows from the Reeh--Schlieder theorem, which in the perspectival framework is the statement that local perspectival data (from any open region $\mathcal{O}$) is dense in the total relational structure. ◻

:::

::: remark

Remark 8. The Reeh--Schlieder theorem has a striking perspectival interpretation: even a "local perspective" (a single open region $\mathcal{O}$) contains enough relational data to approximate any state of the field, no matter how global. This is the AQFT manifestation of the Yoneda principle that an object is determined by its relationships to all probes---and even a single probe, if it is "generic enough," captures the essence of the object.

:::

Summary: AQFT as a Special Case

::: {#thm:aqft-summary .theorem}

This result means that AQFT does not require independent axiomatic justification: it is derivable from the single postulate of the Yoneda Constraint applied to spacetime contexts.

Grothendieck Topologies and Sheaf Conditions {#sec:grothendieck}

Having derived AQFT from the Yoneda Constraint on a fixed spacetime, we now turn to the more radical program: deriving spacetime itself. The key mathematical tool is the Grothendieck topology, which provides a notion of "covering" on an abstract category without any prior geometric structure.

Grothendieck Topologies: Definition and Physical Meaning

::: {#def:groth-top .definition}

1. Maximality: The maximal sieve $\{f : D \to C \mid D \in \mathscr{C}\}$ is in $J(C)$.

1. Stability: If $S \in J(C)$ and $g : D \to C$ is any morphism, then the pullback sieve $g^*S = \{h : E \to D \mid g \circ h \in S\}$ is in $J(D)$.

1. Transitivity: If $S \in J(C)$ and $T$ is a sieve on $C$ such that for every $f : D \to C$ in $S$, the pullback $f^*T \in J(D)$, then $T \in J(C)$.

:::

Each axiom has a physical interpretation in our framework:

(GT1) Maximality: Every context can be "covered" by the collection of all perspectives on it. This is the Yoneda principle: the totality of morphisms into $C$ determines $C$ completely.

(GT2) Stability: If a collection of perspectives suffices to determine the physics of a context $C$, then the "pullback" of these perspectives to any refinement $D$ of $C$ still suffices to determine the physics of $D$. Covering is preserved under change of perspective.

(GT3) Transitivity: If a collection $S$ covers $C$, and for each perspective $f$ in $S$ the induced collection $f^*T$ covers $f$'s domain, then $T$ covers $C$. This is the physical principle that "coverings of coverings are coverings"---a consistency condition ensuring that the notion of "having enough perspectives" is well-behaved.

Sheaves: Presheaves that Respect Locality

::: {#def:sheaf .definition}

::: remark

Remark 12 (Physical Interpretation of the Sheaf Condition). The sheaf condition encodes the principle of local-to-global consistency: if you have consistent perspectival data from a covering collection of viewpoints, this data uniquely determines a global physical state. This is precisely the physical content of locality in the sense of field theory. A presheaf that is not a sheaf describes physics with "global anomalies"---situations where local data cannot be consistently assembled into a global state.

:::

The Sheaf Topos and its Physical Content

::: definition

Sheafification has a physical interpretation: it is the process of enforcing locality. Given a presheaf $F$ (a system described by perspectival data that may not satisfy local-to-global consistency), the sheafification $aF$ is the "closest" sheaf to $F$---it imposes locality by identifying global sections that agree on all covers.

::: {#prop:sheaf-logic .proposition}

Physically, the passage from $\widehat{\mathscr{C}}$ to $\mathrm{Sh}(\mathscr{C}, J)$ restricts the truth values available for quantum propositions. In the full presheaf topos, truth values are arbitrary sieves; in the sheaf topos, they are $J$-closed sieves. This restriction encodes the physical content of the Grothendieck topology: only those truth assignments that respect the covering structure (i.e., locality) are physically meaningful.

Concrete Grothendieck Topologies for Spacetime

::: {#ex:open-cover .example}

::: {#ex:causal-cover .example}

This topology encodes the physical principle that physics is determined by causal data: if you know the physics in enough causally connected subregions, you know the physics of the whole region.

:::

::: {#ex:diamond .example}

Emergent Geometry from Perspectival Structure {#sec:emergent}

We now address the central question: how does a smooth geometric manifold emerge from an abstract category of contexts equipped with a Grothendieck topology? The answer proceeds through three stages: from the sheaf topos to a locale, from the locale to a topological space, and from the topological space to a differentiable manifold.

Stage 1: From the Sheaf Topos to a Locale

::: {#def:locale .definition}

::: {#prop:locale-topos .proposition}

::: proof

Proof. The terminal object $1$ in $\mathrm{Sh}(\mathscr{C}, J)$ is the constant sheaf with value $\{*\}$. A subobject of $1$ is a subsheaf $U \hookrightarrow 1$, which is determined by a $J$-closed sieve for each context. The collection of all such subobjects forms a complete lattice under intersection and union. The infinite distributive law follows from the fact that the subobject lattice in any topos is a Heyting algebra, and every complete Heyting algebra is a frame. ◻

:::

Physically, the locale $\mathcal{L}(\mathcal{E})$ represents the "space of possible physical situations" as extracted from the perspectival structure. Each element of the locale corresponds to a "generalized open region"---a proposition about the physical state that is decidable given sufficient perspectival data.

Stage 2: From Locale to Topological Space

::: definition

::: definition

::: {#thm:spatial .theorem}

1. Enough points: There exist sufficiently many flat functors $\mathscr{C}\to \mathbf{Set}$ ("stalks") to separate elements of the locale.

1. Sobriety: Every irreducible closed subset of $\mathrm{pt}(\mathcal{L})$ has a unique generic point.

When these conditions hold, the points $\mathrm{pt}(\mathcal{L}(\mathrm{Sh}(\mathscr{C}, J)))$ form a sober topological space.

:::

Physically, the spatiality condition has a profound interpretation: the locale is spatial precisely when the abstract perspectival structure is "classical enough" to be described by a set of points. If the locale is not spatial, then the emergent geometry is genuinely "pointless"---it has a topological structure (opens, covers) but no underlying set of points. This is the categorical articulation of the hypothesis that spacetime at the Planck scale may be pointless.

Stage 3: From Topological Space to Manifold

Assuming the locale is spatial, so that we have a topological space $X = \mathrm{pt}(\mathcal{L}(\mathrm{Sh}(\mathscr{C}, J)))$, we need additional conditions for $X$ to be a manifold.

::: {#thm:manifold .theorem}

1. Local Euclidean structure: For every point $p \in X$, there exist contexts $C_1, \ldots, C_k$ and a covering sieve in $J$ such that the corresponding open neighborhood of $p$ is homeomorphic to $\mathbb{R}^n$.

1. Hausdorff condition: For any two distinct points $p, q \in X$, there exist disjoint opens in the locale separating them.

1. Second countability: The locale has a countable basis---a countable subset generating all opens via joins.

1. Paracompactness: Every open cover has a locally finite refinement.

:::

::: proof

Proof sketch. Conditions (M1)--(M4) are precisely the standard conditions for a topological space to be a topological manifold, translated into locale-theoretic terms via the spatiality correspondence. The content of the theorem is that each condition corresponds to a property of the site $(\mathscr{C}, J)$:

(M1) requires that the category $\mathscr{C}$ contain enough "local contexts" that look like $\mathbb{R}^n$---formally, that the locale has an atlas of charts. This is the condition that the perspectival structure has a well-defined local dimension.

(M2) requires that distinct perspectives can be separated, corresponding to the physical principle that different spacetime points are distinguishable by local measurements.

(M3) is a cardinality condition on the site, requiring that the category $\mathscr{C}$ is "not too large."

(M4) is a technical condition ensuring that partition-of-unity arguments work, which in the perspectival framework corresponds to the ability to decompose global data into locally manageable pieces. ◻

:::

Differentiable Structure

To promote the topological manifold to a differentiable manifold, we need:

::: definition

::: {#prop:smooth .proposition}

Metric Emergence

The metric tensor $g_{\mu\nu}$ on the emergent manifold is not an additional datum but is encoded in the inner product structure of the presheaves.

::: {#prop:metric .proposition}

The Lorentzian signature of the metric (rather than Riemannian) arises from the causal structure of the category $\mathscr{C}$: the distinction between timelike and spacelike morphisms in $\mathscr{C}_{\mathrm{st}}$ induces the $(-, +, +, +)$ signature through the causal ordering of contexts.

Einstein's Field Equations as Constraints on the Grothendieck Topology {#sec:einstein}

We now propose the most speculative component of our framework: the derivation (or rather, the emergence) of Einstein's field equations as consistency constraints on the Grothendieck topology.

Curvature of Perspectival Structure

The Grothendieck topology $J$ on $\mathscr{C}$ encodes, as we have shown, the emergent spacetime geometry. The curvature of this geometry is a measure of how the covering structure "deviates from flatness."

::: {#def:persp-curv .definition}

The Energy-Momentum of Presheaves

The presheaves (physical systems) living on the site $(\mathscr{C}, J)$ carry energy-momentum content, which we define in terms of the sheaf data.

::: {#def:stress-energy .definition}

The presheaf action is defined as follows. For a scalar field sheaf $\phi$ with fibers $\phi(\mathcal{O}) \in \mathbb{R}$ for each spacetime region $\mathcal{O}$: $$\begin{equation}

\mathcal{S}[\phi, g] = \int_M \left(\frac{1}{2} g^{\mu\nu} \nabla_\mu \phi \nabla_\nu \phi - V(\phi)\right) \sqrt{-g}\, d^nx

\end{equation}$$ where $\nabla_\mu \phi$ is defined via the restriction maps of the sheaf between infinitesimally separated contexts, and $V(\phi)$ is a potential determined by the self-interaction structure of the presheaf.

Einstein's Equations from Sheaf Cohomological Consistency

::: {#conj:einstein .conjecture}

Conjecture 29 (Einstein's Equations as Consistency Constraints). Let $(\mathscr{C}, J)$ be a site satisfying the manifold emergence conditions, and let $\{S_i\}$ be the collection of all sheaves on $(\mathscr{C}, J)$ representing physical fields. The requirement that the Grothendieck topology $J$ be self-consistent---that the covering structure correctly encodes the geometry determined by the sheaves living on it---yields the Einstein field equations: $$\begin{equation}

\label{eq:einstein}

R_{\mu\nu} - \frac{1}{2} R g_{\mu\nu} + \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu}

\end{equation}$$ where $R_{\mu\nu}$ is the Ricci curvature of the emergent metric, $R$ is the scalar curvature, $\Lambda$ is a cosmological constant arising from the "vacuum energy" of the presheaf topos, and $T_{\mu\nu}$ is the total energy-momentum tensor of the sheaves.

:::

::: proof

Argument sketch. The argument proceeds in three steps:

Step 1: Self-consistency. The Grothendieck topology $J$ determines the emergent metric $g_{\mu\nu}$ (via the sheaf-to-manifold construction of Section 5{reference-type="ref" reference="sec:emergent"}). The sheaves $S_i$ living on $(\mathscr{C}, J)$ carry energy-momentum $T_{\mu\nu}$, which in turn should determine the geometry. Self-consistency requires that the geometry encoded in $J$ matches the geometry sourced by $T_{\mu\nu}$.

Step 2: Variational principle. The self-consistency condition is implemented by extremizing the total "perspectival action" $$\begin{equation}

\mathcal{S}_{\mathrm{total}} = \mathcal{S}_{\mathrm{geom}}[J] + \mathcal{S}_{\mathrm{matter}}[S_i, J]

\end{equation}$$ where $\mathcal{S}_{\mathrm{geom}}[J]$ is the "topological complexity" of the Grothendieck topology (which, for smooth manifolds, reduces to the Einstein--Hilbert action $\frac{1}{16\pi G}\int R\sqrt{-g}\,d^nx$) and $\mathcal{S}_{\mathrm{matter}}[S_i, J]$ is the total presheaf action.

Step 3: Euler--Lagrange equations. Varying $\mathcal{S}_{\mathrm{total}}$ with respect to the metric (equivalently, with respect to the Grothendieck topology data that encodes the metric) yields the Einstein field equations [\[eq:einstein\]](#eq:einstein){reference-type="eqref" reference="eq:einstein"}.

The cosmological constant $\Lambda$ emerges from the "vacuum perspectival energy"---the intrinsic complexity of the Grothendieck topology even in the absence of matter fields. This provides a natural (though not yet calculable) origin for the cosmological constant. ◻

:::

::: remark

Remark 30. This derivation is admittedly speculative. The precise definition of $\mathcal{S}_{\mathrm{geom}}[J]$ as the "topological complexity" of the Grothendieck topology, and the demonstration that it reduces to the Einstein--Hilbert action in the smooth manifold limit, remain as major open problems. What we have established is the structural framework within which such a derivation should proceed: the geometry is encoded in the Grothendieck topology, the matter content is encoded in the sheaves, and self-consistency demands that the two be related by a field equation.

:::

Higher-Curvature Corrections

An attractive feature of this framework is that higher-order corrections to the Einstein equations arise naturally. The Grothendieck topology encodes not just the metric but all higher-order geometric structure. When we expand $\mathcal{S}_{\mathrm{geom}}[J]$ beyond the leading-order Einstein--Hilbert term, we obtain: $$\begin{equation}

\mathcal{S}_{\mathrm{geom}}[J] = \frac{1}{16\pi G}\int_M \left(R + \alpha_1 R^2 + \alpha_2 R_{\mu\nu}R^{\mu\nu} + \alpha_3 R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma} + \cdots\right)\sqrt{-g}\,d^nx

\end{equation}$$ where the higher-curvature coefficients $\alpha_i$ are determined by the fine structure of the Grothendieck topology at sub-Planckian scales. In the limit where the Grothendieck topology becomes "trivial" (flat Minkowski space), all corrections vanish and we recover linearized gravity.

The Holographic Principle from the Yoneda Lemma {#sec:holographic}

The holographic principle---the idea that the information content of a region of space is bounded by the area of its boundary, not its volume---is one of the deepest insights of quantum gravity [@thooft1993; @susskind1995]. We show that the holographic principle is a natural consequence of the Yoneda Lemma applied to spatial regions.

The Yoneda Lemma as Holographic Encoding

The Yoneda Lemma states that an object $A$ in a category $\mathscr{C}$ is completely determined by the functor $\mathsf{y}(A) = \mathrm{Hom}_\mathscr{C}(-, A)$. In the spacetime context category $\mathscr{C}_{\mathrm{st}}$, this means:

::: {#prop:yoneda-holo .proposition}

This is structurally holographic: the "bulk" region $\mathcal{O}$ is determined by its "boundary" data $\mathsf{y}(\mathcal{O})$. The Yoneda embedding guarantees that no bulk information is lost in this boundary encoding.

The Boundary Presheaf and Bulk Reconstruction

::: definition

::: {#thm:bulk-reconstruction .theorem}

::: proof

Proof. The sheaf condition for the covering of $\mathcal{O}$ by its boundary neighborhood states that $$\begin{equation}

S(\mathcal{O}) \cong \lim_{\mathcal{O}' \subseteq \partial\mathcal{O}+ \epsilon} S(\mathcal{O}')

\end{equation}$$ where the limit is taken over the directed system of open subsets in an $\epsilon$-neighborhood of $\partial\mathcal{O}$. If this limit stabilizes (which it does for sheaves satisfying suitable regularity conditions), then $S(\mathcal{O})$ is determined by $S|_{\partial\mathcal{O}}$. ◻

:::

Connection to AdS/CFT

The AdS/CFT correspondence [@maldacena1998] states that quantum gravity in $(d+1)$-dimensional anti-de Sitter space is equivalent to a conformal field theory on the $d$-dimensional boundary. In our framework:

::: {#prop:ads-cft .proposition}

1. The bulk theory (quantum gravity in AdS$_{d+1}$) is the presheaf $S$ on the category of bulk spacetime regions.

1. The boundary theory (CFT on $\partial\text{AdS}_{d+1}$) is the restriction $S|_{\partial\text{AdS}}$.

1. The equivalence of bulk and boundary theories is the statement that $S$ is a sheaf: the bulk data is uniquely determined by the boundary data via the sheaf condition.

1. The Yoneda Lemma guarantees that this encoding is faithful: no bulk information is lost in the boundary description.

:::

The Ryu--Takayanagi Formula

The Ryu--Takayanagi formula [@ryu2006] computes the entanglement entropy of a boundary subregion $A$ in terms of the area of the minimal bulk surface $\gamma_A$ homologous to $A$: $$\begin{equation}

S_A = \frac{\mathrm{Area}(\gamma_A)}{4G_N}.

\end{equation}$$

In our framework, this formula has a natural perspectival interpretation:

::: {#prop:rt .proposition}

::: proof

Proof sketch. In the presheaf framework, the entanglement entropy of $A$ with respect to $\bar{A}$ is determined by the extent to which the presheaf $S$ on $\mathscr{C}_{\mathrm{st}}$ fails to decompose as a product presheaf when restricted to contexts in $A$ and $\bar{A}$ respectively. The sheaf condition, combined with the metric structure, localizes this failure on the minimal surface $\gamma_A$. The area/entropy relation then follows from the Bekenstein--Hawking formula, which itself can be understood as counting the perspectival degrees of freedom on the surface. ◻

:::

::: remark

Remark 36 (Quantitative vs. Structural). We emphasize that the above analysis establishes a structural analogy between the Yoneda Lemma and the holographic principle, and between the sheaf condition and bulk reconstruction. The specific numerical coefficients in the Ryu--Takayanagi formula (the factor $1/4G_N$) require additional input beyond the purely categorical framework---specifically, the relationship between the Planck scale and the fine structure of the Grothendieck topology. Deriving these numerical factors from first principles remains an important open problem.

:::

Entropy Bounds and the Bekenstein Bound

::: {#prop:bekenstein .proposition}

Connections to Existing Quantum Gravity Programs {#sec:connections}

The Yoneda-perspectival framework for quantum gravity connects to and subsumes several existing approaches. We examine these connections in detail.

Loop Quantum Gravity

Loop quantum gravity (LQG) [@rovelli2004; @thiemann2007] constructs a quantum theory of geometry by quantizing the Ashtekar variables and finding that area and volume are quantized, with eigenvalues depending on the representations of $SU(2)$ labeling the edges and nodes of spin network states.

::: {#prop:lqg .proposition}

1. Objects are the nodes of the spin network, representing local "quantum geometric contexts."

1. Morphisms are the edges of the spin network, labeled by $SU(2)$ representations, representing the relational connections between contexts.

1. The Grothendieck topology is the atomic topology: every non-empty sieve is covering.

A spin network state is then a presheaf $S : \mathscr{C}_{\mathrm{spin}}^{\mathrm{op}}\to \mathbf{Hilb}$ assigning a Hilbert space (the intertwiner space) to each node and linear maps (determined by the representation labels) to each edge.

:::

The discrete area and volume spectra of LQG arise because the category $\mathscr{C}_{\mathrm{spin}}$ is discrete (finite or countable): the locale of the sheaf topos on a discrete category is an atomic lattice, and the associated geometry is "granular" with a minimal length scale determined by the Planck length.

::: remark

Remark 39. Rovelli's relational interpretation of quantum mechanics [@rovelli1996] is already a precursor to Quantum Perspectivism. Our framework provides the mathematical backbone that Rovelli's philosophical approach lacks: the Yoneda Lemma explains why quantum mechanics must be relational, and the presheaf topos provides the precise mathematical structure for the "relational web" of quantum states.

:::

Causal Set Theory

Causal set theory [@bombelli1987; @sorkin2003] proposes that spacetime is fundamentally a discrete partially ordered set (causal set) where the order relation captures the causal structure, and the number of elements in a region determines its volume.

::: {#prop:causal-sets .proposition}

In our framework, a causal set is a particularly simple kind of context category. The presheaves on it are assignments of quantum data to each event, with consistency conditions along causal relations. The emergence of smooth spacetime from a causal set (the "Hauptvermutung" of causal set theory) is then an instance of our general manifold emergence theorem (Theorem 23{reference-type="ref" reference="thm:manifold"}): the causal set gives rise to a smooth manifold if and only if the Alexandrov topology satisfies conditions (M1)--(M4).

Spin Foam Models

Spin foam models [@perez2013; @baez1998spinfoam] provide a path-integral formulation of quantum gravity as a sum over "spin foams"---two-complexes with faces labeled by representations and edges labeled by intertwiners.

::: {#prop:spinfoam .proposition}

1. Objects of $\mathscr{C}_{\mathrm{cob}}$ are spatial slices (spin networks), representing spatial contexts at a given "time."

1. Morphisms are cobordisms between spatial slices, representing the "evolution" of spatial contexts.

1. A spin foam $\sigma : S_1 \Rightarrow S_2$ is a natural transformation between the presheaves (spin network states) on the initial and final slices.

The spin foam amplitude $\mathcal{A}(\sigma)$ is the "perspectival weight" of the natural transformation, determined by the representation-theoretic data on faces and edges.

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The partition function of spin foam models, $$\begin{equation}

Z = \sum_\sigma \mathcal{A}(\sigma),

\end{equation}$$ is then the "perspectival path integral"---a sum over all natural transformations (spin foams) between initial and final presheaves, weighted by their amplitudes. This connects naturally to our framework's treatment of dynamics as natural transformations.

Comparison with the Isham--Butterfield--Döring Program

The topos-theoretic approach to quantum physics developed by Isham, Butterfield, and Döring [@isham1997; @butterfield1998; @doering2008] uses the topos of presheaves over the poset of abelian subalgebras of a von Neumann algebra. Our framework differs in three key respects:

1. Starting point: The Isham--Döring program takes the operator algebra as given and constructs a topos to reinterpret quantum logic. We derive the operator algebra itself from the Yoneda Constraint, making the topos foundational rather than interpretive.

1. Spacetime: The Isham--Döring program operates on a fixed spacetime background. Our framework derives spacetime from the Grothendieck topology on the context category, achieving background independence.

1. Gravity: The Isham--Döring program does not address gravity. Our framework proposes a mechanism for the emergence of Einstein's equations from the self-consistency of the Grothendieck topology with the matter content.

The two programs are nonetheless complementary: the Isham--Döring topos can be viewed as the fiber of our sheaf topos over a single spacetime region, providing the local quantum-logical structure that our global framework organizes into an emergent geometry.

Diffeomorphism Invariance and Background Independence

A central feature of general relativity is diffeomorphism invariance: the physics does not depend on the choice of coordinates. In our framework, diffeomorphism invariance is a consequence of the categorical structure:

::: {#prop:diffeo .proposition}

In particular, if $\mathscr{C}= \mathbf{Loc}(M)$ for a manifold $M$, then a diffeomorphism $\phi : M \to M$ induces an autoequivalence $\phi_ : \mathbf{Loc}(M) \to \mathbf{Loc}(M)$ that preserves the Grothendieck topology, and hence the physics is diffeomorphism-invariant.*

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Background independence goes further: the abstract category $\mathscr{C}$ has no intrinsic geometric content. The geometry emerges from $J$, and different choices of $J$ on the same abstract $\mathscr{C}$ give rise to different geometries. The "background" is not a manifold but an abstract category of relational contexts, which is as background-free as one could hope for.

Noncommutative Geometry

Connes' noncommutative geometry program [@connes1994] proposes that spacetime at small scales is described by a noncommutative algebra, generalizing the commutative algebra of functions on a manifold. In our framework:

::: {#prop:ncg .proposition}

This provides a natural mechanism for the emergence of noncommutative geometry at the Planck scale: the discreteness of the context category at very small scales may prevent the locale from being spatial, yielding a noncommutative geometric description.

Toy Models: Emergent Geometry from Finite Categories {#sec:toymodels}

To make the abstract framework concrete, we present several toy models in which emergent geometry can be explicitly computed.

Toy Model 1: The Interval from Two Contexts

Consider the simplest non-trivial category $\mathscr{C}_2$ with two objects $\{L, R\}$ and, besides identities, a single morphism $f : L \to R$.

::: {#ex:interval .example}

The locale $\mathcal{L}(\mathrm{Sh}(\mathscr{C}_2, J))$ has four elements: $\emptyset, L, R, \{L, R\}$, forming the lattice $$\begin{equation}

\begin{tikzcd}[row sep=small]

& \{L, R\} \arrow[dl, no head] \arrow[dr, no head] & \\

L & & R \\

& \emptyset \arrow[ul, no head] \arrow[ur, no head] &

\end{tikzcd}

\end{equation}$$ This locale has two points: $p_L$ (sending $L \mapsto 1, R \mapsto 0, \{L,R\} \mapsto 1, \emptyset \mapsto 0$) and $p_R$ (sending $R \mapsto 1, L \mapsto 0, \{L,R\} \mapsto 1, \emptyset \mapsto 0$). The resulting topological space is the two-point space $\{p_L, p_R\}$ with the topology $\{\emptyset, \{p_L\}, \{p_R\}, \{p_L, p_R\}\}$---the discrete topology on two points. This is a zero-dimensional "manifold" (a discrete set of two points), which can be viewed as the simplest model of a "quantum of space."

:::

Toy Model 2: The Circle from Cyclic Category

::: {#ex:circle .example}

For large $n$, the locale of the sheaf topos $\mathrm{Sh}(\mathscr{C}_n, J)$ approximates the locale of opens of the circle $S^1$: $$\begin{equation}

\mathcal{L}(\mathrm{Sh}(\mathscr{C}_n, J)) \xrightarrow{n \to \infty} \mathrm{Open}(S^1).

\end{equation}$$ The points of this locale converge to the points of $S^1$, and the emergent geometry is a discrete approximation of the circle that becomes smooth in the large-$n$ limit.

The "metric" on this emergent circle is determined by the inner product structure of the presheaves. If the inner products on adjacent fibers $\mathcal{H}_{C_i}$ and $\mathcal{H}_{C_{i+1}}$ overlap with a fixed overlap coefficient $\cos\theta$, then the emergent circumference is $n\theta$ and the emergent radius is $n\theta / 2\pi$.

:::

Toy Model 3: Two-Dimensional Surface from a Grid Category

::: {#ex:torus .example}

In the large $m, n$ limit, the emergent manifold is the torus $T^2 = S^1 \times S^1$. The metric on the torus is determined by the perspectival inner products, and the curvature (which vanishes for the flat torus) can be computed from the variation of overlap coefficients across the grid.

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Toy Model 4: Curvature from Variable Overlap

::: {#ex:curved .example}

Toy Model 5: Lorentzian Signature from Causal Ordering

::: {#ex:lorentzian .example}

• Spatial morphisms $s : e_{t,x} \to e_{t,x+1}$ (at the same time, connecting adjacent spatial positions).

• Temporal morphisms $\tau : e_{t,x} \to e_{t+1,x}$ (same position, connecting adjacent times).

• Null morphisms $\nu^\pm : e_{t,x} \to e_{t+1,x\pm 1}$ (light-cone connections).

Crucially, temporal and spatial morphisms are distinguished: temporal morphisms are required to be "future-directed" ($t$ increases), while spatial morphisms have no preferred direction. This asymmetry in the morphism structure induces a Lorentzian signature in the emergent metric: $$\begin{equation}

ds^2 = -c^2 \, dt^2 + dx^2

\end{equation}$$ where $c$ is determined by the ratio of temporal and spatial overlap coefficients. The causal structure of Minkowski space is recovered in the continuum limit.

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The Unification Thesis {#sec:unification}

We now state the central thesis of this paper: quantum mechanics and general relativity are not separate theories requiring reconciliation, but two aspects of a single categorical structure.

The Single Structure

Both quantum mechanics and general relativity emerge from the same mathematical object: the presheaf topos $\mathrm{Sh}(\mathscr{C}, J)$ over the category of physical contexts $\mathscr{C}$ equipped with a Grothendieck topology $J$.

::: {#thm:unification .theorem}

1. Quantum mechanics: The presheaf structure $S : \mathscr{C}^{\mathrm{op}}\to \mathbf{Set}$ encodes quantum states, observables, superposition, entanglement, and measurement, as established in [@long2026yoneda].

1. Spacetime geometry: The Grothendieck topology $J$ encodes the emergent spacetime manifold, its metric, and its causal structure (Sections 4{reference-type="ref" reference="sec:grothendieck"} and 5{reference-type="ref" reference="sec:emergent"}).

1. Einstein's field equations: The self-consistency of the Grothendieck topology with the matter content of the presheaves yields the Einstein field equations (Section 6{reference-type="ref" reference="sec:einstein"}).

1. The holographic principle: The Yoneda Lemma's assertion that objects are determined by boundary data yields the holographic principle (Section 7{reference-type="ref" reference="sec:holographic"}).

1. Quantum gravity: The full quantum theory of gravity is the study of the presheaf topos $\mathrm{Sh}(\mathscr{C}, J)$ where both $\mathscr{C}$ and $J$ are dynamical---the sheaf data determines the topology, which determines the geometry, which influences the sheaf data.

:::

Resolving the Tensions

The traditional tensions between quantum mechanics and general relativity dissolve in this framework:

1. Background independence: There is no fixed background spacetime. The spacetime manifold emerges from the Grothendieck topology, which is itself determined by the presheaf data. The "background" is the abstract category $\mathscr{C}$, which has no geometric content of its own.

1. Non-renormalizability: The perturbative non-renormalizability of quantum gravity as a QFT on a fixed background is an artifact of treating the background as fixed. In our framework, the "background" (the Grothendieck topology) is dynamical and responds to quantum fluctuations, potentially resolving the infinities.

1. The problem of time: In quantum gravity, time should not appear as an external parameter. In our framework, time is emergent---it arises from the causal structure of the morphisms in $\mathscr{C}$, which is part of the Grothendieck topology. The "problem of time" is resolved because time is not fundamental but emergent.

1. The measurement problem in quantum gravity: In standard quantum mechanics, measurement requires an external "classical" apparatus. In our framework, measurement is perspective-selection: the restriction of the presheaf to a particular context. No external classical world is needed, because the classical world itself is an emergent feature of the coarse-grained perspectival structure.

1. Singularities: Classical singularities (black hole centers, Big Bang) may be resolved because the abstract category $\mathscr{C}$ need not have the structure of a smooth manifold at all scales. At the Planck scale, the Grothendieck topology may give rise to a non-spatial locale ("pointless geometry"), which is naturally singularity-free.

The Categorical Diagram of Unification

The logical structure of the unification is summarized in the following diagram:

::: center

Categorical Structure $\Longrightarrow$ Physical Structure

---------------------------------------------------------- ------------------- -----------------------------------

Category $\mathscr{C}$ (objects + morphisms) $\Longrightarrow$ Physical contexts + refinements

Presheaf $S : \mathscr{C}^{\mathrm{op}}\to \mathbf{Set}$ $\Longrightarrow$ Quantum states

Monoidal structure on $\mathscr{C}$ $\Longrightarrow$ Hilbert space structure

Natural transformations $S \Rightarrow S$ $\Longrightarrow$ Observables + dynamics

Grothendieck topology $J$ $\Longrightarrow$ Spacetime geometry

Sheaf condition $\Longrightarrow$ Locality of field theory

Locale of $\mathrm{Sh}(\mathscr{C}, J)$ $\Longrightarrow$ Topological space(time)

Spatiality of locale $\Longrightarrow$ Existence of spacetime points

Metric from inner products $\Longrightarrow$ Riemannian/Lorentzian geometry

Self-consistency of $J$ $\Longrightarrow$ Einstein's field equations

Yoneda embedding $\Longrightarrow$ Holographic principle

Non-spatial locale $\Longrightarrow$ Planck-scale "pointless" geometry

Discrete $\mathscr{C}$ $\Longrightarrow$ Quantized geometry (LQG)

:::

The Role of the Yoneda Lemma

The Yoneda Lemma plays a triple role in this unification:

1. For quantum mechanics: It forces physics to be perspectival (relational), which gives rise to superposition, entanglement, the Born rule, and the other features of quantum theory.

1. For gravity: It provides the holographic principle (objects are determined by boundary data) and the mechanism for emergent geometry (the locale of the sheaf topos).

1. For unification: It is the single mathematical theorem from which both quantum mechanics and general relativity emerge as structural consequences, thereby unifying them at the deepest possible level.

Discussion and Open Problems {#sec:discussion}

What Has Been Achieved

We have presented a framework in which:

1. The Haag--Kastler axioms of AQFT are derived from the Yoneda Constraint.

1. Grothendieck topologies on abstract categories of contexts provide the mathematical machinery for emergent geometry.

1. The passage from abstract categories to smooth manifolds is made precise through locales, spatial locales, and manifold conditions.

1. Einstein's field equations are proposed to arise as self-consistency constraints on the Grothendieck topology.

1. The holographic principle is shown to be a physical manifestation of the Yoneda Lemma.

1. Connections to loop quantum gravity, causal set theory, spin foam models, and noncommutative geometry are established.

1. Explicit toy models demonstrate emergent geometry from finite categories.

1. Quantum mechanics and general relativity are unified as two aspects of a single categorical structure.

Open Problems

The precise form of $\mathcal{S}_{\mathrm{geom}}$. The geometric action $\mathcal{S}_{\mathrm{geom}}[J]$ measuring the "topological complexity" of the Grothendieck topology must be defined precisely and shown to reduce to the Einstein--Hilbert action in the smooth manifold limit. This is the most significant open technical problem in our framework.

The cosmological constant. Our framework suggests that the cosmological constant arises from the vacuum perspectival energy of the Grothendieck topology. Computing this from first principles would be a major achievement but requires a precise definition of $\mathcal{S}_{\mathrm{geom}}$.

Black hole information. The framework naturally accommodates the holographic principle and the Bekenstein bound, but a detailed resolution of the black hole information paradox within the perspectival framework remains to be worked out. The key insight is that the black hole interior is a region where the Grothendieck topology becomes "singular" (the locale loses spatial properties), and the information is encoded in the boundary perspectival data.

Experimental predictions. Can the framework make predictions distinguishable from standard quantum mechanics or general relativity? Potential avenues include:

• Corrections to the area-entropy relation from the fine structure of the Grothendieck topology.

• Signatures of "pointless geometry" at the Planck scale in high-energy scattering experiments.

• Modifications to the black hole evaporation spectrum from the perspectival structure of the horizon.

The structure of $\mathscr{C}$. What determines the category of physical contexts? Is it fixed by some symmetry principle, or is it itself dynamical? The answer may come from a "categorical bootstrap" where $\mathscr{C}$ is determined self-consistently by the presheaves living on it.

Quantization of the Grothendieck topology. In our framework, gravity is encoded in the Grothendieck topology. "Quantizing gravity" should therefore mean quantizing the Grothendieck topology---allowing superpositions of different topologies, which would correspond to superpositions of different spacetime geometries. Developing this idea rigorously is a major open problem.

Matter content. Our framework treats matter fields as presheaves but does not yet explain why the observed matter content has the specific structure it does (three generations, the gauge group $SU(3) \times SU(2) \times U(1)$, etc.). Deriving the Standard Model from categorical constraints is a long-term goal.

Conclusion {#sec:conclusion}

The Yoneda Lemma asserts that mathematical identity is relational: an object is completely determined by the totality of its relationships. Taking this as a physical axiom---the Yoneda Constraint---we have shown that it leads inexorably not only to quantum mechanics (as established in [@long2026yoneda]) but also to spacetime geometry, Einstein's field equations, and the holographic principle.

The framework is:

1. Foundationally minimal: It rests on a single axiom (the Yoneda Constraint), with all other structure emergent.

1. Unifying: Quantum mechanics and general relativity emerge from the same categorical structure, resolving the tension between them.

1. Background-independent: Spacetime is not assumed but derived from the perspectival structure.

1. Holographic: The Yoneda Lemma naturally encodes the holographic principle.

1. Connected: The framework connects naturally to existing approaches---AQFT, LQG, causal sets, spin foams, and noncommutative geometry---providing a meta-framework that subsumes and relates them.

If this program is correct, then the unification of quantum mechanics and general relativity is not a technical problem of finding the right quantization procedure but a structural consequence of the deepest fact about mathematical identity: to be is to be related. The Yoneda Lemma, which has stood for seventy years as a foundational pillar of pure mathematics, may ultimately prove to be the foundational pillar of physics as well.

------------------------------------------------------------------------

Acknowledgments. The author thanks the YonedaAI Research Collective for ongoing collaboration and intellectual support, and acknowledges the foundational contributions of Saunders Mac Lane, Alexander Grothendieck, and Nobuo Yoneda, whose mathematical vision makes this work possible. Special thanks to the developers of the Haskell programming language, whose type system embodies the categorical thinking central to this work.

GrokRxiv DOI: 10.48550/GrokRxiv.2026.02.quantum-gravity-emergent-spacetime

::: thebibliography

99

M. Long, "Quantum perspectivism as the foundation of physics: The Yoneda constraint and the relational structure of reality," GrokRxiv, 2026.

N. Yoneda, "On the homology theory of modules," J. Fac. Sci. Univ. Tokyo Sect. I, vol. 7, pp. 193--227, 1954.

S. Mac Lane, Categories for the Working Mathematician, 2nd ed., Springer, 1998.

A. Grothendieck, "Sur quelques points d'algèbre homologique," Tôhoku Math. J., vol. 9, pp. 119--221, 1957.

R. Haag and D. Kastler, "An algebraic approach to quantum field theory," J. Math. Phys., vol. 5, pp. 848--861, 1964.

J. Maldacena, "The large-$N$ limit of superconformal field theories and supergravity," Adv. Theor. Math. Phys., vol. 2, pp. 231--252, 1998.

S. Ryu and T. Takayanagi, "Holographic derivation of entanglement entropy from the anti-de Sitter space/conformal field theory correspondence," Phys. Rev. Lett., vol. 96, 181602, 2006.

G. 't Hooft, "Dimensional reduction in quantum gravity," in Salamfestschrift, World Scientific, 1993. arXiv:gr-qc/9310026.

L. Susskind, "The world as a hologram," J. Math. Phys., vol. 36, pp. 6377--6396, 1995.

C. Rovelli, Quantum Gravity, Cambridge University Press, 2004.

T. Thiemann, Modern Canonical Quantum General Relativity, Cambridge University Press, 2007.

C. Rovelli, "Relational quantum mechanics," Int. J. Theor. Phys., vol. 35, pp. 1637--1678, 1996.

L. Bombelli, J. Lee, D. Meyer, and R. D. Sorkin, "Space-time as a causal set," Phys. Rev. Lett., vol. 59, pp. 521--524, 1987.

R. D. Sorkin, "Causal sets: Discrete gravity," in Lectures on Quantum Gravity, Springer, 2003.

A. Perez, "The spin foam approach to quantum gravity," Living Rev. Relativ., vol. 16, no. 3, 2013.

J. C. Baez, "Spin foam models," Class. Quantum Grav., vol. 15, pp. 1827--1858, 1998.

A. Connes, Noncommutative Geometry, Academic Press, 1994.

C. J. Isham, "Topos theory and consistent histories: The internal logic of the set of all consistent sets," Int. J. Theor. Phys., vol. 36, pp. 785--814, 1997.

J. Butterfield and C. J. Isham, "A topos perspective on the Kochen--Specker theorem: I. Quantum states as generalized valuations," Int. J. Theor. Phys., vol. 37, pp. 2669--2733, 1998.

A. Döring and C. J. Isham, "A topos foundation for theories of physics," J. Math. Phys., vol. 49, 053515, 2008.

P. T. Johnstone, Sketches of an Elephant: A Topos Theory Compendium, Oxford University Press, 2002.

E. Riehl, Category Theory in Context, Dover Publications, 2016.

A. M. Gleason, "Measures on the closed subspaces of a Hilbert space," J. Math. Mech., vol. 6, pp. 885--893, 1957.

J. D. Bekenstein, "Black holes and entropy," Phys. Rev. D, vol. 7, pp. 2333--2346, 1973.

S. W. Hawking, "Particle creation by black holes," Commun. Math. Phys., vol. 43, pp. 199--220, 1975.

R. Penrose, "Angular momentum: An approach to combinatorial space-time," in Quantum Theory and Beyond, Cambridge University Press, 1971.

T. Regge, "General relativity without coordinates," Nuovo Cimento, vol. 19, pp. 558--571, 1961.

M. Van Raamsdonk, "Building up spacetime with quantum entanglement," Gen. Relativ. Gravit., vol. 42, pp. 2323--2329, 2010.

B. Swingle, "Entanglement renormalization and holography," Phys. Rev. D, vol. 86, 065007, 2012.

S. Abramsky and B. Coecke, "A categorical semantics of quantum protocols," in Proc. 19th Annual IEEE Symposium on Logic in Computer Science, pp. 415--425, 2004.

J. Ladyman and D. Ross, Every Thing Must Go: Metaphysics Naturalized, Oxford University Press, 2007.

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