The interactive examples in this post require installing Wolfram CDF player, which is free and works on most desktop computers using Firefox, Safari and Internet Explorer, but not Chrome. The source code is the Mathematica Notebook algebra1.nb, which is available for free use under a Creative Commons Attribution-ShareAlike 2.5 License. The notebook can be read by CDF Player if you cannot make the embedded versions in this post work.

In Graph-Based Logic and Sketches, Atish Bagchi and I needed to construct a lot of cones based on fairly complicated diagrams. We generally show the base diagram and left the reader to imagine the cone. This post is an experiment in presenting such a diagram in 3D, with its cone and other constructions based on it.

To understand this post, you need a basic understanding of categories, functors and limit cones (see References).

The notebook and CDF files that generate this display may be downloaded from here:

• CompositeDiagram.cdf
• CompositeDiagramSource.nb

These files may be used and modified as you wish according to the Creative Commons rule listed under “Permissions” (at the top of the window).

The sketch for categories: composition

A finite-limit sketch (FL sketch) is a category with finite limits given by specifying certain  nodes and arrows, commutative diagrams using these nodes and arrows, and limit cones based on diagrams using the given nodes and arrows.  A model of an FL sketch is a finite-limit-preserving functor from the FL sketch into some category .  Detailed descriptions of FL-sketches are  in References [1], [2] and [3] (below).

Categories themselves may be sketched by FL-sketches. Here I will present the part of the sketch that constructs (in a model) the object of composites of two arrows.  This is the specification for composite:

1. The composite of two arrows and is defined if and only if .
2. The composite is denoted by .
3. The domain of is and the codomain is .

We start with a diagram in the FL sketch for categories that gives the data corresponding to two arrows that may be composed.  This diagram involves nodes ob and ar, which in a model become the object of objects and the object of arrows of the category object in .  (Suppose is the category of sets; then the model is simply a small category.  The node ob goes to the set of objects of the small category and ar goes to the set of arrows.)  The arrows labeled dom and cod take (in a model) an arrow to its domain and codomain respectively. Here is the diagram:

You can move the diagram around in three dimensions to see it from different perspectives. (Of course it isn’t really in three dimensions. Your eyes-to-brain module reconstructs the illusion of three dimensions when you twirl the diagram around.)

Note that this is a diagram, not a directed graph (digraph). (In the paper, Atish and I, like most category theorists, say “graph” instead of “digraph”.) It has an underlying digraph (see Chapter 2 of Graph-Based Logic and Sketches), but the labeling of several different nodes of the underlying digraph by the name of the same node of the sketch is meaningful. 

Here, the key fact is that in the diagram there are two arrows, one labeled dom and the other cod, to the same node labeled ob, and two other arrows to two different nodes labeled ob. 

Now click c1.

This shows a cone over the diagram.  One of the nodes in the sketch must be cp (in other words given beforehand; that is, we are specifying not only that the blue stuff is a limit cone but that the limit is the node cp.)   In a model, this cone must become a limit cone.  It follows from the properties of limits that the elements of cp in the model in Sets are pairs of arrows with the property that one has a codomain that is the same as the domain of the other.  The label “cp” stands for “compatible pairs”.

Now click c2.

The green stuff is a diagram showing two arrows from the node labeled ar to the left and right nodes labeled ob in the original black diagram.  This is not a cone; it is just a diagram.  In a model, any arrow in the vertex must have domain the same as the domain of one of the arrows in the compatible pair, and codomain the same as the codomain of the other arrow of the pair.  Thus in the model, an arrow living in the set labeled with “ar” in green must satisfy requirement 3 in the specification for composition given above.

Note that the requirement that the green diagram be commutative in a model is vacuous, so it doesn’t matter whether we specify it specifically as a diagram in the sketch or not.

Now click c3.

The arrow labeled comp must be specified as an arrow in the sketch.  We want its value to be the composite of an element of cp in a model, in other words a compatible pair of arrows.  At this point that will not necessarily be true.  But all can be saved:

Now click c4.

We must specify that the diagram given by the thick arrows must be a diagram of the sketch.  The fact that it must become commutative in a model means exactly that the red arrow comp from cp to ar takes a compatible pair to an arrow that satisfies requirements 1–3 of the specification of composite shown above.

References

1. Peter T. Johnstone, Sketches of an Elephant: A Topos Theory Compendium, Volume 2 (Oxford Logic Guides 44), by Oxford University Press, ISBN 978-0198524960.
2. Michael Barr and Charles Wells, Category theory for computing science (1999).    (This is the easiest to start with but it doesn’t get very far.)
3. Michael Barr and Charles Wells, Toposes, Triples and Theories (2005).  Reprints in Theory and Applications of Categories 1.