The Grand Dictionary of Galois Correspondences

In the Language of Adjoint Functors

Categorical Concept	Field Galois Theory (\(L/K\))	Hilbert’s Nullstellensatz (\(k[x_1,\dots,x_n]\))	Grothendieck’s Scheme Theory	Covering Space Theory (\(\text{Cov}(X)\))	Pointless Topology (Locales)	Stone Duality (Boolean \(\leftrightarrow\) Stone)	Functional Analysis (Banach Duals)	Convex Geometry (Polar Polytopes)	Pontryagin Duality (LCA Groups)
Dualizing Object \(\Omega\) & Pairing	Ambient field \(L\) via action \(\sigma(x)=x\)	Field \(k\) via evaluation \(f(x)=0\)	Affine line scheme \(\mathbb{A}^1_{\mathbb{Z}} = \text{Spec}(\mathbb{Z}[t])\)	Universal cover \(\widetilde{X}\) as a \(\pi_1(X)\)-torsor	Sierpiński space \(\{0,1\}\) (open/closed point)	Discrete 2-point space \(\mathbf{2}=\{0,1\}\)	Scalar field \(\mathbb{R}/\mathbb{C}\) via \(\langle \varphi, x \rangle = 0\)	Bound \(\mathbb{R}\) via inequality \(\langle x,y \rangle \le 1\)	Circle group \(S^1 = \mathbb{R}/\mathbb{Z}\) via \(\chi(g) = 1\)
Category \(\mathcal{C}\)	Intermediate fields \(K \subseteq E \subseteq L\) (\(\subseteq\))	Ideals \(J \subseteq R\) (\(\subseteq\))	Commutative Rings \(\mathbf{CRing}\)	Covering spaces \(\text{Cov}(X)\) over \(X\)	Topological Spaces \(\mathbf{Top}\)	Stone Spaces \(\mathbf{Stone}\) (or \(\mathbf{Top}\))	Subspaces \(Y \subseteq X\) of Banach space (\(\subseteq\))	Euclidean subsets \(A \subseteq \mathbb{R}^n\) (\(\subseteq\))	Subgroups \(H \le G\) of LCA group (\(\subseteq\))
Category \(\mathcal{D}\)	Subgroups \(H \le G\) (\(\supseteq\))	Variety subsets \(S \subseteq \mathbb{A}^n_k\) (\(\supseteq\))	Locally Ringed Spaces \(\mathbf{LRS}^{\text{op}}\)	\(\pi_1(X, x_0)\)-Sets	Locales \(\mathbf{Loc} = \mathbf{Frm}^{\text{op}}\)	Opposite of Booleans \(\mathbf{Bool}^{\text{op}}\)	Dual subspaces \(W \subseteq X^*\) (\(\supseteq\))	Dual subsets \(B \subseteq (\mathbb{R}^n)^*\) (\(\supseteq\))	Dual subgroups \(\Lambda \le \widehat{G}\) (\(\supseteq\))
Morphism in \(\mathcal{C}\)	Field inclusion \(E_1 \subseteq E_2\)	Ideal inclusion \(J_1 \subseteq J_2\)	Ring homomorphism \(\varphi: R \to S\)	Continuous map of covers \(E_1 \to E_2\)	Continuous map \(f: X \to Y\)	Continuous map \(f: X \to Y\)	Subspace inclusion \(Y_1 \subseteq Y_2\)	Subset inclusion \(A_1 \subseteq A_2\)	Subgroup inclusion \(H_1 \subseteq H_2\)
Morphism in \(\mathcal{D}\)	Subgroup containment \(H_1 \supseteq H_2\)	Subset containment \(S_1 \supseteq S_2\)	Morphism \((Y, \mathcal{O}_Y) \to (X, \mathcal{O}_X)\)	\(\pi_1\)-equivariant map \(S_1 \to S_2\)	Frame map \(L_2 \to L_1\) (preimages)	Boolean map \(B_2 \to B_1\)	Dual containment \(W_1 \supseteq W_2\)	Dual containment \(B_1 \supseteq B_2\)	Dual containment \(\Lambda_1 \supseteq \Lambda_2\)
Left Adjoint \(F: \mathcal{C} \to \mathcal{D}\)	Galois group \(\text{Gal}(L/E)\)	Vanishing locus \(\mathbb{V}(J)\)	Spectrum functor \(\text{Spec}(R)\)	Fiber functor \(\text{Fib}_{x_0}: E \mapsto p^{-1}(x_0)\)	Open-set frame \(\mathcal{O}(X)\)	Clopen algebra \(\text{Clopen}(X)\)	Annihilator \(Y^\perp\)	Polar set \(A^\circ = \{y \mid \langle x,y \rangle \le 1\}\)	Annihilator \(H^\perp = \{\chi \mid \chi\vert{}_H = 1\}\)
Right Adjoint \(G: \mathcal{D} \to \mathcal{C}\)	Fixed field \(L^H\)	Vanishing ideal \(\mathbb{I}(S)\)	Global sections \(\Gamma(X, \mathcal{O}_X)\)	Associated bundle \((\widetilde{X} \times S)/\pi_1(X)\)	Point spectrum \(\text{pt}(L)\)	Ultrafilter space \(\text{Ult}(B)\)	Pre-annihilator \({}^\perp W\)	Polar set \(B^\circ = \{x \mid \langle x,y \rangle \le 1\}\)	Inverse annihilator \({}^\perp \Lambda\)
Hom-Set Bijection \(\text{Hom}_{\mathcal{D}}(F(c),d) \cong \text{Hom}_{\mathcal{C}}(c,G(d))\)	\(H \subseteq \text{Gal}(L/E) \iff E \subseteq L^H\)	\(S \subseteq \mathbb{V}(J) \iff J \subseteq \mathbb{I}(S)\)	\(\text{Hom}_{\mathbf{LRS}}(X, \text{Spec}(R)) \cong \text{Hom}(R, \Gamma(X))\)	\(\text{Hom}_{\pi_1}(p^{-1}(x_0), S) \cong \text{Hom}(E, \widetilde{X} \times_{\pi_1} S)\)	\(\text{Hom}(L, \mathcal{O}(X)) \cong \text{Hom}(X, \text{pt}(L))\)	\(\text{Hom}(B, \text{Clopen}(X)) \cong \text{Hom}(X, \text{Ult}(B))\)	\(W \subseteq Y^\perp \iff Y \subseteq {}^\perp W\)	\(B \subseteq A^\circ \iff A \subseteq B^\circ\)	\(\Lambda \subseteq H^\perp \iff H \subseteq {}^\perp \Lambda\)
Monad \(T = GF\)	Subfield Galois closure \(E \mapsto L^{\text{Gal}(L/E)}\)	Radical ideal \(J \mapsto \mathbb{I}(\mathbb{V}(J)) = \sqrt{J}\)	Identity functor \(\text{Id}_{\mathbf{CRing}}\) (\(R \cong \Gamma(\text{Spec} R)\))	Identity functor \(\text{Id}_{\text{Cov}(X)}\)	Soberification \(X \mapsto \text{pt}(\mathcal{O}(X))\)	Stone–Čech completion \(X \mapsto \text{Ult}(\text{Clopen} X)\)	Norm-closure of span \(\overline{\text{span}(Y)}^{\Vert{}\cdot\Vert{}}\)	Closed convex hull with \(0\): \(A^{\circ\circ}\)	Topological closure \(\overline{H}\) in \(G\)
Unit \(\text{Id}_{\mathcal{C}} \xrightarrow{\eta} T = GF\)	Field inclusion \(E \subseteq L^{\text{Gal}(L/E)}\)	Ideal inclusion \(J \subseteq \sqrt{J}\)	Canonical isomorphism \(R \stackrel{\sim}{\to} \Gamma(\text{Spec} R)\)	Canonical map \(E \stackrel{\sim}{\to} \widetilde{X} \times_{\pi_1} p^{-1}(x_0)\)	Canonical map \(x \mapsto \mathcal{O}_x\) to \(\text{pt}(\mathcal{O}(X))\)	Canonical evaluation map \(X \to \text{Ult}(\text{Clopen}(X))\)	Subspace inclusion \(Y \subseteq {}^\perp(Y^\perp)\)	Hull inclusion \(A \subseteq A^{\circ\circ}\)	Subgroup inclusion \(H \subseteq \overline{H}\)
Comonad \(W = FG\)	Krull subgroup closure \(H \mapsto \text{Gal}(L/L^H)\)	Zariski closure \(S \mapsto \mathbb{V}(\mathbb{I}(S)) = \overline{S}^{\text{Zar}}\)	Affinization \((X, \mathcal{O}_X) \mapsto \text{Spec}(\Gamma(X))\)	Identity functor \(\text{Id}_{\pi_1\text{-Set}}\)	Spatialization \(L \mapsto \mathcal{O}(\text{pt}(L))\)	Identity functor \(\text{Id}_{\mathbf{Bool}^{\text{op}}}\)	Weak\(^*\)-closure of span \(\overline{\text{span}(W)}^{w^*}\)	Closed convex hull with \(0\): \(B^{\circ\circ}\)	Topological closure \(\overline{\Lambda}\) in \(\widehat{G}\)
Counit \(W = FG \xrightarrow{\varepsilon} \text{Id}_{\mathcal{D}}\)	Subgroup inclusion \(\text{Gal}(L/L^H) \supseteq H\)	Subset inclusion \(\overline{S}^{\text{Zar}} \supseteq S\)	Canonical map \(X \to \text{Spec}(\Gamma(X))\)	Canonical isomorphism \(p^{-1}(x_0) \stackrel{\sim}{\to} S\)	Frame map \(u \mapsto \{p \mid u \in p\}\)	Stone isomorphism \(B \stackrel{\sim}{\to} \text{Clopen}(\text{Ult}(B))\)	Weak\(^*\)-inclusion \(({}^\perp W)^\perp \supseteq W\)	Dual hull inclusion \(B^{\circ\circ} \supseteq B\)	Dual subgroup inclusion \(\overline{\Lambda} \supseteq \Lambda\)
Triangle Identities \(F \xrightarrow{F\eta} FGF \xrightarrow{\varepsilon F} F\) \(G \xrightarrow{\eta G} GFG \xrightarrow{G\varepsilon} G\) Both composites are identities	\(\text{Gal}(L/L^{\text{Gal}(L/E)}) = \text{Gal}(L/E)\)	\(\mathbb{V}(\mathbb{I}(\mathbb{V}(J))) = \mathbb{V}(J)\)	\(\Gamma(\text{Spec}(\Gamma(X))) \cong \Gamma(X)\)	Adjunction triangle identities hold	\(\mathcal{O}(\text{pt}(\mathcal{O}(X))) \cong \mathcal{O}(X)\)	\(\text{Clopen}(\text{Ult}(\text{Clopen} X)) \cong \text{Clopen} X\)	\(({}^\perp(Y^\perp))^\perp = Y^\perp\)	\(A^{\circ\circ\circ} = A^\circ\) and \(B^{\circ\circ\circ} = B^\circ\)	\(({}^\perp(H^\perp))^\perp = H^\perp\)
Eilenberg–Moore \(\mathcal{C}^T\) (Fixed Points)	Galois-Closed Fields (\(L^{\text{Gal}(L/E)} = E\))	Radical Ideals (\(\sqrt{J} = J\))	All Commutative Rings \(\mathbf{CRing}\)	All Covering Spaces \(\text{Cov}(X)\)	Sober Spaces \(\mathbf{Sob}\)	Stone Spaces \(\mathbf{Stone}\)	Norm-Closed Subspaces of \(X\)	Closed Convex Sets with \(0 \in A\)	Closed Subgroups of \(G\)
Co-Eilenberg–Moore \(\mathcal{D}_W\) (Fixed Points)	Closed Subgroups in Krull topology	Affine Varieties (\(\overline{V}^{\text{Zar}} = V\))	Affine Schemes \(\mathbf{AffSch}^{\text{op}}\)	All \(\pi_1(X, x_0)\)-Sets	Spatial Locales \(\mathbf{SpatLoc}\)	All Boolean Algebras \(\mathbf{Bool}^{\text{op}}\)	Weak\(^*\)-Closed Subspaces of \(X^*\)	Closed Convex Sets with \(0 \in B\)	Closed Subgroups of \(\widehat{G}\)
Restricted Equivalence \(\mathcal{C}^T \simeq \mathcal{D}_W\)	Fundamental Theorem of Galois Theory	Hilbert’s Nullstellensatz	Grothendieck Equivalence (\(\mathbf{CRing} \simeq \mathbf{AffSch}^{\text{op}}\))	Poincaré–Grothendieck Equivalence	Isbell Duality (\(\mathbf{Sob} \simeq \mathbf{SpatLoc}\))	Stone’s Representation Theorem	Hahn–Banach Duality	Minkowski Bipolar / Farkas Lemma	Pontryagin Subgroup Duality
LAPC / Colimit Preservation \(F(\operatorname{colim}_i c_i) \cong \operatorname{colim}_i F(c_i)\)	\(\text{Gal}(L/E_1 E_2) = \text{Gal}(L/E_1) \cap \text{Gal}(L/E_2)\)	\(\mathbb{V}(J_1 + J_2) = \mathbb{V}(J_1) \cap \mathbb{V}(J_2)\)	\(R \otimes_A S \longleftrightarrow \text{Spec} R \times_{\text{Spec} A} \text{Spec} S\)	Coproduct of covers \(\longleftrightarrow\) Disjoint union of fibers	\(\coprod X_i \mapsto \prod \mathcal{O}(X_i)\)	Disjoint union of spaces \(\leftrightarrow\) Product of algebras	\((Y_1 + Y_2)^\perp = Y_1^\perp \cap Y_2^\perp\)	\((A_1 \cup A_2)^\circ = A_1^\circ \cap A_2^\circ\)	\(\overline{H_1 H_2}^\perp = H_1^\perp \cap H_2^\perp\)
RAPL / Limit Preservation \(G(\lim_i d_i) \cong \lim_i G(d_i)\)	\(L^{\langle H_1, H_2 \rangle} = L^{H_1} \cap L^{H_2}\)	\(\mathbb{I}(S_1 \cup S_2) = \mathbb{I}(S_1) \cap \mathbb{I}(S_2)\)	Global sections of glued schemes = Pullback of rings	Fiber product of covers \(\longleftrightarrow\) Pullback of \(\pi_1\)-sets	Limit of spaces \(\longleftrightarrow\) Colimit of frames	Colimit of spaces \(\longleftrightarrow\) Limit of algebras	\({}^\perp(W_1 + W_2) = {}^\perp W_1 \cap {}^\perp W_2\)	\(\overline{\text{conv}}(B_1 \cup B_2)^\circ = B_1^\circ \cap B_2^\circ\)	\({}^\perp(\overline{\Lambda_1 \Lambda_2}) = {}^\perp \Lambda_1 \cap {}^\perp \Lambda_2\)
Initial / Terminal Duality \(F(0_{\mathcal{C}}) \cong 0_{\mathcal{D}}\) \(G(1_{\mathcal{D}}) \cong 1_{\mathcal{C}}\)	\(K \mapsto G\), \(\{e\} \mapsto L\)	\((0) \mapsto \mathbb{A}^n_k\), \(\emptyset \mapsto R=(1)\)	\(\mathbb{Z} \mapsto \text{Spec}(\mathbb{Z})\), \(\{0\} \mapsto \emptyset\)	\(X \leftrightarrow \{*\}\), \(\widetilde{X} \leftrightarrow \pi_1(X)\)	\(\emptyset \leftrightarrow \{0=1\}\), \(\{*\} \leftrightarrow \{0,1\}\)	\(\emptyset \leftrightarrow \{0=1\}\), \(\{*\} \leftrightarrow \mathbf{2}\)	\(\{0\} \mapsto X^*\), \(\{0\} \mapsto X\)	\(\{0\} \mapsto (\mathbb{R}^n)^*\), \(\{0\} \mapsto \mathbb{R}^n\)	\(\{e\} \mapsto \widehat{G}\), \(\{1\} \mapsto G\)