The model structure on sSet for quasi-categories (the Joyal model structure) is not enriched over sSet with the Quillen model structure, so ordinary internal Hom of simplicial sets is not a correct homotopy function complex of quasi-categories. We can give a correct definition using either cosimplicial or simplicial resolutions, which can be defined by the small object argument as usual. But we can give a more explicit description of such resolutions.


For example, we can define a cosimplicial frame on the terminal simplicial set by taking nerves of groupoids $\{ 0 \simeq \ldots \simeq n \}$. Then a cosimplicial frame on a simplicial set can be defined as the cartesian product with this cosimplicial object. This construction can be found, for example, in [1].


The problem with this construction is that it produces too large function complex. For example, a 1-simplex of $\mathrm{hMap}(\Delta^0,X)$ is a 1-simplex of $X$ together with an inverse and higher simplices that guarantee that they are inverses. It seems that there should be a definition of a homotopy function complexes such that 1-simplices in the example above are just 1-simplices of $X$ which are equivalences.


The obvious way to define such homotopy function complex is to take the largest Kan complex contained in the internal Hom. It seems that we also can define a simplicial resolution which produces the same homotopy function complex. Can we construct homotopy function complexes in such a way? Were such constructions described in the literature?


[1] *Daniel Dugger and David I. Spivak*, MR 2764043 [**Mapping spaces in quasi-categories**](http://dx.doi.org/10.2140/agt.2011.11.263), *Algebr. Geom. Topol.* **11** (2011), no. 1, 263--325.



Homotopy function complex for quasi-categories
**Yes**, you can compute the mapping spaces in ∞-categories by taking the biggest Kan subcomplex of the internal hom.


The trick is not to use the Joyal model structure, but instead the model structure on marked simplicial sets defined in Higher Topos Theory, proposition 3.1.3.7 (in the case $S=\Delta^0$). By proposition 3.1.4.1 the fibrant objects are exactly the quasicategories with the equivalences marked, by corollary 3.1.4.4 it is a simplicial model category and its mapping spaces are defined exactly as you describe and by theorem 3.1.5.1 this is Quillen equivalent to the Joyal model structure in a way that does the obvious thing on fibrant objects.


