# Universal polygraphic factorization of strict ω-categories relative to a cobase

Recall from 1 that a cofibration of strict ω-categories is a retract of relative $I$-cell complexes, where $I$ denotes the set of boundary inclusions $\partial D^n \hookrightarrow D^n$, where $D_n$ denotes the free-standing $n$-cell (the $n$-disk or the $n$-globe). The class of trivial fibrations is the class of maps with the right lifting property with respect to $I$.

It is a theorem of Métayer that the cofibrant strict ω-categories are those strict ω-categories generated by polygraphs. Indeed, we have a comonadic cofibrant replacement $Q$, called the standard polygraphic resolution in 2, associated with the adjunction $$\mathbf{Pol} \rightleftarrows \mathbf{\omega\operatorname{-}Cat},$$ which gives a terminal factorization of every map $\emptyset \to X$ as a cofibration followed by a trivial fibration. That is, there is a terminal cofibrant strict ω-category in the category of cofibrant strict ω-categories trivially fibrant over any strict ω-category.

What I'd like to know is if there is a cofibrant replacement for strict ω-categories under a fixed strict ω-category $X$.

That is, can we find a universal/terminal factorization of any strict ω-functor $X\to Y$ as a pair $X\to Q_X Y \to Y$ of a cofibration followed by a trivial fibration, where $Q_X$ is some comonad given by an adjunction $$\mathbf{Pol}_{X/} \rightleftarrows \mathbf{\omega\operatorname{-}Cat}_{X/},$$ where $\mathbf{Pol}_{X/}$ is some category of "relative polygraphs under $X$"?

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