Trying to understand sheaves through constructions

Recently I have been reading some Algebraic Geometry notes by Andreas Gathmann - available here. While I can accept and broadley understand the definition of a (pre-)sheaf, I wanted to get some idea of what a sheaf might look like in the wild. I found this paper which gives a construction of a sheaf on a graph. This is the example which elucidated sheaves for me finally. But also a sentence in the paper caught my attention: “the only structure you need to construct a sheaf is a partial order.” I thought about this and I wanted to explore this further so I am going to go through constructing a sheaf on just a partial order.

(Pre-)Sheaves quickly

In this note I am not going to distinguish between pre-sheaves and sheaves. The distinction will not be useful here, just keep in mind that sometimes I may say sheaf and mean presheaf.

Now the definition in full.

So essentially this is saying that for each component of our object , whatever it may be, we assign it a new space, and that we require these spaces to behave nicely. To quote Agrios directly - “Think about it like the mathematical object is a plot of land and a sheaf is like a garden on top of it.”

Now lets get to building a simple sheaf which should hopefully illuminate this further.

Sheaf on a partial order

So let’s assume we have a partial order where and let the partial order be and . We can define a topology (The Alexandrov topology) on this by defining the open sets to be the upper sets for this partial order, i.e. a subset is open if and then . Let’s quickly confirm that this is a topology. For our case, the open sets are

Clearly any intersection or union of these upper (open) sets is also an upper set, and hence open.

This topology can be generated by the basic open sets . In our case we have

Note that if then :

Let then and since we have . So and .

Lets now define our sheaf . We have to define two things, to each element of our space we must assign new a space, and then define restriction maps between these spaces.

So for the spaces we can define the following:

We can define the restriction maps between them to just be the following:

with

Let’s calculate the sections of this sheaf, recall the sections are the elements of the spaces we have defined, so we will calculate the elements of for each of the basic open sets.

So now we have spaces assigned to each element of our poset and restriction maps between them - hence a sheaf - and we have calculated the space of sections of this sheaf. The Sheaf axioms weren’t explicitly checked, but they are fairly easy to confirm if you wanted.