This is an example for the page about sheaves which is here.
|In model theory (see this page) a 'theory' is a set of first-order sentences, for instance:
In this diagram we have a box for each set of these sentances:
The arrows mean implies. A set with more sentances can imply a set with less sentances because we are using Intuitionistic logic so we don't use the excluded middle rule. So if a sentance is not included in a set it doesn't mean its false it just means we are not saying anything about it.
I have left out arrows if they are given by composing the above arrows.
|There is more we can imply from these sentances because α and γ implies β so we can add the red arrows:
|In a similar way we can imply α from β and γ. We can't however imply γ from β and α.
|If there is an implication arrow going in both directions we can treat the sets as being the same. So here I have put them in the same box:
We can now draw this as a presheaf.
Each entry in set represents a different value for the varable 'y'. We can also have every possible value for the varable 'x' but, to avoid complicating things I have not done this yet.
Equalities as a Shape
Each equation can be a loop in some space.
What is a 'variable' in a space?
Equalities as a Group or Groupoid
A 'group presentation' represents a group as a set of equations (see page here).