Proposition 4.7.3.1.2 (02SU): Properties of Bijective Functions — The Clowder Project

Table of Contents
Part 2: Sets
Chapter 4: Constructions With Sets
Section 4.7: Injective, Surjective, and Bijective Functions
Subsection 4.7.3: Bijective Functions
Proposition 4.7.3.1.2: Properties of Bijective Functions (cite)

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4.7.3 Bijective Functions

Let $A$ and $B$ be sets.

Definition 4.7.3.1.1 ▶ Bijective Functions

A function $f\colon A\to B$ is bijective if it is injective and surjective.

Proposition 4.7.3.1.2 ▶ Properties of Bijective Functions

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

1.
Characterisations. The following conditions are equivalent:
1. (a)
  The function $f$ is bijective.
2. (b)
  The function $f$ is an isomorphism in $\mathsf{Sets}$.
3. (c)
  The function $f$ is a monomorphism and an epimorphism in $\mathsf{Sets}$.
4. (d)
  We have $f_{!}=f_{*}$.
5. (e)
  The direct image function
  
  \[ f_{!}\colon \mathcal{P}(A)\to \mathcal{P}(B) \]
  
  associated to $f$ is bijective.
6. (f)
  The inverse image function
  
  \[ f^{-1}\colon \mathcal{P}(B)\to \mathcal{P}(A) \]
  
  associated to $f$ is bijective.
7. (g)
  The codirect image function
  
  \[ f_{*}\colon \mathcal{P}(A)\to \mathcal{P}(B) \]
  
  associated to $f$ is bijective.
8. (h)
  The direct image functor
  
  \[ f_{!}\colon (\mathcal{P}(A),\subset )\to (\mathcal{P}(B),\subset ) \]
  
  associated to $f$ is an equivalence of categories.
9. (i)
  The inverse image functor
  
  \[ f^{-1}\colon (\mathcal{P}(B),\subset )\to (\mathcal{P}(A),\subset ) \]
  
  associated to $f$ is an equivalence of categories.
10. (j)
  The codirect image functor
  
  \[ f_{*}\colon (\mathcal{P}(A),\subset )\to (\mathcal{P}(B),\subset ) \]
  
  associated to $f$ is an equivalence of categories.
11. (k)
  We have
  That is, we have
  
  \begin{align*} f^{-1}(f(a)) & = \left\{ a\right\} ,\\ f_{!}(f^{-1}(b)) & = \left\{ b\right\} \end{align*}
  
  for each $a\in A$ and each $b\in B$.
12. (l)
  We have
13. (m)
  We have
2.
Two-Out-of-Three. Let
be a diagram in $\mathsf{Sets}$. If two out of the three morphisms among $f$, $g$, and $g\circ f$ are bijections, then so is the third.

Proof of Proposition 4.7.3.1.2.

Item 1: Characterisations

We proceed by showing:

•
Step 1: Item 1a$\iff $Item 1b.
•
Step 2: Item 1a$\iff $Item 1c.
•
Step 3: Item 1a$\iff $Item 1d.
•
Step 4: Item 1a$\iff $Item 1e.
•
Step 5: Item 1a$\iff $Item 1f.
•
Step 6: Item 1a$\iff $Item 1g.
•
Step 7: Item 1a$\iff $Item 1k.
•
Step 8: Item 1a$\iff $Item 1l.
•
Step 9: Item 1a$\iff $Item 1m.
•
Step 10: Item 1e$\iff $Item 1h.
•
Step 11: Item 1f$\iff $Item 1i.
•
Step 12: Item 1g$\iff $Item 1j.

Step 1: Item 1a$\iff $Item 1b

Omitted.

Step 2: Item 1a$\iff $Item 1c

This follows from Item 1b of Item 1 of Proposition 4.7.1.1.2 and Item 1b of Item 1 of Proposition 4.7.2.1.2.

Step 3: Item 1a$\iff $Item 1d

This follows from Item 19 and Item 20 of Proposition 4.6.3.1.7.

Step 4: Item 1a$\iff $Item 1e

This follows from Item 1b of Item 1 of Proposition 4.7.1.1.2 and Item 1e of Item 1 of Proposition 4.7.2.1.2.

Step 5: Item 1a$\iff $Item 1f

This follows from Item 1e of Item 1 of Proposition 4.7.1.1.2 and Item 1d of Item 1 of Proposition 4.7.2.1.2.

Step 6: Item 1a$\iff $Item 1g

This follows from Item 1f of Item 1 of Proposition 4.7.1.1.2 and Item 1f of Item 1 of Proposition 4.7.2.1.2.

Step 7: Item 1a$\iff $Item 1k

This follows from Item 1l of Item 1 of Proposition 4.7.1.1.2 and Item 1k of Item 1 of Proposition 4.7.2.1.2.

Step 8: Item 1a$\iff $Item 1l

This follows from Item 1m of Item 1 of Proposition 4.7.1.1.2 and Item 1l of Item 1 of Proposition 4.7.2.1.2.

Step 9: Item 1a$\iff $Item 1l

This follows from Item 1n of Item 1 of Proposition 4.7.1.1.2 and Item 1m of Item 1 of Proposition 4.7.2.1.2.

Step 10: Item 1e$\iff $Item 1h

This follows from the fact that $\mathcal{P}(B)$ is locally posetal.

Step 11: Item 1f$\iff $Item 1i

This follows from the fact that $\mathcal{P}(A)$ is locally posetal.

Step 12: Item 1g$\iff $Item 1j

This follows from the fact that $\mathcal{P}(B)$ is locally posetal.

Item 2: Two-Out-of-Three

This is a special case of Chapter 11: Categories, Item 1 of Proposition 11.4.1.1.3.

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