On this page, we work work through several of the key examples of colimits in the category Set. This is part of a bigger project: Understanding Constructions in Categories.
Recall that a colimit, or universal cocone, over a diagram is a cocone over such that, given any cocone , there is a unique cone function? from to .
An initial object is a universal cocone over the empty diagram. In this section, we demonstrate how this leads us to the statement:
The empty set is the initial object in .
To demonstrate, first note that a cocone over an empty diagram is just a set and a corresponding cocone function is just a function. Therefore, we are looking for a “universal set” such that for an other set , there is a unique function
The empty set fills the bill because we have the empty function from to for all .
Therefore the empty set is an initial object in .
Binary coproducts correspond to disjoint unions in .
Under Construction
arbitrary (but small) coproducts
Last revised on June 20, 2021 at 05:28:20. See the history of this page for a list of all contributions to it.