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 (usual definition), then the construction is due to Boardman and Vogt,^[1] 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 $\tau (C)$ , where $\tau$ is a left adjoint of the nerve functor.^[2]

For example, the singular complex of a (reasonable) topological space X is a Kan complex and the homotopy category of it is the fundamental groupoid of X.^[3]

Boardman–Vogt construction

Let C be an ∞-category. If $f,g:x\to y$ are morphisms (1-simplexes) in C, then we write $f\sim g$ if there is a 2-simplex $\sigma :\Delta ^{2}\to C$ such that $\sigma (0\to 1)=f,\,\sigma (0\to 2)=g,\,\sigma (1\to 2)=\operatorname {id} _{y}.$ Then by Joyal's work, the relation $\sim$ turns out to be an equivalence relation.^[4] Hence, we can take the quotient

[x,y]=\operatorname {Hom} _{C}(x,y)/\sim .

Then the homotopy category $\tau (C)$ in the sense of Boardman–Vogt is the category where $\operatorname {obj} (\tau (C))=\operatorname {obj} (C)$ , $\operatorname {Hom} _{\tau (C)}(x,y)=[x,y]$ and the composition is given by $[f]\circ [g]=[h]$ when $h$ exhibits some composition of $f,g$ .^[5]

Let $\pi _{0}$ be a left adjoint to the inclusion of the category of sets into the category of simplicial sets.^[6] If $K$ is a Kan complex, then $\pi _{0}K$ coincides with the set of simplicial homotopy classes of maps $\Delta ^{0}\to K$ .^[7] Then