§ Colimits examples with small diagram categories
- Given a colimit, compute the value as taking the union of all objects, and imposing the relation for all arrows and all .
- A colimit of the form is computed by taking and then imposing the relation . This is entirely useless.
- A colimit of the form is computed by taking and then imposing the relation as well as . Thus, this effectively imposes . If we choose , then we get . So we can create quotients by taking the colimit of an arrow with the identity.
- A colimit of the form will construct and impose the relations and . Thus, we take and we glue and along via . Imagine gluing the upper and lower hemispheres of a sphere by a great circle.