The Clowder Project

Let $A$ and $B$ be sets and let $R\colon A\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}B$ be a relation from $A$ to $B$.

1. Functoriality I. The assignment $R\mapsto \mathbf{Coll}\webleft (R\webright )$ defines a functor¹
   \[ \mathbf{Coll}\colon \mathbf{Rel}\webleft (A,B\webright )\to \mathsf{Pos}_{/\Delta ^{1}}\webleft (A,B\webright ), \]

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

     ²

2. Equivalence. The functor of Item 1 is an equivalence of categories.

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¹Here $\mathsf{Pos}_{/\Delta ^{1}}\webleft (A,B\webright )$ is the category defined as the pullback

\[ \mathsf{Pos}_{/\Delta ^{1}}\webleft (A,B\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\mathsf{pt}\operatorname*{\mathbin {\times }}_{\webleft [A\webright ],\mathsf{Pos},\mathrm{fib}_{0}}\mathsf{Pos}_{/\Delta ^{1}}\operatorname*{\mathbin {\times }}_{\mathrm{fib}_{1},\mathsf{Pos},\webleft [B\webright ]}\mathsf{pt}, \]

as in the diagram

Explicitly, an object of $\mathsf{Pos}_{/\Delta ^{1}}\webleft (A,B\webright )$ is a pair $\webleft (X,\phi _{X}\webright )$ consisting of

• A poset $X$;
• A morphism $\phi _{X}\colon X\to \Delta ^{1}$;

such that $\phi ^{-1}_{X}\webleft (0\webright )=A$ and $\phi ^{-1}_{X}\webleft (0\webright )=B$, with morphisms between such objects being morphisms of posets over $\Delta ^{1}$.

²Note that this is indeed a morphism of posets: if $x\preceq _{\mathbf{Coll}\webleft (R\webright )}y$, then $x=y$ or $x\sim _{R}y$, so we have either $x=y$ or $x\sim _{S}y$ (as $R\subset S$), and thus $x\preceq _{\mathbf{Coll}\webleft (S\webright )}y$.