In every $n$-category (weak or strict) can be defined the concept of equivalence via a recursive definition:
\* an equivalence in a set ($0$-category) is just an identity;
\* for each $n \in \mathbb N$ an equivalence between two object (or $0$-cell), let say $a$ and $b$, in a $n+1$-category is just a $1$-cell $f \colon a \to b$ such that exist a $1$-cell $g \colon b \to a$ and two $2$-cells $\alpha \colon g \circ f \to 1\_a$ and $\beta \colon f \circ g \to 1\_b$ which are equivalence into the $n$-categories $\hom(g\circ f, 1\_a)$ and $\hom(f \circ g, 1\_b)$ respectively.
There is a good formal definition of equivalence also for $\infty$-category?



Equivalence in $\infty$-categories
For some precise definitions and results, see this [paper](http://www.cheng.staff.shef.ac.uk/duals/duals1103.pdf) by Eugenia Cheng. 


