While an algebraic definition of strict ∞-categories is comparatively straightforward, algebraic definitions of fully weak $\omega$-categories (aka $\infty$-categories) are difficult to work with (although some definitions exist, such as those of Batanin?, Trimble, Leinster, and Penon).
However, just like strict $\omega$-categories have a simplicial nerve – a complicial set – induced by the orientals, and just like the category of strict $\omega$-categories is equivalent to the category of complicial sets, one expects that every weak $\omega$-category naturally has a simplicial nerve and that the theory of algebraically defined weak $\omega$-categories is equivalent to the theory of the simplicial sets that arise as their nerves. In fact, one can hope that the theory is simpler in the weak case: complicial sets are simplicial sets equipped with extra structure (a stratification representing the chosen ‘strict’ composites), while in the nerve of a weak $\omega$-category this structure might be reducible to a property.
In the context of simplicial models for weak $\omega$-categories the goal is to characterize those simplicial sets (or stratified simplicial sets) which should arise as nerves of algebraically defined weak $\omega$-categories and thus provide a geometric definition of higher category generalizing the familiar simplicial description of $\omega$-groupoids as Kan complexes and of $(\omega,1)$-categories as quasi-categories to general higher categories.
In effect, the goal is to define a weak $\omega$-category to be a certain sort of (stratified) simplicial set. One could then hope to prove that these are precisely the nerves of weak $\omega$-categories defined in some other way.
To distinguish the study of weak $\omega$-categories in terms of their presumed nerves from the study of their would-be algebraic descriptions people speak of simplicial weak ∞-categories, see in particular the articles by Dominic Verity referenced below. One should just beware that in this context simplicial weak ∞-category? is not meant as simplicial object in the category of weak $\omega$-categories.
This program was originally begun by Ross Street and has been carried forward by Dominic Verity with the theory of weak complicial sets. It is expected that the (nerves of) weak $\omega$-categories will be weak complicial sets satisfying an extra “saturation” condition ensuring that “every equivalence is thin.”
Dominic Verity, Weak complicial sets, a simplicial weak omega-category theory. Part I: basic homotopy theory (arXiv)
Dominic Verity, Weak complicial sets, a simplicial weak omega-category theory. Part II: nerves of complicial Gray-categories (pdf)
Discussion on a previous version of this entry:
Mike: This term is kind of unfortunate; simplicial weak $\omega$-category could also mean a simplicial object in weak $\omega$-categories. I don’t suppose we can do anything about that?
Urs: my impression is that what Dominic Verity mainly wants to express with the term is “simplicial model for weak $\omega$-category”. Maybe we could/should use a longer phrase like that?
Mike: That would make me happier.
Urs: okay, I changed it. Let me know if this is good now.
Toby: But what about ‘globular $\omega$-category’ and things like that? Doesn't ‘simplicial $\omega$-category’ fit right into that framework? This page title sounds like an entire framework for defining $\omega$-category rather than a single $\omega$-category simplicially defined.
Urs: i am open to suggestions – but notice that it does indeed seem to me that Dominic Verity wants to express “an entire framework for defining $\omega$-category”, namely the framework where one skips over the attempt to define $\omega$-categories and instead tries to find a characterization of what should be their nerves.
Toby: OK, that fits in with most of what's written here, but not the beginning
Simplicial models for weak $\omega$-categories – sometimes called simplicial weak ∞-categories – are [] Maybe that was just poorly written, but it threw me off. Should it be A simplicial model for weak $\omega$-categories – which are then sometimes called simplicial weak ∞-categories? – is [] or even A simplicial model for weak $\omega$-categories is [] and only later mention simplicial weak ∞-categories??
Mike: You’re right that ‘simplicial $\omega$-category’ it fits into ‘globular $\omega$-category’ and ‘opetopic $\omega$-category’ and so on. It seems more problematic in this case, though, since simplicial objects of random categories are a good deal more prevalent than globular ones and opetopic ones. But perhaps I should just live with it.
Urs: I have now expanded the entry text on this point, trying to make very clear to the reader what’s going on here.
Toby: Thanks, that's much clearer. And if Verity's definition is at weak complicial set, then we may not really need anything at simplicial weak ∞-category?, so no need to offend Mike's sensibilities (^_^) either.