headings=small

# A note on coCartesian fibrations

Hoang Kim Nguyen

We prove properness of (co)Cartesian fibrations as well as a straightening and
unstraightening equivalence, which is compatible with cartesian products, when
the base is the nerve of a small category.

###### Contents

  1. 1 Introduction
  2. 2 Reminder on the coCartesian model structure
  3. 3 Invariance properties of the (co)Cartesian model structure
  4. 4 Properness of (co)Cartesian fibrations
  5. 5 Straightening and unstraightening

##  1 Introduction

######  1.1.

An essential tool in higher category theory is the theory (co)Cartesian
fibrations; the higher categorical analogue of Grothendieck (op)fibrations. By
Lurie’s straightening equivalence, coCartesian fibrations correspond to
functors to the $\infty$-category of $\infty$-categories and manipulating
coCartesian fibrations is often the preferred way to construct such functors.

######  1.2.

The first proof of the straightening equivalence was given by Lurie in [Lur09]
and relies on a comparison with simplicial categories. Lurie then uses the
straightening equivalence to prove important fundamental properties of
coCartesian fibrations. An efficient streamlined proof has appeared in
[HHR21]. This proof also uses the comparison with simplicial categories but
proves the fundamental properties of coCartesian fibrations first and derives
the equivalence from there.

######  1.3.

In upcoming joint work with Denis-Charles Cisinski we provide a new
construction of the $\infty$-category of $\infty$-categories inspired by the
construction of universes in semantic models of type theory. The straightening
equivalence is then derived from a property of the universe of
$\infty$-categories: _directed univalence_.

######  1.4.

The present article serves as backbone to the above mentioned work on directed
univalence. Similar to [HHR21], we will prove fundamental properties of
coCartesian fibrations from scratch using only methods from (marked)
simplicial sets. As such, there is some overlap with [HHR21] and we indicate
whenever this is the case. Since the purpose of this article is to lay the
technical foundations to prove a straightening theorem, we are careful to
prove everything we need without using a straightening theorem.

######  1.5.

The main contributions of this article are the following:

  * •

A proof that pullback along (co)Cartesian fibrations preserves Joyal
equivalences (Theorem 4.7). This has been proven by Lurie [Lur09, Corollary
3.3.1.2] using the full power of his straightening equivalence. Although we
don’t need this result, a weaker assertion which is much easier to prove
suffices for us (Proposition 2.10), we are not aware of a proof of this result
using only elementary methods, so we include here.

  * •

A proof of the straightening equivalence for nerves of categories (Theorem
5.2). We can’t completely avoid straightening, but this version avoids
simplicial categories. This is not the full straightening equivalence as it
only applies when the base is the nerve of a small category. The upshot
however is that the proof is relatively short and we also prove a
compatibility with cartesian products.

### Acknowledgements

The author would like to thank Denis-Charles Cisinski for many useful
discussions and for his encouragement. Most of this work has been completed
when the author was a member of the _SFB 1085 Higher Invariants_ funded by the
_Deutsche Forschungsgesellschaft (DFG)_.

##  2 Reminder on the coCartesian model structure

######  2.1.

We denote $\mathbf{sSet}^{+}$ the category of marked simplicial sets. It’s
objects are given by pairs $(K,E_{K})$ where $K$ is a simplicial set and
$E_{K}$ is a set of 1-simplices of $K$ containing all the degenerate
1-simplices, called _marked edges_. It’s morphisms are maps of simplicial sets
preserving the marked edges. In general we will denote a marked simplicial set
by $K^{+}$.

######  2.2.

The forgetful functor

$\mathbf{sSet}^{+}\to\mathbf{sSet}$

has both a left and a right adjoint. We denote the left adjoint by

$(\cdot)^{\flat}\colon\mathbf{sSet}\to\mathbf{sSet}^{+}$

Given a simplicial set $A$, the marked simplicial set $A^{\flat}$ has
precisely the degenerate 1-simplices marked. The right adjoint will be denoted
by

$(\cdot)^{\sharp}\colon\mathbf{sSet}\to\mathbf{sSet}^{+}$

Given a simplicial set $B$, the marked simplicial set $B^{\sharp}$ has all
1-simplices marked.

######  2.3.

The functor $(\cdot)^{\sharp}$ has a further right adjoint, denoted by

$\mu\colon\mathbf{sSet}^{+}\to\mathbf{sSet}$

Given a marked simplicial set $A^{+}$, the simplicial set $\mu(A^{+})$ is the
simplicial subset of $A$ spanned by the marked edges.

######  Definition 2.4.

We define the class of _marked left anodyne extensions_ to be the smallest
saturated class containing the morphisms

  * (A1)

$(\Lambda^{n}_{k})^{\flat}\to(\Delta^{n})^{\flat}$ for $n\geq 2$ and $0<k<n$,

  * (A2)

$J^{\flat}\to J^{\sharp}$,

  * (B1)

$(\Delta^{1})^{\sharp}\times(\Delta^{1})^{\flat}\cup\\{0\\}\times(\Delta^{1})^{\sharp}\to(\Delta^{1})^{\sharp}\times(\Delta^{1})^{\sharp}$,

  * (B2)

$(\Delta^{1})^{\sharp}\times(\partial\Delta^{n})^{\flat}\cup\\{0\\}\times(\Delta^{n})^{\flat}\to(\Delta^{1})^{\sharp}\times(\Delta^{n})^{\flat}$.

######  Definition 2.5.

A map $X^{+}\to A^{+}$ is called a _marked left fibration_ if it has the right
lifting property with respect to the class of marked left anodyne extensions.
A marked simplicial set $X^{+}$ is called _marked left fibrant_ if the map
$X^{+}\to\Delta^{0}$ is a marked left fibration.

######  2.6.

There are dual notions of marked _right_ anodyne extensions and marked _right_
fibrations. The class of marked right anodyne extensions has generators (A1)
and (A2) of Definition 2.4 and the classes

  * (B1’)

$(\Delta^{1})^{\sharp}\times(\Delta^{1})^{\flat}\cup\\{1\\}\times(\Delta^{1})^{\sharp}\to(\Delta^{1})^{\sharp}\times(\Delta^{1})^{\sharp}$,

  * (B2’)

$(\Delta^{1})^{\sharp}\times(\partial\Delta^{n})^{\flat}\cup\\{1\\}\times(\Delta^{n})^{\flat}\to(\Delta^{1})^{\sharp}\times(\Delta^{n})^{\flat}$.

The marked right fibrations have the right lifting property against the marked
right anodyne extensions.

######  Theorem 2.7.

Let $A^{+}$ be a marked simplicial set. Then there is a unique model structure
on the category $\mathbf{sSet}^{+}/A^{+}$ with

  1. 1.

cofibrations given by maps whose underlying map of simplicial sets is a
monomorphism,

  2. 2.

fibrant objects given by marked left fibrations $X^{+}\to A^{+}$.

Moreover, the fibrations between fibrant objects are precisely the marked left
fibrations.

###### Proof.

See [Ngu19, Theorem 4.29] for this precise statement. An alternative proof of
the existence of this model structure when $A^{+}=A^{\sharp}$ is in [Lur09,
Proposition 3.1.3.7]. ∎

######  Definition 2.8.

We will call the model structure on $\mathbf{sSet}^{+}/A^{+}$ the _coCartesian
model structure_. We denote its homotopy category by $\mathbf{coCart}(A^{+})$.
The dual model structure will be called the _Cartesian model structure_ and
its homotopy category is denoted by $\mathbf{Cart}(A^{+})$.

######  2.9.

A useful property of coCartesian fibrations is that pullback along them
preserves _cellular marked right anodyne extensions_ , which are those marked
right anodyne extensions lying in the saturated class generated by the sets
(B1) and (B2) in Definition 2.4.

######  Theorem 2.10.

Consider a pullback square of marked simplicial sets

${X^{+}}$${Y^{+}}$${A^{+}}$${B^{+}}$$\scriptstyle{j}$$\scriptstyle{p}$$\scriptstyle{q}$$\scriptstyle{i}$

where $p$ and $q$ are marked left fibrations and $i$ is a cellular marked
right anodyne extension. Then $j$ is a marked right anodyne extension. The
dual statement for marked right fibrations and cellular marked left anodyne
extensions also holds.

###### Proof.

See [Ngu19, 4.45]. ∎

######  2.11.

One of the main goals of this article is to extend this Theorem to general
marked right anodyne extensions, see section 4. The main difficulty is to
prove that pulling back along (co)Cartesian fibrations preserves Joyal
equivalences. A proof of this fact using straightening/unstraightening can be
found in [Lur17, Appendix B.3]. However, the goal of this note is to prove
properties of (co)Cartesian fibrations without straightening/unstraightening
and instead (eventually) derive it as a consequence. A proof for left/right
fibrations without straightening/unstraightening has appeared in [Cis19,
Proposition 5.3.5] and we will make use of this fact in the following
Proposition.

######  Proposition 2.12.

In the Theorem above, if the underlying map of simplicial sets $q\colon Y\to
B$ is a left (resp. right) fibration and $i$ is a marked right (resp. left)
anodyne extension, then the map $j$ is a marked right (resp. left) anodyne
extension.

###### Proof.

It suffices to show this when the map $i\colon A^{+}\to B^{+}$ belongs to the
class of generators for marked right anodyne extensions.

We first show this for the class (A1). We need to show that for any diagram of
pullback squares of the form

${Y^{+}}$${X^{+}}$${(\Lambda^{n}_{k})^{\flat}}$${(\Delta^{n})^{\flat}}$$\scriptstyle{j}$$\scriptstyle{i}$

where the vertical maps are marked left fibrations with underlying map of
simplicial sets being left fibrations and $i$ is inner horn inclusion, the map
$j$ is a marked right anodyne extension. We observe that the marked simplicial
sets $Y^{+}$ and $X^{+}$ have precisely the equivalences marked. In particular
$X^{+}$ is fibrant over the point. Since the underlying map of simplicial sets

$Y\to X$

is a Joyal trivial cofibration by [Cis19, Proposition 5.3.5], the map

$Y^{\flat}\to X^{\flat}$

is a trivial cofibration over the point. We have a square

${Y^{\flat}}$${X^{\flat}}$${Y^{+}}$${X^{+}}$

Here, the vertical maps are marked left anodyne, since they are given by
marking equivalences and the upper horizontal map is a trivial cofibration.
Thus the lower horizontal map is a trivial cofibration over the point. Since
$X^{+}$ is fibrant, this map is in fact marked right anodyne by [Ngu19, 2.31].

For the class (A2) it suffices to show that for a pullback square

${Y^{+}}$${X^{\sharp}}$${J^{\flat}}$${J^{\sharp}}$$\scriptstyle{i}$

the map $i$ is marked left anodyne. Clearly, the map $i$ is the identity on
underlying simplicial sets and $i$ is obtained by marking equivalences, thus
is marked left anodyne.

The classes (B1’) and (B2’) follow from Theorem 2.10 since they are cellular
marked right anodyne. ∎

##  3 Invariance properties of the (co)Cartesian model structure

######  3.1.

The goal of this section is to show that for any Joyal equivalence $A\to B$ we
obtain a Quillen equivalence of coCartesian (resp. Cartesian) model structures
$\mathbf{sSet}^{+}/(A^{\sharp})\to\mathbf{sSet}^{+}/(B^{\sharp})$.

######  Theorem 3.2.

Let $i\colon A\to B$ be inner anodyne. Then the induced functor

$i_{!}\colon\mathbf{sSet}^{+}(A^{\sharp})\to\mathbf{sSet}^{+}(B^{\sharp})$

is a Quillen equivalence for the coCartesian and Cartesian model structures.

###### Proof.

We only prove the coCartesian case as the Cartesian case is analogous. It is
clear that $i_{!}$ is left Quillen, thus it suffices to show that the left
derived functor $\mathbf{L}i_{!}$ is an equivalence of categories. Since $i$
is a bijection on objects, the right derived functor $\mathbf{R}i^{*}$ is
conservative, hence it suffices to show that $\mathbf{L}i_{!}$ is fully
faithful.

Consider a commutative square

${X^{\natural}}$${Y^{\natural}}$${A^{\sharp}}$${B^{\sharp}}$$\scriptstyle{j}$$\scriptstyle{p}$$\scriptstyle{q}$$\scriptstyle{i}$

in which $p$ and $q$ are coCartesian fibrations and $j$ is marked left
anodyne. We prove that the induced map on fibers is an equivalence (in the
(co)Cartesian model structure over the point). Consider a point
$a\colon\Delta^{0}\to A$. Choose a commutative square

${\Delta^{0}}$${E^{\sharp}}$${A^{\sharp}}$${B^{\sharp}}$$\scriptstyle{u}$$\scriptstyle{f}$

in which $u$ is cellular marked right anodyne and $f$ is a marked right
fibration and denote $F^{\sharp}:=A^{\sharp}\times_{B^{\sharp}}E^{\sharp}$.
Note that the induced map $\Delta^{0}\to D^{\sharp}$ is cellular marked right
anodyne by [Cis19, Theorem 5.2.14] and $i$ is a marked left anodyne extension.
It follows again from [Cis19, Theorem 5.2.14] that the pullback square

${F^{\sharp}}$${E^{\sharp}}$${A^{\sharp}}$${B^{\sharp}}$

has horizontal arrows cellular marked right fibrations and vertical arrows
marked right fibrations. Pulling back the maps $p$ and $q$ along this pullback
square, we obtain a commutative diagram of pullback squares

${X^{\natural}_{a}}$${Y^{\natural}_{a}}$${X^{\natural}_{F}}$${Y^{\natural}_{E}}$${X^{\natural}}$${Y^{\natural}}$

Here we define

$X^{\natural}_{F}:=F^{\sharp}\times_{A^{\sharp}}X^{\natural},\quad
Y^{\natural}_{E}:=E^{\sharp}\times_{B^{\sharp}}Y^{\natural}$

The maps

$X^{\natural}_{F}\to X^{\natural},\quad Y^{\natural}_{E}\to Y^{\natural}$

are marked right fibrations whose underlying map of simplicial sets are right
fibrations. Since $X^{\natural}\to Y^{\natural}$ was assumed to be marked left
anodyne it follows from Proposition 2.12 that

$X^{\natural}_{F}\to Y^{\natural}_{E}$

is marked left anodyne. Also by Theorem 2.10 the maps

$X^{\natural}_{a}\to X^{\natural}_{F},\quad Y^{\natural}_{a}\to
Y^{\natural}_{E}$

are marked right anodyne extensions. Since marked right anodyne and marked
left anodyne extensions are in particular weak equivalences in the
(co)Cartesian model structure over the point, it follows by 2-out-of-3 that
the map

$X^{\natural}_{a}\to Y^{\natural}_{a}$

is an equivalence. This shows that the derived unit is an isomorphism and
hence $\mathbf{L}i_{!}$ is fully faithful. ∎

######  Corollary 3.3.

Let

${Y^{\natural}}$${X^{\natural}}$${A^{\sharp}}$${B^{\sharp}}$$\scriptstyle{j}$$\scriptstyle{i}$

be a pullback square of marked simplicial sets. Suppose $i$ is inner anodyne
and $X^{\natural}\to B^{\sharp}$ is a marked left (resp. right) fibration.
Then the map $j$ is a marked left (resp. right) anodyne extension.

###### Proof.

We show this for marked left fibrations. Since marked left anodyne extensions
are saturated, it suffices to show this for inner horn inclusions. By the
previous Theorem 3.2, the functor

$i^{\ast}\colon\mathbf{sSet}^{+}/(\Delta^{n})^{\sharp}\to\mathbf{sSet}^{+}/(\Lambda_{k}^{n})^{\sharp}$

is a Quillen equivalence for the coCartesian model structures. Hence the map
$j$ is a trivial cofibration over $B^{\sharp}$ with fibrant target, thus by
[Ngu19, 2.31] a marked left anodyne extension. ∎

######  Corollary 3.4.

Let $i\colon A\to B$ be a Joyal equivalence. Then the induced functor

$i_{!}\colon\mathbf{sSet}^{+}/A^{\sharp}\to\mathbf{sSet}^{+}/B^{\sharp}$

is a Quillen equivalence when both categories are endowed with the
(co)Cartesian model structure.

###### Proof.

Let $W$ be the class of morphisms for which $i_{!}$ is a Quillen equivalence.
We want to show that the class $W$ contains the Joyal equivalences. According
to [Cis19, 3.6.2] it suffices to show that

  1. 1.

$W$ is closed under 2-out-of-3,

  2. 2.

$W$ contains the inner anodyne extensions

  3. 3.

$W$ contains the trivial fibrations.

The first assertion follows from the fact that Quillen equivalences are closed
under 2-out-of-3. The second assertion follows from the previous Theorem and
the third assertion is clear. ∎

######  Remark 3.5.

A proof along similar lines has appeared as [HHR21, Theorem 5.15].

##  4 Properness of (co)Cartesian fibrations

######  4.1.

In this section we prove that the pullback of an inner anodyne map along a
(co)Cartesian fibration is a Joyal equivalence. This has first appeared in the
literature in [Lur09, Proposition 3.3.1.3], but the proof uses the
straightening/unstraightening equivalence. Our proof will only use elementary
properties of (locally) coCartesian fibrations. Together with Theorem 2.10,
this generalizes Proposition 2.12.

######  4.2.

Recall that a morphism of simplicial sets $i\colon A\to B$ is called _final_ ,
if for any morphism $f\colon B\to C$, the induced morphism
$i\colon(A,fi)\to(B,f)$ in the slice $\mathbf{sSet}/C$ is a Contravariant
equivalence. A monomorphism is final if and only if it is right anodyne, see
[Cis19, Corollary 4.1.9].

Furthermore recall that a morphism $p\colon X\to Y$ is called _proper_ , if
for any diagram

${A^{\prime}}$${B^{\prime}}$${X}$${A}$${B}$${Y}$$\scriptstyle{i^{\prime}}$$\scriptstyle{p}$$\scriptstyle{i}$

in which the squares are pullbacks and the map $i$ is final, it follows that
$i^{\prime}$ is final. Examples of proper morphisms are left fibrations, see
[Cis19, Proposition 4.4.11] and coCartesian fibrations, see [Ngu19, Corollary
4.46] (or [Lur09, Proposition 4.1.2.15] using straightening).

######  Theorem 4.3.

Let $p\colon X\to Y$ be an inner fibration of $\infty$-categories. Then $p$ is
proper if and only if in any diagram

${A^{\prime}}$${B^{\prime}}$${X}$${\\{1\\}}$${\Delta^{1}}$${Y}$$\scriptstyle{i^{\prime}}$$\scriptstyle{p^{\prime}}$$\scriptstyle{p}$

in which the squares are pullbacks, the map $i^{\prime}$ is final.

###### Proof.

This is [Cis19, 4.4.36]. ∎

######  Corollary 4.4.

Locally coCartesian fibrations between $\infty$-categories are proper.

###### Proof.

In the diagram of the theorem, if $p$ is locally coCartesian then the pullback
$p^{\prime}$ is a coCartesian fibration. Since coCartesian fibrations are
proper, the assertion follows. ∎

######  Lemma 4.5.

Suppose we have a diagram

${X}$${Y}$${S}$$\scriptstyle{i}$$\scriptstyle{p}$$\scriptstyle{q}$

in which $p$ is a left fibration, $i$ is a trivial cofibration of the Joyal
model structure and $q$ is a Joyal fibration. Then $q$ is also a left
fibration.

###### Proof.

Choose a factorization

${Y}$${Z}$${S}$$\scriptstyle{j}$$\scriptstyle{q}$$\scriptstyle{r}$

where $j$ is left anodyne and $r$ is a left fibration. Since Joyal
equivalences are cofinal [Cis19, Proposition 5.3.1], the composition $ji$ is
cofinal. Since this determines a Covariant equivalence between the left
fibrations $p$ and $r$, it follows that $ji$ is in fact a Joyal equivalence.
Since $i$ was assumed to be a Joyal equivalence, it follows that $j$ is also a
Joyal equivalence. By the Retract Lemma, the map $q$ is a retract of $r$ and
thus a left fibration. ∎

######  Lemma 4.6.

Consider the commutative diagram

${E^{\natural}}$${X^{\natural}}$${Y^{\natural}}$${(\Lambda^{n}_{k})^{\sharp}}$${(\Delta^{n})^{\sharp}}$$\scriptstyle{e}$$\scriptstyle{p}$$\scriptstyle{j}$$\scriptstyle{q}$$\scriptstyle{i}$

in which the lower square is a pullback, each vertical arrow is a marked left
fibration and the map of simplicial sets $i$ is inner anodyne. Assume
furthermore that the underlying map of $e$ is a left fibration of simplicial
sets. Choose a factorization

${E^{\natural}}$${F^{\natural}}$${X^{\natural}}$${Y^{\natural}}$$\scriptstyle{e}$$\scriptstyle{k}$$\scriptstyle{f}$$\scriptstyle{j}$

where $k$ is marked left anodyne and $f$ is a marked left fibration. Then the
underlying map of $f$ is a locally coCartesian fibration. Moreover, for each
0-simplex $x\colon\Delta^{0}\to X$, the induced map on fibers $E_{x}\to F_{x}$
is a Joyal equivalence.

###### Proof.

Note that the markings on $E^{\natural}$ correspond to the $pe$-coCartesian
edges and that markings on $F^{\natural}$ correspond to the $qf$-coCartesian
edges. First we observe that the base-change map

$E^{\natural}\to\left(\Lambda^{n}_{i}\right)^{\sharp}\times_{\left(\Delta^{n}\right)^{\sharp}}F^{\natural}$

is marked left anodyne by Corollary 3.2. Consequently, for each object
$m\in\Delta^{n}$, the induced map on fibers

$E_{m}\to F_{m}$

is a Joyal equivalence.

Next we observe that $f$ is an isofibration, since the marked edges in
$Y^{\natural}$ are precisely the $q$-coCartesian edges and thus in particular
the equivalences are marked. We obtain a commutative diagram

${E_{m}}$${F_{m}}$${X_{m}}$$\scriptstyle{k_{m}}$$\scriptstyle{e_{m}}$$\scriptstyle{f_{m}}$

where $k_{m}$ is a trivial cofibration of the Joyal model structure, $e_{m}$
is a right fibration (by assumption) and $f_{m}$ is an isofibration between
$\infty$-categories, thus a Joyal fibration. By Lemma 4.5 the map $f_{m}$ is
thus a left fibration. In other words, the map $F\to X$ induces for each
$m\in\Delta^{n}$ a left fibration on fibers $F_{m}\to X_{m}$. Now since
$k_{m}$ is a Joyal equivalence, it is in particular a Covariant equivalence
between the left fibrations $e_{m}$ and $f_{m}$ and thus a fiberwise
equivalence.

It remains to show that $f\colon F\to Y$ is locally coCartesian. By
construction, we have a commutative diagram

${F}$${Y}$${\Delta^{n}}$$\scriptstyle{f}$$\scriptstyle{qf}$$\scriptstyle{q}$

with $qf$ and $q$ being coCartesian fibrations and $f$ sending
$qf$-coCartesian edges to $q$-coCartesian edges. Thus by [Lur09, 2.4.2.11] the
map $f$ is locally coCartesian since the maps $f_{m}\colon F_{m}\to Y_{m}$ are
in fact left fibrations. ∎

######  Theorem 4.7.

Suppose we have a pullback square

${X}$${Y}$${A}$${B}$$\scriptstyle{p}$$\scriptstyle{j}$$\scriptstyle{q}$$\scriptstyle{i}$

in which $p$ and $q$ are coCartesian fibrations and $i$ is inner anodyne. Then
$j$ is a Joyal equivalence.

###### Proof.

It suffices to show the assertion for squares of the form

${X}$${Y}$${\Lambda^{n}_{k}}$${\Delta^{n}}$$\scriptstyle{p}$$\scriptstyle{j}$$\scriptstyle{q}$$\scriptstyle{i}$

with $i$ an inner horn inclusion.

According to [Cis19, 5.2.16] the map $j$ is a Joyal equivalence if and only if
it induces an essentially surjective functor on homotopy categories and it
induces a fully faithful functor

$\mathbf{L}j_{!}\colon\mathbf{LFib}(X)\to\mathbf{LFib}(Y)$

where $\mathbf{LFib}(X)$ denotes the homotopy category of the covariant model
structure on simplicial sets over $X$. Since $j$ is a pullback of an inner
anodyne map, it is a bijection on objects, thus clearly essentially
surjective. For the second condition, we need to show that the derived counit

$id\to\mathbf{R}j^{\ast}\mathbf{L}j_{!}$

is an isomorphism in $\mathbf{LFib}(X)$. Let $e\colon E\to X$ be a left
fibration. To prove that the derived counit is an isomorphism, we construct a
particular fibrant replacement of the composition $E\to X\to Y$ in the
covariant model structure over $Y$.

To start off, we have a pullback square in marked simplicial sets

${X^{\natural}}$${Y^{\natural}}$${(\Lambda^{n}_{k})^{\sharp}}$${(\Delta^{n})^{\sharp}}$$\scriptstyle{p}$$\scriptstyle{j}$$\scriptstyle{q}$$\scriptstyle{i}$

in which $p$ and $q$ are marked left fibrations. Since $e$ is a left
fibration, we have a marked left fibration $e\colon E^{\sharp}\to X^{\sharp}$
and pulling back along the inclusion $X^{\natural}\to X^{\sharp}$, we obtain a
marked left fibration $E^{\natural}\to X^{\natural}$. We thus have a diagram

${E^{\natural}}$${X^{\natural}}$${Y^{\natural}}$${(\Lambda^{n}_{k})^{\sharp}}$${(\Delta^{n})^{\sharp}}$$\scriptstyle{e}$$\scriptstyle{p}$$\scriptstyle{j}$$\scriptstyle{q}$$\scriptstyle{i}$

where the markings on $E^{\natural}$ correspond to the $pe$-coCartesian edges.

Now complete the diagram as follows,

${E^{\natural}}$${F^{\natural}}$${X^{\natural}}$${Y^{\natural}}$${(\Lambda^{n}_{k})^{\sharp}}$${(\Delta^{n})^{\sharp}}$$\scriptstyle{e}$$\scriptstyle{k}$$\scriptstyle{f}$$\scriptstyle{p}$$\scriptstyle{j}$$\scriptstyle{q}$$\scriptstyle{i}$

with $k$ marked left anodyne and $f$ a marked left fibration. The markings on
$F^{\natural}$ thus correspond to the $qf$-coCartesian edges. The underlying
map of simplicial sets $f\colon F\to Y$ is in general not a coCartesian
fibration, but the previous Lemma 4.6 shows that it is a locally coCartesian
fibration and that for any $x\in X$ the induced map $E_{x}\to F_{x}$ is a
Joyal equivalence (between Kan complexes).

Now find a factorization

${F}$${G}$${Y}$$\scriptstyle{l}$$\scriptstyle{f}$$\scriptstyle{g}$

with $l$ left anodyne and $g$ a left fibration. By Lemma 4.4 the map $f$ is
proper and thus by [Cis19, Corollary 4.4.28] the induced map on fibers
$F_{y}\to G_{y}$ is cofinal for any $y\in Y$.

To summarize, we have constructed a commutative diagram

${E}$${F}$${G}$${X}$${Y}$$\scriptstyle{k}$$\scriptstyle{e}$$\scriptstyle{l}$$\scriptstyle{f}$$\scriptstyle{g}$$\scriptstyle{j}$

in which $k$ and $l$ are left anodyne and $g$ is a left fibration, thus $g$ is
a fibrant replacement for the composition $je$. Now for any $x\in X$ we have
induced maps on fibers

$E_{x}\xrightarrow{k_{x}}F_{x}\xrightarrow{l_{x}}G_{x}$

where $k_{x}$ is cofinal by Lemma 4.6 and $l_{x}$ is cofinal by the above
arguments. In particular this implies that the induced map

$E\to X\times_{Y}G$

is cofinal which in turn implies that the derived counit is an isomorphism.
Thus $X\to Y$ is fully faithful and this finishes the proof. ∎

######  4.8.

As a consequence we show that the coCartesian model structure is functorial
with respect to coCartesian fibrations.

######  Theorem 4.9.

Let $p\colon X^{\natural}\to A^{\sharp}$ be a Cartesian fibration (i.e. a
marked right fibration). Then the pullback functor

$p^{\ast}\colon\mathbf{sSet}^{+}/A^{\sharp}\to\mathbf{sSet}^{+}/X^{\natural}$

is a left Quillen functor when each category is endowed with the coCartesian
model structure.

###### Proof.

Since marked simplicial sets are locally cartesian closed, the functor
$p^{\ast}$ is a left adjoint. To show that $p^{\ast}$ is left Quillen, it
suffices to show that it preserves marked left anodyne extensions. In
particular, it suffices to show that $p^{\ast}$ preserves the generating
marked left anodyne extensions of Definition 2.4.

We first show this for set (A1). We need to show that for any diagram of
pullback squares of the form

${X^{\natural}_{\Lambda^{n}_{k}}}$${X^{\natural}_{\Delta^{n}}}$${X^{\natural}}$${(\Lambda^{n}_{k})^{\flat}}$${(\Delta^{n})^{\flat}}$${A^{\sharp},}$$\scriptstyle{j}$$\scriptstyle{p}$$\scriptstyle{i}$

where $i$ is inner horn inclusion, the map $j$ is a trivial cofibration in
$\mathbf{sSet}^{+}/X^{\natural}$. We observe that the marked simplicial sets
$X^{\natural}_{\Lambda^{n}_{k}}$ and $X^{\natural}_{\Delta^{n}}$ have
precisely the equivalences in their fibers over $(\Lambda^{n}_{k})^{\flat}$
and $(\Delta^{n})^{\flat}$ marked. In particular $X^{\natural}_{\Delta^{n}}$
is fibrant over the point. Since the underlying map of simplicial sets

$X_{\Lambda^{n}_{k}}\to X_{\Delta^{n}}$

is a Joyal trivial cofibration by Theorem 4.7, the map

$X^{\flat}_{\Lambda^{n}_{k}}\to X^{\flat}_{\Delta^{n}}$

is a trivial cofibration over the point. We have a square

${X_{\Lambda^{n}_{k}}^{\flat}}$${X_{\Delta^{n}}^{\flat}}$${X_{\Lambda^{n}_{k}}^{\natural}}$${X^{\natural}_{\Delta^{n}}}$

Here, the vertical maps are marked left anodyne, since they are given by
marking equivalences and the upper horizontal map is a trivial cofibration.
Thus the lower horizontal map is a trivial cofibration over the point. Since
$X^{\natural}_{\Delta^{n}}$ is fibrant, this map is in fact marked left
anodyne by [Ngu19, 2.31] and thus a trivial cofibration in
$\mathbf{sSet}^{+}/X^{\natural}$. ∎

##  5 Straightening and unstraightening

######  5.1.

This section proves a straightening/unstraightening equivalence for
coCartesian fibrations. For this we assume that $A$ is the nerve of a small
category. We will prove the following Theorem.

######  Theorem 5.2.

For any simplicial set $B$ there is a Quillen equivalence

$\mathrm{Fun}(A,\mathbf{sSet}^{+}/B^{\sharp})\simeq\mathbf{sSet}^{+}/A^{\sharp}\times
B^{\sharp}$

where the right hand side is endowed with the Cartesian model structure and
the left hand side is endowed with the projective Cartesian model structure.

######  5.3.

Contrary to the existing literature [Lur09], [HHR21], our proof does not
involve simplicial categories. Our equivalence is also _not_ the full
straightening/unstraightening equivalence as our base category is assumed to
be the nerve of a category. Our proof follows ideas from [HM15].

######  5.4.

We first define the functors involved. Let $X^{+}\to A^{\sharp}$ be a map and
$B^{+}$ be a marked simplicial set. We have a functor

$\mathbf{sSet}^{+}/B^{+}\to\mathbf{sSet}^{+}/A^{\sharp}\times B^{+}$

given by sending a map $Y^{+}\to B^{+}$ to the product $X^{+}\times Y^{+}\to
A^{\sharp}\times B^{+}$. This functor has a right adjoint

$\mathrm{Map}^{B}(X^{+},-):\mathbf{sSet}^{+}/A^{\sharp}\times
B^{+}\to\mathbf{sSet}^{+}/B^{+}$

By the universal property a map

${K^{+}}$${\mathrm{Map}^{B}(X^{+},W^{+})}$${B^{+}}$

is thus given by a commutative triangle

${X^{+}\times K^{+}}$${W^{+}}$${A^{\sharp}\times B^{+}}$

Note that $\mathrm{Map}^{B}(X^{+},W^{+})$ is (contravariantly) functorial in
$X^{+}\to A^{+}$. This defines a functor

$\rho:\mathbf{sSet}^{+}/A^{\sharp}\times
B\to\mathbf{Fun}(A,\mathbf{sSet}^{+}/B^{+}),\quad
W^{+}\mapsto(a\mapsto\mathrm{Map}^{B}((a/A)^{\sharp},W^{+})$

Given a map $p:W^{+}\to A^{\sharp}\times B^{+}$ The marked simplicial set
$\mathrm{Map}^{B^{+}}(X^{+},W^{+})$ can be described as the pullback

${\mathrm{Map}^{B^{+}}(X^{+},W^{+})}$${\mathrm{Hom}^{+}(X^{+},W^{+})}$${B^{+}}$${\mathrm{Hom}^{+}(X^{+},A^{\sharp}\times
B^{+})}$$\scriptstyle{p_{\ast}}$

where the bottom map is given by the product of the fixed map $X^{+}\to
A^{\sharp}$ and the identity on $B^{+}$.

######  5.5.

This functor has a left adjoint

$\lambda:\mathbf{Fun}(A,\mathbf{sSet}^{+}/B^{+})\to\mathbf{sSet}^{+}/A^{\sharp}\times
B^{+}.$

Given a functor $F:A\to\mathbf{sSet}^{+}/B^{+}$ we obtain the functor

$A^{op}\times A\to\mathbf{sSet}^{+}/A^{\sharp}\times
B^{+},\quad(a,a)\mapsto((a/A)^{\sharp}\times F(a)^{+}\to A^{\sharp}\times
B^{+})$

The value of the left adjoint is then given by taking the coend

$\lambda(F)=\int^{A}(a/A)^{\sharp}\times F(a)^{+}$

######  Example 5.6.

Let $X^{+}\to B^{+}$ be a map of marked simplicial sets and let $a$ be an
object of $A$. Let $a\otimes X^{+}$ be the functor given by left Kan extension
along the inclusion $\\{a\\}\to A$. Then we have

$\lambda(a\otimes X^{+})=\leavevmode\hbox to63.67pt{\vbox
to54.19pt{\pgfpicture\makeatletter\hbox{\hskip 31.83316pt\lower-27.0972pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{\offinterlineskip{}{}{{{}}{{}}}{{{}}}{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-31.83316pt}{-27.0972pt}\pgfsys@invoke{
}\hbox{\vbox{\halign{\pgf@matrix@init@row\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding&&\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding\cr\hfil\hskip
31.83316pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-27.52762pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${(a/A)^{\sharp}\times
X^{+}}$} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
31.83316pt\hfil\cr\vskip 18.00005pt\cr\hfil\hskip
20.3656pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-16.06006pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${A^{\sharp}\times
B^{+}}$} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
20.3656pt\hfil\cr}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}{{{{}}}{{}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} { {}{}{}}{}{ {}{}{}} {{{{{}}{
{}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{0.0pt}{3.14032pt}\pgfsys@lineto{0.0pt}{-14.05977pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{-1.0}{1.0}{0.0}{0.0pt}{-12.6153pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$

######  Proposition 5.7.

For any coCartesian equivalence $K^{+}\to L^{+}$ over $A^{\sharp}$ and any
marked left fibration $X^{+}$ over $A^{\sharp}\times B^{+}$ the induced map

$\mathrm{Map}^{B^{+}}(L^{+},X^{+})\to\mathrm{Map}^{B^{+}}(K^{+},X^{+})$

is a coCartesian equivalence over $B^{+}$.

###### Proof.

We have a pullback square

${\mathrm{Map}^{B^{+}}(L^{+},X^{+})}$${\underline{\mathrm{Hom}}^{+}(L^{+},X^{+})}$${\mathrm{Map}^{B^{+}}(K^{+},X^{+})}$${\underline{\mathrm{Hom}}^{+}(K^{+},X^{+})\times_{\underline{\mathrm{Hom}}^{+}(K^{+},A^{\sharp}\times
B^{+})}\underline{\mathrm{Hom}}^{+}(L^{+},A^{\sharp}\times B^{+})}$

Since the right hand vertical map is a trivial fibration whenever $K^{+}\to
L^{+}$ is marked left anodyne, the functor
$\mathrm{Map}_{A^{\sharp}}^{B^{+}}(-,X^{+})$ sends trivial cofibrations to
weak equivalences in the opposite of the cocartesian model structure on
$\mathbf{sSet^{+}}/B^{+}$. ∎

######  Corollary 5.8.

Suppose $X^{+}\to A^{\sharp}\times B^{+}$ is a marked left fibration. Let
$X^{+}_{a}$ be the pullback

${X^{+}_{a}}$${X^{+}}$${\\{a\\}\times B^{+}}$${A^{\sharp}\times B^{+}}$

Then there is a coCartesian equivalence

$X^{+}_{a}\simeq\mathrm{Map}_{A^{\sharp}}^{B^{+}}((a/A)^{\sharp},X^{+})$

over $B^{+}$.

###### Proof.

The map $\\{a\\}\to(a/A)^{\sharp}$ is marked left anodyne. ∎

######  Proposition 5.9.

The right adjoint preserves fibrations between fibrant objects.

###### Proof.

Let

${X^{+}}$${Y^{+}}$${A^{\sharp}\times B^{+}}$

be a fibration between marked left fibrations. In particular by [Ngu19] it is
a marked left fibration. We have a pullback square

${\mathrm{Map}_{A^{\sharp}}^{B^{+}}(K^{+},X^{+})}$${\underline{\mathrm{Hom}}^{+}(K^{+},X^{+})}$${\mathrm{Map}_{A^{\sharp}}^{B^{+}}(K^{+},Y^{+})}$${\underline{\mathrm{Hom}}(K^{+},Y^{+})}$

The right hand side is a marked left fibration hence the pullback is a marked
left fibration between marked left fibrations over $B^{+}$. ∎

######  Corollary 5.10.

The functors determine a Quillen adjunction

$\lambda\colon\mathrm{Fun}(A,\mathbf{sSet}^{+}/B^{+})\leftrightarrow\mathbf{sSet}^{+}/A^{\sharp}\times
B^{+}\colon\rho$

######  5.11.

In order to prove that this defines a Quillen equivalence, we show that this
Quillen adjunction respects evaluation at a point in $A$. It is easy to see
that the following square commutes for each object $a$ of $A$:

${\mathrm{Fun}(A,\mathbf{sSet}^{+}/B^{+})}$${\mathbf{sSet}^{+}/A^{\sharp}\times
B^{+}}$${\mathbf{sSet}^{+}/B^{+}}$${\mathbf{sSet}^{+}/B^{+}}$$\scriptstyle{\lambda}$$\scriptstyle{ev_{a}}$$\scriptstyle{a^{*}}$$\scriptstyle{id}$
(1)

######  Proposition 5.12.

The induced transformation

$ev_{a}\rho\Rightarrow a^{*}$

is a coCartesian equivalence for each fibrant object of
$\mathbf{sSet}^{+}/A^{\sharp}\times B^{+}$.

###### Proof.

Let $W^{+}\to A^{\sharp}\times B^{+}$ be a coCartesian fibration. We need to
show that the counit

$\lambda\rho(W^{+})\to W^{+}$

induces a coCartesian equivalence after taking fibers at the object $a$ of
$A$. The map

$\lambda\rho(W^{+})_{a}\to W^{+}_{a}$

can be written as

$\mathrm{Map}^{B^{+}}((a/A)^{\sharp},W^{+})\to W_{a}^{+}$

and an explicit computation shows that this coincides with the map of
Corollary 5.8, hence is a trivial fibration. ∎

######  5.13.

One of the key observations is that the pullback functor preserves homotopy
colimits, see also [HHR21, Lemma 3.5].

######  Proposition 5.14.

Let $f\colon K\to L$ be a map of simplicial sets. Then the induced functor

$f^{\ast}\colon\mathbf{sSet}^{+}/L^{\sharp}\to\mathbf{sSet}^{+}/K^{\sharp}$

preserves homotopy colimits.

###### Proof.

We won’t give a full proof here, as we cannot improve on the proof of [HHR21]
or offer a different viewpoint. The idea is to reduce to the case when
$K\cong\Delta^{0}$ and thus $f\colon\Delta^{0}\to L$ is the specification of
an object in $L$. Then one observes that the pullback $f^{\ast}$ is weakly
equivalent to pulling back along a marked left fibration. By Theorem 4.9
pulling back along a marked left fibration is left Quillen, thus it preserves
homotopy colimits and the assertion follows. ∎

######  Corollary 5.15.

Let $a$ be an object of $A$. Then the functor

$\mathbf{R}a^{\ast}:\mathbf{sSet}^{+}/A^{\sharp}\times
B^{+}\to\mathbf{sSet}^{+}/B^{+}$

preserves homotopy colimits.

######  Corollary 5.16.

The functor $ev_{a}$ preserves homotopy colimits.

###### Proof.

This follows immediately from the commutative diagram (1) and the previous
Corollary. ∎

######  Corollary 5.17.

The functor $\rho$ preserves homotopy colimits.

###### Proof.

We show that the map

$hocolim_{I}\mathbf{R}\rho\to\mathbf{R}\rho\ hocolim_{I}$

is an equivalence. Since equivalences are computed point wise, it is enough to
show that

$\mathbf{R}ev_{a}hocolim_{I}\mathbf{R}\rho\to\mathbf{R}ev_{a}\mathbf{R}\rho\
hocolim_{I}$

is an equivalence. By the previous corollary $ev_{a}$ commutes with homotopy
colimits and by Proposition 5.12 we have

$\mathbf{R}ev_{a}\mathbf{R}\rho\simeq\mathbf{R}a^{\ast}$

Thus the assertion follows from Proposition 5.15. ∎

######  Lemma 5.18.

Let $X^{\natural}\to A^{\sharp}$ be a cartesian fibration and $Y^{+}\to
B^{\sharp}$ be a map. Let $Z^{\natural}\to A^{\sharp}\times B^{\sharp}$ be a
cartesian fibration. Then a map

${X^{\natural}\times Y^{+}}$${Z^{\natural}}$${A^{\sharp}\times
B^{\sharp}}$$\scriptstyle{f}$

is a cartesian equivalence if and only if for all points $a$ of $A$ the map

${X^{\natural}_{a}\times
Y^{+}}$${Z^{\natural}_{a}}$${B^{\sharp}}$$\scriptstyle{f_{a}}$

is a cartesian equivalence.

###### Proof.

Choose a factorization

$Y^{+}\to W^{\natural}\to B^{\sharp}$

into a marked right anodyne extension followed by a cartesian fibration. We
get an induced factorization

$X^{\natural}\times Y^{+}\to X^{\natural}\times W^{\natural}\to
A^{\sharp}\times B^{\sharp}$

into marked right anodyne followed by cartesian fibration. We find a solution
to the lifting problem

${X^{\natural}\times Y^{+}}$${Z^{\natural}}$${X^{\natural}\times
W^{\natural}}$${A^{\sharp}\times
B^{\sharp}}$$\scriptstyle{f}$$\scriptstyle{g}$

since the left hand side is marked right anodyne and the right hand side is a
cartesian fibration by assumption. Thus, the map $f$ is a cartesian
equivalence if and only if the map $g$ is. Now let us take fibers at the
inclusion

$\\{a\\}\times B^{\sharp}\to A^{\sharp}\times B^{\sharp}.$

We obtain

${X^{\natural}_{a}\times Y^{+}}$${Z^{\natural}_{a}}$${X^{\natural}_{a}\times
W^{\natural}}$${B^{\sharp}}$$\scriptstyle{f_{a}}$$\scriptstyle{g_{a}}$

Again, the left hand side is marked right anodyne hence $f_{a}$ is a cartesian
equivalence if and only if $g_{a}$ is a cartesian equivalence. It thus
suffices to show that $g$ is a cartesian equivalence if and only if $g_{a}$
is. This now follows from the fact that cartesian equivalences between
cartesian fibrations are detected pointwise. ∎

######  Proposition 5.19.

A map $\lambda(a\otimes X)\to W^{+}$ in $\mathbf{sSet}^{+}/A^{\sharp}\times
B^{+}$ is a weak equivalence if and only if the adjoint map $a\otimes X\to\rho
W^{+}$ is.

###### Proof.

It suffices to show that for an object $a$ of $A$, a map $X^{+}\to B^{\sharp}$
and a cartesian fibration $Z^{\natural}\to A^{\sharp}\times B^{\sharp}$, a map

${\lambda(a\otimes p)}$${Z^{\natural}}$${A^{\sharp}\times B^{\sharp}}$

is a cartesian equivalence if and only if the adjoint map

${a\otimes p}$${\rho(Z^{\natural})}$

is a pointwise cartesian equivalence. In the first case we have the map

${A/a^{\sharp}\times X^{+}}$${Z^{\natural}}$${A^{\sharp}\times B^{\sharp}}$

which is a cartesian equivalence if and only if for any point $a^{\prime}$ of
$A$ the induced map

${\mathrm{Hom}(a^{\prime},a)\times
X^{+}}$${Z_{a^{\prime}}^{\natural}}$${B^{\sharp}}$

is a cartesian equivalence. In the second case, we have a pointwise cartesian
equivalence if and only if for any point $a^{\prime}$ of $A$ the induced map

${\mathrm{Hom}(a^{\prime},a)\times
X^{+}}$${\mathrm{Map}_{A}(A/a^{\prime\sharp},Z^{\natural})}$${B^{\sharp}}$

is a cartesian equivalence. Since we have a cartesian equivalence

$\mathrm{Map}_{A}(A/a^{\prime\sharp},Z^{\natural})\simeq
Z^{\natural}_{a^{\prime}}$

over $B^{\sharp}$, this is equivalent to the first case. ∎

###### Proof of Theorem 5.2.

Since weak equivalences between Cartesian fibrations are computed fiberwise,
it follows from Proposition 5.12 that $\rho$ reflects weak equivalences
between fibrant objects. It thus suffices to show that the derived unit is a
weak equivalence. It follows from Proposition 5.19 that the derived unit is a
weak equivalence for each object of the form $a\otimes X$. Since any functor
is a homotopy colimit of objects of this form and the functor $\rho$ preserves
homotopy colimits by Corollary 5.17, it follows that the derived unit is in
fact a weak equivalence. ∎

## References

  * [Cis19] Denis-Charles Cisinski.  Higher Categories and Homotopical Algebra, volume 180 of Cambridge Studies in Advanced Mathematics.  Cambridge University Press, 2019.
  * [HHR21] Fabian Hebestreit, Gijs Heuts, and Jaco Ruit.  A short proof of the straightening theorem, 2021.  arXiv:2111.00069.
  * [HM15] Gijs Heuts and Ieke Moerdijk.  Left fibrations and homotopy colimits.  Math. Z., 279(3-4):723–744, 2015.
  * [Lur09] Jacob Lurie.  Higher topos theory, volume 170 of Annals of Mathematics Studies.  Princeton University Press, 2009.
  * [Lur17] Jacob Lurie.  Higher algebra.  http://www.math.harvard.edu/ lurie/papers/HA.pdf, 2017.
  * [Ngu19] Hoang Kim Nguyen.  Covariant and Contravariant Homotopy Theories, 2019.  arXiv:1908.06879.


# Hidden Zeros in Scaffolded General Relativity and Exceptional Field Theories

Yang<EMAIL_ADDRESS>Diederik<EMAIL_ADDRESS>Tonnis ter
<EMAIL_ADDRESS>1Van Swinderen Institute for Particle
Physics and Gravity,
University of Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlands
2Department of Physics and Astronomy, Macalester College, Saint Paul MN
55105-1899, U.S.A.

###### Abstract

It was recently discovered by Arkani-Hamed et al that the colour-ordered
scattering amplitudes of Tr$(\Phi^{3})$, the NLSM and scaffolded YM vanish at
specific loci. We build on this observation and demonstrate that, beyond
colour ordering, scattering amplitudes can display multiple instances of these
hidden zeros. A first example are the flavour-ordered amplitudes of scaffolded
General Relativity, corresponding to graviton exchange. The other two cases
are multi-flavour DBI and the special Galileon. These are the natural
generalisations of the NLSM with multiple instances of Adler’s zero. The zero
loci naturally define factorisation limits; in all three cases these include a
metric-coupled cubic scalar theory. We comment on the general picture of
hidden zeros and the relation to the double copy.

##  I Introduction

Positive geometry has been shown to encode rich properties of scattering
amplitudes that are hidden from the Lagrangian prospective. For instance, the
canonical form Arkani-Hamed  _et al._ (2017) of the kinematic associahedron
Arkani-Hamed  _et al._ (2018) was found to be directly related to bi-adjoint
scalar (BAS) amplitudes. Furthermore, its $\alpha^{\prime}$-generalization
gives rise to amplitudes of stringy nature Arkani-Hamed  _et al._ (2021). Much
more recently, it was found that the amplitudes of ${\rm Tr}(\Phi^{3})$ theory
(that are contained in those of BAS) have a surprising property with a
beautiful geometric structure: at all-loop and of any multiplicity, these
follow from compact geometric rules on surfaces Arkani-Hamed  _et al._ (2023a,
b). This framework of surfaceology has led to a series of interesting follow-
up works Arkani-Hamed  _et al._ (2023c, 2024a, 2024b); Arkani-Hamed and
Figueiredo (2024).

Additionally, the tree-level amplitudes of ${\rm Tr}(\Phi^{3})$ vanish on
specific hypersurfaces in the space of Mandelstam variables Arkani-Hamed  _et
al._ (2023d) (as also found much earlier in string theory D’Adda  _et al._
(1971)). These loci naturally arise as ’causal diamonds’ in a two-dimensional
kinematic mesh Arkani-Hamed  _et al._ (2022), that is defined with respect to
a specific colour ordering (and applies to the corresponding partial
amplitude). On top of this, the loci give rise to new factorisation channels
into lower-point amplitudes of the same theory.

Moreover, these minor miracles carry over to more intricate and interesting
theories. Two of these were identified in Arkani-Hamed  _et al._ (2023d) and
involve colour ordering as well. The first is the scattering of pions, as
described by the chiral non-linear sigma model (NLSM). The non-linear symmetry
of these famously gives rise to the Adler zero in the soft limit Adler (1965).
Interestingly, the zeros of Arkani-Hamed et al include and extend this soft-
limit behaviour. The second theory describes external scalars interacting via
gluon exchange. This is referred to as ’scaffolded Yang-Mills (YM)’ as these
include the full YM amplitudes when stripping off the external scalars Arkani-
Hamed  _et al._ (2023c).

In this letter, we will identify three theories that naturally generalise
these results. The first two are higher-order variations of the Adler zero:
the amplitudes of multi-Dirac-Born Infeld (DBI) and the special Galileon (SG)
vanish in the soft limit as $p^{\sigma}$ with $\sigma=2,3$, respectively
Cheung  _et al._ (2017). Similar to the Adler zero, we will demonstrate that
their amplitudes vanish at multiple loci that include and extend these soft-
limit behaviours. The third theory describes flavoured scalars interacting via
graviton exchange, and can thus be seen as scaffolded General Relativity (GR).

Together, these six theories naturally span the set of scalar field theories
that are related via the double copy procedure Bern  _et al._ (2008, 2010,
2019, 2022). One can think of ${\rm Tr}(\Phi^{3})$ as the zeroth copy,
scaffolded YM and the NLSM as single copies, completed by scaffolded GR,
multi-DBI and the special Galileon as double copies. We use the KLT
formulation of these relations to demonstrate how the single-copy zeros carry
over to the double copies, thus completing the web of exceptional scalar field
theories with zeros in their amplitudes.

Note added -  Upon completion of this manuscript, the preprints Cao  _et al._
(2024); Bartsch  _et al._ (2024) appeared whose results overlap in parts with
ours.

##  II Colour ordering & single zeros

We first recap the results of Arkani-Hamed  _et al._ (2023d) on the structure
of hidden zeros in colour-ordered partial amplitudes.

Tr$(\Phi^{3})$ theory - Starting at four-point (4-pt) and requiring the
colour-ordered partial amplitude to vanish at $u=0$, there are two poles
available for exchange interactions: one can include contributions via the
$s$\- and $t$-channel. At lowest order in derivatives, the unique possibility
is

$\displaystyle A^{\Phi^{3}}_{4}\sim\frac{1}{s}+\frac{1}{t}\sim\frac{u}{st}\,.$
(1)

One can identify this as the partial amplitude (with alphabetic colour
ordering) of the Tr$(\Phi^{3})$ theory.

The same applies at higher multiplicities. For instance, the six-point (6-pt)
partial amplitude again vanishes along a number of loci defined by the colour
ordering. We will distinguish these based on the shape of their ’causal
diamond’ in the kinematic mesh specific to that colour ordering Arkani-Hamed
_et al._ (2023d). At 6-pt, there are two possibility, either a ’skinny
rectangle’ of height one or a square of height two. Starting with the latter,
the amplitude vanishes on the locus $s_{\alpha,\beta}=0$ with

$\displaystyle\alpha=(1,2)\,,\quad\beta=(4,5)\,,$ (2)

and similar for cyclic permutations. When only taking three equal to zero, the
amplitude factorises and reads

$\displaystyle A_{6}^{\Phi^{3}}\to$
$\displaystyle\left(\frac{1}{X_{T}}+\frac{1}{X_{B}}\right)A^{\Phi^{3},\,\rm{down}}_{4}A^{\Phi^{3},\,\rm{up}}_{4}\,,$
(3)

and thus factorises into the product of two 4-pt amplitudes. $X_{T}$ and
$X_{B}$ are the planar variables corresponding to the top and bottom vertices
of the causal diamond; see Arkani-Hamed  _et al._ (2023d). For skinny
rectangles, instead, the corresponding Mandelstams are $s_{\alpha,\beta}$ with

$\displaystyle\alpha=(1)\,,\quad\beta=(3,4,5)\,.$ (4)

In order to vanish, all terms in the partial amplitude must be at least linear
in one of these non-adjacent Mandelstam variables. When taking all but one to
zero, it factorises as

$\displaystyle A_{6}^{\Phi^{3}}\to$
$\displaystyle(\frac{1}{X_{T}}+\frac{1}{X_{B}})A^{\Phi^{3}}_{3}A^{\Phi^{3}}_{5}\,,$
(5)

into a product of three- and five-point amplitudes of the Tr$(\Phi^{3})$
theory.

Scaffolded Yang-Mills -  Adding two derivatives to the 4-pt partial amplitude
with two channels, one can also consider

$\displaystyle A^{\rm
scYM}_{4}\sim\frac{t}{s}-\frac{s}{t}\sim\frac{u(s-t)}{st}\,.$ (6)

This structure arises from gluon exchange by adjoint scalars with an
additional flavour structure, with the additional requirement that the flavour
structures is aligned with the colour ordering444This restriction is
automatically enforced when extracting Yang-Mills amplitude from the residue
as considered in Arkani-Hamed  _et al._ (2023c). As both have zeros, we will
refer to both this amplitude and its residue as scaffolded YM.: it matches
flavour pairs of subsequent colour indices, that is, for the alphabetic colour
ordering the flavour structure is $\delta_{ab}\delta_{cd}$ or
$\delta_{bc}\delta_{ad}$. The above amplitude structure also requires a
quartic interaction that follows from the reduction of higher-dimensional
Yang-Mills theory Arkani-Hamed  _et al._ (2023c, d) and hence can be written
in terms of scattering equations Cachazo  _et al._ (2015).

At 6-pt, the partial amplitudes with flavour alignment (that is, restricted to
the flavour structures $\delta_{ab}\delta_{cd}\delta_{ef}$ and
$\delta_{bc}\delta_{de}\delta_{af}$) vanish at exactly the same loci (2) and
(4). Moreover, in the factorisation limits defined by these loci, the
amplitudes read

$\displaystyle A^{\text{scYM}}_{6}\to$
$\displaystyle\left(\frac{1}{X_{B}}+\frac{1}{X_{T}}\right)\times
A_{4}^{\text{scYM},\;{\rm up}}\times A_{4}^{\text{scYM},\;{\rm down}}\,,$
$\displaystyle A^{\text{scYM}}_{6}\to$
$\displaystyle\frac{X_{T}+X_{B}}{X_{B}}\times A_{5}^{\text{ext scYM}}\,,$ (7)

where

$\displaystyle A_{5}^{\text{ext
scYM}}=\frac{1}{s_{3,4}}\left(\frac{s_{2,3}+s_{3,4}}{s_{2,3,4}}+\frac{s_{3,5}}{s_{3,4,5}}\right)\,.$
(8)

This extension involves a novel ${\rm Tr}(\Phi^{3})$ sector,

$\displaystyle{\cal L}^{\rm
ext}=-\tfrac{1}{2}D_{\mu}\Phi^{a}D^{\mu}\Phi^{a}+{\rm Tr}(\Phi^{3})\,.$ (9)

that contributes via minimal coupling to the gluons.

NLSM - Finally, in the absence of poles, the simplest 4-pt amplitude that
vanishes at the locus is

$\displaystyle A_{4}^{\rm NLSM}\sim u\,.$ (10)

This is the partial amplitude of the chiral NLSM, and thus provides the
natural generalisation of (1) and (6) to contact interactions.

Its higher-multiplicity amplitudes also vanish at the loci defined by the
causal diamonds. Note that the vanishing of the partial amplitude at the
skinny rectangle locus (4) foreshadows the Adler zero: in the absence of poles
at $s_{1,2}$ or $s_{1,6}$ (guaranteed by the absence of cubic vertices - in
contrast to the previous two cases), the vanishing of the partial amplitude at
this locus implies the vanishing of the full amplitude in the soft limit
$p_{1}\to 0$. Thus the linearity in the soft Mandelstam variables of the
skinny rectangle (4) goes beyond the Adler zero precisely by excluding
adjacent soft Mandelstams. Moreover, the amplitudes of the NLSM factorise
close to these loci as

$\displaystyle A^{\text{NLSM}}_{6}\to$
$\displaystyle\left(\frac{1}{X_{B}}+\frac{1}{X_{T}}\right)\times
A_{4}^{\text{NLSM},\;{\rm up}}\times A_{4}^{\text{NLSM},\;{\rm down}}\,,$
$\displaystyle A^{\text{NLSM}}_{6}\to$
$\displaystyle\left(X_{T}+X_{B}\right)\times A_{5}^{\text{ext NLSM}}\,,$ (11)

where

$\displaystyle A_{5}^{\text{ext NLSM}}=s_{3,4}A_{5}^{\text{ext scYM}}\,.$ (12)

The latter amplitude is generated by a similar extension as (9), but now with
the covariant derivative defined in terms of pions rather than gluons.

##  III Flavour ordering & double zeros

Going beyond colour-ordered partial amplitudes, we will demonstrate that
scattering amplitudes can display multiple instance of the hidden zeros of
Arkani-Hamed  _et al._ (2023d). As a first generalisation, we will retain some
structure for the external scalars and require that they belong to a non-
trivial group representation. Similar to the gluon exchange case, the full
amplitudes at any multiplicity then splits up into different ’flavour
structures’ that pair up the external scalars via a product of Kronecker delta
functions. The coefficients of these structures will be referred to as
flavour-ordered amplitudes. Without loss of generality, we will focus on the
$\delta_{ab}\delta_{cd}\cdots$ flavour-ordered amplitude in this section.

Scaffolded GR - As a first step, consider 4-pt amplitude structures that
vanish when imposing $t=0$ or $u=0$ separately. This leaves only the
$s$-channel for exchange. The lowest order structure with two zeros is

$\displaystyle A_{4}^{\rm scGR}\sim\frac{tu}{s}\,.$ (13)

This is the 4-pt amplitude of flavoured scalars coupled via graviton exchange.

At 6-pt, the flavour-ordered amplitude again vanishes at loci defined by
causal diamonds of heights one and two. Starting with the latter squares, one
such locus is (2). In the absence of colour ordering, the symmetry of the
flavour structures then implies that it should also vanish at the loci defined
by555This is the natural generalisation of the 4-pt loci $s_{\alpha,\beta}$
with $\alpha=(\\{1,2\\}_{|1|})$ and $\beta=(\\{3,4\\}_{|1|})$ \- that is, $t$
or $u$, and allows for similar extension for higher multiplicities.

$\displaystyle
s_{\alpha,\beta}\,,\quad\alpha=(1,2)\,,\quad\beta=(\\{3,4\\}_{|1|},\\{5,6\\}_{|1|})\,,$
(14)

where $\\{\cdots\\}_{|i|}$ is shorthand for all the length-$i$ subsets of
$\\{\cdots\\}$. Near this zero locus, the amplitudes becomes product of lower-
point amplitudes multiplied by a prefactor, exactly in the way described by
the kinematic mesh; for instance, if we set $s_{2,5},s_{2,6},s_{3,5}=0$ and
$s_{3,6}\neq 0$, the amplitude becomes

$\displaystyle A_{6}^{\rm scGR}\to$
$\displaystyle\left(\frac{1}{s_{1,2,3}}+\frac{1}{s_{1,5,6}}\right)\times\frac{s_{1,3}s_{2,3}}{\left(s_{1,3}+s_{2,3}\right)}\times\frac{\left(s_{4,6}+s_{5,6}\right)s_{4,6}}{s_{5,6}}$
$\displaystyle=$
$\displaystyle\left(\frac{1}{X_{B}}+\frac{1}{X_{T}}\right)\times A_{4}^{\rm
up,\;scGR}\times A_{4}^{\rm down,\;scGR}$ (15)

The two 4-pt amplitudes are the pure scalar amplitudes with exchanged gluons.

The amplitude of graviton exchange also vanishes at the locus defined by the
skinny rectangle. Again, flavour symmetry then implies that the same amplitude
structure (with the alphabetic pairing) vanishes at all loci defined by
setting the Mandelstams with

$\displaystyle\alpha=(1)\,,\quad\beta=(\\{3,4,5,6\\}_{|3|})\,,$ (16)

to zero. In order to vanish for all choices, the flavour-ordered amplitude
needs to be at least bilinear in this set (16) of Mandelstams. Under the near-
zero condition $s_{1,3},s_{1,4}=0,s_{1,5}\neq 0$, the amplitude factorizes
into a product of 5-pt mixed amplitude and a prefactor

$\displaystyle A_{6}^{\rm scGR}\to$
$\displaystyle\frac{X_{T}(X_{B}+X_{T})}{X_{B}}\times A_{5}^{{\rm
ext\;scGR}}\,,$ (17)

with

$\displaystyle A_{5}^{{\rm ext\;scGR}}=$
$\displaystyle\frac{s_{2,3}\left(s_{2,3}+s_{3,4}\right)}{s_{3,4}s_{2,3,4}}+\frac{s_{3,5}\left(s_{3,4}+s_{3,5}\right)}{s_{3,4}s_{3,4,5}}$
$\displaystyle+\frac{s_{3,6}\left(s_{3,4}+s_{3,6}\right)}{s_{3,4}s_{3,4,6}}\,.$
(18)

The Lagrangian producing this mixed amplitude is provided by the extension

$\displaystyle{\cal
L}^{\text{ext}}=\sqrt{-g}(-\tfrac{1}{2}\partial^{\mu}\Phi\partial_{\mu}\Phi+\Phi^{3})\,,$
(19)

where $g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$, and $\Phi$ is colorless. The
kinematics of this 5-pt amplitude and distribution of scaffolded gravitons and
$\Phi$ follows the pattern of Arkani-Hamed  _et al._ (2023d).

Multi-DBI - Moving beyond exchange interactions with zeros at $t=0$ or $u=0$
for 4-pt, the lowest-order contact amplitude structure with this property is

$\displaystyle A^{\rm DBI}_{4}\sim tu\,,$ (20)

which is the flavour amplitude of multi-DBI. This theory realises a
$(D+N)$-dimensional Poincaré algebra on $N$ scalars $\varphi^{a}$ in $D$
dimensions via non-linear transformations. Its central object is the induced
metric

$\displaystyle
g_{\mu\nu}=\eta_{\mu\nu}+\partial_{\mu}\varphi_{a}\partial_{\nu}\varphi^{a}\,,$
(21)

which transforms covariantly under the higher-dimensional space-time symmetry.

Moving on to 6-pt, the multi-DBI amplitude is a rather large expression.
However, it is simple to verify that it again vanishes along a number of loci.
These include the different squares defined by (14). When only taking three
equal to zero, the amplitude factorises into

$\displaystyle A_{6}^{\rm DBI}\to$
$\displaystyle\left(\frac{1}{s_{1,2,3}}+\frac{1}{s_{1,5,6}}\right)s_{1,3}s_{2,3}\left(s_{4,6}+s_{5,6}\right)s_{4,5}$
$\displaystyle=$
$\displaystyle\left(\frac{1}{X_{B}}+\frac{1}{X_{T}}\right)A_{4}^{{\rm
up,DBI}}\times A_{4}^{{\rm down,DBI}}$ (22)

in terms of two 4-pt DBI amplitudes.

Turning to the skinny rectangles, the corresponding Mandelstams are three out
of the set of four Mandelstam variables (16). Again the DBI flavour amplitudes
vanish at all these loci. Moreover, DBI amplitudes are even, and therefore
have no poles on single Mandelstams such as $s_{1,2}$. Bilinearity then
implies that the full amplitude vanishes as

$\displaystyle A_{6}^{\rm DBI}\sim p_{1}^{2}\,,$ (23)

in the soft limit of $p_{1}\to 0$. Therefore, similar to the single zero
underlying the Adler zero of the NLSM, we find that the multiple loci of the
DBI theory give rise to its extended soft limit.

In the near-zero locus $s_{1,3},s_{1,4}\to 0$, the DBI 6-pt flavour-ordered
amplitude factorizes into

$\displaystyle A^{\rm DBI}_{6}\to$ $\displaystyle
X_{T}\left(X_{B}+X_{T}\right)A_{5}^{{\rm ext\;DBI}}\,,$ (24)

with

$\displaystyle A_{5}^{{\rm ext\;DBI}}=$ $\displaystyle s_{3,4}A_{5}^{{\rm
ext\;scGR}}\,.$ (25)

The Lagrangian of the mixed theory is given by the same form (19) but now with
the DBI covariant metric (21) instead.

##  IV Single scalar & triple zeros

Finally, one can consider amplitudes that vanish when imposing $s=0$, $t=0$ or
$u=0$ separately. This leaves us with no channels, and hence there can only be
contact interactions.

Special Galileon - The lowest-order contact interaction with three zeros is

$\displaystyle A_{4}^{\rm SG}\sim stu\,,$ (26)

which is the amplitude of the special Galileon. This is the unique single-
scalar field theory that realises a non-linear symmetry that is quadratic in
coordinates Hinterbichler and Joyce (2015) and has an interesting geometric
interpretation Novotny (2017); Roest (2021).

Given the absence of colour and flavour structures, the special Galileon 6-pt
amplitude vanishes in any causal diamond spanned by a square, with

$\displaystyle\alpha=(i,j)\,,\quad\beta=(k,l)\,,$ (27)

for any pair of pairs $(i,j),(k,l)$ (of which there are 15). Moreover, when
only taking three out of the four equal to zero, e.g.
$s_{2,6},s_{3,5},s_{2,5}\to 0$, the amplitude becomes very simple and reads

$\displaystyle A_{6}^{\rm SG}\to$
$\displaystyle\left(\frac{1}{s_{1,2,3}}+\frac{1}{s_{1,5,6}}\right)s_{1,3}s_{2,3}\left(s_{1,3}+s_{2,3}\right)$
$\displaystyle\hskip 56.9055pt\times
s_{1,6}s_{5,6}\left(s_{1,6}+s_{5,6}\right)$ $\displaystyle=$
$\displaystyle\left(\frac{1}{X_{B}}+\frac{1}{X_{T}}\right)A_{4}^{{\rm
up,SG}}\times A_{4}^{{\rm down,SG}}\,.$ (28)

This can be interpreted as a product of 4-pt SG amplitudes.

Looking at the skinny rectangles, the corresponding Mandelstams are an
arbitrary selection of three variables from the soft Mandelstams with

$\displaystyle\alpha=(1)\,,\quad\beta=(\\{2,3,4,5,6\\}_{|3|})\,.$ (29)

The requirement that the amplitude must vanish at all these loci can only be
satisfied by being trilinear in the above set. Combined with the evenness of
the theory that ensures there are no odd poles, this implies that the
amplitude scales with third power in the soft limit $p_{1}\to 0$:

$\displaystyle A_{6}^{\rm SG}\sim p_{1}^{3}\,.$ (30)

This was first found by Cheung et al Cheung  _et al._ (2015) and later shown
to follow from the non-linear symmetry Hinterbichler and Joyce (2015). From
the present perspective, we find however that the special Galileon has yet
more special amplitudes: they vanish at a large number of kinematic loci.

At a near-zero locus, $s_{1,3},s_{1,4}=0$ and $s_{1,5}\neq 0$, the amplitude
becomes

$\displaystyle A_{6}^{\rm SG}\to s_{1,5}s_{1,2}s_{1,6}\times A_{5}^{{\rm
ext\;SG}}\,,$ (31)

where

$\displaystyle A_{5}^{{\rm ext\;SG}}=$ $\displaystyle s_{3,4}^{2}A_{5}^{{\rm
ext\;scGR}}\,.$ (32)

Note that this is exactly the mixed amplitude
$A_{5}(\varphi,\varphi,\varphi,\Phi,\Phi)$ from Cachazo  _et al._ (2016) (up
to permutation). It can also be generated from the extended Lagrangian with
(19) in terms of the special Galileon covariant metric
$g_{\mu\nu}=\eta_{\mu\nu}+\partial_{\mu}\partial_{\rho}\varphi\partial^{\rho}\partial_{\nu}\varphi$.

##  V Zeros under the double copy

We have seen in the previous sections that the single zeros of the colour-
ordered theories of Arkani-Hamed  _et al._ (2023d) can be extended to multiple
zeros in three theories that lack colour ordering. In this section we will
provide an explanation for this, by demonstrating that the hidden zeros are
compatible with the double copy and thus are inherited from the colour-ordered
theory.

The six theories identified above are very closely related to those arising in
the double copy triangle of different scalar EFTs, with the triangle spanned
by adjoint colour, fundamental flavour and singlet kinematic structures de
Neeling  _et al._ (2022); for instance, the BCJ numerators Bern  _et al._
(2008, 2010, 2019, 2022) for flavour and kinematic structures at
4point666Extensions to higher derivatives can be found in e.g. Carrasco  _et
al._ (2023, 2022); Brown  _et al._ (2023); Li  _et al._ (2023). are

$\displaystyle n^{\rm flav}_{s}$
$\displaystyle=\delta_{ab}\delta_{cd}(t-u)-(\delta_{ac}\delta_{bd}-\delta_{bc}\delta_{ad})s\,,$
$\displaystyle n^{\rm kin}_{s}$ $\displaystyle=s(t-u)\,.$ (33)

Together with colour, this leads to six possibilities that are closely related
to the theories with zeros, as illustrated in figure 1:

  * •

The coloured theories include the NLSM, as well as Yang-Mills-scalar (YMS) and
the bi-adjoint scalar. The latter contain scaffolded YM and ${\rm
Tr}(\Phi^{3})$ theory as subsectors.

  * •

Similarly, in the flavoured sector, one finds multi-DBI, as well as gravity-
coupled scalars that span a non-chiral $SO(M,N)/(SO(M)\times SO(N))$ NLSM.
This theory includes a sector that corresponds to graviton exchange only
(referred to as scaffolded GR), and these amplitude structures display the
double zeros.

  * •

Finally, with only kinematics, the remaining possibility is the special
Galileon.

Note that the subsectors generally have one zero more than the full theory;
for instance, YMS has none while scaffolded YM has single zeros.

Figure 1: The different theories with zeros, arranged in terms of their soft
degree $\sigma$. The underlined theories have no odd vertices and hence the
zeros imply their soft behaviours. The vertical dotted lines indicate the
increase in zeros for the DC-related theories.

Our approach to zeros and the DC will be based on the KLT formalism Kawai  _et
al._ (1986) that combines two amplitudes $A_{n}$ and $\widetilde{A}_{n}$ into
a third one $M_{n}$. In particular we will employ the formulation Bjerrum-Bohr
_et al._ (2011)

$\displaystyle M_{n}=$ $\displaystyle\sum_{\sigma\in
S_{n-3}}\sum_{\begin{subarray}{c}\alpha\in S_{i-1},\\\ \beta\in
S_{n-2-i}\end{subarray}}S[\alpha(\sigma(2,\dots,i))|\sigma(2,\dots,i)]_{k_{1}}$
$\displaystyle\times
S[\beta(\sigma(i+1,\dots,n-2))|\sigma(i+1,\dots,n-2)]_{k_{n-1}}$
$\displaystyle\times A_{n}(1,\sigma(2,\dots,n-2),n-1,n)$
$\displaystyle\times\widetilde{A}_{n}(\alpha(\sigma(2,\dots,i)),1,n-1,\beta(\sigma(i+1,\dots,n-2)),n)\,,$
(34)

where the KLT kernel is defined as

$\displaystyle
S[a_{1},\dots,a_{k}|b_{1}\dots,b_{k}]_{p}=\prod_{t=1}^{k}\left(p\cdot
k_{a_{t}}+\sum_{q>t}\theta(a_{t},a_{q})k_{a_{t}}\cdot k_{a_{q}}\right)\,,$
(35)

and $\theta(a_{t},a_{q})=0(1)$ if the relative ordering of $a_{t},a_{q}$ is
the same (opposite) in $(a_{1},\dots,a_{k})$ and $(b_{1},\dots,b_{k})$.

This formulation involves in general three sums over permutations: one ’long’
permutation $\sigma$ and two shorter ones $\alpha,\beta$. The latter two can
be seen to split the former one, with $i$ governing where the split takes
place. This formulation is particularly suitable to show that the zeros of the
$i\times(n-2-i)$ causal diamonds carry over from the single copy
$\widetilde{A}$ to the double copy $M$. Without loss of generality, a causal
diamond of height $i$ can be defined by all Mandelstam variables that link the
first set $\\{1,\ldots,i\\}$ with the second set $\\{i+1,\ldots,n-2\\}$ (while
cyclicity generalises this to all causal diamonds of the same shape). At this
locus, the above formula can be massaged into a more compact form by the
following observation.

The long permutation, that permutes all but $1$ of these, either respects the
above separation into the two sets (and just permutes the indices within those
two sets), or it permutes elements from one set with the other (on top of
permutations within each set separately). In the latter case, the KLT matrix
$S[\alpha|\sigma]_{k_{1}}$ vanishes. This can be seen as follows: consider in
the permutation $\alpha(\sigma(2,\dots,j-1))$ the index of the second set that
appears last. The corresponding factor in (35) then consists of a sum of
Mandelstam variables that link the first and second sets, and hence vanish on
the causal diamond. The long permutation $\sigma$ therefore splits up into two
shorter permutations. The expression above then simplifies to

$\displaystyle M_{n}=$ $\displaystyle\sum_{\begin{subarray}{c}\alpha_{1,2}\in
S_{i-1},\\\ \beta_{1,2}\in
S_{n-2-i}\end{subarray}}S[\alpha_{1}|\alpha_{2}]_{k_{1}}\times
S[\beta_{1}|\beta_{2}]_{k_{n-1}}$ $\displaystyle\times
A_{n}(1,\alpha_{1},\beta_{1},n-1,n)\times\widetilde{A}_{n}(\alpha_{2},1,n-1,\beta_{2},n)\,,$
(36)

where $\alpha_{1,2}$ and $\beta_{1,2}$ are pairs of permutations of
$(2,\ldots,i,)$ and $(i+1,\ldots,n-2)$, respectively.

In this form of the KLT formula, the remaining permutations of $\widetilde{A}$
do not affect this causal diamond; in other words, all partial amplitudes
$\widetilde{A}$ that are summed over vanish at this causal diamond. These
zeros therefore carry over from $\widetilde{A}$ to $M$, provided the other
factor $A$ does not introduce any singularities at this locus. The latter is
guaranteed when taken to be the NLSM on account of having only even
amplitudes.

The other possibility, with Yang-Mills-scalar as one of the factors, requires
more discussion as it has singularities and flavour structures; at e.g. 4-pt,
its partial amplitude is given by

$\displaystyle A_{4}(1,2,3,4)\sim u(\frac{\delta_{ab}\delta_{cd}}{s}+{\rm
cyclic})\,.$ (37)

which vanishes at $u=0$ for the two colour-compatible flavour structures
$\delta_{ab}\delta_{cd}$ and $\delta_{bc}\delta_{ad}$ (and not for the third
one). At higher points, we will require a slightly stronger condition: at a
given (colour-compatible) locus, the only non-vanishing terms have flavours
structures that link the first and second set of indices (i.e. are
proportional to $\delta_{1\beta_{1}}$ or $\delta_{\alpha_{1}\beta_{1}}$). We
have checked this property at 6point and 8point; it would be interesting to
investigate it at higher multiplicity.

Under this assumption, it is straightforward to derive the zeros of DBI from
the KLT formalism with $A^{\rm NLSM}$ and $\widetilde{A}^{\rm YMS}$: again all
partial amplitudes $\widetilde{A}$ of the sum are only non-vanishing when the
flavour structures link the first and second sets that define the locus. The
resulting DBI amplitude therefore has the same property and hence vanishes
along the complementary flavour structures.

The final possibility is to take both single copy factors to be Yang-Mills-
scalar. For the resulting doubly flavoured theory, it is convenient to define
a trace notation for the nested delta structures. For example, at 4-pt,
$[AB][CD]:=\delta_{ab}\delta_{\tilde{a}\tilde{b}}\delta_{cd}\delta_{\tilde{c}\tilde{d}}$
while
$[ABCD]:=\delta_{ab}\delta_{\tilde{b}\tilde{c}}\delta_{cd}\delta_{\tilde{a}\tilde{d}}$.
The gravitational interaction only gives ’short traces’ pairing up two
scalars, effectively reducing to a single flavour structure777This is similar
to the truncation from BAS to ${\rm Tr}(\Phi^{3})$ ., and switches off the
non-chiral NLSM.

Under the KLT sum (36), the zeros of YMS carry over to the NLSMgrav amplitudes
that do not pair up the first and second sets that define the locus888A
possible pitfall in this construction lies in the presence of singularities in
the other factor $A$. Indeed this can have a singularity at some
$s_{\alpha_{1},\beta_{1}}=0$; however, the resulting term will then be
proportional to $\delta_{\alpha_{1}\beta_{1}}$ and hence outside of the
flavour structures that should exhibit the zero at this locus.. For the 6-pt
skinny rectangle (4), this implies that flavour structures like $[ABCDEF]$ and
$[ABCF][DE]$ have single zeros, while e.g. $[ABCD][EF]$ does not. Moreover,
the short trace sector (for instance $[AB][CD][EF]$) corresponds to graviton
exchange and hence has double zeros.

##  VI Conclusions

In this letter, we have shown that three double copy theories inherit
amplitude zeros from their single copies: scaffolded GR, multi-DBI and the
special Galileon. For the latter two, these zeros underlie their enhanced soft
behaviours. Moreover, we have identified the extended Lagrangians that
generate the mixed amplitudes from the near-zero factorization as covariantly
coupled colorless $\Phi^{3}$, where the minimal coupling is to different
metrics for the three cases. Finally, we prove that the inherited zeros of the
double copy theories can be explained via the most general version of the KLT
formula.

Independently of the double copy relations, one could wonder whether the
different causal diamonds of Arkani-Hamed  _et al._ (2023d) correspond to
independent zeros and factorisations, or whether they imply each other.
Indeed, the zero of skinny rectangles combined with the near-zero
factorization imply the vanishing of height-two ($i=2)$ causal diamonds. To
see this, mentally split the height-two causal diamond into two strips of
length $n-4$. One of these can be seen as the near-zero limit of an $n$-point
skinny rectangle, and hence implies the factorisation into an $(n-1)$-point
amplitude. The second strip of the original causal diamond is then the full
skinny rectangle for this $(n-1)$-point amplitude, which therefore has to
vanish.

Unfortunately, there appears to be an obstacle to generalising this argument
to $i=3$ and higher. When splitting these rectangles into smaller ones, the
resulting shapes are $i-1$ Mandelstam variables short of defining a lower
causal diamond. For $i\geq 3$, one therefore misses more than one Mandelstam
variables and the near-zero factorisation of Arkani-Hamed  _et al._ (2023d)
does not apply. Possibly, this issue can be remedied by applying the new
insights from Cao  _et al._ (2024). In this work, it was found that the ${\rm
Tr}(\Phi^{3})$ amplitude factorizes under a 2-split of Mandelstam variables,
which is exactly one row shorter than the entire zero locus. This might allow
for an extension of the previous argument to all $i$; we leave a further
investigation for future work.

Another interesting topic for future research is to look for a possible
stringy generalization of the special Galileon. The SG inherits all the zeros
from the NLSM via the KLT double copy, which arises from the closed/open
string relation. We have checked via some examples that the mod square
generalization Arkani-Hamed  _et al._ (2021) of the ${\rm Tr}(\Phi^{3})$
stringy amplitude also admits zeros. For scaffolded GR, the stringy integral
is the mod square generalisation of that of scaffolded YM. There is no known
UV completion or string theory embedding of the SG, although higher-derivative
corrections up to finite ${\cal O}(\alpha^{\prime})$ has been found Elvang
_et al._ (2019); Bartsch  _et al._ (2024). It would be interesting to see
whether the framework of surfaceology provides a new way to approach such a
stringy generalization of the SG.

Finally, it is interesting to consider invariant polynomials that feature zero
loci in a covariant manner as building blocks for amplitudes. Such a program
will make manifest to what extent amplitudes are determined by their zero loci
and pole structure.

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