4 - Limits and colimits in general

Summary

In Chapter 2, Sections 2.3 to 2.7 we looked at some simple examples of limits and colimits. These are brought together in Table 2.1 which is repeated here as Table 4.1. In this chapter we generalize the idea.

Before we begin the details it is useful to outline the five steps we go through together with the associated notions for each step. After that we look at each step in more detail.

Template

This is the shape ∇ that a particular kind of diagram can have. It is a picture consisting of nodes (blobs) and edges (arrows). The central column of Table 4.1 lists a few of the simpler templates. Technically, a template is often a directed graph or more generally a category.

Diagram

This is an instantiation of a particular template ∇ in a category C. Each node of ∇ is instantiated with an object of C, and each edge is instantiated with an arrow of C. There are some obvious source and target restrictions that must be met, and the diagram may require that some cells commute. Thus we sometimes use a category as a template.

Posed problem

Each diagram in a category C poses two problems, the left (blunt end) problem and the right (sharp end) problem. We never actually say what the problem is (which is perhaps the reason why it is rarely mentioned) but we do say what a solution is.