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category theory, math, proof, representations

Proofs using diagrams

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Introduction

This post gives a proof of an easy theorem in category theory using the graph-based logic approach of Graph based logic and sketches, (GBLS) by Atish Bagchi and me.

Formal logic is typically defined in terms of formulas and terms, defined recursively as strings of characters, together with rules of inference. GBLS proposes a new approach to logic where diagrams are used instead of strings of characters. The exposition here spells out the proof in more detail than GBLS does and uses various experimental ways of drawing diagrams using Mathematica.

To follow this proof, you need to be familiar with basic category theory. Most special definitions that are needed are defined in this post where they are first used. Section 1 of GBLS also gives the definitions you need with more context.

The theorem

The Theorem to be proved (it is Theorem 8.3.1 of GBLS) says that, in any category, if the triangles in the diagram below commute, then the outside square commutes. This is easy using the associative law: If $xf=h$ and $kx=g$, then $kh=k(xf)=(kx)f=gf$.

Subject Diagram

So what?

This theorem is not interesting. The point of this post is to present a new approach to proving such theorems, using diagrams instead of strings. The reason that exhibiting the dig
rammatic proof is interesting is that many different kinds of categories have a FL cattheory, including these:

Essentially algebraic string-based logic is described in detail in Partial Horn logic and cartesian categories, by E. Palmgren and Steven Vickers.

Commercial

My concept of form in A generalization of the concept of sketch generalizes sketches to all the categories that can be defined as models of FL cattheories. So the method of proof using diagrams can be applied to theorems about the objects defined by forms.

Concepts needed for the graph-based proof

To prove the theorem, I will make use of $\mathbf{ThCat}$, the FL cattheory for categories.

  • An FL category is a category with all finite limits.
  • GLBS uses the word cattheory for what Category theory for computing science and Toposes, triples and theories call the theory of a sketch.
  • In many books and articles, and in nLab, a “sketch” is what we call the cattheory (or the theory) of a sketch. For us, the sketch is a generating collection of objects, arrows, diagrams, cones and cocones for the cattheory. The category of models of the sketch and the cattheory are equivalent.
  • $\mathbf{ThCat}$ is a category with finite limits freely generated by certain designated objects, arrows, commutative diagrams and limit cones, listed below.
  • A model of $\mathbf{ThCat}$ in $\mathbf{Set}$ (the category of sets, whichever one you like) is an FL functor $\mathfrak{C}:\mathbf{ThCat}\to\mathbf{Set}.$
  • Such a model $\mathfrak{C}$ is a small category, and every small category is such a model. If this statement worries you, read Section 3.4 of GBLS.
  • Natural transformations between models are FL-preserving functors that preserve the structure on the nose.
  • The category of models of $\mathbf{ThCat}$ in $\mathbf{Set}$ is equivalent to the category of small categories and morphisms, which, unlike the category of models, includes functors that don’t preserve things on the nose.
  • $\mathbf{ThCat}$ is an example of the theory of an FL sketch. Chapter 4 of GBLS describes this idea in detail. The theory has the same models as the sketch.
  • The sketch generating $\mathbf{ThCat}$ is defined in detail in section 7.2 of GBLS.

Some objects and arrows of $\mathbf{ThCat}$

I will make use of the following objects and arrows that occur in $\mathbf{ThCat}.$ A formal thing is a construction in $\mathbf{ThCat}$ that becomes an actual thing in a model. So for example a model $\mathfrak{C}$ of $\mathbf{ThCat}$ in $\mathbf{Set}$ is an actual (small) category, and $\mathfrak{C}(\mathsf{ar_2})$ is the set of all composable pairs of arrows in the category $\mathfrak{C}$.

  • $\mathsf{ob}$, the formal set of objects.
  • $\mathsf{ar}$, the formal set of arrows.
  • $\mathsf{ar}_2$, the formal set of composable pairs of arrows.
  • $\mathsf{ar}_3$, the formal set of composable triples of arrows.
  • $\mathsf{unit} : \mathsf{ob}\to \mathsf{ar}$ that formally picks out the identity arrow of an object.
  • $\mathsf{dom},\mathsf{cod} : \mathsf{ar}\to \mathsf{ob}$ that formally pick out the domain and codomain of an arrow.
  • $\mathsf{comp} : \mathsf{ar}_2\to \mathsf{ar}$ that picks out the composite of a composable pair.
  • $\mathsf{lfac}, \mathsf{rfac} :\mathsf{ar}_2\to \mathsf{ar}$ that pick out the left and right factors in a composable pair.
  • $\mathsf{lfac}, \mathsf{mfac},\mathsf{rfac} :\mathsf{ar}_3 \to\mathsf{ar}$ that pick out the left, middle and right factors in a composable triple of arrows.
  • $\mathsf{lass}, \mathsf{rass} : \mathsf{ar}_3 \to \mathsf{ar}_2$: $\mathsf{lass}$ formally takes $\langle{h,g,f}\rangle$ to $\langle{hg,f}\rangle$ and $\mathsf{rass}$ takes it to $\langle{h,gf}\rangle$.

$\mathsf{ob}$, $\mathsf{ar}$, $\mathsf{unit}$, $\mathsf{dom}$, $\mathsf{cod}$ and $\mathsf{comp}$ are given primitives and the others are defined as limits of finite diagrams composed of those objects. This is spelled out in Chapter 7.2 of GBLS. The definition of $\mathbf{ThCat}$ also requires certain diagrams to be commutative. They are all provided in GBLS; the one enforcing associativity is shown later in this post.

Color coding

I will use color coding to separate syntax from semantics.

  • Syntax consists of constructions in $\mathbf{ThCat}.$ The description will always be a commutative diagram in black, with annotations as explained later.
  • The limit of the description will be an object in $\mathbf{ThCat}$ (the form) whose value in a model $\mathfrak{C}$ will be shown in green, because being an element of the value of a model makes it semantics.
  • When a limit cone is defined, the projections (which are arrows in $\mathbf{ThCat}$) will be shown in blue.

Descriptions

In graph-based logic, a type of construction that can be made in a category has a description, which (in the case of our Theorem) is a finite diagram in $\mathbf{ThCat}$. The value of the limit of the description in a model $\mathfrak{C}$ is the set of all instances of that type of construction in $\mathfrak{C}$.

The Subject Diagram

  • This diagram is the subject matter of the Theorem. It is not assumed to be commutative.
  • As in most diagrams in category theory texts, the labels in this diagram are variables, so the diagram is implicitly universally quantified. The Subject Diagram is a generic diagram of its shape.
  • “Any diagram of its shape” includes diagrams in which some of the nodes may represent the same object. An extreme example is the graph in which every node is an object $\mathsf{E}$ and every arrow is its identity arrow. The diagram below is nevertheless an example of the Subject Diagram:

  • Shapes of diagrams are defined properly in Section 2.3 of
    GBLS and in Section 4.1 of Category Theory for Computing Science.

The description of the Subject Diagram

Diagram SDD below shows the Subject Diagram as the limit of its description. The description is the black diagram.


Diagram SDD

Definition of $\mathsf{ar}_2$

The object $\mathsf{ar}_2$ of composable pairs of arrows is defined as a pullback:

In the usual categorical notation this would be shown as

This makes use of the fact that the unnamed blue arrow is induced by the other two projection arrows. In the rest of the post, projection arrows that are induced are normally omitted.

An enrichment of the description

Because $\mathsf{ar}_2$ is defined as a pullback, we can enrich the description of Diagram SDD by adjoining two pullbacks as shown below. This is Diagram 8.10 in GBLS. The enriched diagram has the same limit as the description of Diagram SDD.

Enriched Diagram SDD

Note that the projections from the limit to the two occurrences of $\mathsf{ar}_2$ induce all the other projections. This follows by diagram chasing; remember that the description must be a commutative diagram.

Make the triangles commute

To make the triangles commute, we add two comp arrows to the enriched diagram as shown below. These two arrows are not induced by the description; they are therefore additions to the description — they describe a more restrictive (green) diagram with commutative triangles and so are shown in black.

Diagram TC: The triangles commute

The left comp makes $xf=h$ and the right comp makes $kx=g$.

The outside square commutes

Now we enrich Diagram TC with four objects, <comp,id>, <id,comp> and three comp arrows as shown in bolder black. These objects and arrows already exist in $\mathbf{ThCat}$ and therefore do not change the limit, which must be the same as the limit of Diagram TC.

The outside square commutes

The diagram in bold black is exactly the commutative diagram that requires associativity for these particular objects and arrows, which immediately implies that $gf=kh$, as the Theorem requires.

By the definitions of $\mathsf{ar_2}$ and $\mathsf{ar_3}$, the part of the description in bold black induces the rest of the diagram. Omitting the rest of the diagram would make $\mathsf{ar_2}$ and $\mathsf{ar_3}$ modules in the sense of GBLS, Chapter 7.4. Modules would be vital to deal with proofs more complicated than the one given here.

References

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Thinking about thought

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Modules of the brain

Cognitive neuroscientists have taken the point of view that concepts, memories, words, and so on are represented in the brain by physical systems: perhaps they are individual neurons, or systems of structures, or even waves of discharges. In my previous writing I have referred to these as modules, and I will do that here. Each module is connected to many other modules that encode various properties of the concept, thoughts and memories that occur when you think of that concept (in other words stimulate the module), and so on.

How these modules implement the way we think and perceive the world is not well understood and forms a major research task of cognitive neuroscience. The fact that they are implemented in physical systems in the brain gives us a new way of thinking about thought and perception.

Examples

The grandmother module

There is a module in your brain representing the concept of grandmother. It is likely to be connected to other modules representing your actual grandmothers if you have any memory of them. These modules are connected to many others — memories (if you knew them), other relatives related to them, incidents in their lives that you were told about, and so on. Even if you don’t have any memory of them, you have a module representing the fact that you don’t have any memory of them, and maybe modules explaining why you don’t.

Each different aspect related to “grandmother” belongs to a separate module somehow connected to the grandmother module. That may be hard to believe, but the human brain has over eighty billion neurons.

A particular module connected with math

There is a module in your brain connected with the number $42$. That module has many connections to things you know about it, such as its factorization, the fact that it is an integer, and so on. The module may also have connections to a module concerning the attitude that $42$ is the Answer. If it does, that module may have a connection with the module representing Douglas Adams. He was physically outside your body, but is the number $42$ outside your body?

That has a decidedly complicated answer. The number $42$ exists in a network of brains which communicate with each other and share some ideas about properties of $42$. So it exists socially. This social existence occasionally changes your knowledge of the properties of $42$ and in particular may make you realize that you were wrong about some of its aspects. (Perhaps you once thought it was $7\times 8$.)

This example suggests how I have been using the module idea to explain how we think about math.

A new metaphor for understanding thinking

I am proposing to use the idea of module as a metaphor for thinking about thinking. I believe that it clarifies a lot of the confusion people have about the relation between thinking and the real world. In particular it clarifies why we think of mathematical objects as if they were real-world objects (see Modules and math below.)

I am explicitly proposing this metaphor as a successor to previous metaphors drawn from science to explain things. For example when machines became useful in the 18th century many naturalists used metaphors such as the Universe is a Machine or the Body is a Machine as a way of understanding the world. In the 20th century we fell heavily for the metaphor that the Mind Is A Computer (or Program). Both the 18th century and the 20th century metaphors (in my opinion) improved our understanding of things, even though they both fell short in many ways.

In no way am I claiming that the ways of thinking I am pushing have anything but a rough resemblance to current neuroscientists’ thinking. Even so, further discoveries in neuroscience may give us even more insight into thinking that they do now. Unless at some point something goes awry and we have to, ahem, think differently again.

For thousands of years, new scientific theories have been giving us new metaphors for thinking about life, the universe and everything. I am saying here is a new apple on the tree of knowledge; let’s eat it.

The rest of this post elaborates my proposed metaphor. Like any metaphor, it gets some things right and some wrong, and my explanations of how it works are no doubt full of errors and dubious ideas. Nevertheless, I think it is worth thinking about thought using these ideas with the usual correction process that happens in society with new metaphors.

Our theory of the world

We don’t have any direct perception of the “real world”; we have only the sensations we get from those parts of our body which sense things in the world. These sensations are organized by our brain into a theory of the world.

  • The theory of the world says that the world is “out there” and that our sensory units give us information about it. We are directly aware of our experiences because they are a function of our brain. That the experiences (many of them) originate from outside our body is a very plausible theory generated by our brain on the bases of these experience.
  • The theory is generated by our brain in a way that we cannot observe and is out of our control (mostly). We see a table and we know we can see in in daytime but not when it is dark and we can bump into it, which causes experiences to occur via our touch and sound facilities. But the concept of “table” and the fact that we decide something is or is not a table takes place in our brain, not “out there”.
  • We do make some conscious amendments to the theory. For example, we “know” the sky is not a blue shell around our world, although it looks like it. That we think of the apparent blue surface as an artifact of our vision processing comes about through conscious reasoning. But most of how we understand the world comes about subconsciously.
  • Our brain (and the rest of our body) does an enormous amount of processing to create the view of the world that we have. Visual perception requires a huge amount of processing in our brain and the other sensory methods we use also undergo a lot of processing, but not as much as vision.
  • The theory of the world organizes a lot of what we experience as interaction with physical objects. We perceive physical objects as having properties such as persistence, changing with time, and so on. Our brains create the concept of physical object and the properties of persistence, changing, and particular properties an individual object might have.
  • We think of the Mississippi River as an object that is many years old even though none of its current molecules are the same as were in the river a decade ago. How is it one thing when its substance is constantly changing? This is a famous and ancient conundrum which becomes a non-problem if you realize that the “object” is created inside your brain and imposed by your thinking on your understanding of the world.
  • The notion that semantics is a connection between our brain and the outside world has also become a philosophical conundrum that vanishes if we understand that the connection with the outside world exists entirely inside our theory, which is entirely within our brain.

Society

Our brain also has a theory of society We are immersed in a world of people, that we have close connections with some of them and more distant connections with many other via speech, stories, reading and various kinds of long-distance communications.

  • We associate with individual people, in our family and with our friend. The communication is not just through speech: it involves vision heavily (seeing what The Other is thinking) and probably through pheromones, among other channels. For one perspective on vision, see The vision revolution, by Mark Changizi. (Review)
  • We consciously and unconsciously absorb ideas and attitudes (cultural and otherwise) from the people around us, especially including the adults and children we grow up with. In this way we are heavily embedded in the social world, which creates our point of view and attitudes by our observation and experience and presumably via memes. An example is the widespread recent changes in attitudes in the USA concerning gay marriage.
  • The theory of society seems to me to be a mechanism in our brain that is separate from our theory of the physical world, but which interacts with it. But it may be that it is better to regard the two theories as modules in one big theory.

Modules and math

The module associated with a math object is connected to many other modules, some of which have nothing to do with math.

  • For example, they may have have connections to our sensory organs. We may get a physical feeling that the parabola $y=x^2$ is going “up” as $x$ “moves to the right”. The mirror neurons in our brain that “feel” this are connected to our “parabola $y=x^2$” module. (See Constructivism and Platonism and the posts it links to.)
  • I tend to think of math objects as “things”. Every time I investigate the number $111$, it turns out to be $3\times37$. Every time I investigate the alternating group on $6$ letters it is simple. If I prove a new theorem it feels as if I have discovered the theorem. So math objects are out there and persistent.
  • If some math calculation does not give the same answer the second time I frequently find that I made a mistake. So math facts are consistent.
  • There is presumably a module that recognizes that something is “out there” when I have repeatable and consistent experiences with it. The feeling originates in a brain arranged to detect consistent behavior. The feeling is not evidence that math objects exist in some ideal space. In this way, my proposed new way of thinking about thought abolishes all the problems with Platonism.
  • If I think of two groups that are isomorphic (for example the cyclic group of order $3$ and the alternating group of rank $3$), I picture them as in two different places with a connection between the two isomorphic ones. This phenomenon is presumably connected with modules that respond to seeing physical objects and carrying with them a sense of where they are (two different places). This is a strategy my brain uses to think about objects without having to name them, using the mechanism already built in to think about two things in different places.

Acknowledgments

Many of the ideas in this post come from my previous writing, listed in the references. This post was also inspired by ideas from Chomsky, Jackendoff (particularly Chapter 9), the Scientific American article Brain cells for Grandmother by Quian Quiroga, Fried and Koch, and the papers by Ernest and Hersh.


References

Previous posts

In reverse chronological order

Abstractmath articles

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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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An Introduction to Forms

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In 2009, I wrote a sequence of posts on this blog explaining the concept of form that I introduced in [1].  I have now updated and combined them into an article [2].  The posts no longer exist on the blog. The article contains links to other papers on forms.

[1] A generalization of the concept of sketch, Theoretical Computer Science 70, 1990.

[2] An Introduction to forms.

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