In mathematics, especially
category theory
Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Category theory ...
, the homotopy category of an
∞-category
In mathematics, more specifically category theory, a quasi-category (also called quasicategory, weak Kan complex, inner Kan complex, infinity category, ∞-category, Boardman complex, quategory) is a generalization of the notion of a Category (ma ...
''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
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric, and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. A simpler example is equ ...
.
If an ∞-category is defined as a
weak Kan complex
In mathematics, more specifically category theory, a quasi-category (also called quasicategory, weak Kan complex, inner Kan complex, infinity category, ∞-category, Boardman complex, quategory) is a generalization of the notion of a category. Th ...
(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
, where
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 In algebraic topology, the fundamental groupoid is a certain topological invariant of a topological space. It can be viewed as an extension of the more widely-known fundamental group; as such, it captures information about the homotopy type of a to ...
of ''X''.
Boardman–Vogt construction
Let ''C'' be an ∞-category. If
are morphisms (1-simplexes) 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
,