Maths - Sheaf Example - Equalities

This is an example for the page about sheaves which is here.

Theory

 In model theory (see this page) a 'theory' is a set of first-order sentences, for instance: α: x.(x+0=x) β: x.(0+x=x) γ: xy.(x+y=y+x)
 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?

 x=x+0 <=> 0=0+0,1=1+0,2=2+0,3=3+0...

Equalities as a Group or Groupoid

A 'group presentation' represents a group as a set of equations (see page here).