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, 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.

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

$[x,y]={\mathrm{Hom}}_C(x,y)/\sim .$

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

Let ${\pi }_0$ be a left adjoint to the inclusion of the category of sets into the category of simplicial sets. If $K$ is a Kan complex, then ${\pi }_{0}K$ coincides with the set of simplicial homotopy classes of maps ${\mathrm{\Delta }}^0\to K$. Then

${\mathrm{Hom}}_{\tau (C)}(x,y)\simeq {\pi }_0\mathrm{Map}(x,y)$

for each objects $x,y$ in $C$.

See also

• Weak equivalence between simplicial sets

References

• Cisinski, Denis-Charles (2023) (in en). Higher Categories and Homotopical Algebra. Cambridge University Press. ISBN 978-1108473200. https://cisinski.app.uni-regensburg.de/CatLR.pdf. 

Further reading

• "homotopy category of an (infinity,1)-category in nLab". https://ncatlab.org/nlab/show/homotopy+category+of+an+%28infinity%2C1%29-category.