LECTURE . DISTRIBUTIVE VIRTUAL WALDHAUSEN ∞-CATEGORIES
ω
.. Definition. A presheaf X ∈ P(Wald∞
) is distributive if for every compact Waldhausen quasicategory C and
every integer m ≥ 0, the exact functors Em and Jm induce functors
X (Fm (C )) .
X (C )
X (Fm (C )) .
and
X (Sm (C ))
that together exhibit X (Fm (C )) as the product of X (C ) and X (S (C )m ).
ω
ω
.. Lemma. A presheaf X ∈ P(Wald∞
) is distributive only if X carries direct sums in Wald∞
to products — that
is, only if X is a virtual Waldhausen quasicategory.
Proof. Suppose C and D two compact Waldhausen ∞-categories. Consider the retract diagrams
C.
C ⊕. D
. E1
C ⊕. D
E1 ⊕ J1
C.
F1 (C . ⊕ D)
I1,0 ⊕ F1
C ⊕. D
and
D.
C ⊕. D
. J1
C ⊕. D
E1 ⊕ J1
D.
F1 (C . ⊕ D)
Here I1,0 is the functor induced by the morphism 0 .
X , we have an induced retract diagram
I1,0 ⊕ F1
C ⊕. D.
0. For any distributive virtual Waldhausen quasicategory
X (C .⊕ D)
X (F1 (C. ⊕ D))
.
X (C .⊕ D)
X (C ) ×. X (D)
X (C ⊕ D) ×. X (C ⊕ D)
X (C ) ×. X (D).
(..)
Since the center vertical map is an equivalence, and since equivalences are closed under retracts, so are the outer
vertical maps.
□
.. Notation. Denote by Vadd Wald∞ the full subcategory of VWald∞ spanned by the distributive functors.
Since sied colimits in VWald∞ commute with products, we deduce the following.
.. Lemma. e subcategory Vadd Wald∞ ⊂ VWald∞ is stable under sied colimits.
.. Note that representable presheaves are typically not distributive. Consequently, the obvious fully faithful incluω
sion Wald∞
. VWald∞ does not factor through Vadd Wald∞ ⊂ VWald∞ .
.. Proposition. e inclusion functor admits a le adjoint
Ladd : VWald∞ .
Vadd Wald∞ ,
which exhibits Vadd Wald∞ as an accessible localization of VWald∞ .

Proof. For any compact Waldhausen quasicategory C and every integer m ≥ 0, consider the exact functor
Em ⊕ Jm : C ⊕ Sm (C ) .
Fm (C );
ω
let S be the set of morphisms of VWald∞ of this form; let S be the strongly saturated class it generates. Since Wald∞
is essentially small, the class S is of small generation. Hence we may form the accessible localization S−1 VWald∞ .
ω,op
Since virtual Waldhausen ∞-categories are functors X : Wald∞
. Kan that preserve products, one sees that
−1
S VWald∞ coincides with the full subcategory Vadd Wald∞ ⊂ VWald∞ .
□
e fully faithful inclusion Vadd Wald∞ . VWald∞ preserve finite products, and its le adjoint Ladd preserve
finite coproducts, whence we deduce the following.
... Corollary. e quasicategory Vadd Wald∞ is compactly generated and admits finite direct sums, which are preserved by the inclusion Vadd Wald∞ . VWald∞
A large portion of the usefulness of the quasicategory Vadd Wald∞ of distributive virtual Waldhausen ∞-categories
is derived from the relationship it bears to the endofunctor S on VWald∞ . e first indication of such a close relationship is the following result.
.. Proposition. e diagram
S
VWald
. ∞
VWald
. ∞
.
Ladd
Vadd Wald
. ∞
Ladd
ΣVadd Wald∞
Vadd Wald
. ∞
commutes (up to homotopy), where ΣVadd Wald∞ is the suspension endofunctor on Vadd Wald∞ .
Proof. We have a square
Ladd
.
0.
(..)
.
Ladd .◦ F
F
Ladd ◦. S ,
of natural transformations between functors VWald∞ . Vadd Wald∞ . Since F is essentially constant with value
the zero object, this gives rise to a natural transformation Σ ◦ Ladd . Ladd ◦ S . To see that this natural transformation is an equivalence, it suffices to consider its value on a compact Waldhausen quasicategory C . Now for any
m ∈ NΔop , we have a diagram
Ladd S. 0 (C )
J0
E′m
Ladd S.m (C )
Jm
Ladd F. 0 (C )
Em .
Ladd F.m (C )
F0
Ladd S. 0 (C )
E′m
Fm
Ladd S.m (C )
of Waldhausen ∞-categories in which the horizontal composites are equivalences. Since S0 (C ) is a zero object, the
le-hand square is a pushout by definition; hence the right-hand square is as well. e geometric realization of the
right-hand square is precisely the value of the square (..) on C .
□
.. Construction. Suppose m ≥ 0 an integer, and suppose 0 ≤ k ≤ m. Write ik : 0 . m for the morphism
of Δ that carries 0 ∈ 0 to k ∈ m. For any Waldhausen quasicategory C , write Im,k for the induced functor
Fm (C ) . F0 (C ), and write I′m,k for the induced functor Sm (C ) . S0 (C ).

For future reference, we note that we may now contemplate diagrams (Δ2 /Δ{0,2} ) × (Δ2 /Δ{0,2} ) .
of the form
J0
S0 (C
. )
E′m
(..)
F0
F0 (C
. )
S0 (C
. )
E′m
Em
Sm (C
. )
Jm
I′m,k
S0 (C
. )
J0
Fm (C
.. )
Fm
Sm (C
. )
Im,k
F0 (C
. )
Wald∞
Im,k
S0 (C
. ).
F0
We observe that F0 (C ) ≃ C and that S0 (C ) is a zero object. Only the upper right square of (..) is functorial
in m.
We may now apply the localization functor Ladd to (..). In the resulting diagram
Ladd S. 0 (C )
J0
Ladd F. 0 (C )
E′m
(..)
F0
Ladd S. 0 (C )
E′m
Em
Ladd S.m (C )
I′m,k
Ladd S. 0 (C )
Ladd F..m (C )
Jm
Fm
Ladd S.m (C )
Im,k
Ladd F.
J0
Im,k
0 (C )
F0
Ladd S.0 (C ),
the square in the upper le corner is a pushout, whence every square is a pushout.
.. Proposition. For any virtual Waldhausen ∞-category X , the virtual Waldhausen ∞-category S X is distributive.
Proof. Select a Waldhausen cocartesian fibration Y . NΔop such that X is equivalent to |Y |NΔop . e claim is
that for any compact Waldhausen ∞-category C and any integer m ≥ 0, the map
H(Fm (C ), (S Y /NΔop × NΔop )) .
H(C , (S Y /NΔop × NΔop )) × H(Sm (C ), (S Y /NΔop × NΔop ))
is an equivalence. Since NΔop is sied and geometric realization commutes with products, we reduce to the case in
which Y is a single Waldhausen ∞-category.
Let us now use Joyal’s ∞-categorical variant of uillen’s eorem A (see HTT, ...). For any object
((p, α), (q, β)) ∈ H(C , (S Y /NΔop )) × H(Sm (C ), (S Y /NΔop )),
write J((p, α), (q, β)) for the pullback
J((p, α),. (q, β))
H(C , (S Y /NΔop )) × H(S
. m (C ), (S Y /NΔop ))
.
H(Fm (C ), (S
. Y /NΔop ))
(H(C , (S Y /NΔop )) × H(Sm .(C ), (S Y /NΔop )))((p,α),(q,β))/
We may identify J((p, α), (q, β)) with a quasicategory whose objects are tuples (r, γ, μ, ν, σ, τ), where r ≥ 0 is
an integer, γ : Sm (C ) . Sr Y is an exact functor, μ : [r] . [p] and ν : [r] . [q] are morphisms of Δ, and
σ : μ∗ α . ∼ γ|C and τ : ν∗ β . ∼ γ|Sm (C ) are equivalences of exact functors.
Denote by κ the constant functor J((p, α), (q, β)) . J((p, α), (q, β)) at the object
(0, 0, {0} .
[p], {0} .
[q], 0, 0).
To prove that J((p, α), (q, β)) is contractible, we construct an endofunctor λ and natural transformations
id .
λ.

κ.
For any integer r ≥ 0, consider 1 + r as the category [r] ; we define the functor λ by
λ(r, γ, μ, ν, σ, τ) := (1 + r, s0 ◦ γ, μ′ , ν′ , σ ′ , τ ′ ),
where μ′ |[r] = μ and μ′ (−∞) = 0, ν′ |[r] = ν and ν′ (−∞) = 0, and σ ′ and τ ′ are the obvious extensions of σ and
τ. e inclusion r . 1 + r induces a natural transformation λ . id, and the inclusion {−∞} . 1 + r induces
a natural transformation λ . κ.
□
... Corollary. If X is a virtual Waldhausen quasicategory, then the virtual Waldhausen quasicategory S (X ) is
distributive.
We thus have the following enhancement of Pr. ..
... Corollary. e diagram
VWald
. ∞
Ladd
.
S
.
. Vadd Wald
. ∞
ΣVadd Wald∞
Vadd Wald
. ∞
.
commutes (up to homotopy), where ΣVadd Wald∞ is the suspension endofunctor on Vadd Wald∞ .
