# Maths - Oidification

## Oidification

Oidification allows us to start with a single object and one (or more) arrows back to itself and generalise this to multiple objects. This new structure does not need to have arrows between every pair of objects but composition always exists and every object has an identity arrow.

Here the words 'object' and 'arrow' are used in the category theory sense as described on the page here.

Here are some examples:

Single Object Multiple Objects

Setoid

This imposes an external equivalence relation and internalises it.

The single object needs to have the reflexive relation, that is, be equal to itself.

An example would be a set with a set with a fixed number of elements.

Since the arrows are reflexive, symmetric and transitive this gives an equivalence relation.

Groupoid

This can be thought of as a weakening of a setiod. Instead of equivalences we have isomorphisms, that is the permutations show how objects are equal in multiple ways.

Category

Just to confuse things the naming conventions have changed here. Monoid has an 'oid' but this is the single object case. A Monoidoid is a category.