The Clowder Project

where

• Action on Objects. For each $R\in \text{Obj}\webleft (\mathbf{Rel}\webleft (A,B\webright )\webright )$, we have
  \[ \webleft [\mathbf{Coll}\webright ]\webleft (R\webright ) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft (\mathbf{Coll}\webleft (R\webright ),\phi _{R}\webright ) \]

  for each $R\in \mathbf{Rel}\webleft (A,B\webright )$, where

  • The poset $\mathbf{Coll}\webleft (R\webright )$ is the collage of $R$ of Definition 7.3.13.1.1;
  • The morphism $\phi _{R}\colon \mathbf{Coll}\webleft (R\webright )\to \Delta ^{1}$ is given by
    \[ \phi _{R}\webleft (x\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\begin{cases} 0 & \text{if $x\in A$,}\\ 1 & \text{if $x\in B$} \end{cases} \]

    for each $x\in \mathbf{Coll}\webleft (R\webright )$;

• Action on Morphisms. For each $R,S\in \text{Obj}\webleft (\mathbf{Rel}\webleft (A,B\webright )\webright )$, the action on $\textup{Hom}$-sets
  \[ \mathbf{Coll}_{R,S}\colon \textup{Hom}_{\mathbf{Rel}\webleft (A,B\webright )}\webleft (R,S\webright ) \to \mathsf{Pos}\webleft (\mathbf{Coll}\webleft (R\webright ),\mathbf{Coll}\webleft (S\webright )\webright ) \]

  of $\mathbf{Coll}$ at $\webleft (R,S\webright )$ is given by sending an inclusion

  \[ \iota \colon R\subset S \]

  to the morphism

  \[ \mathbf{Coll}\webleft (\iota \webright )\colon \mathbf{Coll}\webleft (R\webright )\to \mathbf{Coll}\webleft (S\webright ) \]

  of posets over $\Delta ^{1}$ defined by

  \[ \webleft [\mathbf{Coll}\webleft (\iota \webright )\webright ]\webleft (x\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}x \]

  for each $x\in \mathbf{Coll}\webleft (R\webright )$.

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