The Clowder Project

In detail, by , , the relation $\mathord {\sim }$ of Definition 2.2.4.1.1 is given by declaring $a\sim b$ iff one of the following conditions is satisfied:

1. We have $a,b\in A$ and $a=b$.
2. We have $a,b\in B$ and $a=b$.
3. There exist $x_{1},\ldots ,x_{n}\in A\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}B$ such that $a\sim 'x_{1}\sim '\cdots \sim 'x_{n}\sim 'b$, where we declare $x\sim 'y$ if one of the following conditions is satisfied:
   1. There exists $c\in C$ such that $x=\webleft (0,f\webleft (c\webright )\webright )$ and $y=\webleft (1,g\webleft (c\webright )\webright )$.
   2. There exists $c\in C$ such that $x=\webleft (1,g\webleft (c\webright )\webright )$ and $y=\webleft (0,f\webleft (c\webright )\webright )$.

   In other words, there exist $x_{1},\ldots ,x_{n}\in A\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}B$ satisfying the following conditions:

   1. There exists $c_{0}\in C$ satisfying one of the following conditions:
      1. We have $a=f\webleft (c_{0}\webright )$ and $x_{1}=g\webleft (c_{0}\webright )$.
      2. We have $a=g\webleft (c_{0}\webright )$ and $x_{1}=f\webleft (c_{0}\webright )$.
   2. For each $1\leq i\leq n-1$, there exists $c_{i}\in C$ satisfying one of the following conditions:
      1. We have $x_{i}=f\webleft (c_{i}\webright )$ and $x_{i+1}=g\webleft (c_{i}\webright )$.
      2. We have $x_{i}=g\webleft (c_{i}\webright )$ and $x_{i+1}=f\webleft (c_{i}\webright )$.
   3. There exists $c_{n}\in C$ satisfying one of the following conditions:
      1. We have $x_{n}=f\webleft (c_{n}\webright )$ and $b=g\webleft (c_{n}\webright )$.
      2. We have $x_{n}=g\webleft (c_{n}\webright )$ and $b=f\webleft (c_{n}\webright )$.