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Stream: learning: questions

Topic: faces of convex sets

David Egolf (Feb 03 2022 at 17:34):

We talked about ordered monoids and quantales recently at #learning: questions > are measures functors?, in the context of an algebra of sets. I was reading about "faces" of convex sets (in "Basic Algebraic Topology" by Shastri) and was noticing some similarities.

A subset $F$ of a convex set $S$ is called a face of $S$ if:

• $F$ is convex and
• for any two distinct points $y,z$ in $S$, if $(y,z) \cap F \neq \empty$ then $[y,z] \subseteq F$.

Let $\mathcal{F}(S)$ denote the set of all faces of $S$. We can form a monoid using $\mathcal{F}(S)$ where the morphisms are the faces of $S$ and composition corresponds to taking the intersection, so $A \circ B = A \cap B$ for faces $A,B$. Since $S$ is itself a face, we have an identity morphism.

We can order the element of $\mathcal{F}(S)$ by inclusion. The resulting poset apparently forms a complete lattice, where the $\inf$ of a collection of faces is their intersection, and the $\sup$ of a collection of faces is the intersection of all the faces that contains the collection.

The monoid composition operation is compatible with the ordering, as $A \leq B \implies A \circ C \leq B \circ C$ corresponds to $A \subseteq B \implies A \cap C \subseteq B \cap C$, for faces $A,B,C$. So, I think we have an ordered monoid of faces.

Does this form a quantale? We need the monoid operation to preserve sups (colimits) for this to be the case. That is, I think we need $A \circ \sup(C_1, C_2) = \sup(A \circ C_1, A \circ C_2)$ to hold, for faces $A,C_1,C_2$, among other equations.