Computing the truth values in $\mathbf{Set}^2$.

category-theorytopos-theory

This is part of Exercise 4.5.2 of Goldblatt's, "Topoi: A Categorial Analysis of Logic".

Context:

Here is an old question of mine on the preceding exercise:

Verifying a Construction Satisfies the $\Omega$-axiom.

I have read Goldblatt's book before but didn't do all of its exercises. After struggling with the second set of exercises in Mac Lane and Moerdijk's, "Sheaves in Geometry and Logic: [. . .]," I have returned to Goldblatt (and this time I have a study buddy).

The Question:

Compute the truth values in $\mathbf{Set}^2$.

Thoughts:

I remember doing this exercise before; from what I recall, with $\top:\{0\}\to 2=\{0,1\}$ being ${\rm true}:0\mapsto 0$ in $\mathbf{Set}$ and – because ${\rm false}$ is not yet defined in general in Goldblatt's book – the "other" truth value in $\textbf{Set}$ being given by $\bot: \{0\}\to 2, 0\mapsto 1$, because the product arrow $\langle \top, \top\rangle: \langle \{0\}, \{0\}\rangle\to \langle 2,2\rangle$ is the subobject classifier of $\mathbf{Set}^2$, and $\langle \top, \bot\rangle\simeq \langle \bot, \top\rangle^\dagger$, the truth values of $\mathbf{Set}^2$ are, up to isomorphism, the following:

  • $\langle \top, \top\rangle$,

  • $\langle \bot, \top\rangle$, and

  • $\langle \bot, \bot\rangle$.

But I'm not sure. I'm confused about the details. How do I prove that the candidate truth values are what I'm after?

The exercise in question was easy enough for me the first time I read Goldblatt, as indicated by the very next question I asked being

Epic-monic factorisation in $\mathbf{Set}$.

about Exercise 5.2.1 just three days after Exercise 4.5.1.

I think I'm losing my mind.

I'm aware that

$${\rm Sub}(1_{\mathbf{Set}^2})\cong \mathbf{Set}^2(1_{\mathbf{Set}^2}, \Omega_{\mathbf{Set}^2}),$$

where the latter is the set of truth values of $\mathbf{Set}^2$.

I think what I have so far is nonsense.

Please help 🙂

$\dagger:$ Is this right?

Best Answer

Since you're working in $\text{Set}^2$ elements are essentially pairs of set elements in usual math. Therefore the truth values or $\text{Set}^2(\{(0,0)\}, \{0,1\}^2) \simeq \{(0,0), (0,1), (1,0), (1,1)\}$ (granted this isomorphism is in $\textbf{Set}$, but I guess you can identify objects & arrows in $\textbf{Set}^2$ with a full subcategory of $\textbf{Set}$). No need to check for isomorphisms between the truth values according to Goldblatt's book.

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