Homotopy category of an ∞-category

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

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

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

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

${\displaystyle \operatorname {Hom} _{\tau (C)}(x,y)\simeq \pi _{0}\operatorname {Map} (x,y)}$

for each objects ${\displaystyle x,y}$ in ${\displaystyle C}$.^[8]

• Weak equivalence between simplicial sets