# ct.class principle – Which free strict \$omega\$-categories are too free as feeble \$(infty,infty)\$-categories? retort

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## ct.class principle – Which free strict \$omega\$-categories are too free as feeble \$(infty,infty)\$-categories?

If you do not mind, I’ll speak about strinct $$infty$$-categories, however feeble $$(infty,n)$$-category to keep away from discussing the ‘drawback’ relating to the non uniqueness of the acceptation of $$(infty,infty)$$-categories talked about Here.

too I do not arbitrator what follows utterly retort the query, as Harry mentioned within the remark, this can be a fairly launch drawback.

So, I arbitrator, a really unostentatious, and ‘primarily’ unique, path to safe such standing is to assemble mannequin classes for $$(infty,n)$$-category, that are classes of both presheaf of clique or presheaf of areas over a diminutive class $$C$$ of ‘diagrams’. (You can typically bounce from an outline as presheaf of units to an outline as presheaf of areas, utilizing some variant of the simplicial completion strategies as illustrated for example within the illustration of the class $$Theta$$ Here.)

Obviously, if you happen to maintain such a mannequin then the objects of your class $$C$$ maintain all of the properties you anticipate. too notes that typically presheaf over $$C$$, or no less than some presheaves over $$C$$, quiet corresponds to some kindly of diagrams and so that you purchase this systematize of building not justs for objects of $$C$$ however too for extra frequent diagrams construct out of objects of $$C$$. (principally, the cofibrant unprejudiced of your mannequin construction).

(And of passage mannequin that aren’t precisely presheaves can too give some partial retort to your query so long as they purchase some subcategories of cofibrant objects that may breathe considered diagrams.)

While producing a rigorous dispute for the discourse will breathe troublesome, it positively feels too correct: as quickly as you maintain a prosperous sufficient class of diagrams $$C$$ with the systematize of properties you’re requiring one ought to breathe in a position to show some systematize of Nerve theorem (an $$infty$$-categorical model of monads with arities or Nervous monads) to array that $$(infty,n)$$-categories can breathe represented as presehaves of areas on $$C$$ satisfying some segal kindly circumstances, for which you may breathe in a position to construct a projective/injective mannequin construction on $$[C^{op},sSet]$$. This mannequin construction will too usually (particularly if $$C$$ is prosperous sufficient) maintain a “simplicial decompletion” on the class of presheaf of units of $$C$$.

So it stays to contemplate at examples of such mannequin for $$(infty,n)$$-categories… But right here mighty labor is left to breathe performed (and to breathe utterly trustworthy that is one thing that I’m very concerned with and actively engaged on) One instinct for that is that folks maintain typically tried to method up with diminutive and unostentatious fashions, whereas right here we query about very immense fashions.

The first that involves mind is clearly the class $$Theta_n$$ (graze for example Dimitri’s paper talked about above) which clearly matches into this painting, and verisimilitude complicial clique fashions which does the job for Street Orientals, with the one drawback that this mannequin hasn’t been in comparison with others ones.

But possibly probably the most fascinating occasion already labored out relating to your query is the so-called “gargantuan model” constructed in Barwick and Schommer-Pries’ paper On the unicity of the speculation of upper classes, which does this for a large class of gaunt classes. But I’ll maintain to learn once more this paper earlier than I can say one thing extra require right here.

Finally my avow labor on polygraphs and the Simpson surmise (right here and enter hyperlink description right here) is basicaly an try to show this consequence for the very sizable class of all “non-unital polygraphs”.

Here the await is that $$(infty,n)$$-category can breathe represented by a mannequin buildings on presehaves of units and or area over the class of “plex” (the representable within the class of polygraphs) most likely with some “stratification” within the spirit of the complicial mannequin. So far I maintain been focusing totally on modeling $$infty$$-groupoids for simplicity, however I anticipate the extention to $$infty$$-category is not going to breathe the toughest sever.
airplane within the groupoid illustration I’m caught for technical instinct for frequent polygraphs, and I can solely make the speculation works for “regular polygraphs”, however that is quiet a reasonably sizable class (containing $$Theta, Delta$$ and lots of different issues and closed beneath the Gray tensor merchandise) and I’ve proven that common polygraphs design a presheaf class and carries a unaffected mannequin construction that fashions all $$infty$$-groupoids. I endure extending this to a mannequin construction on “stratified regular polygraphs” modeling feeble $$(infty,n)$$-categories ought to breathe workable with a little bit of labor (I denote by that: it’s most likely a whole lot of labor, however very possible if somebody wish to expend a while on it) and this is able to represent a really wonderful retort to your query.

too notes that in every little thing I maintain mentioned above the functor “$$F$$” of your query isn’t actually expecte to breathe absolutely dependable. I do not know if this can be a requirement you maintain or in case you are happy staying with “polygraphic morphisms”.

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