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, 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 τ ( C ) {\displaystyle \tau (C)} , where τ {\displaystyle \tau } is a left adjoint of the nerve functor. 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.
Boardman–Vogt construction Let C be an ∞-category. If f , g : x → y {\displaystyle f,g:x\to y} are morphisms (1-simplexes) in C, then we write f ∼ g {\displaystyle f\sim g} if there is a 2-simplex σ : Δ 2 → C {\displaystyle \sigma :\Delta ^{2}\to C} such that σ ( 0 → 1 ) = f , σ ( 0 → 2 ) = g , σ ( 1 → 2 ) = id y . {\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. Hence, we can take the quotient
[ x , y ] = Hom C ( x , y ) / ∼ . {\displaystyle [x,y]=\operatorname {Hom} _{C}(x,y)/\sim .}
Then the homotopy category τ ( C ) {\displaystyle \tau (C)} in the sense of Boardman–Vogt is the category where obj ( τ ( C ) ) = obj ( C ) {\displaystyle \operatorname {obj} (\tau (C))=\operatorname {obj} (C)} , Hom τ ( C ) ( x , y ) = [ x , y ] {\displaystyle \operatorname {Hom} _{\tau (C)}(x,y)=[x,y]} and the composition is given by [ f ] ∘ [ g ] = [ h ] {\displaystyle [f]\circ [g]=[h]} when h {\displaystyle h} exhibits some composition of f , g {\displaystyle f,g} . Let π 0 {\displaystyle \pi _{0}} be a left adjoint to the inclusion of the category of sets into the category of simplicial sets. If K {\displaystyle K} is a Kan complex, then π 0 K {\displaystyle \pi _{0}K} coincides with the set of simplicial homotopy classes of maps Δ 0 → K {\displaystyle \Delta ^{0}\to K} . Then
Hom τ ( C ) ( x , y ) ≃ π 0 Map ( x , y ) {\displaystyle \operatorname {Hom} _{\tau (C)}(x,y)\simeq \pi _{0}\operatorname {Map} (x,y)}
for each objects x , y {\displaystyle x,y} in C {\displaystyle C} .
See also Weak equivalence between simplicial sets
Notes
References Cisinski, Denis-Charles (2023). Higher Categories and Homotopical Algebra (PDF). Cambridge University Press. ISBN 978-1108473200.
Further reading "homotopy category of an (infinity,1)-category in nLab". ncatlab.org. "1.4.5 The Homotopy Category of an ∞-Category". Kerodon.
