threeorders don't work

freyd-multicategories.tex

@@ -220,41 +220,70 @@ categories to get preorderd Freyd (multi)categories.
 %
 Further, we will likely (for the models' sake) need to develop
 ``double'' freyd multicategories as well.
+%
 
-To get that, we can use categories \emph{internal} to Mono-categories.
-
-\begin{definition}[Double Mono-Category]
-  A double Mono Category is a category internal to the category of
-  Mono categories.
+I've had several false starts here.
+%
+Here's 2 options that are \emph{close}, but no cigar.
 
-  I \emph{conjecture} that it is equivalent to a double category of
-  ``loose'' vertical and horizontal arrows, and a sub-double-category
-  with the same objects.
-\end{definition}
+First, in analogy with double categories, we could just take
+categories internal to Mono-categories, so a span of mono-categories
+with identity and unit.
+%
+However, a span of Mono-categories is not quite right.
+%
+You have an object mono-category, representing objects and vertical
+morphisms, but then your arrows are also a mono-category, so you have
+only a \emph{set} of arrows, and tight and loose \emph{squares} and
+source and target take tight squares to tight vertical morphisms.
+%
+So this is definitely not right
 
-To get a preordered Mono category, there are two places to modify the
-definition: at Mono, or at category internal to.
+Second, instead of looking at $\Set^{\to}$ and its refinement to Mono,
+we could start with $\text{Preorder}^{\to}$ or $\text{Cat}^{\to}$ and
+refine those somehow.
 %
-To get the right double category structure for the multicategory
-stuff, this is the right order:
+I think something here \emph{includes} our categories, but is also too
+big. Need to look into this more.
 
-\begin{definition}[Threeorder/3-Valued Preorder/Inclusion of Preorders]
-  A 3-valued Preorder or inclusion of preorders is a category enriched
-  in the ordinal 3: $0 \leq 1 \leq 2$, which is a reflective
-  subcategory of $\text{Set}^{\to}$ defined by taking a set $X$ to the
-  truth value $\exists x\in X. \top$.
+%% To get that, we can use categories \emph{internal} to Mono-categories.
 
-  They are equivalent to an object with a ``tight'' and ``loose''
-  ordering, where tight ordering implies loose ordering.
-\end{definition}
+%% \begin{definition}[Double Mono-Category]
+%%   A double Mono Category is a category internal to the category of
+%%   Mono categories.
 
-\begin{definition}[Threeorder Category/Wide Double Subcategory]
-  A threeorder category is a category internal to threeorders.
+%%   I \emph{conjecture} that it is equivalent to a double category of
+%%   ``loose'' vertical and horizontal arrows, and a sub-double-category
+%%   with the same objects.
+%% \end{definition}
 
-  It is equivalent (I think) to a pair of preorder category structures
-  on the same set of objects, where the preorder structures form a
-  threeorder, and the category structures form a Mono category.
-\end{definition}
+%% To get a preordered Mono category, there are two places to modify the
+%% definition: at Mono, or at category internal to.
+%% %
+%% To get the right double category structure for the multicategory
+%% stuff, this is the right order:
+
+%% \begin{definition}[Threeorder/3-Valued Preorder/Inclusion of Preorders]
+%%   A 3-valued Preorder or inclusion of preorders is a category enriched
+%%   in the ordinal 3: $0 \leq 1 \leq 2$, which is a reflective
+%%   subcategory of $\text{Set}^{\to}$ defined by taking a set $X$ to the
+%%   truth value $\exists x\in X. \top$.
+
+%%   They are equivalent to an object with a ``tight'' and ``loose''
+%%   ordering, where tight ordering implies loose ordering.
+%% \end{definition}
+
+%% \begin{definition}[Threeorder Category/Wide Double Subcategory]
+%%   A threeorder category is a category internal to threeorders.
+
+%%   It is equivalent (I think) to a pair of preorder category structures
+%%   on the same set of objects, where the preorder structures form a
+%%   threeorder, and the category structures form a Mono category.
+%% \end{definition}
+
+\subsection{Kleisli Construction}
+
+Basically, take Coreflection(Kl(C,T))
 
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