Proposition 2.1.4.1.5 (001C): Properties of Pullbacks of Sets

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Proposition 2.1.4.1.5 ▶ Properties of Pullbacks of Sets

Let $A$ , $B$ , $C$ , and $X$ be sets.

1. Functoriality. The assignment $(A, B, C, f, g) \mapsto A \times_{f, C, g} B$ defines a functor
$-_{1} \times_{-_{3}} -_{1} : Fun (P, Sets) \to Sets,$

where $P$ is the category that looks like this:

In particular, the action on morphisms of $-_{1} \times_{-_{3}} -_{1}$ is given by sending a morphism

in $Fun (P, Sets)$ to the map $ξ : A \times_{C} B \overset{\exists!}{\to} A^{'} \times_{C^{'}} B^{'}$ given by

$ξ (a, b) \overset{def}{=} (ϕ (a), ψ (b))$

for each $(a, b) \in A \times_{C} B$ , which is the unique map making the diagram

commute.
2. Adjointness. We have adjunctions
witnessed by bijections
$\begin{aligned} {Sets}_{/ X} (A \times_{X} B, C) & ≅ {Sets}_{/ X} (A, {Sets}_{/ X} (B, C)), \\ {Sets}_{/ X} (A \times_{X} B, C) & ≅ {Sets}_{/ X} (B, {Sets}_{/ X} (A, C)), \end{aligned}$

natural in $(A, ϕ_{A}), (B, ϕ_{B}), (C, ϕ_{C}) \in Obj ({Sets}_{/ X})$ , where ${Sets}_{/ X} (A, B)$ is the object of ${Sets}_{/ X}$ consisting of (see , ):
- The Set. The set ${Sets}_{/ X} (A, B)$ defined by
  
  ${Sets}_{/ X} (A, B) \overset{def}{=} \underset{x \in X}{∐} Sets (ϕ_{A}^{- 1} (x), ϕ_{Y}^{- 1} (x))$
- The Map to $X$ . The map
  
  $ϕ_{{Sets}_{/ X} (A, B)} : {Sets}_{/ X} (A, B) \to X$
  
  defined by
  
  $ϕ_{{Sets}_{/ X} (A, B)} (x, f) \overset{def}{=} x$
  
  for each $(x, f) \in {Sets}_{/ X} (A, B)$ .
3. Associativity. Given a diagram

in $Sets$ , we have isomorphisms of sets

$(A \times_{X} B) \times_{Y} C ≅ (A \times_{X} B) \times_{B} (B \times_{Y} C) ≅ A \times_{X} (B \times_{Y} C),$

where these pullbacks are built as in the diagrams
4. Interaction With Composition. Given a diagram

in $Sets$ , we have isomorphisms of sets

$\begin{aligned} X \times_{K}^{f \circ ϕ, g \circ ψ} Y & ≅ (X \times_{A}^{ϕ, q_{1}} (A \times_{K}^{f, g} B)) \times_{A \times_{K}^{f, g} B}^{p_{2}, p_{1}} ((A \times_{K}^{f, g} B) \times_{B}^{q_{2}, ψ} Y) \\ ≅ X \times_{A}^{ϕ, p} ((A \times_{K}^{f, g} B) \times_{B}^{q_{2}, ψ} Y) \\ ≅ (X \times_{A}^{ϕ, q_{1}} (A \times_{K}^{f, g} B)) \times_{B}^{q, ψ} Y \end{aligned}$

where

$\begin{aligned} q_{1} & = {pr}_{1}^{A \times_{K}^{f, g} B}, \\ p_{1} & = {pr}_{1}^{(A \times_{K}^{f, g} B) \times_{Y}^{q_{2}, ψ}}, \\ p & = q_{1} \circ {pr}_{1}^{(A \times_{K}^{f, g} B) \times_{B}^{q_{2}, ψ} Y}, \end{aligned} \begin{aligned} q_{2} & = {pr}_{2}^{A \times_{K}^{f, g} B}, \\ p_{2} & = {pr}_{2}^{X \times_{A \times_{K}^{f, g} B}^{ϕ, q_{1}} (A \times_{K}^{f, g} B)}, \\ q & = q_{2} \circ {pr}_{2}^{X \times_{A}^{ϕ, q_{1}} (A \times_{K}^{f, g} B)}, \end{aligned}$

and where these pullbacks are built as in the following diagrams:
5. Unitality. We have isomorphisms of sets
natural in $(A, f) \in Obj ({Sets}_{/ X})$ .
6. Commutativity. We have an isomorphism of sets
natural in $(A, f), (B, g) \in Obj ({Sets}_{/ X})$ .
7. Distributivity Over Coproducts. Let $A$ , $B$ , and $C$ be sets and let $ϕ_{A} : A \to X$ , $ϕ_{B} : B \to X$ , and $ϕ_{C} : C \to X$ be morphisms of sets. We have isomorphisms of sets
$\begin{aligned} δ_{ℓ}^{{Sets}_{/ X}} & : A \times_{X} (B ∐ C) \tilde{⇢} (A \times_{X} B) ∐ (A \times_{X} C), \\ δ_{r}^{{Sets}_{/ X}} & : (A ∐ B) \times_{X} C \tilde{⇢} (A \times_{X} C) ∐ (B \times_{X} C), \end{aligned}$

as in the diagrams
natural in $A, B, C \in Obj ({Sets}_{/ X})$ .
8. Annihilation With the Empty Set. We have isomorphisms of sets
natural in $(A, f) \in Obj ({Sets}_{/ X})$ .
9. Interaction With Products. We have an isomorphism of sets
10. Symmetric Monoidality. The 8-tuple $({Sets}_{/ X}$ , $\times_{X}$ , $X$ , ${Sets}_{/ X}$ , $α^{{Sets}_{/ X}}$ , $λ^{{Sets}_{/ X}}$ , $ρ^{{Sets}_{/ X}}$ , $σ^{{Sets}_{/ X}})$ is a symmetric closed monoidal category.

Proof of Proposition 2.1.4.1.5.

Item 1: Functoriality

This is a special case of functoriality of co/limits,

, with the explicit expression for

ξ

following from the commutativity of the cube pullback diagram.

Item 2: Adjointness

This is a repetition of

, and is proved there.

Item 3: Associativity

We have

\begin{aligned} (A \times_{X} B) \times_{Y} C & ≅ {((a, b), c) \in (A \times_{X} B) \times C | h (b) = k (c)} \\ ≅ {((a, b), c) \in (A \times B) \times C | f (a) = g (b) and h (b) = k (c)} \\ ≅ {(a, (b, c)) \in A \times (B \times C) | f (a) = g (b) and h (b) = k (c)} \\ ≅ {(a, (b, c)) \in A \times (B \times_{Y} C) | f (a) = g (b)} \\ ≅ A \times_{X} (B \times_{Y} C) \end{aligned}

and

\begin{aligned} (A \times_{X} B) \times_{B} (B \times_{Y} C) & ≅ {((a, b), (b^{'}, c)) \in (A \times_{X} B) \times (B \times_{Y} C) | b = b^{'}} \\ ≅ {((a, b), (b^{'}, c)) \in (A \times B) \times (B \times C) | \begin{aligned} f (a) = g (b), b = b^{'}, \\ and h (b^{'}) = k (c) \end{aligned}} \\ ≅ {(a, (b, (b^{'}, c))) \in A \times (B \times (B \times C)) | \begin{aligned} f (a) = g (b), b = b^{'}, \\ and h (b^{'}) = k (c) \end{aligned}} \\ ≅ {(a, ((b, b^{'}), c)) \in A \times ((B \times B) \times C) | \begin{aligned} f (a) = g (b), b = b^{'}, \\ and h (b^{'}) = k (c) \end{aligned}} \\ ≅ {(a, ((b, b^{'}), c)) \in A \times ((B \times_{B} B) \times C) | \begin{aligned} f (a) = g (b) and \\ h (b^{'}) = k (c) \end{aligned}} \\ ≅ {(a, (b, c)) \in A \times (B \times C) | f (a) = g (b) and h (b) = k (c)} \\ ≅ A \times_{X} (B \times_{Y} C), \end{aligned}

where we have used Item 5 for the isomorphism

B \times_{B} B ≅ B

Item 4: Interaction With Composition

By Item 3, it suffices to construct only the isomorphism

X \times_{K}^{f \circ ϕ, g \circ ψ} Y ≅ (X \times_{A}^{ϕ, q_{1}} (A \times_{K}^{f, g} B)) \times_{A \times_{K}^{f, g} B}^{p_{2}, p_{1}} ((A \times_{K}^{f, g} B) \times_{B}^{q_{2}, ψ} Y) .

We have

\begin{aligned} (X \times_{A}^{ϕ, q_{1}} (A \times_{K}^{f, g} B)) & \overset{def}{=} {(x, (a, b)) \in X \times (A \times_{K}^{f, g} B) | ϕ (x) = q_{1} (a, b)} \\ \overset{def}{=} {(x, (a, b)) \in X \times (A \times_{K}^{f, g} B) | ϕ (x) = a} \\ ≅ {(x, (a, b)) \in X \times (A \times B) | ϕ (x) = a and f (a) = g (b)}, \\ ((A \times_{K}^{f, g} B) \times_{B}^{q_{2}, ψ} Y) & \overset{def}{=} {((a, b), y) \in (A \times_{K}^{f, g} B) \times Y | q_{2} (a, b) = ψ (y)} \\ \overset{def}{=} {((a, b), y) \in (A \times_{K}^{f, g} B) \times Y | b = ψ (y)} \\ ≅ {((a, b), y) \in (A \times B) \times Y | b = ψ (y) and f (a) = g (b)}, \end{aligned}

so writing

\begin{aligned} S & = (X \times_{A}^{ϕ, q_{1}} (A \times_{K}^{f, g} B)) \\ S^{'} & = ((A \times_{K}^{f, g} B) \times_{B}^{q_{2}, ψ} Y), \end{aligned}

we have

\begin{aligned} S \times_{A \times_{K}^{f, g} B}^{p_{2}, p_{1}} S^{'} & \overset{def}{=} {((x, (a, b)), ((a^{'}, b^{'}), y)) \in S \times S^{'} | p_{1} (x, (a, b)) = p_{2} ((a^{'}, b^{'}), y)} \\ \overset{def}{=} {((x, (a, b)), ((a^{'}, b^{'}), y)) \in S \times S^{'} | (a, b) = (a^{'}, b^{'})} \\ ≅ {((x, a, b, y)) \in X \times A \times B \times Y | ϕ (x) = a, ψ (y) = b, and f (a) = g (b)} \\ ≅ {((x, a, b, y)) \in X \times A \times B \times Y | f (ϕ (x)) = g (ψ (y))} \\ \overset{def}{=} X \times_{K} Y . \end{aligned}

This finishes the proof.

Item 5: Unitality

We have

\begin{aligned} X \times_{X} A & ≅ {(x, a) \in X \times A | f (a) = x}, \\ A \times_{X} X & ≅ {(a, x) \in X \times A | f (a) = x}, \end{aligned}

which are isomorphic to $A$ via the maps $(x, a) \mapsto a$ and $(a, x) \mapsto a$ . The proof of the naturality of $λ^{{Sets}_{/ X}}$ and $ρ^{{Sets}_{/ X}}$ is omitted.

Item 6: Commutativity

We have

\begin{aligned} A \times_{C} B & \overset{def}{=} {(a, b) \in A \times B | f (a) = g (b)} \\ = {(a, b) \in A \times B | g (b) = f (a)} \\ ≅ {(b, a) \in B \times A | g (b) = f (a)} \\ \overset{def}{=} B \times_{C} A . \end{aligned}

The proof of the naturality of $σ^{{Sets}_{/ X}}$ is omitted.

Item 7: Distributivity Over Coproducts

We have

\begin{aligned} A \times_{X} (B ∐ C) & \overset{def}{=} {(a, z) \in A \times (B ∐ C) | ϕ_{A} (a) = ϕ_{B ∐ C} (z)} \\ = {(a, z) \in A \times (B ∐ C) | z = (0, b) and ϕ_{A} (a) = ϕ_{B ∐ C} (z)} \\ \cup {(a, z) \in A \times (B ∐ C) | z = (1, c) and ϕ_{A} (a) = ϕ_{B ∐ C} (z)} \\ = {(a, z) \in A \times (B ∐ C) | z = (0, b) and ϕ_{A} (a) = ϕ_{B} (b)} \\ \cup {(a, z) \in A \times (B ∐ C) | z = (1, c) and ϕ_{A} (a) = ϕ_{C} (c)} \\ ≅ {(a, b) \in A \times B | ϕ_{A} (a) = ϕ_{B} (b)} \\ \cup {(a, c) \in A \times C | ϕ_{A} (a) = ϕ_{C} (c)} \\ \overset{def}{=} (A \times_{X} B) \cup (A \times_{X} C) \\ ≅ (A \times_{X} B) ∐ (A \times_{X} C), \end{aligned}

with the construction of the isomorphism

δ_{r}^{{Sets}_{/ X}} : (A ∐ B) \times_{X} C \tilde{⇢} (A \times_{X} C) ∐ (B \times_{X} C)

being similar. The proof of the naturality of $δ_{ℓ}^{{Sets}_{/ X}}$ and $δ_{r}^{{Sets}_{/ X}}$ is omitted.

Item 8: Annihilation With the Empty Set

We have

\begin{aligned} A \times_{X} Ø & \overset{def}{=} {(a, b) \in A \times Ø | f (a) = g (b)} \\ = {k \in Ø | f (a) = g (b)} \\ = Ø, \end{aligned}

and similarly for $Ø \times_{X} A$ , where we have used Item 7 of Proposition 2.1.3.1.3. The proof of the naturality of $ζ_{ℓ}^{{Sets}_{/ X}}$ and $ζ_{r}^{{Sets}_{/ X}}$ is omitted.

Item 9: Interaction With Products

We have

\begin{aligned} A \times_{pt} B & \overset{def}{=} {(a, b) \in A \times B |!_{A} (a) =!_{B} (b)} \\ \overset{def}{=} {(a, b) \in A \times B | ⋆ = ⋆} \\ = {(a, b) \in A \times B} \\ = A \times B . \end{aligned}

Item 10: Symmetric Monoidality

Omitted.

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