# Property of $n$-morphism in "weak sense" in higher category theory

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.
• the example with $n=1$ confuses me a bit: ncatlab.org/nlab/show/equivalence tells me exactly that what you said: in $1$-category every equivalence (=weak invertibleness) is nothing but an isomorphism in "usual" sense. on the other hand ncatlab.org/nlab/show/n-category says that in an $n$-category all $k$-morphisms with $k >n$ are equivalences (in formal sense that the corresponding "homotopy" from layer above is again an equivalence and so on).
• I'm not sure how these two viewpoints don't violate each other. the first implies that in a $n$-category all $k$-morphisms with $k >n$ are only identities in usual sense while the explanations from ncatlab.org/nlab/show/n-category says that they are equivalences (in weak sense). isn't there something fishy?
• @katalaveino There are a few different terminologies. The usage at the nLab link is about $(\infty,n)$-categories, whereas $(n,n)$-categories proper should have no $k$-morphisms at all for $k>n$, or equivalently only identities. All this is quite imprecise so far, because there are a number of definitions which coincide perfectly or nearly in certain contexts. Sep 15, 2019 at 22:38