completely distrib note

P1-Polynomials.tex

@@ -681,7 +681,7 @@ \section{Expanding products of sums} \label{sec.poly.rep-sets.expand}
 In general, we refer to a category having this property as follows.
 
 \begin{definition}[Completely distributive category]
-    A category $\Cat{C}$ with all small products and coproducts is \emph{completely distributive} if products distribute over coproducts as in \eqref{eqn.set_completely_distributive}; that is, for any set $I$, sets $(J(i))_{i\in I}$, and objects $(X(i,j))_{i\in I,j\in J(i)}$ from $\Cat{C}$, we have a natural isomorphism
+    A category $\Cat{C}$ with all small products and coproducts is \emph{completely distributive}\footnote{While our terminology generalizes that of a completely distributive lattice, which has the additional requirement that the category be a poset, it is unfortunately not standard: a completely distributive category refers to a different concept in some categorical literature. We will not use this other concept, so there is no ambiguity.} if products distribute over coproducts as in \eqref{eqn.set_completely_distributive}; that is, for any set $I$, sets $(J(i))_{i\in I}$, and objects $(X(i,j))_{i\in I,j\in J(i)}$ from $\Cat{C}$, we have a natural isomorphism
     \begin{equation}\label{eqn.cat_completely_distributive}
         \prod_{i\in I}\sum_{j\in J(i)}X(i,j)
         \iso