LECTURE . FILTERED OBJECTS
.. Notation. Denote by M the ordinary category whose objects of M are pairs (m, i) consisting of an object m ∈ Δ
and an element i ∈ m and whose morphisms (n, j) . (m, i) are maps φ : m . n of Δ such that j ≤ φ(i). is
category comes equipped with a natural projection M . Δop .
e nerve NM can be endowed with a pair structure by setting
(NM)† := NM ×NΔop ιNΔop .
Put differently, an edge of M is ingressive just in case it covers an equivalence of Δop . It is easy to see that the projection
M . Δop is a Grothendieck fibration, and so the projection π : NM . NΔop is a pair cartesian fibration; the
functor NΔ . Pair∞ classified by π assigns to any object m ∈ NΔ the pair (Δm )♯ .
.. Construction. Fix a quasicategory S. For any morphism of pairs X . S♭ , define the simplicial set F (X /S)
as the simplicial set over NΔop × S satisfying the following universal property. We require, for any simplicial set K
and any map σ : K . NΔop × S, a bijection
Mor/(NΔop ×S) (K, F (X /S)) ∼
= MorsSet(2)/(S,ιS) ((K ×NΔop NM, K ×NΔop (NM)† ), (X , X† )),
functorial in σ. Here, (NM, (NM)† ) is the pair introduced in ., and the category sSet(2) is the one defined in ??.
For any object m ∈ Δ, write Fm (X /S) for the fiber of the projection F (X /S) . NΔop over m. When
S = Δ0 , write F (X ) for F (X /S), and for any object m ∈ Δ, write Fm (X ) for the fiber of the projection
F (X ) . NΔop over m.
.. Definition. Suppose X . S♭ a morphism of pairs. en the vertices of F (X /S) will be called filtered objects
of X over S; more specifically, the vertices of Fm (X /S) will be called m-filtered objects of X over S. e edges will
be called morphisms of filtered objects of X over S.
.. Proposition. Suppose X .
a cocartesian fibration.
S a pair cocartesian fibration. en the structure map F (X /S) .
... Corollary. For any pair cocartesian fibration X .
.. e functor NΔop × S .
NΔop × S is
S, the simplicial set F (X /S) is a quasicategory.
Cat∞ classified by the cocartesian fibration
NΔop × S
F (X /S) .
assigns to any object (m, s) ∈ NΔop × S the quasicategory Fun♭Pair∞ ((Δm )♯ , Xs ) of filtered objects
X0 .
of Xs and to any morphism (φ, f) : (m, s) .
··· .
X1 .
Xm
(n, t) of NΔop × S the functor
Fun♭Pair∞ ((Δn )♯ , Xt )
Fun♭Pair∞ ((Δm )♯ , Xs ) .
that carries the filtered object above to the filtered object
f⋆ (Xφ(0) ) .
f⋆ (Xφ(1) ) .
··· .
f⋆ (Xφ(m) )
We may endow the ∞-categories F (X /S) of filtered objects with a pair structure in a variety of ways, but we
wish to focus on one pair structure that will retain good formal properties when we pass to the subcategory of totally
filtered objects.

.. Definition. Suppose C a Waldhausen quasicategory, and consider the functors
s : Fun♭Pair∞ ((Δ1 )♯ , C ) .
C
and t : Fun♭Pair∞ ((Δ1 )♯ , C ) .
C.
Clearly s is a cocartesian fibration. We may now endow the quasicategory Fun♭Pair∞ ((Δ1 )♯ , C ) with a pair structure
by letting Fun♭Pair∞ ((Δ1 )♯ , C )† be the smallest subcategory of Fun♭Pair∞ ((Δ1 )♯ , C ) containing the following classes
of edges:
(..) any edge φ such that both s(φ) is an equivalence and t(φ) is ingessive and
(..) any s-cocartesian edge that covers a cofibration.
Now suppose p : X . S a Waldhausen cocartesian fibration. We may now endow the quasicategory F (X /S)
with a pair structure in the following manner. We let F (X /S)† ⊂ F (X /S) be the smallest subcategory containing all equivalences as well as any edge φ : Δ1 . X that covers a degenerate edge id(m,s) of NΔop × S — whence
it can be identified as a functor φ : Δ1 . Fun♭Pair∞ ((Δm )♯ , Xs ) — such that for any edge η : Δ1 . Δm , the edge
Δ1 .
φ
Fun♭Pair∞ ((Δm )♯ , Xs ) .
η⋆
Fun♭Pair∞ ((Δ1 )♯ , Xs )
is ingressive in the sense above.
.. Proposition. Suppose p : X . S a Waldhausen cocartesian fibration. e functor F (p) : F (X /S) .
is a Waldhausen cocartesian fibration.
NΔop × S
Proof. With the structure on F (X /S) described above, F (p) is easily seen to be a pair cocartesian fibration.
We claim that for any vertex (m, s) ∈ NΔop × S, the pair Fun♭Pair∞ ((Δm )♯ , Xs ) is a Waldhausen quasicategory.
Note that since Xs admits a zero object, so does Fun♭Pair∞ ((Δm )♯ , Xs ). For the remaining two axioms, one reduces
immediately to the case where m = 1.
Now, to see that pushouts along cofibrations exist, one may note that cofibations of Fun♭Pair∞ ((Δ1 )♯ , Xs ) are
in particular cofibrations of O(C ), for which the existence of pushouts is clear. Second, to see that a pushout of a
cofibration is again a cofibration, it suffices to see that a pushout of any edge of either of the classes (..) or (..) is
of the same class. For the class (..), this follows from the fact that pushouts in Fun♭Pair∞ ((Δ1 )♯ , Xs ) are computed
pointwise. A pushout of a morphism of the class (..) is a cube
X : (Δ1 )♯ × (Δ1 )♯ × (Δ1 )♭ .
Xs
in which the faces
X|((Δ1 )♯ × (Δ1 )♯ × Δ{0} ),
X|((Δ1 )♯ × Δ{0} × (Δ1 )♭ ),
and
X|((Δ1 )♯ × Δ{0} × (Δ1 )♭ )
are all pushouts. By uetzalcoatl, the face X|((Δ1 )♯ × (Δ1 )♯ × Δ{1} ) must be a pushout as well; this is precisely the
claim that the pushout is s-cocartesian.
For any m ∈ Δ and any edge f : s . t of S, since the functor fX ,! : Xs . Xt is exact, it follows directly that the
functor fF ,! : Fun♭Pair∞ ((Δm )♯ , Xs ) . Fun♭Pair∞ ((Δm )♯ , Xt ) is exact as well. Now for any fixed vertex s ∈ S0 and
any simplicial operator φ : n . m of Δ, the functor φF ,! : Fun♭Pair∞ ((Δm )♯ , Xs ) . Fun♭Pair∞ ((Δn )♯ , Xs ) visibly
carries cofibrations to cofibrations, and it preserves zero objects as well as any pushouts that exist, since limits and
colimits are formed pointwise.
□
... Corollary. e assignment (X /S) .
F (X /S) defines a functor
F : Waldcocart
.
∞
covering the endofunctor S .
Waldcocart
∞
NΔop × S of Cat∞ .
Now we wish to isolate a certain class of filtered object.
.. Definition. Suppose (X /S) a Waldhausen cocartesian fibration. A filtered object X : (Δm )♯ .
said to be totally filtered if X0 is a zero object of some fiber Xs .

X will be
.. Notation. Suppose (X /S) a Waldhausen cocartesian fibration. Denote by S (X /S) the full subcategory
of F (X /S) spanned by the totally filtered objects, and for any object m of Δ, write Sm (X /S) for the fiber of
S (X /S) . NΔop over m ∈ Δ. When S = Δ0 , write S (X ) for S (X /S), and for any object m of Δ, write
Sm (X ) for the fiber of S (X ) . NΔop over m ∈ Δ.
.. Proposition. Suppose (X /S) a Waldhausen cocartesian fibration. For any integer m ≥ 0, the 0-th face map
defines an equivalence of ∞-categories S1+m (X /S) . Fm (X /S), and the map S0 (X /S) . S is an equivalence.
We may li the pair structure on Fm (X /S) along this equivalence to obtain a pair structure on S1+m (X /S). One
sees that the inclusion
Jm : Sm (X /S) .
Fm (X /S) ≃ S1+m (X /S)
is a strict functor of pairs, and we deduce the following.
... Corollary. Suppose X . S a Waldhausen cocartesian fibration. For any integer m ≥ 0, the 0-th face map
defines an essentially surjective functor S1+m (X /S) . Sm (X /S) that is a le inverse in hPair∞ to the inclusion
Jm .
We now aim to show that for any Waldhausen cocartesian fibration X . S, the functor S (X /S) . NΔop × S
is a Waldhausen cocartesian fibration. For this purpose, it is convenient to study the mapping cylinder M (X /S)
of the functor S (X /S) . F (X /S).
.. Notation. For any Waldhausen cocartesian fibration X . S, write M (X /S) for the full subcategory of
Δ1 × F (X /S) spanned by those pairs (i, X) such that X is totally filtered if i = 1. is quasicategory comes
equipped with an inner fibration
M (X /S) .
Δ1 × NΔop × S.
Let M (X /S)† ⊂ M (X /S) be the subcategory whose edges are maps (i, X) .
is a cofibration of F (X /S).
(j, Y) such that i = j and X .
Y
Our first lemma is obvious by construction.
.. Lemma. For any Waldhausen cocartesian fibration X .
cartesian fibration.
S, the natural projection M (X /S) .
Δ1 is a pair
S, the natural projection M (X /S) .
Δ1 is a pair
Our next lemma, however, is subtler.
.. Lemma. For any Waldhausen cocartesian fibration X .
cocartesian fibration.
Proof. It suffices to show that for any vertex (m, s) ∈ (NΔop × S)0 , the inner fibration
q : Mm (Xs ) .
Δ1
is a pair cocartesian fibration. Note that an edge X . Y of Mm (Xs ) covering the nondegenerate edge σ of Δ1 is
q-cocartesian if and only if it is an initial object of the fiber Mm (Xs )X/ ×Δ1 {σ}. If m = 0, then the map
0/
M0 (Xs )X/ .
Δ10/
is a trivial fibration, so the fiber over σ is a contractible Kan complex. Let us now induct on m; assume that m > 0 and
that the functor p : Mm−1 (Xs ) . Δ1 is a cocartesian fibration. It is easy to see that the inclusion {0, 1, . . . , m − 1} .
induces an inner fibration φ : Mm (Xs ) . Mm−1 (Xs ) such that q = p ◦ φ. It suffices to observe that for any object
X of Mm (Xs ) and any p-cocartesian edge η : φ(X) . Y′ covering σ, there exists a φ-cocartesian edge X . Y of
Mm (Xs ) covering η.

m
We now show that q is a pair cocartesian fibration. Suppose
X.′
X.
.
Y.′
Y.
is a square of Mm (Xs ) in which X′ . X and Y′ . Y are q-cocartesian morphisms and X . Y is a cofibration.
We aim to show that for any edge η : Δ{p,q} . Δm , the morphism X′ |Δ{p,q} . Y′ |Δ{p,q} is a cofibration. For this,
we may factor X . Y as
X . Z . Y,
{0,...,p}
{0,...,p}
where Z|Δ
= Y|Δ
, and for any r > p, the edge X|Δ{p,r} . Z|Δ{p,r} is cocartesian. Now choose a
′
cocartesian morphism Z . Z as well. e proof is now completed by the following observations.
(..) Since the morphism X|Δ{p,q} . Z|Δ{p,q} is of type (..), it follows by uetzalcoatl X′ |Δ{p,q} . Z′ |Δ{p,q}
is of type (..) as well.
(..) e morphism Z|Δ{p,q} . Y|Δ{p,q} is of type (..) and the morphism Z′p . X′p is an equivalence; so
again by uetzalcoatl, the morphism Z′ |Δ{p,q} . Y′ |Δ{p,q} is of type (..).
□
Together, these lemmas exhibit, for any Waldhausen cocartesian fibration X .
F : F (X /S) .
S, an adjunction
S (X /S) : J
over NΔ × S in which both F and J are functors of pairs.
op
.. eorem. Suppose X . S a Waldhausen cocartesian fibration. en the functor S (X /S) .
Waldhausen cocartesian fibration.
NΔop × S is a
Proof. We first show that the functor S (X /S) . NΔop × S is a cocartesian fibration by proving the stronger
assertion that the inner fibration
p : M (X /S) . Δ1 × NΔop × S
is a cocartesian fibration. By ., the map
(..)
Δ{0} ×Δ1 M (X /S) .
Δ{0} × NΔop × S
is a cocartesian fibration. By ., for any vertex (m, s) ∈ (NΔop × S)0 , the map
(..)
M (X /S) ×NΔop ×S {(m, s)} .
Δ1 × {(m, s)}
is a cocartesian fibration. For any m ∈ Δ, the map
(Δ{1} × {m}) ×Δ1 ×NΔop M (X /S) .
Δ{1} × {m} × S
is easily seen to be a cocartesian fibration. Now one may complete the proof that p is a cocartesian fibration by showing
that for any vertex s ∈ S0 , any simplicial operator φ : n . m, and any totally m-filtered object X of Xs , then there
exists a p-cartesian morphism (1, X) . (1, Y) of F (X /S) covering (id1 , φ, ids ). [Exercise!]
From . and . it follows that the fibers of S (X /S) . NΔop × S are all Waldhausen ∞-categories. For any
m ∈ Δ and any edge f : s . t of S, the functor fX ,! : Xs . Xt is exact, whence it follows by . that the functor
fS ,! : Sm (Xs ) ≃ Fun♭Pair∞ ((Δm−1 )♯ , Xs ) .
Fun♭Pair∞ ((Δm−1 )♯ , Xt ) ≃ Sm (Xt )
is exact, just as in the proof of .. Now for any fixed vertex s ∈ S0 and any simplicial operator φ : n .
functor φS ,! : Sm (Xs ) . Sn (Xs ) is by construction the composite
Sm (Xs ) .
Jm,s
φF ,!
Fm (Xs ) .
Fn (Xs ) .
Fn,s
m of Δ, the
Sn (Xs ),
and as φF ,! is an exact functor (.), we are reduced to checking that the functors of pairs Jm,s and Fn,s are each exact
functors.

For this, it is clear that Jm,s and Fn,s each carry zero objects to zero objects, and as Fn,s is a le adjoint, it preserves
any pushout squares that exist in Fn (Xs ). Moreover, a pushout square in Sm (Xs ) is nothing more than a pushout
square in Fm (Xs ) of totally m-filtered objects; hence Jm,s preserves pushouts along cofibrations.
□
f(X ) : NΔop × S . Wald∞ for the functor classiFor any Waldhausen cocartesian fibration X . S, write S
f(X ) : NΔop × S . Wald∞
fied by the Waldhausen cocartesian fibration S (X /S) . NΔop × S, and write F
for the functor classified by the Waldhausen cocartesian fibration F (X /S) . NΔop × S. An instant consequence
of the construction of the functoriality of S in the proof above is the following.
... Corollary. e functors Fm : Fm (X /S) .
Sm (X /S) assemble to a natural transformation
f(X ) .
F: F
f(X ).
S
Note, however, that it is not the case that the functors Jm assemble to a natural transformation of this kind.
.. For any Waldhausen cocartesian fibration X . S, the functor NΔop × S . Wald∞ classified by the Waldhausen cocartesian fibration S (X /S) . NΔop × S assigns to any object (m, s) the quasicategory of totally filtered
objects
0 ≃ X0 . X1 . · · · . Xm
of Xs . For any morphism (φ, f) : (m, s) . (n, t) of NΔop × S, the induced functor carries the totally filtered object
X above to a representative of the totally filtered object
0 ≃ f⋆ (Xφ(0) /Xφ(0) ) .
··· .
f⋆ (Xφ(1) /Xφ(0) ) .
f⋆ (Xφ(n) /Xφ(0) ).
Giving a definition directly in this style requires solving certain homotopy-coherence problems. Waldhausen showed
that the solving these coherence problems amounts to making compatible choices of successive quotients. We have
chosen instead to avoid these issues by means of the theory of fibrations.
e assignments
(F (X /S)/(NΔop × S))
(X /S) .
and
(X /S) .
(S (X /S)/(NΔop × S))
define endofunctors of Waldcocart
over the endofunctor S . NΔop × S of Cat∞ . We may descend these functors
∞
to endofunctors of the quasicategory of virtual Waldhausen ∞-categories.
.. Lemma. e functors Wald∞ .
C.
Waldcocart
∞/NΔop given by
(F (C )/NΔop )
and
C.
and
S : Wald∞ .
(S (C )/NΔop )
preserve filtered colimits.
.. Construction. One may compose the functors
F : Wald∞ .
Wald∞,/NΔop
Wald∞,/NΔop
with the functor |·|NΔop ; the results are models for the functors Wald∞ . VWald∞ that assign to any Waldhausen
quasicategory the geometric realizations of the simplicial virtual Waldhausen ∞-categories F∗ (C ) and S∗ (C ). In
particular, these composites are ω-continuous functors Wald∞ . VWald∞ , whence one obtains essentially unique
endofunctors F and S of VWald∞ that preserve sied colimits such that the squares
Wald
. ∞
F
.
j
VWald
. ∞
Waldcocart
.
op
∞,/NΔ
F
| · |NΔop
Wald
. ∞
and
VWald
. ∞

Waldcocart
.
op
∞,/NΔ
.
j
VWald
. ∞
commute via a specified homotopy.
S
S
| · |NΔop
VWald
. ∞
.. Now the natural transformation F from Cor. .. also descends to a natural transformation F : F .
endofunctors of VWald∞ .
As it happens, the functor F : VWald∞ .
S of
VWald∞ is not particularly exciting:
.. Proposition. For any virtual Waldhausen quasicategory X , the virtual Waldhausen quasicategory F (X ) is
terminal.
Proof. For any Waldhausen quasicategory C , the virtual Waldhausen quasicategory |F (C )|NΔop is by definition a
ω
functor Wald∞
. Kan that assigns to any compact Waldhausen quasicategory Y the geometric realization of the
simplicial space
Δ
m . Wald∞
(Y , Fm (C )).
Δ
By ., this is the path space of the simplicial space m . Wald∞
(Y , Sm (C )).
□
is result permits us to regard the virtual Waldhausen quasicategory F (X ) as a cone on the virtual Waldhausen
quasicategory X . With this perspective, we will view the induced morphism F : F (X ) . S (X ) induced by the
functor F as a suitable “quotient” of F (X ) that identifies S (X ) as a suspension of X in a suitable quasicategory.
We shall return to this point in the next section.
e essential unicity of the extensions F and S to VWald∞ now implies the following.
.. Proposition. If S is a small sied quasicategory, then the squares
Wald.cocart
∞,/S
F
Waldcocart
. op ×S
∞,/NΔ
.
| · |S
VWald
. ∞
F
Wald.cocart
∞,/S
and
| · |NΔop ×S
S
.
| · |S
VWald
. ∞
VWald
. ∞
Waldcocart
. op ×S
∞,/NΔ
S
| · |NΔop ×S
VWald
. ∞
commute via a specified homotopy.
Of course this is no surprise for F : VWald∞ .
functor whose value at any object is terminal.
VWald∞ , as we have already seen that F is an essentially constant
.. Construction. Suppose C a Waldhausen category. For any integer m ≥ 0, one has, corresponding to the unique
edge m . 0 of Δ, exact functors
Em : C ≃ F0 (C ) .
Fm (C ) and E′m : 0 ≃ S0 (C ) .
Sm (C ).
We have a commutative square
F0 (C
. )
F0
S0 (C
. )
.
Em
Fm (C
. )
Fm
E′m
Sm (C
. )
e functor Em may equivalently be described as the m-th component of the counit of an adjunction
Waldcocart
∞/NΔop : R,
C : Wald∞ .
where C is the “constant” functor C . C × NΔop ; hence Em is natural in C and functorial in m ∈ NΔop . Consequently there is an induced natural transformation E : id . F of endofunctors of VWald∞ , and the square above
descends to a square of natural transformations
id.
(..)
E
F.
0.
.
F
S. ,
where 0 is the essentially constant endofunctor of VWald∞ whose value at any object is a zero object.
