Limits, Products, and Equalizers

\[ \newcommand{\tensor}{\otimes} \newcommand{\Arr}{\mathrm{Arr}} \newcommand{\Ob}{\mathrm{Ob}} \newcommand{\cat}[1]{\mathtt{#1}} \newcommand{\RR}{\mathbb{RR}} \]

As we have previously discussed, we can construct limits of diagrams in a category \(\cat{C}\), see the post about universal_properties.

The following are extra special diagrams, because they show up very often:

Figure 1: An equalizer diagram, which is \(X \rightrightarrows^f_g Y\).

Figure 2: A product diagram, i.e. only objects \(A_i\) for \(i \in I\), where only the identity arrows exist.

We can now consider the universal cones/i.e. limits over these diagrams, which will result in:

Figure 3: Two cones \(N_1\) and \(N_2\) over the equalizer diagram from Figure 1

Figure 4: Two cones \(N_1\) and \(N_2\) over the product diagram from Figure 2

Warning

From here on, I still need to finish this post… Specifically, we wish to show

“Theorem 3.41: In any category, if products and equalizers exist, then limits according to any data type can be constructed” (Lawvere and Rosebrugh, p. 73)