These are some notes relating to my talk about convex spaces and probability theory.
Intuitively, a convex space is a space closed under convex combinations. Precisely, we consider a set $X$ equipped with a mixture operation $\mu: \Delta_X \to X$ from the space $\Delta_X \triangleq \left\{ f:X\to\RR_{\geq 0}: |\text{supp}(f)|\lt\infty\,\,\,\text{and}\,\,\,\sum_x f(x)=1 \right\}$ of formal finitary convex combinations of $X$'s elements to $X$. We say that $X$ is a convex space provided that $\mu(x \mapsto \delta_{x,y}) = y$ for each $y\in Y$ and that $ \mu(\sum_{i\in I} c_i f_i) = \mu(x \mapsto \sum_{i\in I} c_i \delta_{x,\mu(f_i)}) $ for all finite $I$ and all sequences of $c_i\in \RR_{\geq 0}$ and $f_i$ for which the two sides are defined. Each valid $\mu$ is determined by its restriction to functions with size-$2$ support, so we will sometimes conflate $\mu$ and this restriction, and for convenience we will sometimes write $ \mu(x \mapsto \alpha \delta_{x,v} + (1-\alpha) \delta_{x,w}) \in X $ as $ \alpha v + (1-\alpha) w \in X $ when $v, w \in X$.
The morphisms in the category of convex spaces are those that preserve $\mu$. More precisely, we observe that $\Delta$ is an endofunctor on the category of Sets, and we declare the morphisms from $(X,\mu_X)$ to $(Y, \mu_Y)$ to be those functions $T:X \to Y$ such that $T \circ \mu_X = \mu_Y \circ \Delta(T)$. We have defined a category. We could have defined the category in fewer words --- as the Eilenberge Moore category for $\Delta$ --- but we choose here to be very concrete.
Not every convex space is a subspace of some $\RR^n$. A source of examples comes from the semilattices, i.e. the posets $X$ with binary joins $\vee$: a canonical mixture operation on $X$ is determined by is determined by the rule $\alpha v + (1-\alpha) w \triangleq v \vee w$ as $v,w$ range through $X$ and $\alpha$ ranges through $(0,1)$. For example, say $X = \{\text{no}, \text{maybe}, \text{yes}\}$, where $\text{no},\text{yes}\lt\text{maybe}$. Then we have $(1/3) \text{no} + (2/3) \text{yes} = \text{maybe}$ and $(4/7) \text{no} + (3/7) \text{maybe} = \text{maybe}$. This is a space not of probabilities but of possibilities! Likewise, if $X = \{\text{zero}, \text{poly}, \text{exp}\}$ with $\text{zero}\lt\text{poly}\lt\text{exp}$, then we have $(2/5) \text{zero} + (3/5) \text{poly} = \text{poly}$ and $(5/6) \text{poly} + (1/6) \text{exp} = \text{exp}$: we have a space of sizes.
I figured out this stuff by myself and for fun. As usual, I found out that others had thought of isomorphic ideas before me:
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