The n-Category Café

Recap

Let’s just run through what the machine does. Start with sets $G$ and $M$ with a relation $I$ between them. This gives rise to a Galois connection between the partially ordered sets of subsets of $G$ and $M$: $$I^{\ast }\colon \mathcal{P}(G)\rightleftarrows \mathcal{P}(M)^{\mathrm{op}}\colon I_{\ast }.$$ We then get closure operations on both powersets: $$I_{\ast }I^{\ast }\colon \mathcal{P}(G)\to \mathcal{P}(G),\quad I^{\ast }I_{\ast }\colon \mathcal{P}(M)\to \mathcal{P}(M).$$ The closed subsets of $G$ are those that are invariant under the closure operation, and the set of closed subsets is denoted $\mathcal{P}_{\mathrm{cl}}(G)$ — in formal concept analysis these are the extents. Similarly, we have $\mathcal{P}_{\mathrm{cl}}(M)$ — in formal concept analysis these are called the intents.

The Galois connection restricts to a Galois correspondence, i.e. a duality, between the closed subsets of $G$ and the closed subsets of $M$. $$I^{\ast }\colon \mathcal{P}_{\mathrm{cl}}(G)\cong \mathcal{P}_{\mathrm{cl}}(M)^{\mathrm{op}}\colon I_{\ast }.$$

The concepts are then pairs $(A,B)$ consisting of a closed subset of $G$ and the corresponding closed subset of $M$. So the set of concepts is isomorphic to both the set of closed subsets of $G$ and the set of closed subsets of $M$.

The answers

Algebraic geometry Here the concepts are affine varieties. A concept consists of the pair $(X,A)$ where $X\subseteq \mathbb{C}^{n}$ is the set of points of the variety and $A\subseteq \mathbb{C}[x_{1},\dots x_{n}]$ is the ideal of functions vanishing on $X$. This gives the classic duality between affine varieties and radical ideals.

Number theory This is, of course, ‘the’ Galois correspondence. The one in Galois theory. The closed subsets of $L$ are the intermediate field extensions between $K$ and $L$ and the closed subsets of the set of automorphisms are the subgroups.

Linear algebra The closed sets of $V$ are precisely the linear subsets and the concepts are of the form $(W,W^{\circ })$, where $W^{\circ }$ is the annihilator of $W$.

Logic Here concepts are theories. A concept $(S,\alpha )$ consists of $\alpha$ a set of sentences which is closed under logical consequence and $S$ the set of models of the theory. The duality in this case is Lawvere’s adjunction between semantics and syntax.

Convex geometry The closed subsets of $\mathbb{A}$ are the closed convex sets, the closure of a set is (the closure of) its convex hull.

Analysis The concepts here are Dedekind cuts, i.e. real numbers (together with $\pm \infty$)! A concept consists of a pair $(A,B)$ of sets of rational numbers, where $A$ is everything less than or equal to everything in $B$ and $B$ is everything greater than or equal to everything in $A$.

There are undoubtably, other interesting examples.