Definition 3.1.7.1.1 (01NY): The Symmetry of $\times $ — The Clowder Project

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Table of Contents
Part 1: Sets
Chapter 3: Monoidal Structures on the Category of Sets
Section 3.1: The Monoidal Category of Sets and Products
Subsection 3.1.7: The Symmetry
Definition 3.1.7.1.1: The Symmetry of $\times$ (cite)

Statistics Cite

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Definition 3.1.7.1.1 ▶ The Symmetry of $\times$

The symmetry of the product of sets is the natural isomorphism

whose component

σ_{X, Y}^{Sets} : X \times Y \tilde{⇢} Y \times X

at $X, Y \in Obj (Sets)$ is defined by

σ_{X, Y}^{Sets} (x, y) \overset{def}{=} (y, x)

for each $(x, y) \in X \times Y$ .

Proof of Definition 3.1.7.1.1.

Invertibility

The inverse of

σ_{X, Y}^{Sets}

is the morphism

σ_{X, Y}^{Sets, - 1} : Y \times X \tilde{⇢} X \times Y

defined by

σ_{X, Y}^{Sets, - 1} (y, x) \overset{def}{=} (x, y)

for each $(y, x) \in Y \times X$ . Indeed:

Invertibility I. We have

$\begin{aligned} [σ_{X, Y}^{Sets, - 1} \circ σ_{X, Y}^{Sets}] (x, y) & \overset{def}{=} σ_{X, Y}^{Sets, - 1} (σ_{X, Y}^{Sets} (x, y)) \\ \overset{def}{=} σ_{X, Y}^{Sets, - 1} (y, x) \\ \overset{def}{=} (x, y) \\ \overset{def}{=} [{id}_{X \times Y}] (x, y) \end{aligned}$

for each $(x, y) \in X \times Y$ , and therefore we have

$σ_{X, Y}^{Sets, - 1} \circ σ_{X, Y}^{Sets} = {id}_{X \times Y} .$
Invertibility II. We have

$\begin{aligned} [σ_{X, Y}^{Sets} \circ σ_{X, Y}^{Sets, - 1}] (y, x) & \overset{def}{=} σ_{X, Y}^{Sets, - 1} (σ_{X, Y}^{Sets} (y, x)) \\ \overset{def}{=} σ_{X, Y}^{Sets, - 1} (x, y) \\ \overset{def}{=} (y, x) \\ \overset{def}{=} [{id}_{Y \times X}] (y, x) \end{aligned}$

for each $(y, x) \in Y \times X$ , and therefore we have

$σ_{X, Y}^{Sets} \circ σ_{X, Y}^{Sets, - 1} = {id}_{Y \times X} .$

Therefore

σ_{X, Y}^{Sets}

is indeed an isomorphism.

Naturality

We need to show that, given functions

\begin{aligned} f & : X \to A, \\ g & : Y \to B \end{aligned}

the diagram

commutes. Indeed, this diagram acts on elements as

and hence indeed commutes, showing

σ^{Sets}

to be a natural transformation.

Being a Natural Isomorphism

Since

σ^{Sets}

is natural and

σ^{Sets, - 1}

is a componentwise inverse to

σ^{Sets}

, it follows from Chapter 9: Categories, Item 2 of Proposition 9.9.7.1.2 that

σ^{Sets, - 1}

is also natural. Thus

σ^{Sets}

is a natural isomorphism.

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