Weak $n$-Category I'm trying to obtain an intuition for weak n-categories based on explanations from: 
https://en.wikipedia.org/wiki/Higher_category_theory#Weak_higher_categories
https://ncatlab.org/nlab/show/n-category
https://ncatlab.org/nlab/show/equivalence
Remark: I'm sure that there are some other approaches to this topic but from my point of view the links above might provide a nice intuition (modulo following questions).
My attempt:
So if I visualize a $k$-morphism as a $k$-cell then can I interpret a weak n-category $W$ in following way:
When we consider the  "$k$-layer" $W_k$ (=so the $k$-morphisms of $W$) with $k \le n$ of $W$ then the "weak associativity" means just that that for compatible $a,b,c \in W_k$ there exist a $k+1$ morphism $d \in W_{k+1}$(in geometrical language a $k+1$-cell ) between $(a * b)*c$ and $a*(b*c)$, right? 
Futhermore two $k$-morphisms $a,b \in W_k$ are identified iff there exist $k+1$-morphism $d \in W_{k+1}$ between $a$ and $b$.
On the other hand for $k >n$ all $k$-morphisms are identical (in the sense that the exist a $k+1$-morphism between them. So a morphism from "next level or layer".
Is this a correct way to understand the concept of equivalence given in https://ncatlab.org/nlab/show/equivalence
Now the actual QUESTION is if what I explained above is a correct intuition behind the weak n-categories and the concept  of equivalence or is my attempt to describe my intuition misleading the essentiality of the topic?
 A: Well, the $n$th layer is a standard category. 
We can even omit layers above $n$, or say they all contain only the identity cells.
What you wrote rather fits for (weak) $n$-groupoids (such as the fundamental groupoids of a topological space), where all cells are invertible.
In general, two $(n-1)$-cells $\alpha$ and $\beta$ are equivalent ($\alpha\simeq\beta$) in an $n$-category, if there is an invertible $n$-cell $\varphi:\alpha\to\beta$, i.e. there's also a $\psi:\beta\to\alpha$ such that $\psi\varphi=1_\alpha$ and $\varphi\psi=1_\beta$.
If $\alpha, \beta$ are $(n-2)$-cells, then we require $\simeq$ in the above equations in place of $=$. 
And so on..
To see a specific example, consider the bicategory of rings and bimodules (with bimodules ${}_AM_B$ as arrows $A\to B$, tensor product as composition, and bimodule morphisms as 2-cells).
Here two bimodules are equivalent iff they are isomorphic, and two rings are equivalent iff there are bimodules between them, which are inverses to each other w.r.t tensor product (which is called they are Morita-equivalent).
Associativity is required on each layer up to (equivalence on that layer). 
In the above example it's just $(M\otimes N)\otimes P\cong M\otimes (N\otimes P) $. Note that these entities are indeed not identical, for the same reason as $(A\times B)\times C\ne A\times(B\times C)$ for sets, but there's a natural isomorphism between them. 
However, this condition for associativity in itself proved to be not sufficient, and further coherence conditions had to be posed, see e.g. the definition of a bicategory. 
In weak higher categories these coherence issues are highly nontrivial. 
