Does the left adjoint of the homotopy coherent nerve have a name?

Question

The simplicial nerve $N: \operatorname{Sset-Cat} \to \operatorname{Sset}$ has the left adjoint $\mathfrak{C}: \operatorname{Sset} \to \operatorname{Sset-Cat}.$ Wherever I see it, it is always referred by its symbol (and even that is not consistent: here it is called "$S$"). That's awkward and weird. Is there a (n at least proposed) name for it?

Answer 1

There is no standard name for this functor.

One could call it the rigidification functor, since rigidification (or strictification) is a recognized name for this type of constructions.

However, such a name can refer to any of the several weakly equivalent models for this functor, see, for example, Rigidification of quasi-categories by Dugger and Spivak.

Answer 2

Emily Riehl calls it (e.g. in Homotopy coherent structures), homotopy coherent realization, due to "formal similarity" with the realization functor $\operatorname{Top} \to \operatorname{Sset}.$

Images of standard simplices are called homotopy coherent simplices. There is a related functor $\mathfrak{C}: \operatorname{Cat} \to \operatorname{Sset-Cat}$ which makes a "homotopy coherent" indexing category out of an ordinary one (enabling one to define a homotopy coherent diagram in the evident way), which is called the free resolution functor. A homotopy coherent simplex can be seen as a free resolution of the category $\mathfrak{n}$, or as the homotopy coherent realization of of the standard simplex $\Delta^n$.

The formal similarity mentioned above seems to be that homotopy coherent diagram of a simplicial set is glued from homotopy coherent simplices in the same way the realization of a simplicial set is glued from standard simplices - as a colimit over the category of elements of this simplicial set.