Functors (between \(\infty \)-categories) that aren't strictly unital (Q721972)

A model for \(\infty\)-categories, called \textit{quasi-categories} by \textit{A. Joyal} [J. Pure Appl. Algebra 175, No. 1-3, 207--222 (2002; Zbl 1015.18008)], is given by simplicial sets which satisfy the \textit{restricted Kan condition} of \textit{J. M. Boardman and R. M. Vogt} [Homotopy invariant algebraic structures on topological spaces, Lect. Notes Math. 347 (1973; Zbl 0285.55012)] -- that is, in which each \textit{inner} horn can be filled. (Through the nerve functor, a usual small category is the same as a simplicial set in which each inner horn can be filled \textit{in a unique way}.) The usual notion of functor between quasi-categories is a simplicial set map between them. But this notion is \textit{strict}; in particular, the commutation to degeneracies has to be rigid (not up to homotopy). To preserve degeneracies is a kind of unitality (for usual categories, the degeneracies let appear identities). It is natural to try to relax this condition; it is both a theoretic question and a pratical one, being here motivated by research of the author on Fukaya categories. To see what happens when forgetting degeneracies, one looks at the underlying \textit{semi-simplicial} set of a quasi-category (remind that a semi-simplicial set is a contravariant functor from the subcategory of \textit{injections} in the simplex category \(\mathbf{\Delta}\) to sets). The functor sending a simplicial set to its underlying semi-simplicial set has a left adjoint; the first main result (\textit{Theorem 1.1}) of the article asserts that the counit of this adjunction behaves well on quasi-categories. From this is deduced \textit{Theorem 1.4} which reachs the aim of the work: it shows that a map of the underlying \textit{semi-}simplicial sets of two quasi-categories which preserves ``units'' (that is, degeneracies) only in a very lax sense is homotopy equivalent to a strict functor (that is a simplical map). A complementary (and independent) result to strictify as quasi-categories some kind of lax \(\infty\)-categories (here at the level of objects: one looks at a \textit{semi-}simplicial set satisfying the restricted Kan condition and wonders whether it is equivalent to a quasi-category) has been recently obtained by \textit{W. Steimle} [J. Homotopy Relat. Struct. 13, No. 4, 703--714 (2018; Zbl 1432.55018)].