LECTURE . EXCISION AND WALDHAUSEN’S FIBRATION THEOREM
By analyzing the Goodwillie derivative of the Yoneda embedding, we now find that a every distributive virtual Waldhausen ∞-category one step away from being an infinite loop object. is implies that the ∞-category
Vadd Wald∞ can be said to admit a much stronger form of the Blakers–Massey excision theorem than the ∞-category
of spaces. Armed with this, we give an easy necessary and sufficient criterion for a morphism of virtual Waldhausen
∞-categories to induce an equivalence on every additive theory.
ω
P(Wald∞
) carries finite products to finite products; hence
.. Notation. e Yoneda embedding y : Wald∞ .
it factors through a pre-additive theory
ω
P∗ (Wald∞
)
y : Wald∞ .
ω
and its le derived functor Y : VWald∞ . P∗ (Wald∞
), which is simply the canonical inclusion. Consequently,
thanks to Cor. ??, the additivization of y is now given by the formula
Dy ≃ Ω ◦ Y ◦ S ◦ j.
Let us give some equivalent descriptions of the functor Dy. First, for any Waldhausen ∞-category C , the object
ω,op
Dy(C ) is naturally equivalent to ΩS C , where S (C ) is regarded as a functor Wald∞
. Kan∗ . Since F (C ) is
contractible, a model for this loop object can be obtained as the fiber product
Dy(C ) ≃ F (C ) ×S (C ) F (C ).
ω,op
Alternately, since suspension in Vadd Wald∞ is given by S , the functor Dy(C ) : Wald∞
.
scribed by the formula
Kan∗ can be de-
Dy(C )(D) ≃ MapVWald∞ (S D, S C ).
ω
Lastly, since Vadd Wald∞ ⊂ P∗ (Wald∞
) is stable under limits, we may describe Dy as the composite functor
Wald∞ .
S
Vadd Wald∞ .
Ω
Vadd Wald∞ .
ω
P∗ (Wald∞
)
Ω
Vadd Wald∞ .
ω
P∗ (Wald∞
)
From this discussion, we record the following.
.. Proposition. e composite functor
Wald∞ .
S
Vadd Wald∞ .
is an additive theory.
ω
), it follows that ΩΣVadd Wald∞ is an excisive
Since Ω preserves sied colimits of connected objects of P∗ (Wald∞
ω
functor Vadd Wald∞ . P∗ (Wald∞ ). In particular, ΩΣVadd Wald∞ ≃ Ω∞ Σ∞
Vadd Wald∞ . We may regard this as
a very strong form of Blakers–Massey excision. More generally, we can attempt to study the circumstances under
which a sequence of virtual Waldhausen ∞-categories gives rise to a fiber sequence under any additive functor. In
this direction we have Pr. . below, which is an analogue of Waldhausen’s fibration theorem.
.. Notation. Suppose ψ : B . A an exact functor of Waldhausen ∞-categories. Write K (ψ) for the realization of the Waldhausen cocartesian fibration F A ×S A S B . NΔop .

.. Proposition (Fibration eorem I). Suppose ψ : B . A an exact functor of Waldhausen ∞-categories. en
for any additive theory φ : Wald∞ . E∗ with le derived functor Φ, there is a diagram
φ(B)
.
φ(A
. )
0.
.
0.
Φ(K. (ψ))
Φ(S (B)).
.
of E∗ in which each square is a pullback.
Proof. For any vertex m ∈ NΔop , there exist functors
s := (Em ⊕ Sm (ψ), pr2 ) : F0 (A ) ⊕ Sm (B) .
Fm (A ) ×Sm (A ) Sm (B)
and
p := (Im,0 ◦ pr1 ) ⊕ pr2 : Fm (A ) ×Sm (A ) Sm (B) .
F0 (A ) ⊕ Sm (B).
Clearly p ◦ s ≃ id; we claim that φ(s ◦ p) ≃ φ(id) in E∗ . is follows from additivity applied to the functor
Fm (A ) ×Sm (A ) Sm (B) .
F1 (Fm (A ) ×Sm (A ) Sm (B))
given by the cofibration of functors (Em ◦ Im,0 ◦ pr1 , 0) . id. us φ(Fm (A ) ×Sm (A ) Sm (B)) is exhibited as
the product φ(F0 (A )) × φ(Sm (B)).
us we may consider the following commutative diagram of E∗ :
φ(F0.(B))
F0 (ψ)
φ(F0 (A
. ))
F0
Em
φ(Fm.(B))
.
(0, Fm )
φ(Fm (A ) ×S.m (A ) Sm (B))
pr2 .
φ(Fm.(A ))
Fm
Im,0
.
φ(F0 (A
. ))
F0
φ(S0.(B))
pr1
E′m
φ(Sm.(B))
Sm (ψ)
φ(Sm.(A ))
I′m,0
φ(S0 (A
. )).
e lower right-hand square is a pullback square by additivity; hence, in light of the identification above, all the
squares on the right hand side are pullbacks as well. Again by additivity the wide rectangle of the top row is carried
to a pullback square under φ, whence all the squares of this diagram are carried to pullback squares.
Since φ is additive, so is Φ ◦ S . Hence we obtain a commutative diagram in E∗ :
Φ(S F. 0 (B))
Φ(S F. 0 (A ))
Φ(S S. 0 (B))
Φ(S F.m (B))
. (A ) Sm (B))
Φ(S Fm (A ) ×.S
m
Φ(S S.m (B))
.
Φ(S F.m (A ))
Φ(S S.m (A )),

in which every square is a pullback. All the squares in this diagram are functorial in m, and since the objects that
appear are all connected, it followsthat the squares of the colimi t diagram
Φ(S F. 0 (B))
Φ(S F. 0 (A ))
Φ(S S. 0 (B))
Φ(S F
. B)
.. (ψ))
Φ(S K
Φ(S S
. B)
.
Φ(S F
. A)
Φ(S S
. C ),
are all pullbacks. Applying the loopspace functor ΩE to this diagram now produces a diagram equivalent to the
diagram
φ(F0.(B))
φ(F0 (A
. ))
φ(S0.(B))
Φ(F. B)
Φ(K.. (ψ))
Φ(S. B)
.
Φ(F. A )
Φ(S. A ),
□
in which every square again is a pullback.
.. Given an exact functor ψ : B . A , the virtual Waldhausen ∞-category K (ψ) is the geometric realization
of the simplicial Waldhausen ∞-category whose m-simplices consist of a totally filtered object
0.
U1 .
U2 .
... .
Um
X2 .
... .
Xm
of B, a filtered object
X0 .
X1 .
of A , and a diagram
X.0
X.1
X.2
.
. ...
X.m
0.
ψ(U. 1 )
ψ(U. 2 )
. .. .
ψ(U. m )
of A in which every square is a pushout.
e object K (ψ) is not itself the corresponding fiber product of virtual Waldhausen ∞-categories; however, for
any additive functor φ : Wald∞ . E∗ with le derived functor Φ, the proof of the proposition above shows that
Φ(K (ψ)) is the fiber of Φ(S A ) . Φ(S B). is observation yields the following characterization of those
exact functors that are inverted by all additive theories.
.. Proposition. e following are equivalent for an exact functor ψ : B . A of Waldhausen ∞-categories.
(..) For any ∞-topos E and any φ ∈ Add(E ) with le derived functor Φ : VWald∞ . E∗ , the induced morphism Φ(ψ) : Φ(B) . Φ(A ) is an equivalence of E∗ .
(..) For any ∞-topos E and any φ ∈ Add(E ) with le derived functor Φ : VWald∞ . E∗ , the object Φ(K (ψ))
is contractible.
(..) e virtual Waldhausen ∞-category S K (ψ) is contractible.
Proof. In light of Pr. ., if (..) holds, then the induced morphism ΩS (ψ) : ΩS (B) . ΩS (A ) is an
ω
),
equivalence of virtual Waldhausen ∞-categories. Since S (B) and S (A ) are connected objects of P∗ (Wald∞
this in turn implies (using, say, [?, Cor. ...]) that the induced morphism S (ψ) : S (B) . S (A ) is an equivalence and therefore by Pr. . that (..) holds.
Now if (..) holds, then in particular, ΩS K (ψ) is contractible. Since S K (ψ) is connected, it is contractible, yielding (..).

at the last condition implies the first now follows immediately from Pr. ..

□