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$\require{AMScd}$In the arXiv article arXiv:2011.13070v1 by J. M. Barrett, the following is claimed:

The arrow category $\mathbf{Set}^\to$ is that whose objects are functions $f:A\to B$ (in $\mathbf{Set}$), and morphisms $f:A\to B$ to $g:C\to D$ are commutative square:

$$\begin{CD} A @>{\,}>> C\\ @V{f}VV @VV{g}V \\ B @>>{\,}> D \end{CD}$$

$\mathbf{Set}^\to$ forms a topos whose internal logic is three-valued. The three truth values have a natural interpretation as a time-like logic: fixing a point in time $t_0$, the truth values are always true ($T$), always false ($F$), and false before $t_0$, true afterwards ($C$). The truth tables for the Boolean connectives are as follows (the row is the first argument).

$$\begin{array}{c|ccc} \land & T & C & F\\ \hline T & T & C & F \\ C & C & C & F\\ F & F & F & F \end{array}$$

$$\begin{array}{c|ccc} \lor & T & C & F\\ \hline T & T & T & T \\ C & T & C & C\\ F & T & C & F \end{array}$$

$$\begin{array}{c|ccc} \to & T & C & F\\ \hline T & T & C & F \\ C & T & T & F\\ F & T & T & T \end{array}$$

The Question:

What is the reason behind this interpretation; that is, is there a proof? Please share it if so.

Context:

I have "read" Goldblatt's book once all the way through a long time ago, though I began skimming from Chapter 14 onwards. Since then, I have made a few attempts to start the book again.

One of the exercises in Goldblatt's book is to find the truth values in $\mathbf{Set}^\to$. I have done that exercise at least twice, although spaced far apart.

I don't think I can answer this question on my own. It requires, I believe, a knowledge of nonclassical logic I just don't have (i.e., interpretations of a three-valued logic).

The "time-like logic" baffles me; however, I asked this question eleven years ago now (which makes me feel old!) about a subobject classifier with an interpretation of "time until truth". I guess a similar thing is going on, but I just don't see it yet.

Please pitch your answers at a rusty intermediate person's borderline novice level.

Martin Brandenburg
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Shaun
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    This is just describing the Heyting algebra of subterminal objects in $\mathbf{Set}^\to$, isn't it? See https://ncatlab.org/nlab/show/poset+of+subobjects and https://ncatlab.org/nlab/show/subterminal+object – Naïm Camille Favier Dec 19 '24 at 10:42
  • Not quite, @NaïmFavier; I'm interested in its interpretation. – Shaun Dec 19 '24 at 10:44
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    Well, that's not a precise claim, so it can't be proven; it's more of an intuition. One can think of presheaf logic as the logic of knowledge: some facts may become known/proven at a later point, and then they can't be "unlearned" or "unproven", so they persist to all future worlds. See also this blog. – Naïm Camille Favier Dec 19 '24 at 11:54
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    This "time-like" nature is more pronounced on the topos of trees, i.e. the presheaf category on natural mumbers regarded as a partial order regarded as a category. – Trebor Dec 19 '24 at 15:48

1 Answers1

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The set of truth values is $\hom(1,\Omega) \cong \mathrm{Sub}(1)$. This becomes a Heyting algebra with join and meet of subobjects. For $\mathbf{Mor}(\mathbf{Set})$ the terminal object is the identity $1 \to 1$. Its subobjects are

  • $T$ defined by the identity $1 \to 1$
  • $F$ defined by the identity $0 \to 0$
  • $C$ defined by the unique map $! : 0 \to 1$

You can easily work out that the Heyting algebra structure is the one in the tables. In fact, this is just the partial order

$F < C < T$

Also, from the very definition of $C$ we get the interpretation of it as "from a false statement we can deduce a true statement", a special case of ex falso quodlibet. The time interpretation is also possible but I don't know why it should be useful.

Martin Brandenburg
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