Is '=' or 'equals to' a relationship between Mathematical objects. The Wikipedia definiton of equality gives it as a 'relationship between two expressions'
This confuses me as when we define mathematical expressions like $2+2=4$ it makes no sense to say that '=' or 'equals to' relates the two expressions as it would mean that '$2+2$' and '$4$' are representing themselves as expressions as opposed to denoting or naming the objects.
I think in mathematical contexts when we use expressions it is always to unambiguously name an object, for example: '$2∈N$' is meaningless if '$2$' represents itself as an expression.
I understand there is a relation between the expressions whose value are the same but is defining this relationship as 'equality' and saying it is denoted by '=' correct? It seems that '=' should denote a relationship an object has with itself. If the expressions act as names, then placing the symbol between the names means we discuss the objects, not their names.
Is there an explanation of  whether 'equality' is at object or expression level?
 A: The $=$ sign is overloaded so we need to be careful in which sense we're using it. The first is as a definition when we give a name to some other thing. A familiar example would be $y=mx+b$ which is us defining $y$ to be $mx+b$. Sometimes the notation $:=$ will be used for definitions like this and I prefer it.
The second is as a statement. I might make the claim the $2+2=4$ and want to evaluate the truth of the statement as true or false. I could also make the statement that $2+2=5$ and want to see if that's true or false as well. This is often used in programming languages to made decisions about branching or looping and they will typically have some notational implementation to distinguish the case when it's being used as a definition and when it's being used as a logical statement such as writing $2+2==4$ instead.
Finally, and most importantly, is as an equivalence relation. To understand exactly what they are we first nee to know what a relation is but thankfully it's pretty simple. A relation $R$ on a set $A$ is a subset $ R \subset A \times A$. We say $a \in A$ is related to $b \in A$ when $(a,b) \in R$. This is a very general construction and can be used to create important relations like orderings. In a practical sense evaluating a statement about relationships like $2+2=4$ can sometimes be done by seeing if the element $(2+2,4)$ is in $R$. Lets look at what makes equivalence relations special.
An equivalence relation has three properties, which are reflexivity, symmetry and transitivity. Reflexivity means that $(a,a) \in R$ for all $A \in A$, which just means $a=a$. Symmetry means if $(a,b) \in R$ then $(b,a) \in R$ or if $a=b$ then $b=a$. Finally is transitivity. This means if $(a,b) \in R$ and $(b,c) \in R$ then $(a,c) \in R$. Alternatively if $a=b$ and $b=c$ then $a=c$.
This relation has another structure which is given by the fundamental theorem of equivalence relations, which is as a partition of a set. A partition of a set $A$ is for some indexing set $I$ we for sets $A_i \subset A$ for $i \in I$ $\cup_{i \in I}A_i = A$ and $A_i \cap A_k = \emptyset$ for all $i,k \in I$ with $i \neq k$. Basically you split the set up into bins labeled $A_i$. I don't think it's an understatement to say this is among the most important theorem in modern mathematics so it's worth knowing. It has applications everywhere.
