# 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).

 metadata block see also: Michael Robinson - Youtube from two-day short course on Applied Sheaf Theory: Lecture 1 Lecture 2 - about simplicies & topology (sound on video is not good - quiet and delayed) Lecture 3 Lecture 4 Lecture 5 Lecture 6 Lecture 7 Lecture 8 Correspondence about this page Book Shop - Further reading . Where I can, I have put links to Amazon for books that are relevant to the subject, click on the appropriate country flag to get more details of the book or to buy it from them. Introduction to Topological Manifolds (Graduate Texts in Mathematics S.) Other Books about Curves and Surfaces

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