Preliminaries
A Topos-Theoretic Bridge for Holographic Entanglement Entropy
Definition 1 A small category is a category \(\mathcal{C}\) whose class of objects \(\operatorname{Ob}(\mathcal{C})\) and class of morphisms \(\operatorname{Mor}(\mathcal{C})\) are both sets (i.e. elements of the universe of sets with which we work). Thus one can safely form functor categories such as
\[ \widehat{\mathcal{C}}:=\operatorname{PSh}(\mathcal{C})=\left[\mathcal{C}^{\mathrm{op}}, \mathbf{S e t}\right] \]
without running into size issues.
1 Holographic Principle
The holographic principle was first articulated in the context of black-hole unitarity by ’t Hooft (1993) (Hooft 1993).
2 Holographic Duality
2.1 Topos and site
Definition 2 An adjunction consists of a pair of functors \(L \colon \mathcal{C} \rightleftarrows \mathcal{D} \colon R\) together with a natural isomorphism \(\mathcal{D}(LA, B) \cong \mathcal{C}(A, RB)\).
Definition 3 A group is a set \(G\) with a binary operation \(\cdot\) satisfying: 1. Associativity: \((a\cdot b)\cdot c=a\cdot(b\cdot c)\); 2. Identity: \(\exists e\in G\) such that \(e\cdot a=a\cdot e=a\); 3. Inverses: \(\forall a\in G\ \exists a^{-1}\in G\) such that \(a\cdot a^{-1}=a^{-1}\cdot a=e\).
Definition 4 A category \(\mathcal{C}\) consists of the following data: 1. A class \(\text{ob}(\mathcal{C})\) whose elements are called objects; 2. For every pair of objects \(A, B \in \text{ob}(\mathcal{C})\), a set \(\text{hom}_{\mathcal{C}}(A, B)\) (called morphisms from \(A\) to \(B\)); 3. For every triple of objects \(A, B, C \in \text{ob}(\mathcal{C})\), a composition function: \[\text{hom}_{\mathcal{C}}(A, B) \times \text{hom}_{\mathcal{C}}(B, C) \to \text{hom}_{\mathcal{C}}(A, C), \quad (f, g) \mapsto g \circ f\] satisfying three axioms: - (Disjointness) Morphism sets are disjoint (each morphism uniquely determines its domain and codomain); - (Associativity) For all \(f \in \text{hom}(A,B)\), \(g \in \text{hom}(B,C)\), \(h \in \text{hom}(C,D)\), \((h \circ g) \circ f = h \circ (g \circ f)\); - (Identity) For every object \(A\), there exists a unique identity morphism \(\text{id}_A \in \text{hom}(A,A)\) such that \(f \circ \text{id}_A = f\) (for all \(f \in \text{hom}(A,B)\)) and \(\text{id}_A \circ g = g\) (for all \(g \in \text{hom}(C,A)\)).
Definition 5 The dual category \(\mathcal{C}^{\text{op}}\) of a category \(\mathcal{C}\) has: - The same objects as \(\mathcal{C}\) (i.e., \(\text{ob}(\mathcal{C}^{\text{op}}) = \text{ob}(\mathcal{C})\)); - Morphisms \(\text{hom}_{\mathcal{C}^{\text{op}}}(A, B) = \text{hom}_{\mathcal{C}}(B, A)\) (all morphisms are reversed); - Composition \(\circ_{\text{op}}\) defined by \(f \circ_{\text{op}} g = g \circ f\) (where \(\circ\) is composition in \(\mathcal{C}\)).
Definition 6 Let \(F, G: \mathcal{C} \to \mathcal{D}\) be two functors between categories \(\mathcal{C}\) and \(\mathcal{D}\). A natural transformation \(\eta: F \Rightarrow G\) consists of: 1. For every object \(X \in \text{ob}(\mathcal{C})\), a component morphism \(\eta_X \in \text{hom}_{\mathcal{D}}(F(X), G(X))\); 2. A commutativity condition: For every morphism \(f \in \text{hom}_{\mathcal{C}}(X, Y)\) in \(\mathcal{C}\), the diagram \[\begin{CD} F(X) @>{\eta_X}>> G(X) \\ @V{F(f)}VV @VV{G(f)}V \\ F(Y) @>{\eta_Y}>> G(Y) \end{CD}\] commutes (i.e., \(\eta_Y \circ F(f) = G(f) \circ \eta_X\)).