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exposition, math, Sketches and forms, Uncategorized

Idempotents by sketches and forms

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This post provides a detailed description of an example of a mathematical structure presented as a sketch and as a form.  It is a supplement to my article An Introduction to forms.  Most of the constructions I mention here are given in more detail in that article.
 
It helps in reading this post to be familiar with the basic ideas of category, including commutative diagram and limit cone, and of the concepts of logical theory and model in logic.
 

Sketches and forms

A sketch of a mathematical structure is a collection of objects and arrows that make up a digraph (directed graph), together with some specified cones, cocones and diagrams in the digraph.  A model of the sketch is a digraph morphism from the digraph to some category that takes the cones to limit cones, the cocones to colimit cocones, and the diagrams to commutative diagrams.  A morphism of models of a sketch from one model to another in the same category is a natural transformation.  Sketches can be used to define all kinds of algebraic structures in the sense of universal algebra, and many other types of structures (including many types of categories).  

There are many structures that sketches cannot sketch.  Forms were first defined in [4].  They can define anything a sketch can define and lots of other things.  [5] gives a leisurely description of forms suitable for people who have a little bit of knowledge of categories and [1] gives a more thorough description.  

An idempotent is a very simple kind of algebraic structure.  Here I will describe both a sketch and a form for idempotents. In another post I will do the same for binops (magmas).

Idempotent

An idempotent is a unary operation $u$ for which $u^2=u$.

  • If $u$ is a morphism in a category whose morphisms are set functions, a function $u:S\to S$ is an idempotent if $u(u(x))=u(x)$ for all $x$ in the domain.  
  • Any identity element in any category is an idempotent.
  • A nontrivial example is the function $u(x,y):=(-x,0)$ on the real plane.  

Any idempotent $u$ makes the following diagram commute

and that diagram can be taken as the definition of idempotent in any category.

The diagram is in green.  In this post (and in [5]) diagrams in the category of models of a sketch or a form are shown in green.

A sketch for idempotents

The sketch for idempotents contains a digraph with one object and one arrow from that object to itself (above left) and one diagram (above right).  It has no cones or cocones.  So this is an almost trivial example.  When being expository (well, I can hardly say "when you are exposing") your first example should not be trivial, but it should be easy.  Let's call the sketch $\mathcal{S}$.

  • The diagram looks the same as the green diagram above.  It is in black, because I am showing things in syntax (things in sketches and forms) in black and semantics (things in categories of models) in green.
  • The green diagram is a commutative diagram in some category (unspecified).  
  • The black diagram is a diagram in a digraph. It doesn't make sense to say it is commutative because digraphs don't have composition of arrows.
  • Each sketch has a specific digraph and lists of specific diagrams, cones and cocones.  The left digraph above is not in the list of diagrams of $\mathcal{S}$ (see below).

The definition of sketch says that every diagram in the official list of diagrams of a given sketch must become a commutative diagram in a model.  This use of the word "become" means in this case that a model must be a digraph morphism $M:\mathcal{S}\to\mathcal{C}$ for some category $\mathcal{C}$ for which the diagram below commutes.

This sketch generates a category called the Theory ("Cattheory" in [5]) of the sketch $\mathcal{S}$, denoted by $\text{Th}(\mathcal{S})$.  It is roughly the "smallest" category containing $f$ and $C$ for which the diagrams in $\mathcal{S}$ are commutative.  
 
This theory contains the generic model $G:\mathcal{S}\to \text{Th}(\mathcal{S})$ that takes $f$ and $C$ to themselves.
  • $G$ is "generic" because anything you prove about $G$ is true of every model of $\mathcal{S}$ in any category.
  • In particular, in the category $\text{Th}(\mathcal{S})$, $G(f)\circ G(f)=G(f)$.  
  • $G$ is a universal morphism in the sense of category theory: It lifts any model $M:\mathcal{S}\to\mathcal{C}$ to a unique functor $\bar{M}=M\circ G:\text{Th}(\mathcal{S})\to\mathcal{C}$ which can therefore be regarded as the same model.  See Note [2].
SInce models are functors, morphisms between models are natural transformations.  This gives what you would normally call homomorphisms for models of almost any sketchable structure.  In [2] you can find a sketch for groups, and indeed the natural transformations between models are group homomorphisms.

Sketching categories

You can sketch categories with a sketch CatSk containing diagrams and cones, but no cocones.  This is done in detail in [3]. The resulting theory $\text{Th}(\mathbf{CatSk})$ is required to be the least category-with-finite-limits generated by $\mathcal{S}$ with the diagrams becoming commutative diagrams and the cones becoming limit cones.  This theory is the FL-Theory for categories, which I will call ThCat (suppressing mention of FL).  

Doctrines

In general the theory of a particular kind of structure contains a parameter that denotes its doctrine. The sketch $\mathcal{S}$ for idempotents didn't require cones, but you can construct theories $\text{Th}(\mathcal{S})$, $\text{Th} (\text{FP},\mathcal{S})$ and $\text{Th}(\text{FL},\mathcal{S})$ for idempotents (FP means it is a category with finite products).  

In a strong sense, all these theories have the same models, namely idempotents, but the doctrine of the theory allows you to use more mechanisms for proving properties of idempotents.  (The doctrine for $\text{Th}(\mathcal{S})$ provides for equational proofs for unary operations only, a doctrine which has no common name such as FP or FS.)  The paper [1] is devoted to explicating proof in the context of forms, using graphs and diagrams instead of formulas that are strings of symbols.

Describing composable pairs of arrows

The form for any type of structure is constructed using the FL theory for some type of category, for example category with all limits, cartesian closed category, topos, and so on.  The form for idempotents can be constructed in ThCat (no extra structure needed).  The form for reflexive function spaces (for example) needs the FL theory for cartesian closed categories (see [5]).

Such an FL theory must contain objects $\text{ob}$ and $\text{ar}$ that become the set of objects and the set of arrows of the category that a model produces.  (Since FL theories have models in any category with finite limits, I could have said "object of objects" and "object of arrows".  But in this post I will talk about only models in Set.)

ThCat contains an object  $\text{ar}_2$ that represents composable pairs of arrows.  That requires a cone to define it:

This must become a limit cone in a model.

  • I usually show cones in blue. 
  • $\text{dom}$ and $\text{cod}$ give (in a model) the domain and codomain of an arrow.
  • $\text{lfac}$ gives the left factor and $\text{rfac}$ gives the right factor. It is usually useful to give suggestive names to some of the projections in situations like this, since they will be used elsewhere (where they will be black!).
  • The objects and arrows in the diagram (including $\text{ar}_2$) are already members of the FL theory for categories.
  • This diagram is annotated in green with sample names of objects and arrows that might exist in a model.  Atish and I introduced that annotation system in [1] to help you chase the diagram and think about what it means.

This cone is a graph-based description of the object of composable arrows in a category (as opposed to a linguistic or string-based description).

Describing endomorphisms

Now an idempotent must be an endomorphism, so we provide a cone describing the object of endomorphisms in a category. This cone already exists in the FL theory for categories.

  • $\text{loop}$ is a monomorphism (in fact a regular mono because it is the mono produced by an equalizer) so it is not unreasonable to give the element annotation for $\text{endo}$ and $\text{ar}$ the same name.
  • "$\text{dc}$" takes $f$ to its domain and codomain. 
  • $\text{loop}$ and "$\text{dc}$" were not created when I produced the cone above.  They were already in the FL theory for categories.
 
Since the cone defining $\text{ar}_2$ is a limit cone (in the Theory, not in a model), if you have any other commutative cone (purple) to that cone, a unique arrow (red) $\text{diag}$ automatically is present as shown below:

This particular purple cone is the limit cone defining $\text{endo}$ just defined.  Now $\text{diag}$ is a specific arrow in the FL theory for categories. In a model of the theory (which is a category in Set or in some other category) takes an endomorphism to the corresponding pair of composable arrows.

The object of idempotents

Now using these arrows we can define the object $\text{idm}$ of idempotents using the diagram below. See Note [3].

 

 

 

 

 

Idm is an object in ThCat.  In any category, in other words in any model of ThCat, idm becomes the set of idempotent arrows in that category.

In the terminology of [5], the object idm is the form for idempotents, and the cone it is the limit of is the description of idempotent.  

Now take ThCat and adjoin an arrow $g:1\to\text{idm}$.  You get a new FL category I will call the FL-theory of the form for idempotents.  A model of the theory of the form in Set  is a category with a specified idempotent. A particular example of a model of the form idm in the category of real linear vector spaces is the map $u(x,y):=(-x,0)$ of the (set of points of) the real plane to itself (it is an idempotent endomorphism of $\textbf{R}^2$).  

This example is typical of forms and their models, except in one way:  Idempotents are also sketchable, as I described above.  Many mathematical structures can be perceived as models of forms, but not models of sketches, such as reflexive function spaces as in [5].

Notes

[1] The diagrams shown in this post were drawn in Mathematica.  The code for them is shown in the notebook SketchFormExamples.nb .  I am in the early stages of developing a package for drawing categorical diagrams in Mathematica, so this notebook shows the diagrams defined in very primitive machine-code-like Mathematica.  The package will not rival xypic for TeX any time soon.  I am doing it so I can produce diagrams (including 3D diagrams) you can manipulate.

[2] In practice I would refer to the names of the objects and arrows in the sketch rather than using the M notation:  I might write $f\circ f=f$ instead of $M(f)\circ M(f)=M(f)$ for example.  Of course this confuses syntax with semantics, which sounds like a Grievous Sin, but it is similar to what we do all the time in writing math:  "In a semigroup, $x$ is an idempotent if $xx=x$."  We use same notation for the binary operation for any semigroup and we use $x$ as an arbitrary element of most anything.  Actually, if I write $f\circ f=f$ I can claim I am talking in the generic model, since any statement true in the generic model is true in any model.  So there.

[3] In the Mathematica notebook SketchFormExamples.nb in which I drew these diagrams, this diagram is plotted in Euclidean 3-space and can be viewed from different viewpoints by running your cursor over it.

References

[1] Atish Bagchi and Charles Wells, Graph-Base Logic and Sketches, draft, September 2008, on ArXiv.

[2] Michael Barr and Charles Wells, Category Theory for Computing Science (1999). Les Publications CRM, Montreal (publication PM023).

[3] Michael Barr and Charles Wells, Toposes, Triples and Theories (2005). Reprints in Theory and Applications of Categories 1.

[4] Charles Wells, A generalization of the concept of sketch, Theoretical Computer Science 70, 1990

[5] Charles Wells, An Introduction to forms.

 

 

 

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