@user525966 Usually they mean roughly the same thing in logic.

@LeakyNun Hello! I'm here now. =)

@user21820 hi

@LeakyNun: So do you want to tell me about sheaves?

sure

Ok I'll be ready in 5 minutes.

Ok back.

@user21820 I moved into another timezone but the timers that stick to the messages in the chat did not, do you know how to fix this?

@famesyasd Aren't the timers based on UTC or something?

@user21820 nice, fixed!

@famesyasd Oh I didn't realize that the timestamps shown is my time. But the timestamps on the transcript are UTC.

@LeakyNun: So are we ready to start?

@user21820 sure

So we start with two topological spaces X and Y

For every open subset U of X, denote O(U) by the set of continuous functions from U to Y

If U is a subset of V, then we have a function Res[U,V] : O(V) -> O(U)

ok?

@LeakyNun What does "Res" stand for?

restriction

Ok.

we note that this Res function satisfies some properties

oh, and to be clear, the function Res[U,V] restricts the domain of a function V -> Y to a function U -> Y

Yea.

and we introduce more properties

Locality : Let {U[i] | i in I} be an open cover of U, and f, g in O(U). If Res[U[i],U](f) = Res[U[i],U](g) for every i, then f = g.

Gluing: Let {U[i] | i in I} be an open cover of U, and f[i] in O(U[i]) for each i. Suppose further that for each i and j in I, Res[U[i]∩U[j],U[i]](f[i]) = Res[U[i]∩U[j],U[j]](f[j]). Then, there is f in O(U) such that Res[U[i],U](f) = f[i] for each i.

take some time to digest these properties if you need to.

@LeakyNun Why are we defining Res[U,V] 'backward', instead of from O(U) to O(V)?

given a function from V to Y, we can restrict it to a function from U to Y

because U is a subset of V

so this "restriction" is a function O(V) -> O(U)

Nevermind. It's just that I would define it to be Res[V,U] instead.

fair enough

but, you see, I've already written it like 10 times in the wrong order

Ya I know.

Okay go on.

I know the content of the properties though I didn't check every character. =)

that's ok

Now we generalize this example

X is still a topological space

For each open subset U of X, we still have a set O(U)

If U is a subset of V, we still have a function Res[U,V] : O(V) -> O(U)

but now we leave O(U) undefined

if R1 and R2 are satisfied, we call this a presheaf

What do you mean by "undefined"?

If furthermore, locality and gluing are satisfied, we call this a sheaf

Earlier we defined O(U) to be the set of functions from U to Y

now we allow O(U) to be any set

And so you also leave Res undefined?

Or it's still restriction? No that doesn't make sense...

we also leave Res undefined

Ok.

Why must it be a topology? We could do the same with any set, replacing "open cover" by "cover".

that's a special case, where the topology is the discrete topology

Hmm I'd rather say that the topological case is the special case, and presumably there are results that hold for some special topologies.

So, what's next?

well, another example would be the sheaf of analytic functions

where we take X to be the complex numbers

for each open subset U of X, we let O(U) to be the analytic functions U -> C

@LeakyNun Ok yup that requires the openness.

I don't know what's next, you keep asking me what's next

@LeakyNun I was just going to ask you that again.

Lol.

I was hoping you had some nice wonderful theorem about sheaves coming up.

=)

Though I suspect that apart from order-theoretic results (like the Galois connections), such abstract generalizations would not have deep general theorems. The sheaf of analytic functions, for example, seem to depend crucially on specific knowledge about complex power series.

That's part of the problem I have when reading stuff like Wikipedia articles. The one on sheaves says in the third paragraph that it is versatile and so has many applications, but how much of that is just a matter of language or abstraction, and how much of that comes from something deep, is never made clear.

well sheaf is the language of algebraic geometry

it's more like a language than a deep theorem

That's what I thought, but I never know which rabbit hole to go down to find out haha.. The last paragraph of that article looks like a particularly curious hole.

> It was later discovered that the logic in categories of sheaves is intuitionistic logic (this observation is now often referred to as Kripke–Joyal semantics, but probably should be attributed to a number of authors). This shows that some of the facets of sheaf theory can also be traced back as far as Leibniz.

The first sentence makes it sound like a deep rabbit hole. The second makes it sound like a triviality in hindsight. Lol.