One has a nice "folk" model structure on strict $\infty$-categories due to Yves Lafont, Francois Metayer and Krzysztof Worytkiewicz whose notion of weak equivalences seem to be the notion of weak equivalences for weak $\infty$-category (I.e. a weaker notion than the existence of a strict inverse).

This produces a weak $(\infty,1)$-category of strict $\infty$-categories. My question is: is it expected to be a full subcategory of the category of weak $\infty$-category ?

(Note: I know that strict infinity categories with strict functors between them do not form a full subcategory of weak infinity categories, what I'm asking here is different, essentially because in the model structure mentioned above not all objects are cofibrant)

I'm actually not sure we have satisfying model for general weak $\infty$-category, and I might prefer to avoid the sort of problems mentioned in this answer, so I'll be happy with an answer dealing with $(\infty,n)$-categories defined for exemple as $n$-fold segal spaces, Rezk $\Theta_n$ spaces, Ara $n$-quasicategories or any other reasonable model. Also an answer focusing on $\infty$-groupoid or $(\infty,1)$-category would already be interesting.

Also, as a side question, assuming this is indeed fully faithful, is there any known result about which are the $\infty$-categories (or maybe $\infty$-groupoids) that are representable by strict $\infty$-categories ?

**Edit:** Let me clarify a few things which from what I read in the comments where unclear.

From the model structure of Lafont, Metayer, Worytkiewicz one obtains a notion of weak $\infty$-functor between strict $\infty$-category: as every objects in this model structure is fibrant a weak functor (or a weak anafunctor) from $X$ to $Y$ is a morphism $\widetilde{X} \rightarrow Y$ from some cofibrant replacement $\widetilde{X}$ of $X$, and notion of natural isomorphism of weak functor as morphism $\widetilde{X} \rightarrow PY$ where $PY$ is the path object for $Y$ in this model structure.

One can chose a functorial cofibrant replacement to have something more canonical, or even a comonadic one in order to obtain associative composition, but the choice of the cofibrant replacement does not have any effects on the question I'm asking, and it is possible to formulate it without choosing ones.

My question can be formulated as: does it defines the correct set of equivalence class of weak functor between strict $\infty$-categories if one see these as weak $\infty$-categories (and more generally, the correct space of morphism if one push things a little further).

Also note that I'm only interested in the 'canonical' way of sending strict $\infty$-categories to weak $\infty$-category, by just forgeting their strictness.

I know there is ways to send strict $\infty$-categories to weak $\infty$-groupoids or weak $(\infty,1)$-categories by formally (weakly) inverting all arrows or all $k$-arrow for $k>0$ , but then the image by this construction functor is no longer a strict $\infty$-category, and this construction has absolutely no chance to be fully faithful (it will like asking if the geometric realization functor from categories to the homotopy category of spaces is fully faithful).

The construction I'm refering to has good chances to be fully faithful (which is what I'm asking) but is clearly not essentially surjective even on $\infty$-groupoids: the groupoids in its image have for example trivial whitehead products $\pi_2 \times \pi_2 \rightarrow \pi_3$. The follow up question I asked is about knowing if we do have a good characterization of the image of this functor (for example, by the vanishing for all whitehead product in degree higher than $(2,2)$ or something like that). But please don't try to explain that there is construction which allow to represent all $\infty$-groupoid by strict $\infty$-category.

equivalent toa full subcategory of strict $(\infty,\infty)$-categories, but that equivalence is not the natural inclusion of strict $(\infty,\infty)$-categories into weak $(\infty,\infty)$-categories. $\endgroup$ – Mike Shulman Apr 22 '17 at 4:4816more comments