LECTURE . FROM RELATIVE CATEGORIES TO UASICATEGORIES
In our last lecture, we codified what we mean by a category with a zero object and “a good theory of cofiber
sequences.” is was the notion of a Waldhausen category, which is a particular kind of relative category. But since
the homotopy theory of relative categories is a little delicate, we longed for a homotopy theory of homotopy theories
that resembled the homotopy theory of spaces more closely.
We begin with the nerve functor N : Cat . sSet.
.. Exercise. e nerve of a category C is a Kan complex if and only if C is a groupoid. A morphism G . H of
groupoids is an equivalence if and only if the induced map NG . NH of simplicial sets is a homotopy equivalence.
[Hint: show first that the nerve of any category is 2-coskeletal; now since Λnk . Δn induces an equivalence on
2-skeleta when n ≥ 4, one only has to check the horn-filling conditions explicitly for n = 1, 2, 3.]
Note that the horn fillers in the proof above were all unique. If we relax this condition, then we have the following.
.. Definition. An ∞-groupoid is a Kan complex, i.e., a simplicial set X such that for any positive integer n and any
0 ≤ k ≤ n, the map Mor(Δn , X) . Mor(Λnk , X) is a surjection. A map of ∞-groupoids is an equivalence if and
only if it is a homotopy equivalence.
Also note that when 1 ≤ k ≤ n − 1, the horn fillers still exist as usual. So we can relax this condition to arrive at the
following.
.. Definition. A quasicategory is a simplicial set X such that for any integer n ≥ 2 and any 1 ≤ k ≤ n − 1, the
map Mor(Δn , X) . Mor(Λnk , X) is a surjection. Maps between quasicategories will be called functors.
Our opening exercise guaranteed that the relative category of groupoids is the full relative subcategory of the
relative category of ∞-groupoids, i.e., the category of Kan complexes with homotopy equivalences. To do the same
for quasicategories, we need to work out what we mean by an equivalence of quasicategories.
To start, let us note that, just as we spoke of a homotopy category of a relative category, we may also speak of the
homotopy category of a quasicategory.
.. Proposition. e nerve functor N : Cat .
sSet admits a le adjoint h : sSet .
Cat.
□
Proof. e nerve functor preserves all limits and all filtered colimits.
For a quasicategory X, we declare hX to be the homotopy category of X. e only motivation we offer for this definition is the fact that it’s formally similar to the le adjoint Ho of the inclusion Cat . RelCat. However, a good
justification for this definition comes from the following.
.. Exercise. Suppose X a quasicategory. Define a relation ∼ on X1 by declaring φ ∼ φ′ just in case there exists a
2-simplex σ : Δ2 . X such that d0 σ is degenerate, d1 σ = φ, and d2 σ = φ′ . Show that ∼ is an equivalence relation,
and that hX is equivalent to a category whose objects are vertices of X and whose morphisms are equivalence classes
of 1-simplices of X.
Armed with this, we may now ask which edges in a quasicategory play the role of “isomorphisms.” ese should
be invertible up to coherent homotopy. ere are several ways to formalize this idea, and it’s a pleasant and important
fact that they are all equivalent.
.. eorem ( Joyal). Suppose X a quasicategory. en the following are equivalent for an edge η : Δ1 .
(..) For every integer n ≥ 2 and every map θ :
θ : Δn . X such that θ = θ|Λn0 .
Λn0
.

X such that η = θ|Δ
{0,1}
X.
, there exists an extension
(..) ere exists an extension η : NE . X such that η = η|[1], where E is the unique contractible groupoid with two
objects, 0 and 1.
(..) ere exists an extension η : sk2 NE . X such that η = η|[1].
(..) e corresponding morphism η : [1] . hX is an isomorphism.
Proof. Let’s prove that the first assertion implies the second. We aim to show that the dotted arrow in the diagram
Δ.1
X.
.
NE
.
.
∪
exists. To this end, filter NE as m≥0 Um , where Um is the image of the unique nondegenerate m-simplex σ : Δm .
with σ(0) = 0. e proof is completed by writing the pushout square
Λ.m
0
NE
Um−1
.
.
Δ.m
U.m
for any integer m ≥ 1.
e second statement clearly implies the third, and the previous exercise guarantees that the third statement implies the fourth. e final implication, on the other hand, is tricky and will be easier to prove when we have a little
more theory at our disposal.
An edge of a quasicategory satisfying any (hence all) of these conditions will be called an equivalence.
.. Notation. Suppose X a quasicategory. en denote by ιX the largest simplicial subset whose edges are all equivalences. By the previous theorem, ιX can be described as the fiber product X ×hX (hX)′ , where (hX)′ is the largest
groupoid contained in hX. is defines a functor ι : QCat . Kan that assigns to any quasicategory X the maximal
∞-groupoid ιX contained therein.
.. Proposition. If X is a quasicategory, and K is any simplicial set, then the simplicial set Map(K, X) is a quasicategory.
Proof. It’s clearly enough to check this when K = Δm . e claim is now that for any integer n ≥ 2 and any 1 ≤ k ≤
n − 1, the inclusion Δm × Λnk . Δm × Δn is contained in the smallest class of maps containing the inner horns
Λst . Δs (1 ≤ t ≤ s − 1) that is closed under pushouts, transfinite composition, and retracts.
Let us concentrate on the case when m = 1 and n = 2. In this case, one may obtain Δ1 × Δ2 by gluing first the
two 2-simplices
Δ{0} × Δ2 and Δ{1} × Δ2
along the obvious horns, and then the following three 3-simplices, in order, each along the appropriate horn:
(Δ{0} × Δ2 ) ⋆ (Δ{1} × Δ{0} );
(Δ{0} × Δ{1,2} ) ⋆ (Δ{1} × Δ{0,1} );
and (Δ{0} × Δ{2} ) ⋆ (Δ{1} × Δ2 ).
e higher-dimensional cases are le as an exercise. ey are more combinatorially complicated, but no more difficult
conceptually.
□
is mapping space is what we will call the functor quasicategory between two quasicategories (or even from an arbitrary simplicial set to a quasicategory). So we’ll start denoting it Fun(K, X).
We’ve developed the notion of an equivalence within a quasicategory, and we’ve developed the notion of the functor quasicategory. Together, these are all one needs in order to define the notion of an equivalence of quasicategories.
.. Definition. Suppose X and Y two quasicategories. We shall say that a functor X . Y is an equivalence of quasicategories just in case, for any quasicategory K, the induced functor Fun(K, X) . Fun(K, Y) in turn induces an
equivalence of ∞-groupoids
ι Fun(K, X) . ι Fun(K, Y).

is, then, yields a relative category of quasicategories.
Some remarks are in order. First, note that maps of spaces/∞-groupoids are equivalences if and only if they are
equivalences as functors of quasicategories. Second, note that a functor between ordinary categories is an equivalence
just in case they induce an equivalence of quasicategories on their nerves. us spaces/∞-groupoids and categories
each form a full relative subcategory of the relative category of quasicategories.
In fact we want to go farther:
Gpd
.
Cat
.
.
RelCat
.
Kan
.
QCat
.
sSet. +
Each functor represented here will be an embedding of relative categories, but the three homotopy theories in the
lower right hand corner will turn out to be equivalent.
So let us pass to the category in the lower right-hand corner.
.. Definition. A marked simplicial set is a pair (X, S) consisting of a simplicial set and a set S ⊂ X1 of marked
edges containing the degenerate edges. A map if marked simplicial sets is simply a map of simplicial sets that send
marked edges to marked edges. Write sSet+ for the category of marked simplicial sets.
Before we discuss the relative category of marked simplicial sets, let us first mention some important examples of
them, to get a feel for the role we expect to be played by the marked edges.
.. Example. (..) For any simplicial set X, one has a minimal marking X♭ = (X, s0 (X0 )) and the maximal
marking (X, X1 ). It is easy to see that (−)♭ and (−)♯ define a le and right adjoint to the forgetful functor
sSet+ . sSet.
(..) For any relative category (C, wC), one may mark the edges in the nerve NC that correspond to the weak
equivalences.
(..) For any quasicategory X, one has a canonical marking X♮ = (X, (ιX)1 ), in which the marked edges are
precisely the equivalences.
Here is a theorem we will discuss next time.
.. eorem. ere is a simplicial model structure on the category sSet+ , in which:
(..) e simplicial structure is given by
Map♯ (X, Y)m = Mor(X × (Δm )♯ , Y).
(..) e cofibrations are precisely those marked maps that are monomorphisms on their underlying simplicial sets.
(..) e fibrant objects are precisely the marked simplicial sets of the form X♮ for some quasicategory X.
