# Maths - Category Theory - Comparing Objects

When comparing objects it is often more interesting when two sets are not identical but preserve some form of structure when mapping between them. relative strength type and category notation description
tightest Equality C and D are the same in these terms
Isomorphism
1d = GF
FG = 1c
There must be an arrow and inverse between the objects and when composed these give the identity map for every element.
Equivalence
1d GF
FG 1c For equivalence there is a arrow in both directions between the two objects. It is not necessary that GF and FG are the identity elements but they must be isomorphic to the identity elements.

loosest unit:
μ : 1c =>GF
co-unit:
ξ : FG => 1d

 triangle identities: as 2-cells:  For adjunctions there is a arrow in both directions between the two objects. It is not necessary that GF and FG are the identity elements but only that they have natural transformations to the identity elements.

FG is unit (does not change object - injective followed by surjective) but GF does change object (surjective followed by injective). Like equivalence but in one direction.

Alternative definition: there is an isomorphism:
φ: D(FA , X) -> C(A, GX)
for every A C and X D

## Equivalence Relations

In a less category specific approach to binary relations they may, or may not, have the following properties:

Equivalence Relation Group
x y transitive
 x y y z
x z reflexive x x symmetric
 x y y x ## Comparison of Sets

Here we investigate this comparisons with the simplest concrete categories - sets.

type of
comparison

Sets GF
Equality  Isomorphism
1d = GF
FG = 1c  Equivalence
1d GF
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