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

\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 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

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_y.

Then by Joyal's work, the relation

\sim

turns out to be an equivalence relation. Hence, we can take the quotient :

, y The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...

= \operatorname_C(x, y)/\sim. Then the homotopy category

\tau(C)

in the sense of Boardman–Vogt is the category where

\operatorname(\tau(C)) = \operatorname(C)

\operatorname_(x, y) =

/math> and the composition is given by

\circ = /math> 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 class es of maps \Delta^0 \to K . Then
: \operatorname_(x, y) \simeq \pi_0 \operatorname(x, y) for each objects x, y in C .

Boardman–Vogt construction

See also

References

Further reading