Sheafs appearing in philosophy?

Question

I apologize in advance if I make mistakes in the following construction. I have very recently been introduced to the concept of a sheaf.

I am currently a mathematics major and philosophy minor and have found certain concepts in mathematics to be particularly useful in discussing certain philosophy (the most useful of which being different sizes of infinity, orders on sets, etc). Recently I've been thinking about causality (in philosophy) and sheafs (in mathematics).

If we let $E$ be the set of events the occur in the universe, we can give $\mathcal{P}(E)$, the power set of $E$ a partial ordering $\leq$ that is reflexive. Namely, if $U,V \in \mathcal{P}(E)$, we say $U \leq V$ if $U$ causally implies $V$ or in other words, if all events in $U$ occur, then all of the events in $V$ must occur. So we can construct the category of events with respect to this partial order (that is, Hom$(U,V)=\{(U,V)\}$ if $U\leq V$ and is empty otherwise).

Furthermore, time, or $T$, is traditionally viewed as a subset of $\mathbb{R}$ and thus inherits a subset topology.

Let $F: T \to \mathcal{P}(E)$ be defined so that for every open subset of time $W \subset T$, $F(W)=\{x \in E \mid x$ occurs at time $t_x$ such that $t_x \in W\}$, that is, we associate to a subset of time $W$, the collection of events that happen at any time in $W$.

I claim $F$ is a presheaf. We get natural restriction morphisms since for example, knowing all of the events that occur today is enough for me to know the events that occur between 1 o'clock and 2 o'clock today.

My question is: "Have I made any mistakes?", "Is this also a sheaf?", "Are there any properties of sheafs that can be translated back into language about causality that could be useful? or is this a completely trivial construction?", "Does this tell us anything about the philosophical idea of determinism?".

Answer 1

When you say "We get natural restriction morphisms since for example, knowing all of the events that occur today is enough for me to know the events that occur between 1 o'clock and 2 o'clock today" you seem to suggest that ${\cal P}(E)$ is partially ordered as the opposite of the usual subset ordering. But then your notion of "causal implication" is rather vacuous: all it says is that if some things have occured, then any subset of those things have also ocurred $-$ this is purely logical rather than "causal" in nature. Indeed, there is nothing to connect what events occur at time $t_1$ with what events occur at time $t_2$, for any times $t_1$ and $t_2$, which is a big hole if this is supposed to be a kind of physical model. At any rate it is in fact a presheaf.

I think in differential geometry and special relativity there are notions like timelike, spacelike, null, causal, etc. in regards to directions in spacetime (modelled as a Lorentzian manifold) which are relevant to what you're trying to think about. You may try searching these keywords (and learning some geometry appropriate for the occasion).