Proposition 9.6.2.1.2 (011T): Properties of Full Functors

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Let $F : C \to D$ and $G : D \to E$ be functors.

1. Interaction With Composition. If $F$ and $G$ are full, then so is $G \circ F$ .
2. Interaction With Postcomposition I. If $F$ is full, then the postcomposition functor
$F_{*} : Fun (X, C) \to Fun (X, D)$

can fail to be full.
3. Interaction With Postcomposition II. If, for each $X \in Obj (Cats)$ , the postcomposition functor
$F_{*} : Fun (X, C) \to Fun (X, D)$

is full, then $F$ is also full.
4. Interaction With Precomposition I. If $F$ is full, then the precomposition functor
$F^{*} : Fun (D, X) \to Fun (C, X)$

can fail to be full.
5. Interaction With Precomposition II. If, for each $X \in Obj (Cats)$ , the precomposition functor
$F^{*} : Fun (D, X) \to Fun (C, X)$

is full, then $F$ can fail to be full.
6. Interaction With Precomposition III. If $F$ is essentially surjective and full, then the precomposition functor
$F^{*} : Fun (D, X) \to Fun (C, X)$

is full (and also faithful by Item 4 of Proposition 9.6.1.1.2).
7. Interaction With Precomposition IV. The following conditions are equivalent:
1. (a) For each $X \in Obj (Cats)$ , the precomposition functor
  $F^{*} : Fun (D, X) \to Fun (C, X)$
  
  is full.
2. (b) The functor $F : C \to D$ is a corepresentably full morphism in ${Cats}_{2}$ in the sense of Chapter 11: Types of Morphisms in Bicategories, Definition 11.2.1.1.1.
3. (c) The components
  $η_{G} : G ⟹ {Ran}_{F} (G \circ F)$
  
  of the unit
  
  $η : {id}_{Fun (D, X)} ⟹ {Ran}_{F} \circ F^{*}$
  
  of the adjunction $F^{*} ⊣ {Ran}_{F}$ are all retractions/split epimorphisms.
4. (d) The components
  $ϵ_{G} : {Lan}_{F} (G \circ F) ⟹ G$
  
  of the counit
  
  $ϵ : {Lan}_{F} \circ F^{*} ⟹ {id}_{Fun (D, X)}$
  
  of the adjunction ${Lan}_{F} ⊣ F^{*}$ are all sections/split monomorphisms.
5. (e) For each $B \in Obj (D)$ , there exist:
  - An object $A_{B}$ of $C$ ;
  - A morphism $s_{B} : B \to F (A_{B})$ of $D$ ;
  - A morphism $r_{B} : F (A_{B}) \to B$ of $D$ ;
  satisfying the following condition:
  - For each $A \in Obj (C)$ and each pair of morphisms
    $\begin{aligned} r & : F (A) \to B, \\ s & : B \to F (A) \end{aligned}$
    
    of $D$ , we have
    
    $[(A_{B}, s_{B}, r_{B})] = [(A, s, r \circ s_{B} \circ r_{B})]$
    
    in $\int^{A \in C} h_{F_{A}}^{B^{'}} \times h_{B}^{F_{A}}$ .

Item 1: Interaction With Composition

Since the map

(G \circ F)_{A, B} : {Hom}_{C} (A, B) \to {Hom}_{D} (G_{F_{A}}, G_{F_{B}}),

defined as the composition

{Hom}_{C} (A, B) \overset{F_{A, B}}{\to} {Hom}_{D} (F_{A}, F_{B}) \overset{G_{F (A), F (B)}}{\to} {Hom}_{D} (G_{F_{A}}, G_{F_{B}}),

is a composition of surjective functions, it follows from that it is also surjective. Therefore $G \circ F$ is full.

Item 2: Interaction With Postcomposition I

We follow the proof (completely formalised in cubical Agda!) given by Naïm Camille Favier in [Favier, Postcompose Not Full]. Let

C

be the category where:

Objects. We have $Obj (C) = {A, B}$ .
Morphisms. We have
$\begin{aligned} {Hom}_{C} (A, A) & = {e_{A}, {id}_{A}}, \\ {Hom}_{C} (B, B) & = {e_{B}, {id}_{B}}, \\ {Hom}_{C} (A, B) & = {f, g}, \\ {Hom}_{C} (B, A) & = Ø . \end{aligned}$
Composition. The nontrivial compositions in $C$ are the following:

$\begin{aligned} e_{A} \circ e_{A} & = {id}_{A}, \\ e_{B} \circ e_{B} & = {id}_{B}, \end{aligned} \begin{aligned} f \circ e_{A} & = g, \\ g \circ e_{A} & = f, \end{aligned} \begin{aligned} e_{B} \circ f & = f, \\ e_{B} \circ g & = g . \end{aligned}$

We may picture

C

as follows:

Next, let $D$ be the walking arrow category $1$ of Definition 9.2.5.1.1 and let $F : C \to 1$ be the functor given on objects by

\begin{aligned} F (A) & = 0, \\ F (B) & = 1 \end{aligned}

and on non-identity morphisms by

\begin{aligned} F (f) & = f_{01}, \\ F (g) & = f_{01}, \end{aligned} \begin{aligned} F (e_{A}) & = {id}_{0}, \\ F (e_{B}) & = {id}_{1} . \end{aligned}

Finally, let $X = B Z_{/ 2}$ be the walking involution and let $ι_{A}, ι_{B} : B Z_{/ 2} ⇉ C$ be the inclusion functors from $B Z_{/ 2}$ to $C$ with

\begin{aligned} ι_{A} (∙) & = A, \\ ι_{B} (∙) & = B . \end{aligned}

Since every morphism in $1$ has a preimage in $C$ by $F$ , the functor $F$ is full. Now, for $F_{*}$ to be full, the map

would need to be surjective. However, as we will show next, we have

\begin{matrix} Nat (ι_{A}, ι_{B}) = Ø, \\ Nat (F \circ ι_{A}, F \circ ι_{B}) ≅ pt, \end{matrix}

so this is impossible:

Proof of $Nat (ι_{A}, ι_{B}) = Ø$ : A natural transformation $α : ι_{A} \Rightarrow ι_{B}$ consists of a morphism

$α : \underset{= A}{\underset{⏟}{ι_{A} (∙)}} \to \underset{= B}{\underset{⏟}{ι_{B} (∙)}}$

in $C$ making the diagram

commute for each $e \in {Hom}_{B Z_{/ 2}} (∙, ∙) ≅ Z_{/ 2}$ . We have two cases:
1. (a) If $α = f$ , the naturality diagram for the unique nonidentity element of $Z_{/ 2}$ is given by
  
  However, $e_{B} \circ f = f$ and $f \circ e_{A} = g$ , so this diagram does not commute.
2. (b) If $α = g$ , the naturality diagram for the unique nonidentity element of $Z_{/ 2}$ is given by
  
  However, $e_{B} \circ g = g$ and $g \circ e_{A} = f$ , so this diagram does not commute.
As a result, there are no natural transformations from $ι_{A}$ to $ι_{B}$ .
Proof of $Nat (F \circ ι_{A}, F \circ ι_{B}) ≅ pt$ : A natural transformation

$β : F \circ ι_{A} \Rightarrow F \circ ι_{B}$

consists of a morphism

$β : \underset{= 0}{\underset{⏟}{[F \circ ι_{A}] (∙)}} \to \underset{= 1}{\underset{⏟}{[F \circ ι_{B}] (∙)}}$

in $1$ making the diagram

commute for each $e \in {Hom}_{B Z_{/ 2}} (∙, ∙) ≅ Z_{/ 2}$ . Since the only morphism from $0$ to $1$ in $1$ is $f_{01}$ , we must have $β = f_{01}$ if such a transformation were to exist, and in fact it indeed does, as in this case the naturality diagram above becomes

for each $e \in Z_{/ 2}$ , and this diagram indeed commutes, making $β$ into a natural transformation.