LECTURE . PAIRS AND WALDHAUSEN ∞-CATEGORIES
.. Exercise. Suppose X a quasicategory. Consider the cartesian fibration s : O(X) . X. Show that an edge σ : Δ1 . O(X)
is s-cocartesian just in case the corresponding diagram (Λ20 ) ∼
= Δ1 × Δ1 . X is a colimit. Conclude that s is a cocartesian fibration if and only if X admits all pushout diagrams.
is exercise suggests that cartesian fibrations that are also cocartesian have special significance. is is true!
.. Definition. Suppose p : X . Δ1 a cartesian fibration, and suppose α : C . ∼ X0 and β : D . ∼ X1 are equivalences of quasicategories. en a functor g : D . C is said to be attached to p via α and β just in case there is a map
G : D × Δ1 . X over Δ1 such that the obvious diagram
D ×. Δ1
D.
β
D.
α◦g
G
X.0
X..
X.1
Δ{0}
.
Δ.1
Δ{1}
.
commutes, and, moreover, such that for every vertex x ∈ D0 , the edge G({x} × Δ1 ) is a p-cartesian edge of X.
We will show now that such attached functors exist and are essentially unique.
.. Proposition. Suppose g : D . C a functor between quasicategories.
(..) ere exist a cartesian fibration p : X . Δ1 and equivalences of quasicategories α : C . ∼ X0 and β : D . ∼ X1
such that g is attached to p via α and β.
(..) Suppose p : X . Δ1 and p′ : X′ . Δ1 are cartesian fibrations, and suppose φ : X′ . X an equivalence of
quasicategories over Δ1 . en g is attached to X′ via equivalences α′ : C . ∼ X′0 and β′ : D . ∼ X′1 if and only if
g is attached to X via equivalences α := φ ◦ α′ : C . ∼ X0 and β := φ ◦ β′ : D . ∼ X1 .
(..) Suppose p′ : X′ . Δ1 and p′′ : X′′ . Δ1 are cartesian fibrations. Suppose that g is attached to p′ via equivalences α′ : C . ∼ X′0 and β′ : D . ∼ X′1 , and suppose that g is attached to p′′ via equivalences α′′ : C . ∼ X′′0 and
β′′ : D . ∼ X′′1 . en there exists a cartesian fibration p : X . Δ1 , equivalences α′ : C . ∼ X′0 and β′ : D . ∼ X′1 ,
and equivalences of quasicategories φ : X . X′ and ψ : X . X′′ over Δ1 such that g is associated to p via α and
β, and the diagrams of equivalences
.
X.′0
C.
.
.
X.0
X.′′0
.
D.
.
.
X.′1
X.1
X.′′1
and
commute.
Proof. To prove the first item, use the cartesian model structure on the category sSet+
to factor
/Δ1
(D♮ × (Δ1 )♯ ) ∪D
♮ ×(Δ{0} )♯
C♮ .
(Δ1 )♯
as a trivial cofibration followed by a cartesian fibration.
On to the second item. In one direction, one may simply compose the map D × Δ1 . X′ exhibiting the attachment of g to p′ with φ to obtain a map D × Δ1 . X exhibiting the attachment of g to p. In the other direction, we

aim to extend a map D × ∂Δ1 . X′ to a map D × Δ1 . X′ . Since this can be done in hsSet+
, and since X is
/Δ1
fibrant and everything is cofibrant therein, the proof is complete.
Finally, for the third item, consider the square
(D × Δ1 ) ∪
. D×Δ
.
{0}
X.′
C
Δ1. ,
X.
where the vertical map on the le hand side is the trivial cofibration from the first part, and the top map exhibits the
attachment of g to p′ . One may prove directly that the top map is an equivalence. Now the li X . X′ exists, and
the result is proved.
□
Conversely, we have the following.
.. Proposition. Suppose p : X . Δ1 a cartesian fibration. ere is a functor g : X1 . X0 attached to p via the
identities. Moreover, any other functor g′ : X1 . X0 attached to p via the identities is equivalent to g.
Proof. Consider the square
X♭1 × (Δ
. {1} )♯
.
X.♮
X♭1 × .(Δ1 )♯
(Δ1.)♯ .
e li exists, and the fiber of this li over Δ{0} is a map X1 .
via the identities. We leave the unicity proof as an exercise.
X0 . It’s easy to see that this a functor attached to p
□
.. Definition. Suppose C and D quasicategories. An adjunction
f: C .
D: u
is a map p : X . Δ1 that is both a cartesian fibration and a cocartesian fibration, along with equivalences α : C .
and β : D . X1 such that u is attached to p via α and β, and f op is attached to pop via αop and βop .
Let us now unpack this to explain how we may extract the unit id .
this definition. ere exist maps F : C × Δ1 . X and U : D × Δ1 .
u ◦ f of an adjunction f : C .
X such that:
X0
D : u from
(A) F|(C × Δ{0} ) = idC ;
(B) F|(C × Δ{1} ) = f;
(C) for every object x ∈ C0 , the edge F({x} × Δ1 ) is p-cocartesian;
(D) U|(C × Δ{1} ) = idD ;
(E) U|(C × Δ{0} ) = u;
(F) for every object y ∈ D0 , the edge U({y} × Δ1 ) is p-cartesian.
Now consider the rectangle
C × .Δ{1}
C × Δ.
{0,2}
C × Δ. {1,2}
.
D × Δ. {1,2}
.
X..
F
U
is gives us a map H ′ : C × Λ22 . X with the property that for any x ∈ C0 , the edge H ′ ({x} × Δ{1,2} ) is pcartesian. Hence there is an extension of this map to a map H : C × Δ2 . X. e unit of the adjunction is then the
natural transformation η := H|(C × Δ{0,1} ). You can prove directly for any x ∈ C0 and y ∈ D0 that the composite
MapD ( fx, y) .
MapC (ufx, uy) .

MapC (x, uy)
is an equivalence.
.. Lemma. A cartesian fibration X . S is also a cocartesian fibration if and only if, for every edge s .
attached functor Xt . Xs admits a le adjoint.
t of S, an
Let’s now move along to the study of the sorts of quasicategories that we will input into K-theory. We will at times
hove to return to our foundations in what follows, but we’ve got enough now to get started.
.. Definition. A pair of ∞-categories (C , C† ) is an ∞-category equipped with a subcategory C† ⊂ C that contains ιC . Morphisms of C† will be called ingressive or cofibrations. A functor of pairs (ψ, ψ † ) : (C , C† ) . (D, D† )
is a commutative square
C.†
(..)
ψ†
D.†
.
C.
ψ
D;
.
it is said to be strict if the diagram (..) is a pullback diagram in QCat.
More generally, a functor D . C of quasicategories will be said to exhibit a pair structure on C if it factors as an
equivalence D . E followed by an inclusion E . C of a subcategory such that (C, E) is a pair.
.. Exercise. Show that a functor ψ : D . C of quasicategories exhibits a pair structure on C if and only if it is a
homotopy monomorphism in QCat that induces an equivalence ιD . ιC.
We write Pair∞ for the full subcategory of O(Cat∞ ) spanned by those functors D . C that exhibit a pair
structure on C. We may also describe an ordinary category of pairs of ∞-categories, and declare a functor of pairs
(ψ, ψ † ) : (C , C† ) . (D, D† ) to be a weak equivalence just in case both ψ and ψ † are equivalences.
.. Example. Any quasicategory C can be endowed with a maximal pair structure (X, X) and a minimal pair structure (X, ιX). If we speak of a quasicategory as a pair without more comment, then we will assume that it is the maximal
structure of which we are speaking.
.. Example. Denote by Λ0 Q 2 the pair (Λ20 , Δ{0,1} ⊔ Δ{2} ):
0.
1.
.
2.
.
Denote by Q 2 the pair ((Λ20 ) , Δ{0,1} ⊔ Δ{2,∞} ) ∼
= (Δ1 × Δ1 , (Δ{0} ⊔ Δ{1} ) × Δ1 ):
0.
1.
.
∞.
2.
ere is an obvious inclusion of pairs Λ0 Q 2 .
Q2 .
Roughly speaking, the cofibrations of a pair will be those along which pushouts are reasonable. To make this more
precise, for any pair (C , C† ) be a pair. en the arrow quasicategory O(C ) admits a pair structure as well, given by
a pullback
O† (C
. )
O(C
. )
.
C.†
s
C. .

We are nearly ready to describe the basic inputs of K-theory. We only need one more notion.
.. Definition. A zero object of a quasicategory is an object that is both initial and terminal.
.. Exercise. Show that a quasicategory has a zero object if and only if it has an initial object ∅, a terminal object
∗, and an edge ∗ . ∅.
Now we have our definition.
.. Definition. A Waldhausen ∞-category (C , C† ) is a pair of essentially small ∞-categories such that the following axioms hold.
(..) e ∞-category C contains a zero object.
(..) For any zero object 0, any morphism 0 . X is ingressive.
(..) e source functor s : O† (C ) . C† is a cocartesian fibration, and an edge η ∈ O† (C) is s-cocartesian only
if t(η) is ingressive.
Call a functor of pairs ψ : C . D between two Waldhausen ∞-categories exact if it satisfies the following
conditions.
(..) e underlying functor ψ carries zero objects of C to zero objects of zD.
(..) In the diagram
O† (C
. )
sC
C.†
O† (ψ)
.
ψ†
O† (D)
.
sD
D.† ,
the functor O† (ψ) sends sC -cocartesian edges to sD -cocartesian edges.
A Waldhausen subcategory of a Waldhausen ∞-category C is a subpair D ⊂ C such that D is a Waldhausen
∞-category, and the inclusion D . C is exact.
e conditions demanded of a Waldhausen ∞-category (C , C† ) can be rephrased in the following manner: there
is a zero object in C that is initial in C† , and pushouts of cofibrations exist and are cofibrations. is last point can
again be rephrased as the condition that the morphism of pairs Λ0 Q 2 . Q 2 (.) induces an equivalence of
∞-categories
Colim(Q 2 , C ) ×Fun(Q2 ,C ) FunPair∞ (Q 2 , C ) . FunPair∞ (Λ0 Q 2 , C ).
An exact functor C . D is now one that preserves cofibrations, zero objects, and pushouts along cofibrations.
.. Example. ere are many examples of Waldhausen ∞-categories; here’s a short list.
(..) e ordinary category of pointed finite sets. e cofibrations are just monomorphisms.
(..) e nerve of the relative category of pointed finite CW complexes, with its maximal pair structure.
(..) e category of finitely generated projective R-modules with “admissible monomorphisms,” i.e., monomorphisms whose cokernels are projective.
(..) e nerve of the relative category of perfect complexes with its maximal pair structure.
Right away, we can prove Waldhausen’s approximation theorem in this context.
.. eorem (Approximation). Suppose C and D two ∞-categories that become Waldhausen ∞-categories when
equipped with the maximal pair structure. en an exact functor ψ : C . D is an equivalence if and only if it induces
an equivalence of homotopy categories hC . ∼ hD.
Proof. Since C and D admit all finite colimits and since ψ preserves them, it follows that ψ preserves the tensor
product with any finite space. us for any positive integer n and any morphism η : X . Y of C , the map
π n (MapC (X, Y), η) .
π n (MapD (ψ(X), ψ(Y)), ψ(η))

can be identified with the fiber of the bijection
π 0 MapC (X ⊗ Sn , Y) .
π 0 MapD (ψ(X) ⊗ Sn , ψ(Y))
over [ψ(η)] ∈ π 0 MapD (ψ(X), ψ(Y)). Since this is a bijection, ψ is fully faithful, hence an equivalence.

□