Item 1 (00PG) — The Clowder Project

Table of Contents
Part 3: Relations
Chapter 8: Relations
Section 8.2: Examples of Relations
Subsection 8.2.3: The Inverse of a Function
Item 1 (cite)

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8.2.3 The Inverse of a Function

Let $f\colon A\to B$ be a function.

Definition 8.2.3.1.1 ▶ The Inverse of a Function

The inverse of $f$ is the relation $f^{-1}\colon B\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}A$ defined as follows:

•
Viewing relations from $B$ to $A$ as subsets of $B\times A$, we define

\[ f^{-1}\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\left\{ (b,f^{-1}(b))\in B\times A\ \middle |\ a\in A\right\} , \]

where

\[ f^{-1}(b)\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\left\{ a\in A\ \middle |\ f(a)=b\right\} \]

for each $b\in B$.
•
Viewing relations from $B$ to $A$ as functions $B\times A\to \{ \mathsf{true},\mathsf{false}\} $, we define

\[ [f^{-1}]^{b}_{a}\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\begin{cases} \mathsf{true}& \text{if there exists $a\in A$ with $f(a)=b$,}\\ \mathsf{false}& \text{otherwise} \end{cases} \]

for each $(b,a)\in B\times A$.
•
Viewing relations from $B$ to $A$ as functions $B\to \mathcal{P}(A)$, we define

\[ f^{-1}(b)\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\left\{ a\in A\ \middle |\ f(a)=b\right\} \]

for each $b\in B$.

Proposition 8.2.3.1.2 ▶ Properties of Inverses of Functions

Let $f\colon A\to B$ be a function.

1.
Functoriality. The assignment $A\mapsto A$, $f\mapsto f^{-1}$ defines a functor

\[ (-)^{-1}\colon \mathsf{Sets}\to \mathrm{Rel} \]

where
- •
  Action on Objects. For each $A\in \operatorname {\mathrm{Obj}}(\mathsf{Sets})$, we have
  
  \[ \left[(-)^{-1}\right](A)\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}A. \]
- •
  Action on Morphisms. For each $A,B\in \operatorname {\mathrm{Obj}}(\mathsf{Sets})$, the action on $\operatorname {\mathrm{Hom}}$-sets
  
  \[ (-)^{-1}_{A,B}\colon \mathsf{Sets}(A,B) \to \mathrm{Rel}(A,B) \]
  
  of $(-)^{-1}$ at $(A,B)$ is defined by
  
  \[ (-)^{-1}_{A,B}(f) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\left[(-)^{-1}\right](f), \]
  
  where $f^{-1}$ is the inverse of $f$ as in Definition 8.2.3.1.1.
In particular, the following statements are true:
- •
  Preservation of Identities. We have
  
  \[ \operatorname {\mathrm{id}}^{-1}_{A}=\chi _{A} \]
  
  for each $A\in \operatorname {\mathrm{Obj}}(\mathsf{Sets})$.
- •
  Preservation of Composition. We have
  
  \[ (g\circ f)^{-1}=g^{-1}\mathbin {\diamond }f^{-1} \]
  
  for pair of functions $f\colon A\to B$ and $g\colon B\to C$.

Adjointness Inside $\boldsymbol {\mathsf{Rel}}$. We have an adjunction

in $\boldsymbol {\mathsf{Rel}}$.

Interaction With Converses of Relations. We have

\begin{align*} (f^{-1})^{\dagger } & = \operatorname {\mathrm{Gr}}(f),\\ \operatorname {\mathrm{Gr}}(f)^{\dagger } & = f^{-1}. \end{align*}

Proof of Proposition 8.2.3.1.2.

Item 1: Functoriality

Omitted.

Item 2: Adjointness Inside $\boldsymbol {\mathsf{Rel}}$

This is a repetition of Item 4 of Proposition 8.2.2.1.2 and is proved there.

Item 3: Interaction With Converses of Relations

This is a repetition of Item 5 of Proposition 8.2.2.1.2 and is proved there.

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