There are a number of formalisms available for presenting free strict $\omega$-categories -- Street's parity complexes, Steiner's directed complexes, computads, polygraphs,... Typically one has a certain category $\mathcal C$ of combinatorial data, a notion of "map" from $C \in \mathcal C$ to an $\infty$-category $X$, and a free functor $F: \mathcal C \to \omega Cat$ with an explicit combinatorial description of $F$. The free functor will have the universal property that strict $\omega$-functors $F(C) \to X$ are in bijection with "maps" $C \to X$ whenever $X$ is a strict $\omega$-category.

Depending on one's choice of model, there may still be a clear notion of "map" from $C \in \mathcal C$ to a weak $(\infty,\infty)$-category $Y$. I'm interested to know conditions on $C \in \mathcal C$ guaranteeing that "maps" $C \to Y$ are in bijection with morphisms $i(F(C)) \to Y$, where $i: \omega Cat \to (\infty,\infty)-Cat$ is the inclusion from strict $\omega$-categories to weak $(\infty,\infty)$-categories. Of course, this may be model-dependent. I find myself needing a result of this form in a particular setting, but I'd be interested seeing results of this kind for any choice of $\mathcal C$ and any model of weak $(\infty,\infty)$-categories -- I'd be particularly happy if the model of weak $(\infty,\infty)$-categories is nonalgebraic in nature.

Question: What is an example of

• a category $\mathcal C$ of "presentations of certain strict $\omega$-categories",

• a free functor $F: \mathcal C \to \omega Cat$ from $\mathcal C$ to strict $\omega$-categories with an explicit combinatorial description,

• a 1-category $(\infty,\infty)Cat$ which "models" the $\infty$-category of weak $(\infty,\infty)$-categories (e.g. via a model structure or whatever) with inclusion functor $i: \omega Cat \to (\infty,\infty)Cat$,

• a straightforward notion of "map" from objects $C \in \mathcal C$ to objects of $(\infty,\infty) Cat$,

• and a (not completely vacuous) condition $\Phi$ on the objects of $\mathcal C$

such that

• Objects $C \in \mathcal C$ satisfying $\Phi$ have the property that $Hom(iF(C),Y)$ is naturally isomorphic to the set of maps from $C$ to $Y$, for all (suitably fibrant, perhaps) $Y \in (\infty,\infty)Cat$?

I'm happy to see quite restrictive conditions; in fact in my case I don't need to understand much more than Gray tensor powers of the arrow category $\bullet \to \bullet$.

I suppose I'd also be happy to see examples with "$n$" replacing "$\omega$" and "$(\infty,n)$" replacing "$(\infty,\infty)$".

And let me stress that I'm not looking for some kind of fancy $\infty$-categorical freeness -- when I say $Hom(i(F(C)),Y)$ above, I mean $Hom$ in whatever 1-category is being used to model $(\infty,\infty)Cat$. Although if there are results showing something fancier, that would be interesting to hear about too.

There are a number of formalisms available for presenting free strict $\omega$-categories -- Street's parity complexes, Steiner's directed complexes, computads, polygraphs,... Typically one has a certain category $\mathcal C$ of combinatorial data, a straightforward notion of "map" from $C \in \mathcal C$ to a strict $\omega$-category $X$, and a free functor $F: \mathcal C \to \omega Cat$ with an explicit combinatorial description of $F$. The free functor will have the universal property that strict $\omega$-functors $F(C) \to X$ are in bijection with "maps" $C \to X$ whenever $X$ is a strict $\omega$-category.

Depending on one's choice of model, there may still be a clear notion of "map" from $C \in \mathcal C$ to a weak $(\infty,\infty)$-category $Y$. I'm interested to know conditions on $C \in \mathcal C$ guaranteeing that "maps" $C \to Y$ are in bijection with morphisms $i(F(C)) \to Y$, where $i: \omega Cat \to (\infty,\infty)Cat$ is the inclusion from strict $\omega$-categories to weak $(\infty,\infty)$-categories. Of course, this may be model-dependent. I find myself needing a result of this form in a particular setting, but I'd be interested seeing results of this kind for any choice of $\mathcal C$ and any model of weak $(\infty,\infty)$-categories -- I'd be particularly happy if the model of weak $(\infty,\infty)$-categories is nonalgebraic in nature.

Question: What is an example of

• a category $\mathcal C$ of "presentations of certain strict $\omega$-categories",

• a free functor $F: \mathcal C \to \omega Cat$ from $\mathcal C$ to strict $\omega$-categories with an explicit combinatorial description,

• a 1-category $(\infty,\infty)Cat$ which "models" the $\infty$-category of weak $(\infty,\infty)$-categories (e.g. via a model structure or whatever) with inclusion functor $i: \omega Cat \to (\infty,\infty)Cat$,

• a straightforward notion of "map" from objects $C \in \mathcal C$ to objects of $(\infty,\infty) Cat$,

• and a (not completely vacuous) condition $\Phi$ on the objects of $\mathcal C$

such that

• Objects $C \in \mathcal C$ satisfying $\Phi$ have the property that $Hom(iF(C),Y)$ is naturally isomorphic to the set of maps from $C$ to $Y$, for all (suitably fibrant, perhaps) $Y \in (\infty,\infty)Cat$?

I'm happy to see quite restrictive conditions $\Phi$; in fact in my case I don't need to understand much more than Gray tensor powers of the arrow category $\bullet \to \bullet$. I suspect that something along the lines of "$F(C)$ is gaunt" or "loop-free" or something may often do the trick, but I'd be happy with something more or less restrictive.

I suppose I'd also be happy to see examples with "$n$" replacing "$\omega$" and "$(\infty,n)$" replacing "$(\infty,\infty)$".

And let me stress that I'm not looking for some kind of fancy $\infty$-categorical freeness -- when I say $Hom(i(F(C)),Y)$ above, I mean $Hom$ in whatever 1-category is being used to model $(\infty,\infty)Cat$. Although if there are results showing something fancier, that would be interesting to hear about too.