I'm reading now J. Lurie's "Higher Topos Theory" and have faced some understanding problems with the definition of the $(n,k)$-category. following definition on page 5 the $(n,k)$-category is a $n$-category - i.e. a "$(n-1)$-enriched category - such that as $m$-morphisms with $m > k$ are invertible.
I have some understanding problems with the usage of vocabulary "equivalence of morphisms", "invertibility" and properties of morphisms in "weak sense".
Questions:
1. following https://ncatlab.org/nlab/show/equivalence "equivalence often means a morphism which is invertible in a maximally weak sense (that is, up to all higher equivalences)
What does that mean?
If I try to draw analogy to topology then I would say that intuitively a $m$-morphism $f_m$ (between $(m-1)$-morphisms $A$ and $B$) is invertible if there exist another $m$-morphism $g_m$ and $(m+1)$-morphisms $H_m, K_m$ (which generalises the classical homotopy known from topology) such that $H_m$ maps composition $f_m \circ g_m$ to $id_B$ and respectively $K_m$ maps composition $g_m \circ f_m$ to $id_A$. Is that all?
The point is that must in this game $H_m, K_m$ also be invertible, the $(m+2)$ morphisms between them and so on? what tells me that a certain property of $m$-morphisms is given in "weak sense"?
2. If we forget now the $k$ and think only about $n$-categories then for any two objects $X, Y$ (i.e. $0$-morphisms) the category $Hom(X,Y)$ is an $(n-1)$-category. This requires especially that the $Hom$-sets have to respect the "associativity" and their $Hom$-sets another (more complicated) relations and so on.
Let turn back to the easiest identity: associativity.The question is in which sense has the "associativity" to be fulfilled? Let again $X,Y,Z,W$ be $0$-morphisms and $f_1 \in Hom(X,Y), ..., f_2 \in Hom(Y, Z), f_3= "$ three compatible $1$-morphisms then we expect that there exist some $2$-morphism $A$ which reflects in a meaningful (possible weak) sense the generalisation of associativity:
Therefore $A$ maps $(f_1 \circ f_2) \circ f_3$ to $f_1 \circ (f_2 \circ f_3)$. and the question is which (weak) properties must this $A$ have? of course "being idenetity" might be too strong. invertible? in what sense? maximally weak sense? and here the circle closes: namely what concretely is maximally weak sense?
In an $n$-category, a 1-morphism $f:x\to y$ is an equivalence in the maximally weak sense if there exists $g:y\to x$ such that $gf$ and $\mathrm{id}_x$ are equivalent (in the maximally weak sense) in the $n-1$-category $\mathrm{Hom}(x,x)$ while $fg$ and $\mathrm{id}_y$ are equivalent in $\mathrm{Hom}(y,y)$. This gives an inductive definition with perhaps the most intuitive base case being given at $n=1$, where an equivalence in a $1$-category is simply an isomorphism. It is also possible to give a coinductive definition of equivalence in an $\infty$-category, where the inductive definition would never terminate; basically one ends up with an infinite binary tree of higher and higher morphisms relating a composite to an identity in both directions.
The axioms for an $n$-category in the weak sense are rather difficult to write down explicitly, to the point that it's never been done for $n>4$. This is a significant motivation for quasicategories, the topic of Lurie's book: they package together a highly complex family of axioms for $(\infty,1)$-categories, which indeed we don't even know how to express explicitly and algebraically, into the less explicit but very usable horn-lifting conditions. That said, associativity isn't too hard to understand, given the previous paragraph. Indeed, $A:f(gh)\to (fg)h$ is simply an equivalence, in the weak sense, in the hom $n-1$-category. The problem is coming up with all of the various equivalences necessary to fully axiomatize a weak $n$-category.