I have recently been thinking about some mathematical gadgetry that should together combine into an $(\infty,n)$-category (actually, an $(n,n)$-category) for $n = 4$. I don't know what axioms I need to check in order to promote "should" into "do". My wish with this question is that someone can point me to the appropriate place in the literature where the necessary axioms are written down. Of course, if wishes were research articles....

I understand my gadgetry in terms of a set (actually, a proper class) of "objects" and, between any two objects, an "$(\infty,n-1)$-category" of morphisms. I put that in quotes because to actually have it requires that the answer to my title is affirmative up to dimension $n-1$. I have a pretty good understanding of the putative composition "functors" (but similarly, I don't know what axioms I need to check to guarantee "functoriality"), and can verify many reasonable requests (e.g. I can probably rig up a contractible space of compositions of any tuple of composable morphisms, if requested to do so).

What I don't have good understanding of are the "invertible" morphisms. This is a problem for the following reason. The only axioms for "$(\infty,n)$-category" that I know very well are the axioms of $n$-fold complete Segal space (note that the linked article is incorrect as stated in the opening paragraph; in order to be a model of higher categories, an extra "essential constancy" condition is necessary). But the axiom of "completeness" is not something that's particularly natural for my gadgetry. So I could, if pressed, package my gadgetry into an $n$-fold Segal space, but I don't expect there to be any natural way to satisfy the completeness condition. (Asking for completeness is a homotopical version of asking for your categories to be skeletal; when said this way, it is not surprising that it is unnatural.)

My real wish is that the literature would contain: (1) For any $(\infty,1)$-category $\mathcal S$ satisfying some reasonable axioms, a notion of "$(\infty,1)$-category enriched in $\mathcal S$" consisting of a set of objects and for every pair of objects an object of $\mathcal S$, and some morphisms in $\mathcal S$; (2) there is an $(\infty,1)$-category $\mathrm{Cat}(\mathcal S)$ of these, and it also satisfies said "reasonable axioms"; (3) the 1-category $\mathrm{Set}$ is an example of such $\mathcal S$, and the corresponding enriched categories are just (strict) 1-categories; (4) the definitions unpack such that $\mathrm{Cat}(\mathrm{Cat}(\mathrm{Set}))$ is automatically and effortlessly the collection of bicategories; (5) $\mathrm{Cat}^n(\mathrm{Set})$ is a model of $(n,n)$-categories, and $\mathrm{Cat}^n(\mathrm{Spaces})$ is a model of $(\infty,n)$-categories, where $\mathrm{Spaces}$ is some well-suited $(\infty,1)$-category of homotopy types. But my experience has been that the literature does not contain as many definitions and results as my wishes for it.

Thus: What axioms do I need to check? Where can I look them up? Where can I learn more?

Homotopy theory of higher categories]? He defines higher Segal categories by iterated weak enrichment, so it might be a good fit for your problem. $\endgroup$verynatural condition. Fortunately, there is a universal way to "complete" an incomplete category (which also distinguishes it from skeletality). $\endgroup$6more comments