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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?

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    $\begingroup$ Have you tried turning it off and on... I mean, have you checked Tom Leinster's arxiv.org/abs/math.CT/0107188? $\endgroup$ – David Roberts Apr 22 '14 at 4:32
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    $\begingroup$ Did you have a look at Simpson's [Homotopy theory of higher categories]? He defines higher Segal categories by iterated weak enrichment, so it might be a good fit for your problem. $\endgroup$ – Zhen Lin Apr 22 '14 at 7:13
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    $\begingroup$ There is a certain apparent similarity between completeness and skeletality, but it's misleading to call them similar because of what Karol said. Completeness is actually a very natural condition. Fortunately, there is a universal way to "complete" an incomplete category (which also distinguishes it from skeletality). $\endgroup$ – Mike Shulman Apr 22 '14 at 21:34
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    $\begingroup$ @DavidRoberts I have, of course, read Leinster's foundational paper, but not recently — I'll take another look at it. It was written at a time when the theory of $(\infty,1)$-categories had not yet been settled, and long before comparison results like Barwich–Schommer-Pries were available. Eventually, I would like to apply some known results from the theory of $(\infty,n)$-categories (established via, say, $n$-fold complete Segal spaces) to my gadgetry. So when I ask for "axioms to check", I secretly mean "a model satisfying B–SP's axioms", so that I can transfer the known results. $\endgroup$ – Theo Johnson-Freyd Apr 23 '14 at 1:18
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    $\begingroup$ I suppose the article arxiv.org/abs/1312.3178 contains one approach to your (1)-(5), though that model also involves completion. $\endgroup$ – Rune Haugseng Apr 23 '14 at 6:04
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First, as Rune pointed out in the comments, his paper with David Gepner gives a very general approach to your wish list. However to make it so general that it applies to arbitrary monoidal $(\infty,1)$-categories (whatever those are, ) means that the machinery is arguably very complicated. In particular, it probably doesn't satisfy your (4). It is not "effortless".

However from what you have said I think you should have a serious look as Simpon's model of Segal n-categories, which was also mentioned by Zhen Lin. This is part of a general approach/construction for defining weakly enriched categories, and from what you have said I think there is a very good chance it applies in your case.

I won't give the formal definition. You can look that up in Simpson's book or elsewhere, but the basic flavor is that a weakly enriched $\mathcal{V}$-category $C$ will consist of the following:

  • A set of object $x,y,z, \dots$
  • $C(x,y)$ objects of $\mathcal{V}$ for each pair of objects,
  • $C(x,y,z)$ objects of $\mathcal{V}$ for each triple of objects.
  • $C(w,x,y,z)$ objects of $\mathcal{V}$ for each quadruple of objects.
  • etc.

Plus there are maps which are like the maps of a simplicial set. So for example there are degeneracy/unit maps like $1 \to C(x,x)$ and $C(x,y) \to C(x,x,y)$ which add identities, and there are face/restriction maps like $C(w,x,y,z) \to C(x,y,z)$ or $C(w,x,y,z) \to C(w,y,z)$.

Some derivative maps from these are required to be equivalences (the "Segal maps"). Specifically the maps $$ C(w,x,y,z) \to C(w,x) \times C(x,y) \times C(y,z) $$ and its cousins.

Let me describe the conceptual philosophy in the case that we are building the theory of $(\infty,1)$-categories from $\mathcal{V} = spaces$. Conceptually the $C(x,y)$ are the hom spaces, which parametrize the space of arrows from $x$ to $y$. The $C(x,y,z)$ are spaces which parametrize: (1) a pair of composable arrows, (2) the possible composites of those arrows (3) data witnessing the possible composite as a composite. The higher spaces are similar.

As part of the structure there are maps

$$ C(x,y) \times C(y,z) \leftarrow C(x,y,z) \to C(x,z) $$

The left-hand arrow is required to be a homotopy equivalence of spaces (this is called the Segal condition and there are similar higher conditions). By choosing homotopy inverses to this first map (and data witnessing that it is a homotopy inverse) we get a contractible space of compositions. The Segal category builds in the higher coherence data automatically, without specifying it explicitly.

Now the general case of $\mathcal{V}$ is similar. You need enough structure to mimic this definition. Simpson uses Cartesian model categories, but you can get away with a lot less. A version based on relative categories with products appears in section 5 of this paper. This is pretty minimal. $\mathcal{V}$ must be a category with finite products and a compatible notion of weak equivalence (plus a little bit more extra structure).

In practice if you have a contractible space of compositions already, you can probably construct a Segal n-category. You will simply build that contractible space into the object $C(x,y,z)$. You will have to confront the higher spaces though. Hopefully your construction is natural enough that this isn't an issue.

By the way, when you plug in $\mathcal{V}= Set$, then you essentially get the Simpson/Tamsamani version of weak n-category. The comparison of this with bicategories is described in several places, for example in Leinster's survey. Let's spell it out a bit. We begin with 0-categories which are just sets. An equivalence of sets is a bijection. This means that the Segal maps become isomorphisms. It is a standard exercise that this reproduces the usual notion of category.

Next, on the second iteration, a weak 2-category has a set of object and then for each pair a hom category. There are also categories $C(x,y,z)$, etc. and $C(x,y,z) \to C(x,y) \times C(y,z)$ is an equivalence. To get a bicategory from this we need a composition functor, and to get that we must make choices of inverse equivalences to these maps. By making similar choices for the diagram we get for quadruples of objects, we get an associator, and by looking at the diagram for quintuples, we see that the associator satisfies the pentagon equation.

So you can get a bicategory, but the correspondence is not quite 1-1. However it is up to equivalence (i.e., the bicategories resulting from different choices will be equivalent as bicategories). This is discussed in more detail in this paper by Lack and Paoli.

As you can see from this mental exercise, for the $(n,n)$-categories there isn't really a space floating around, per se. Of course secretly there is a space, but it is hidden. This notion is much more like the classical notion of category and bicategory. Hope this helps!

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