# Week 3 Notes

• EX 2.61: Where $$\mathbf{NMY} := (P,\leq,\texttt{yes},\text{min})$$, recall from D 2.46 that $$\mathbf{NMY}$$-category $$\mathcal{C}$$ consists of

1. a set $$\text{Ob}(\mathcal{C})$$ of objects
2. a hom-object $$\mathcal{C}(x, y) \in \{\texttt{no}, \texttt{maybe}, \texttt{yes}\}$$ for $$x, y \in \text{Ob}(\mathcal{C})$$

satisfying

1. $$\texttt{yes} \leq \mathcal{C}(x, x)$$ for $$x \in \text{Ob}(\mathcal{C})$$, in other words $$\texttt{yes} = \mathcal{C}(x, x)$$
2. $$\text{min}(\mathcal{C}(x, y), \mathcal{C}(y, z)) \leq \mathcal{C}(x, z)$$ for $$x, y, z \in \text{Ob}(\mathcal{C})$$

Condition (a) means all reflexive morphisms have the valuation $$\texttt{yes}$$. Condition (b) means for two valued consecutive morphisms, we can determine the value of the transitive morphism.

$$p \rightarrow q$$ $$q \rightarrow r$$ $$\text{min}(p \rightarrow q, q \rightarrow r)$$ possible values of $$p \rightarrow r$$
$$\texttt{no}$$ $$\texttt{no}$$ $$\texttt{no}$$ $$\{\texttt{no},\texttt{maybe},\texttt{yes}\}$$
$$\texttt{no}$$ $$\texttt{maybe}$$ $$\texttt{no}$$ $$\{\texttt{no},\texttt{maybe},\texttt{yes}\}$$
$$\texttt{no}$$ $$\texttt{yes}$$ $$\texttt{no}$$ $$\{\texttt{no},\texttt{maybe},\texttt{yes}\}$$
$$\texttt{maybe}$$ $$\texttt{no}$$ $$\texttt{no}$$ $$\{\texttt{no},\texttt{maybe},\texttt{yes}\}$$
$$\texttt{maybe}$$ $$\texttt{maybe}$$ $$\texttt{maybe}$$ $$\{\texttt{maybe},\texttt{yes}\}$$
$$\texttt{maybe}$$ $$\texttt{yes}$$ $$\texttt{maybe}$$ $$\{\texttt{maybe},\texttt{yes}\}$$
$$\texttt{yes}$$ $$\texttt{no}$$ $$\texttt{no}$$ $$\{\texttt{no},\texttt{maybe},\texttt{yes}\}$$
$$\texttt{yes}$$ $$\texttt{maybe}$$ $$\texttt{maybe}$$ $$\{\texttt{maybe},\texttt{yes}\}$$
$$\texttt{yes}$$ $$\texttt{yes}$$ $$\texttt{yes}$$ $$\{\texttt{yes}\}$$

Interpretation: the $$\mathbf{NMY}$$-category is where the possible values of $$p \rightarrow r$$ are the upper set (EG 1.54) of $$\text{min}(p \rightarrow q, q \rightarrow r)$$.

• EX 2.75
1. For every $$(x,y) \in \text{Ob}(\mathcal{C} \times \mathcal{D})$$ $\begin{equation}\begin{split} & &&I &&& \\ &= &&I \otimes I &&&\text{(D 2.2 (b))} \\ &\leq &&\mathcal{C}(x,x) \otimes \mathcal{D}(y,y) &&&\text{(D 2.46 (a))} \\ &= &&(\mathcal{C} \times \mathcal{D})((x,y),(x,y)) &&&\text{(D 2.74 (ii))} \\ \end{split}\end{equation}$
2. For every $$(x_1,y_1), (x_2,y_2), (x_3,y_3) \in \text{Ob}(\mathcal{C} \times \mathcal{D})$$ $\begin{equation}\begin{split} & &&(\mathcal{C} \times \mathcal{D})((x_1,y_1),(x_2,y_2)) \otimes (\mathcal{C} \times \mathcal{D})((x_2,y_2),(x_3,y_3)) \\ &= &&(\mathcal{C}(x_1,x_2) \otimes \mathcal{D}(y_1,y_2)) \otimes (\mathcal{C}(x_2,x_3) \otimes \mathcal{D}(y_2,y_3)) &&&\text{(D 2.74 (ii))} \\ &= &&(\mathcal{C}(x_1,x_2) \otimes \mathcal{C}(x_2,x_3)) \otimes (\mathcal{D}(y_1,y_2) \otimes \mathcal{D}(y_2,y_3)) &&&\text{(D 2.2 (b,c))} \\ &\leq &&\mathcal{C}(x_1,x_3) \otimes \mathcal{D}(y_3,y_3) &&&\text{(D 2.46 (b))} \\ &= &&(\mathcal{C} \times \mathcal{D})((x_1,y_1),(x_3,y_3)) &&&\text{(D 2.74 (ii))} \\ \end{split}\end{equation}$
• EX 2.94: $$(\text{P}(S), \subseteq, S, \cap)$$ is a quantale, since it has all joins: for any set of subsets $$A \subseteq \text{P}(S)$$, we define $$\bigvee A := \bigcup A$$ that satisfies

1. $$a \subseteq \bigvee A$$ for all $$a \in A$$
2. if $$b \in \text{P}(S)$$ is any element s.t. $$a \subseteq b$$ for all $$a \in A$$, then $$A \subseteq b$$.

For any two subsets of $$\text{P}(S)$$, say $$X$$ and $$Y$$, $$\bigvee \{X,Y\} = X \cup Y$$, and so satisfies both (a) and (b). For the empty set, (b) requires $$\bigvee \varnothing \subseteq b$$ for all $$b \in \text{P}(S)$$, so $$\bigvee \varnothing = \varnothing$$.

• Ex 2.104:
1. For any sets $$X, Y$$ and $$\mathcal{V}$$-matrix $$M : X \times Y \rightarrow V$$ $\begin{equation}\begin{split} & &&I_X M (x,y) &&& \\ &= &&\bigvee_{x' \in X} I(x,x') \otimes M(x',y) &&&\text{(EQ 2.101)} \\ &= &&(I(x,x) \otimes M(x,y)) \vee \bigvee_{x' \in X, x' \neq x} I(x,x') \otimes M(x',y) &&&\text{(D 1.81)} \\ &= &&(I(x,x) \otimes M(x,y)) \vee \bigvee_{x' \in X, x' \neq x} 0 \otimes M(x',y) &&&\text{(def. of identity \mathcal{V}-matrix)} \\ &= &&(I(x,x) \otimes M(x,y)) \vee \bigvee_{x' \in X, x' \neq x} 0 &&&\text{(}0 \otimes v = 0\text{\quad for all \quad}v \in V\text{)} \\ &= &&(I \otimes M(x,y)) \vee 0 &&&\text{(join of 0 is 0)} \\ &= &&M(x,y) \vee 0 &&&\text{(D 2.2 (b))} \\ &= &&M(x,y) &&&\text{(def. of join)} \\ \end{split}\end{equation}$

It remains to show $$0 \otimes v = 0$$ for all $$v \in V$$ $\begin{equation}\begin{split} & &&0 \otimes v &&& \\ &= &&v \otimes 0 &&&\text{(D 2.2 (d))} \\ &= &&v \otimes \bigvee_{a \in \null) a &&&\text{(def. of join over nothing)} \\ &= &&\bigvee_{a \in \null) (v \otimes a) &&&\text{(P 2.87 (2))} \\ &= &&0 &&&\text{(def. of join over nothing)} \\ \end{split}\end{equation}$

2. For any $$\mathcal{V}$$-matrices $$M : W \times X \rightarrow V$$, $$N : X \times Y \rightarrow V$$, $$P : Y \times Z \rightarrow V$$ $\begin{equation}\begin{split} & &&(MN) P (w,z) &&& \\ &= &&\bigvee_{y \in Y} (MN)(w,y) \otimes P(y,z) &&&\text{(EQ 2.101)} \\ &= &&\bigvee_{y \in Y} (\bigvee_{x \in X} M(w,x) \otimes N(x,y)) \otimes P(y,z) &&&\text{(EQ 2.101)} \\ &= &&\bigvee_{y \in Y,x \in X} M(w,x) \otimes (N(x,y) \otimes P(y,z)) &&&\text{(D 2.2 (c))} \\ &= &&\bigvee_{x \in X} M(w,x) \otimes (\bigvee_{y \in Y} N(x,y) \otimes P(y,z)) &&&\text{(P 2.87 (2), the join} \quad \bigvee_{y \in Y} N(x,y) \otimes P(y,z) \quad \text{exists by EQ 2.101)} \\ &= &&M (NP) (w,z) &&&\text{(EQ 2.101)} \\ \end{split}\end{equation}$

Created: 2020-01-13 Mon 14:36