Subsection 6.3.10 (00M2): 2-Categorical Epimorphisms in $\textbf{Rel}$

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Style 2	`book`	Alegreya Sans	`amsthm`
Style 3	`book`	Arno*	`amsthm`
Style 4	`book`	Computer Modern	`amsthm`

Let $R : A \to | B$ be a relation.

1. Corepresentably Faithful Morphisms in $Rel$ . Every morphism of $Rel$ is a corepresentably faithful morphism.
2. Corepresentably Full Morphisms in $Rel$ . The following conditions are equivalent:
1. (a) The morphism $R : A \to | B$ is a corepresentably full morphism.
2. (b) For each pair of relations $S, T : X ⇉ | A$ , the following condition is satisfied:
  - If $S ⋄ R \subset T ⋄ R$ , then $S \subset T$ .
3. (c) The functor
  $R^{- 1} : (P (B), \subset) \to (P (A), \subset)$
  
  is full.
4. (d) For each $U, V \in P (B)$ , if $R^{- 1} (U) \subset R^{- 1} (V)$ , then $U \subset V$ .
5. (e) The functor
  $R_{- 1} : (P (B), \subset) \to (P (A), \subset)$
  
  is full.
6. (f) For each $U, V \in P (B)$ , if $R_{- 1} (U) \subset R_{- 1} (V)$ , then $U \subset V$ .
3. Corepresentably Fully Faithful Morphisms in $Rel$ . Every corepresentably full morphism of $Rel$ is a corepresentably fully faithful morphism.

Item 1: Corepresentably Faithful Morphisms in

Rel

The relation

R

is a corepresentably faithful morphism in

Rel

iff, for each

X \in Obj (Rel)

, the functor

R^{*} : Rel (B, X) \to Rel (A, X)

is faithful, i.e. iff the morphism

R_{S, T}^{*} : {Hom}_{Rel (B, X)} (S, T) \to {Hom}_{Rel (A, X)} (S ⋄ R, T ⋄ R)

is injective for each $S, T \in Obj (Rel (B, X))$ . However, ${Hom}_{Rel (B, X)} (S, T)$ is either empty or a singleton, in either case of which the map $R_{S, T}^{*}$ is necessarily injective.

Item 2: Corepresentably Full Morphisms in

Rel

We claim Item (a), Item (b), Item (c), Item (d), Item (e), and Item (f) are indeed equivalent:

Item (a) $⟺$ Item (b): This is simply a matter of unwinding definitions: The relation $R$ is a corepresentably full morphism in $Rel$ iff, for each $X \in Obj (Rel)$ , the functor
$R^{*} : Rel (B, X) \to Rel (A, X)$

is full, i.e. iff the morphism

$R_{S, T}^{*} : {Hom}_{Rel (B, X)} (S, T) \to {Hom}_{Rel (A, X)} (S ⋄ R, T ⋄ R)$

is surjective for each $S, T \in Obj (Rel (B, X))$ , i.e. iff, whenever $S ⋄ R \subset T ⋄ R$ , we also have $S \subset T$ .
Item (c) $⟺$ Item (d): This is also simply a matter of unwinding definitions: The functor

$R^{- 1} : (P (B), \subset) \to (P (A), \subset)$

is full iff, for each $U, V \in P (A)$ , the morphism

$R_{U, V}^{- 1} : {Hom}_{P (B)} (U, V) \to {Hom}_{P (A)} (R^{- 1} (U), R^{- 1} (V))$

is surjective, i.e. iff whenever $R^{- 1} (U) \subset R^{- 1} (V)$ , we also necessarily have $U \subset V$ .
Item (e) $⟺$ Item (f): This is once again simply a matter of unwinding definitions, and proceeds exactly in the same way as in the proof of the equivalence between Item (c) and Item (d) given above.
Item (d) $⟹$ Item (f): Suppose that the following condition is true:
- For each $U, V \in P (B)$ , if $R^{- 1} (U) \subset R^{- 1} (V)$ , then $U \subset V$ .
We need to show that the condition
- For each $U, V \in P (B)$ , if $R_{- 1} (U) \subset R_{- 1} (V)$ , then $U \subset V$ .
is also true. We proceed step by step:
1. (c) Suppose we have $U, V \in P (B)$ with $R_{- 1} (U) \subset R_{- 1} (V)$ .
2. (d) By Chapter 7: Constructions With Relations, Item 7 of Proposition 7.4.2.1.3, we have
  
  $\begin{aligned} R_{- 1} (U) & = B ∖ R^{- 1} (A ∖ U), \\ R_{- 1} (V) & = B ∖ R^{- 1} (A ∖ V) . \end{aligned}$
3. (e) By Chapter 2: Constructions With Sets, Item 1 of Proposition 2.3.10.1.2 we have $R^{- 1} (A ∖ V) \subset R^{- 1} (A ∖ U)$ .
4. (f) By assumption, we then have $A ∖ V \subset A ∖ U$ .
5. (g) By Chapter 2: Constructions With Sets, Item 1 of Proposition 2.3.10.1.2 again, we have $U \subset V$ .
Item (f) $⟹$ Item (d): Suppose that the following condition is true:
- For each $U, V \in P (B)$ , if $R_{- 1} (U) \subset R_{- 1} (V)$ , then $U \subset V$ .
We need to show that the condition
- For each $U, V \in P (B)$ , if $R^{- 1} (U) \subset R^{- 1} (V)$ , then $U \subset V$ .
is also true. We proceed step by step:
1. (c) Suppose we have $U, V \in P (B)$ with $R^{- 1} (U) \subset R^{- 1} (V)$ .
2. (d) By Chapter 7: Constructions With Relations, Item 7 of Proposition 7.4.3.1.3, we have
  
  $\begin{aligned} R^{- 1} (U) & = B ∖ R_{- 1} (A ∖ U), \\ R^{- 1} (V) & = B ∖ R_{- 1} (A ∖ V) . \end{aligned}$
3. (e) By Chapter 2: Constructions With Sets, Item 1 of Proposition 2.3.10.1.2 we have $R_{- 1} (A ∖ V) \subset R_{- 1} (A ∖ U)$ .
4. (f) By assumption, we then have $A ∖ V \subset A ∖ U$ .
5. (g) By Chapter 2: Constructions With Sets, Item 1 of Proposition 2.3.10.1.2 again, we have $U \subset V$ .
Item (b) $⟹$ Item (d): Consider the diagram

and suppose that $S ⋄ R \subset T ⋄ R$ . Note that, by assumption, given a diagram of the form

if $R^{- 1} (U) = R ⋄ U \subset R ⋄ V = R^{- 1} (V)$ , then $U \subset V$ . In particular, for each $x \in X$ , we may consider the diagram

for which we have $[x] ⋄ S ⋄ R \subset [x] ⋄ T ⋄ R$ , implying that we have

$S^{- 1} (x) = [x] ⋄ S \subset [x] ⋄ T = T^{- 1} (x)$

for each $x \in X$ , implying $S \subset T$ .
Item (d) $⟹$ Item (b): Let $U, V \in P (B)$ and consider the diagram

By , we have

$\begin{aligned} R^{- 1} (U) & = U ⋄ R, \\ R^{- 1} (V) & = V ⋄ R . \end{aligned}$

Now, if $R^{- 1} (U) \subset R^{- 1} (V)$ , i.e. $U ⋄ R \subset V ⋄ R$ , then $U \subset V$ by assumption.