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?
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For some precise definitions and results, see this paper by Eugenia Cheng. 


A concise coinductive definition can be found, for the case of strict ∞categories, in the paper "A folk model structure on omegacat", arXiv. This can be unraveled in order to become equivalent to Eugenia's more explicit version. 

