nLab Understanding colimits in Set

Idea

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 $F:J\to Set$ is a cocone $T$ over $J$ such that, given any cocone $T'$, there is a unique cone function? from $T$ to $T'$.

Initial Object

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 $\varnothing$ is the initial object in $Set$.

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” $\bullet$ such that for an other set $C$, there is a unique function

$$f:\bullet\to C.$$

The empty set fills the bill because we have the empty function from $\varnothing$ to $C$ for all $C$.

Therefore the empty set is an initial object in $Set$.

Binary Coproducts

Binary coproducts correspond to disjoint unions in $Set$.

Under Construction

Arbitrary Small Coproducts

arbitrary (but small) coproducts

Coequalizers

coequalizers

Pushouts

pushouts

Cofibred Products

cofibred coproducts