Homotopy category of an ∞-category

In mathematics, especially category theory, the homotopy category of an ∞-category C is the category where the objects are those in C but the hom-set from x to y is the quotient of the set of morphisms from x to y in C by an appropriate equivalence relation.
If an ∞-category is defined as a weak Kan complex, then the construction is due to Boardman and Vogt, who also gave the definition of an ∞-category as a weak Kan complex. In this case, the homotopy category of an ∞-category C is equivalent to, where is a left adjoint of the nerve functor.
For example, the singular complex of a topological space X is a Kan complex and the homotopy category of it is the fundamental groupoid of X.

Boardman–Vogt construction

Let C be an ∞-category. If are morphisms in C, then we write if there is a 2-simplex such that Then by Joyal's work, the relation turns out to be an equivalence relation. Hence, we can take the quotient
Then the homotopy category in the sense of Boardman–Vogt is the category where, and the composition is given by when exhibits some composition of.
Let be a left adjoint to the inclusion of the category of sets into the category of simplicial sets. If is a Kan complex, then coincides with the set of simplicial homotopy classes of maps. Then
for each objects in.