nLab Understanding colimits in Set

Contents

Contents

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:JSetF:J\to Set is a cocone TT over JJ such that, given any cocone TT', there is a unique cone function? from TT to TT'.

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 SetSet.

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 CC, there is a unique function

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

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

Therefore the empty set is an initial object in SetSet.

Binary Coproducts

Binary coproducts correspond to disjoint unions in SetSet.

Under Construction

Arbitrary Small Coproducts

arbitrary (but small) coproducts

Coequalizers

coequalizers

Pushouts

pushouts

Cofibred Products

cofibred 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.