# A Very Short Introduction to Topos Theory
(adapted from Prof. Pettigrew’s notes)

Eric Schmid  New York University

## Preface

The original notes by Prof. Pettigrew are invaluable, and Prof. Awodey’s book
is indispensible. My deepest gratitude to Prof. Pettigrew for making his notes
publicly available online and massive thanks to Prof. Awodey for his rigorous
work in category theory. I am extremely grateful for the academic support from
Prof William Chin.

###### Contents

  1. Preface
  2. 1 Motivating Category Theory
    1. 1.1 The Idea Behind Category Theory
    2. 1.2 Traditional vs. Categorical Perspectives
  3. 2 The Definition of a Category
    1. 2.1 Examples of Categories
  4. 3 Slice Categories [3]
    1. 3.1 Sets/I and Sets$(\rightarrow)$ taken from Jacobs
      1. 3.1.1 Sets/I
      2. 3.1.2 Sets$(\rightarrow)$
  5. 4 Monics, Epics, and Isomorphisms
    1. 4.1 Examples in Set
    2. 4.2 Examples in Grp
  6. 5 Diagrams, Cones, Cocones, Limits, Colimits
    1. 5.1 Diagrams
      1. 5.1.1 Examples of Diagrams
    2. 5.2 Cones
    3. 5.3 Limits
      1. 5.3.1 Examples of Limits
    4. 5.4 Cocones
    5. 5.5 Colimits
      1. 5.5.1 Examples of Colimits
    6. 5.6 Equalizers and Coequalizers
      1. 5.6.1 Equalizers
      2. 5.6.2 Coequalizers
    7. 5.7 Products and coproducts
      1. 5.7.1 Products
      2. 5.7.2 Coproducts
    8. 5.8 Pushouts and Pullbacks
      1. 5.8.1 Pullbacks
      2. 5.8.2 Pushouts
  7. 6 Initial and Terminal Objects [1]
    1. 6.1 Definitions
      1. 6.1.1 Examples
    2. 6.2 Properties
  8. 7 Members of Objects [1]
    1. 7.1 Membership
    2. 7.2 Injective and Surjective
  9. 8 Exponential Objects [1]
  10. 9 Subobjects and Their Classifiers [1]
    1. 9.1 Subobjects
    2. 9.2 Subobject Classifiers
      1. 9.2.1 Motivation
      2. 9.2.2 Inverse Images of Subobjects
      3. 9.2.3 Definition
  11. 10 The Definition of a Topos
    1. 10.1 The Definition
    2. 10.2 Examples
    3. 10.3 Fundamental Theorem of Toposes
  12. 11 Algebra of Subobjects
    1. 11.0.1 Subobject Lattices
    2. 11.0.2 Lattice Structure of Subobjects
    3. 11.0.3 Heyting Algebra of Subobjects
    4. 11.0.4 Power Objects in Topos Theory
    5. 11.0.5 Boolean Topos
    6. 11.0.6 Internal Logic and Lawvere-Tierney Topology
  13. 12 Kinds of Topos
    1. 12.1 Non-degenerate Toposes
    2. 12.2 Well-pointed Toposes
    3. 12.3 Bivalent Toposes
    4. 12.4 Boolean Toposes
    5. 12.5 Natural Number Objects
  14. 13 Functors
    1. 13.1 The Definition of a Functor
    2. 13.2 Examples of Functors
    3. 13.3 Properties of Functors
  15. 14 Natural Transformations
    1. 14.1 The Definition of a Natural Transformation
    2. 14.2 Examples of Natural Transformations
    3. 14.3 Properties of Natural Transformations
  16. 15 Adjoint Functors
    1. 15.1 Definition
      1. 15.1.1 Definition via Counit-Unit Adjunction [4]
      2. 15.1.2 Definition via Universal Morphisms [4]
    2. 15.2 Examples Concerning Set
    3. 15.3 Examples Concerning Forgetful Functors

##  Chapter 1 Motivating Category Theory

###  1.1 The Idea Behind Category Theory

Category theory encourages a shift from focusing on the intrinsic properties
of mathematical objects to emphasizing their roles and interactions. This
approach can be summarized by the phrase: ”Ask not what a thing is; ask what
it does” [1]. This shift in perspective helps unify various areas of
mathematics by providing a common framework to describe different mathematical
constructs through their relationships.

###  1.2 Traditional vs. Categorical Perspectives

Traditional mathematics often describes objects by their intrinsic properties:

  * •

Groups: Defined by a set with an operation following group axioms.

  * •

Sets: Collections of distinct elements.

  * •

Topological Spaces: Sets endowed with a topology specifying open sets.

In contrast, category theory interprets these objects through their morphisms:

  * •

Groups: Understood through group homomorphisms.

  * •

Sets: Understood through functions.

  * •

Topological Spaces: Understood through continuous maps.

Category theory provides a more flexible and general approach by focusing on
the relationships between objects rather than their internal structure.

##  Chapter 2 The Definition of a Category

A category $\mathcal{C}$ consists of:

  * •

A collection of objects $\mathrm{Ob}(\mathcal{C})$.

  * •

A collection of morphisms (arrows) $\mathrm{Ar}(\mathcal{C})$.

  * •

Two functions: domain and codomain, assigning to each morphism $f:A\rightarrow
B$ its source $A$ and target $B$.

  * •

A composition function $\circ$ that assigns to each pair of composable
morphisms $f:A\rightarrow B$ and $g:B\rightarrow C$ an arrow $g\circ
f:A\rightarrow C$.

${A}$${B}$${C}$$\scriptstyle{f}$$\scriptstyle{g}$

These components must satisfy the following axioms:

  1. 1.

Associativity: For all composable arrows $f,g,h$, the equation $(h\circ
g)\circ f=h\circ(g\circ f)$ holds.

  2. 2.

Identity: For each object $A$, there exists an identity morphism
$\mathrm{id}_{A}:A\rightarrow A$ such that for any morphisms $f:A\rightarrow
B$ and $g:B\rightarrow A$, we have $\mathrm{id}_{B}\circ f=f$ and
$g\circ\mathrm{id}_{A}=g$.

###  2.1 Examples of Categories

  * •

Set: Objects are sets, and morphisms are functions between sets.

  * •

Grp: Objects are groups, and morphisms are group homomorphisms.

  * •

Top: Objects are topological spaces, and morphisms are continuous maps.

  * •

Vect: Objects are vector spaces, and morphisms are linear transformations.

##  Chapter 3 Slice Categories [3]

###  3.1 Sets/I and Sets$(\rightarrow)$ taken from Jacobs

Consider a family of sets as a function $\varphi:X\rightarrow I$. We often
describe a family of sets as a function $\varphi:X\rightarrow I$ and say that
$X$ is a family over $I$ and that $\varphi$ displays the family $(X_{i})$. In
order to emphasize that we think of such a map $\varphi$ as a family, we often
write it vertically as $\left(\begin{smallmatrix}X\\\ \downarrow\\\
I\end{smallmatrix}\right)$. A constant family is one of the form
$\left(\begin{smallmatrix}I\times X\\\ \downarrow\\\
I\end{smallmatrix}\right)$, where $\pi$ is the Cartesian product projection;
often it is written simply as $I\times X$. Notice that all fibers of this
constant family are (isomorphic to) $X$.

Such families $\left(\begin{smallmatrix}X\\\ \downarrow\\\
I\end{smallmatrix}\right)$ of sets give rise to two categories: the slice
category $\mathbf{Sets}/I$ and the arrow category
$\mathbf{Sets}(\rightarrow)$. The objects of $\mathbf{Sets}/I$ are the
$I$-indexed families, for a fixed set $I$; the objects of
$\mathbf{Sets}(\rightarrow)$ are all the $I$-indexed families, for all
possible $I$.

####  3.1.1 Sets/I

Objects: families $\left(\begin{smallmatrix}X\\\ \downarrow\\\
I\end{smallmatrix}\right)$.

Morphisms: $\left(\begin{smallmatrix}X\\\ \downarrow\\\
I\end{smallmatrix}\right)\xrightarrow{f}\left(\begin{smallmatrix}Y\\\
\downarrow\\\ I\end{smallmatrix}\right)$ are functions $f:X\rightarrow Y$
making the following diagram commute.

${X}$${Y}$${I}$$\scriptstyle{f}$$\scriptstyle{\varphi}$$\scriptstyle{\psi}$

####  3.1.2 Sets$(\rightarrow)$

Objects: families $\left(\begin{smallmatrix}X\\\ \downarrow\\\
I\end{smallmatrix}\right)$, for arbitrary sets $I$.

Morphisms: $\left(\begin{smallmatrix}X\\\ \downarrow\\\
I\end{smallmatrix}\right)\xrightarrow{(u,f)}\left(\begin{smallmatrix}Y\\\
\downarrow\\\ J\end{smallmatrix}\right)$ are pairs of functions
$u:I\rightarrow J$ and $f:X\rightarrow Y$ for which the following diagram
commutes.

${X}$${Y}$${I}$${J}$$\scriptstyle{f}$$\scriptstyle{\varphi}$$\scriptstyle{\psi}$$\scriptstyle{u}$

Hence, objects in the arrow category $\mathbf{Sets}(\rightarrow)$ involve an
extra function $u$ between the index sets. Notice that one can now view $f$ as
a collection of functions $f_{i}:X_{i}\rightarrow Y_{u(i)}$, since for
$x\in\varphi^{-1}(i)$, $f(x)$ lands in $\psi^{-1}(u(i))$. Composition and
identities in $\mathbf{Sets}(\rightarrow)$ are component-wise inherited from
$\mathbf{Sets}$.

We further remark that there is a codomain functor
$\text{cod}:\mathbf{Sets}(\rightarrow)\rightarrow\mathbf{Sets}$; it maps
$\left(\begin{smallmatrix}X\\\ \downarrow\\\ I\end{smallmatrix}\right)\mapsto
I$ and $\left(\begin{smallmatrix}(u,f)\end{smallmatrix}\right)\mapsto u$.

Also, for each $I$, there is a (non-full) inclusion functor
$\mathbf{Sets}/I\rightarrow\mathbf{Sets}(\rightarrow)$.

##  Chapter 4 Monics, Epics, and Isomorphisms

An arrow $f:A\rightarrow B$ in a category $\mathcal{C}$ is:

  * •

_Monic_ (monomorphism) if for all arrows $g,h:C\rightarrow A$, $f\circ
g=f\circ h$ implies $g=h$.

  * •

_Epic_ (epimorphism) if for all arrows $g,h:B\rightarrow C$, $g\circ f=h\circ
f$ implies $g=h$.

  * •

_Isomorphic_ (isomorphism) if there exists an arrow $g:B\rightarrow A$ such
that $g\circ f=\mathrm{id}_{A}$ and $f\circ g=\mathrm{id}_{B}$.

###  4.1 Examples in Set

  * •

Monic: Injective functions.

  * •

Epic: Surjective functions.

  * •

Isomorphisms: Bijective functions.

###  4.2 Examples in Grp

  * •

Monic: Injective group homomorphisms.

  * •

Epic: Surjective group homomorphisms.

  * •

Isomorphisms: Bijective group homomorphisms (isomorphisms of groups).

##  Chapter 5 Diagrams, Cones, Cocones, Limits, Colimits

In category theory, diagrams provide a structured way to visualize and
understand the relationships between objects and morphisms. Cones and cocones
are structures that relate diagrams to limits and colimits, respectively.

###  5.1 Diagrams

A _diagram_ in a category $\mathcal{C}$ is a functor
$D:J\rightarrow\mathcal{C}$ where $J$ is an indexing category. Objects of $J$
are mapped to objects of $\mathcal{C}$, and morphisms of $J$ are mapped to
morphisms in $\mathcal{C}$.

####  5.1.1 Examples of Diagrams

Diagrams can range from simple single-object diagrams to complex networks of
objects and morphisms, such as sequences, commutative squares, or more
intricate structures. They serve as a framework for discussing the
relationships between objects and morphisms.

${A}$${B}$${C}$${D}$$\scriptstyle{f}$$\scriptstyle{g}$$\scriptstyle{h}$$\scriptstyle{k}$

###  5.2 Cones

A _cone_ over a diagram $D:J\rightarrow\mathcal{C}$ consists of an object $L$
and a family of morphisms $\lambda_{X}:L\rightarrow D(X)$ for each object $X$
in $J$ such that for every morphism $f:X\rightarrow Y$ in $J$, the following
diagram commutes:

${L}$${D(X)}$${D(Y)}$$\scriptstyle{\lambda_{X}}$$\scriptstyle{\lambda_{Y}}$$\scriptstyle{D(f)}$

###  5.3 Limits

“Let $F:J\rightarrow\mathcal{C}$ be a diagram of shape $J$ in a category
$\mathcal{C}$. A cone to $F$ is an object $N$ of $\mathcal{C}$ together with a
family $\psi_{X}:N\rightarrow F(X)$ of morphisms indexed by the objects $X$ of
$J$, such that for every morphism $f:X\rightarrow Y$ in $J$, we have
$F(f)\circ\psi_{X}=\psi_{Y}$.

A limit of the diagram $F:J\rightarrow\mathcal{C}$ is a cone $(L,\phi)$ to $F$
such that for every cone $(N,\psi)$ to $F$, there exists a unique morphism
$u:N\rightarrow L$ such that $\phi_{X}\circ u=\psi_{X}$ for all $X$ in $J$.”
[4]

${N}$${F(X)}$${L}$${F(Y)}$$\scriptstyle{\psi_{X}}$$\scriptstyle{\psi_{Y}}$$\scriptstyle{u}$$\scriptstyle{F(f)}$$\scriptstyle{\phi_{X}}$$\scriptstyle{\phi_{Y}}$

“One says that the cone $(N,\psi)$ factors through the cone $(L,\phi)$ with
the unique factorization $u$. The morphism $u$ is sometimes called the
mediating morphism.” [4]

####  5.3.1 Examples of Limits

  * •

Products: The product of two objects $A$ and $B$ in a category $\mathcal{C}$
is a limit of the diagram consisting of $A$ and $B$ with no morphisms between
them.

  * •

Equalizers: An equalizer of two parallel morphisms $f,g:A\rightarrow B$ is a
limit of the diagram formed by $A$ and $B$ with two parallel arrows.

###  5.4 Cocones

A _cocone_ under a diagram $D:J\rightarrow\mathcal{C}$ consists of an object
$L$ and a family of morphisms $\lambda_{X}:D(X)\rightarrow L$ for each object
$X$ in $J$ such that for every morphism $f:X\rightarrow Y$ in $J$, the
following diagram commutes:

${D(X)}$${D(Y)}$${L}$$\scriptstyle{D(f)}$$\scriptstyle{\lambda_{X}}$$\scriptstyle{\lambda_{Y}}$

###  5.5 Colimits

A _colimit_ of a diagram $D:J\rightarrow\mathcal{C}$ is a cocone $(L,\lambda)$
under $D$ that is universal among all such cocones. This means that for any
other cocone $(L^{\prime},\lambda^{\prime})$ under $D$, there exists a unique
morphism $u:L\rightarrow L^{\prime}$ such that $\lambda^{\prime}_{X}\circ
u=\lambda_{X}$ for all $X$ in $J$.

${D(X)}$${D(Y)}$${L}$${L^{\prime}}$$\scriptstyle{D(f)}$$\scriptstyle{\lambda^{\prime}_{X}}$$\scriptstyle{\lambda_{X}}$$\scriptstyle{\lambda_{Y}}$$\scriptstyle{\lambda^{\prime}_{Y}}$$\scriptstyle{u}$

####  5.5.1 Examples of Colimits

  * •

Coproducts: The coproduct of two objects $A$ and $B$ in a category
$\mathcal{C}$ is a colimit of the diagram consisting of $A$ and $B$ with no
morphisms between them.

  * •

Coequalizers: A coequalizer of two parallel morphisms $f,g:A\rightarrow B$ is
a colimit of the diagram formed by $A$ and $B$ with two parallel arrows.

###  5.6 Equalizers and Coequalizers

####  5.6.1 Equalizers

An equalizer of two parallel arrows $f,g:A\rightarrow B$ is an object $E$
together with a morphism $e:E\rightarrow A$ such that $f\circ e=g\circ e$ and
for any object $Z$ with a morphism $z:Z\rightarrow A$ such that $f\circ
z=g\circ z$, there exists a unique morphism $u:Z\rightarrow E$ such that
$e\circ u=z$. This is depicted as follows:

${E}$${A}$${B}$${Z}$$\scriptstyle{e}$$\scriptstyle{f}$$\scriptstyle{g}$$\scriptstyle{u}$$\scriptstyle{z}$

####  5.6.2 Coequalizers

A coequalizer of two parallel arrows $f,g:A\rightarrow B$ in a category
$\mathcal{C}$ is an object $Q$ together with a morphism $q:B\rightarrow Q$,
universal with the property $q\circ f=q\circ g$, as in the following diagram:
[2]

${A}$${B}$${Q}$${Z}$$\scriptstyle{f}$$\scriptstyle{g}$$\scriptstyle{q}$$\scriptstyle{z}$$\scriptstyle{u}$

Given any $Z$ and $z:B\rightarrow Z$, if $z\circ f=z\circ g$, then there
exists a unique $u:Q\rightarrow Z$ such that $u\circ q=z$ [2].

###  5.7 Products and coproducts

####  5.7.1 Products

A product of two objects $A$ and $B$ in a category $\mathcal{C}$ is an object
$P$ together with two morphisms $\pi_{1}:P\rightarrow A$ and
$\pi_{2}:P\rightarrow B$ such that for any object $X$ with morphisms
$f:X\rightarrow A$ and $g:X\rightarrow B$, there exists a unique morphism
$u:X\rightarrow P$ such that $\pi_{1}\circ u=f$ and $\pi_{2}\circ u=g$. This
can be depicted as:

${X}$${A}$${P}$${B}$$\scriptstyle{f}$$\scriptstyle{g}$$\scriptstyle{u}$$\scriptstyle{\pi_{1}}$$\scriptstyle{\pi_{2}}$

####  5.7.2 Coproducts

A coproduct of two objects $A$ and $B$ in a category $\mathcal{C}$ is an
object $C$ together with two morphisms $\iota_{1}:A\rightarrow C$ and
$\iota_{2}:B\rightarrow C$ such that for any object $X$ with morphisms
$f:A\rightarrow X$ and $g:B\rightarrow X$, there exists a unique morphism
$u:C\rightarrow X$ such that $u\circ\iota_{1}=f$ and $u\circ\iota_{2}=g$. This
can be depicted as:

${A}$${C}$${B}$${X}$$\scriptstyle{\iota_{1}}$$\scriptstyle{f}$$\scriptstyle{u}$$\scriptstyle{\iota_{2}}$$\scriptstyle{g}$

###  5.8 Pushouts and Pullbacks

####  5.8.1 Pullbacks

A pullback (also known as a fiber product) of two morphisms $f:X\rightarrow Z$
and $g:Y\rightarrow Z$ in a category $\mathcal{C}$ is an object $P$ together
with two morphisms $p_{1}:P\rightarrow X$ and $p_{2}:P\rightarrow Y$ such that
the following diagram commutes:

${P}$${Y}$${X}$${Z}$$\scriptstyle{p_{2}}$$\scriptstyle{p_{1}}$$\scriptstyle{g}$$\scriptstyle{f}$

Moreover, $P$ must be universal with respect to this property, meaning that
for any other object $Q$ with morphisms $q_{1}:Q\rightarrow X$ and
$q_{2}:Q\rightarrow Y$ making the diagram commute, there exists a unique
morphism $u:Q\rightarrow P$ such that $p_{1}\circ u=q_{1}$ and $p_{2}\circ
u=q_{2}$. This situation is illustrated in the following commutative diagram:

${Q}$${P}$${Y}$${X}$${Z}$$\scriptstyle{q_{2}}$$\scriptstyle{q_{1}}$$\scriptstyle{u}$$\scriptstyle{p_{1}}$$\scriptstyle{p_{2}}$$\scriptstyle{g}$$\scriptstyle{f}$

Explicitly, a pullback of the morphisms $f$ and $g$ consists of an object $P$
and two morphisms $p_{1}:P\rightarrow X$ and $p_{2}:P\rightarrow Y$ for which
the diagram

${P}$${Y}$${X}$${Z}$$\scriptstyle{p_{2}}$$\scriptstyle{p_{1}}$$\scriptstyle{g}$$\scriptstyle{f}$

commutes. Moreover, the pullback $(P,p_{1},p_{2})$ must be universal with
respect to this diagram. That is, for any other such triple $(Q,q_{1},q_{2})$
where $q_{1}:Q\rightarrow X$ and $q_{2}:Q\rightarrow Y$ are morphisms with
$f\circ q_{1}=g\circ q_{2}$, there must exist a unique $u:Q\rightarrow P$ such
that

$p_{1}\circ u=q_{1},\quad p_{2}\circ u=q_{2}.$

This situation is illustrated in the following commutative diagram:

${Q}$${P}$${Y}$${X}$${Z}$$\scriptstyle{q_{2}}$$\scriptstyle{q_{1}}$$\scriptstyle{u}$$\scriptstyle{p_{1}}$$\scriptstyle{p_{2}}$$\scriptstyle{g}$$\scriptstyle{f}$

####  5.8.2 Pushouts

A pushout (also known as a cofiber product) of two morphisms $f:Z\rightarrow
X$ and $g:Z\rightarrow Y$ in a category $\mathcal{C}$ is an object $P$
together with two morphisms $\iota_{1}:X\rightarrow P$ and
$\iota_{2}:Y\rightarrow P$ such that the following diagram commutes:

${Z}$${Y}$${X}$${P}$$\scriptstyle{g}$$\scriptstyle{f}$$\scriptstyle{\iota_{2}}$$\scriptstyle{\iota_{1}}$

Moreover, $P$ must be universal with respect to this property, meaning that
for any other object $Q$ with morphisms $q_{1}:X\rightarrow Q$ and
$q_{2}:Y\rightarrow Q$ making the diagram commute, there exists a unique
morphism $u:P\rightarrow Q$ such that $u\circ\iota_{1}=q_{1}$ and
$u\circ\iota_{2}=q_{2}$.

${Z}$${Y}$${X}$${P}$${Q}$$\scriptstyle{g}$$\scriptstyle{f}$$\scriptstyle{\iota_{2}}$$\scriptstyle{q_{2}}$$\scriptstyle{\iota_{1}}$$\scriptstyle{q_{1}}$$\scriptstyle{u}$

##  Chapter 6 Initial and Terminal Objects [1]

Consider the empty diagram in the category $\mathcal{C}$. The cones and
cocones over this diagram are simply the objects of $\mathcal{C}$. Therefore:

  * •

If the empty diagram has a limit, it is an object $1$ such that, for every
object $A$ in $\mathcal{C}$, there is a unique morphism $1_{A}$ :
$A\rightarrow 1$.

  * •

If the empty diagram has a colimit, it is an object $0$ such that, for every
object $A$ in $\mathcal{C}$, there is a unique morphism $0_{A}$ :
$0\rightarrow A$.

###  6.1 Definitions

  * •

Initial Object: A limit of the empty diagram (if it exists) is called a
terminal object of $\mathcal{C}$.

  * •

Terminal Object: A colimit of the empty diagram (if it exists) is called an
initial object of $\mathcal{C}$.

By Propositions 5.1.3 and 5.2.3, a terminal or initial object is unique up to
isomorphism. By Axiom 2, every topos has initial and terminal objects.

####  6.1.1 Examples

  * •

In Set, the empty set $\emptyset$ is the only initial object, and any
singleton set $\\{a\\}$ is a terminal object.

  * •

In Grp, the trivial group $\\{e_{G}\\}$ is both initial and terminal. Such
objects are called zero objects.

  * •

In a category based on a poset, any minimum element is an initial object (if
it exists), and any maximum element is a terminal object (if it exists).

  * •

If $\mathcal{C}$ is a category, then $Id_{X}:X\rightarrow X$ is a terminal
object of $\mathcal{C}/X$. If $\mathcal{C}$ has an initial object $0$, then
$0_{X}:0\rightarrow X$ is an initial object of $\mathcal{C}/X$.

###  6.2 Properties

  * •

If $1$ is a terminal object and $f:1\rightarrow A$, then the arrow is monic.

  * •

If $1$ is a terminal object, then $1\times A\cong A\cong A\times 1$.

##  Chapter 7 Members of Objects [1]

In traditional set theory, the fundamental notion is the membership relation.
In category theory, the analogous concept involves arrows rather than
elements. In a category $\mathcal{C}$ with a terminal object $1$, the members
of an object $A$ are the morphisms from $1$ to $A$.

###  7.1 Membership

In the context of category theory, the action of picking out a member of a set
$A$ can be understood as a function from a singleton set into $A$. This is
because any singleton is a terminal object and any terminal object is a
singleton in Set.

Member of: If $\mathcal{C}$ is a category with a terminal object $1$ and $A$
is an object of $\mathcal{C}$, then a member of $A$ is an arrow $x$ :
$1\rightarrow A$.

This definition implies that there cannot be ’membership chains’ as members
are arrows, not objects, and arrows cannot have members.

  * •

$1$ has exactly one element.

  * •

If $0$ has an element, then $0\cong 1$.

###  7.2 Injective and Surjective

We can also define injective and surjective arrows in this context.

Injective and Surjective: Given $f:A\rightarrow B$, we say that

  * •

The arrow is injective if, for all $x,y$ : $1\rightarrow A$, if $fx=fy$ then
$x=y$.

  * •

The arrow is surjective if, for all $y:1\rightarrow B$, there is
$x:1\rightarrow A$ such that $fx=y$.

In Set, the injective arrows are the monics, and the surjective arrows are the
epics. However, this is not necessarily true in all categories with terminal
objects.

##  Chapter 8 Exponential Objects [1]

Axiom 2 ensures that all toposes have analogues of the addition and
multiplication operations on sets—they are coproducts and products,
respectively. However, it does not guarantee analogues of power sets $P(A)$ or
function spaces $B^{A}=\\{f:A\rightarrow B\\}$. To address this, we introduce
an axiom guaranteeing these objects.

Consider a (set-theoretical) function $f:A\times C\rightarrow B$. For every
element $c\in C$, the function $f_{c}:a\mapsto f(a,c)$ is a function from $A$
to $B$. The function $\hat{f}:c\mapsto f_{c}$ maps $C$ into $B^{A}$. There is
also a function $\mathrm{ev}:A\times B^{A}\rightarrow B$ that evaluates a
function from $A$ to $B$ at a value in $A$.

Therefore:

$ev(a,f_{c})=f_{c}(a)=f(a,c)$

Exponential: Suppose $\mathcal{C}$ is a category with products. For any
objects $A$ and $B$, an exponential of $A$ and $B$ consists of

  * •

An object $B^{A}$ of category $\mathcal{C}$

  * •

An arrow $ev:A\times B^{A}\rightarrow B$ of category $\mathcal{C}$

such that for any arrow $f:A\times C\rightarrow B$, there is an arrow
$\hat{f}:C\rightarrow B^{A}$ making the following diagram commute:

${A\times B^{A}}$${B}$${A\times
C}$$\scriptstyle{\mathrm{ev}}$$\scriptstyle{\mathrm{Id}_{A}\times\hat{f}}$$\scriptstyle{f}$

Exponentials of $A$ and $B$ are unique up to isomorphism, and anything
isomorphic to an exponential of $A$ and $B$ is itself an exponential of $A$
and $B$.

We can check the validity of the exponential $B^{A}$ by considering whether it
satisfies certain basic conditions stated in terms of the membership relation.
In set theory, an exponential object should have the following property:

$f\text{ is a member of }B^{A}\text{ iff }f:A\rightarrow B$

Although this is not exactly the result we get in category theory, we do
obtain a close analogue: there is a one-to-one correspondence between arrows
$f:A\rightarrow B$ and arrows $g:1\rightarrow B^{A}$.

Name of an arrow: If $A\rightarrow B$ is an arrow, the name of $f$ (written
$\ulcorner f\urcorner$) is the arrow $1\rightarrow B^{A}$ such that the
following diagram commutes:

${A\times B^{A}}$${B}$${A\times
1}$$\scriptstyle{\mathrm{ev}}$$\scriptstyle{\mathrm{Id}_{A}\times\ulcorner
f\urcorner}$$\scriptstyle{f}$

There is a bijection $f\mapsto\ulcorner f\urcorner$ between the set of arrows
$f:A\rightarrow B$ and the set of arrows $\ulcorner f\urcorner:1\rightarrow
B^{A}$.

With the definition of an exponential in hand, we can introduce the third
axiom of a topos.

A topos has exponentials for every pair of objects.

Cartesian Closed Category: A category with limits for all finite diagrams and
exponentials for all pairs of objects is called a Cartesian closed category.

[1].

##  Chapter 9 Subobjects and Their Classifiers [1]

###  9.1 Subobjects

In category theory, the analogue of a subset of a set $A$ is called a
subobject of $A$. A subobject is not another object but an arrow, specifically
a monic arrow from an object $S$ into $A$.

Part or Subobject: Suppose $A$ is an object. Then a subobject of $A$ is a
monic arrow $S\rightarrow A$.

Inclusion: Suppose $S\rightarrow A$ and $T\rightarrow A$ are subobjects of
$A$. We say that $S$ is included in $T$ (written $S\subseteq T$) if there is a
morphism $S\rightarrow T$ such that the following diagram commutes:

${S}$${T}$${A}$

###  9.2 Subobject Classifiers

Next, we introduce an object $\Omega$ in a topos whose members act as truth
values and associate with each subobject $S\rightarrow A$ a characteristic
function $A\rightarrow\Omega$.

####  9.2.1 Motivation

Consider sets where $\Omega=\\{\text{true, false}\\}$. The characteristic
function of a subset $S\subseteq A$ can be seen as the function $\chi_{S}$
mapping $A$ to $\\{\text{true, false}\\}$ such that the inverse image of
$\\{\text{true}\\}$ under $\chi_{S}$ is $S$.

####  9.2.2 Inverse Images of Subobjects

In Set, a pullback for the diagram

${B}$${S}$${A}$$\scriptstyle{f}$$\scriptstyle{i}$

is the set

$\\{(b,s)\mid f(b)=i(s)\\}$

If $S\rightarrow A$ and $i$ is the inclusion map $i:s\mapsto s$ for all $s\in
S$, then a pullback is

$\\{(b,s)\in B\times S\mid f(b)=i(s)=s\\}\cong\\{b\in B\mid f(b)\in
S\\}=f^{-1}(S)$

Inverse Image: If $S\rightarrow A$ and $B\rightarrow A$, then the pullback of
the diagram

${B}$${S}$${A}$$\scriptstyle{f}$$\scriptstyle{i}$

is called an inverse image of $S\rightarrow A$ under $f$ (written
$f^{-1}(S)$).

####  9.2.3 Definition

We define the characteristic function $\chi_{i}$ of a subobject $S\rightarrow
A$ to be the unique function such that $S$ is an inverse image of
$1\rightarrow\Omega$ under $\chi_{i}$.

Subobject Classifier: Suppose $\mathcal{C}$ is a category with a terminal
object $1$. A subobject classifier in $\mathcal{C}$ consists of:

  * •

An object $\Omega$ of category $\mathcal{C}$.

  * •

An arrow $1\rightarrow\Omega$

such that for any object $A$ and subobject $S\rightarrow A$, there is a unique
arrow $\chi_{S}:A\rightarrow\Omega$ such that:

${S}$${1}$${A}$${\Omega}$$\scriptstyle{1}$$\scriptstyle{\chi_{i}}$

is a pullback square.

Subobject classifiers are unique up to isomorphism, and anything isomorphic to
a subobject classifier is itself a subobject classifier.

The final axiom of toposes can now be stated:

A topos has a subobject classifier.

False: The arrow $1\rightarrow\Omega$ is the characteristic function of the
subobject $0\rightarrow 1$.

[1].

##  Chapter 10 The Definition of a Topos

###  10.1 The Definition

Having outlined the necessary axioms, we define a topos:

Topos: A topos is a category with:

  * •

Limits and colimits for all finite diagrams.

  * •

Exponentials for every pair of objects.

  * •

A subobject classifier.

###  10.2 Examples

  * •

Set is a topos.

  * •

Grp is not a topos.

  * •

FinSet is a topos.

###  10.3 Fundamental Theorem of Toposes

“If $\mathcal{E}$ is a topos and X is an object in $\mathcal{E}$, then
$\mathcal{E}$/X is a topos.” [1].

##  Chapter 11 Algebra of Subobjects

In category theory, understanding subobjects and their algebra is crucial for
grasping the structure and properties of categories. Here, we explain
subobject lattices and their relationship to Boolean and Heyting algebras.

####  11.0.1 Subobject Lattices

A subobject of an object $A$ in a category $\mathcal{C}$ is defined as an
equivalence class of monomorphisms $m:S\hookrightarrow A$. Two monomorphisms
$m:S\hookrightarrow A$ and $m^{\prime}:S^{\prime}\hookrightarrow A$ are
considered equivalent if there exists an isomorphism $f:S\rightarrow
S^{\prime}$ such that $m=m^{\prime}\circ f$.

The collection of all subobjects of $A$, denoted as $\text{Sub}(A)$, forms a
partially ordered set (poset) under inclusion. Specifically, for subobjects
$[m:S\hookrightarrow A]$ and $[m^{\prime}:S^{\prime}\hookrightarrow A$, we
have $[m]\leq[m^{\prime}]$ if there exists a morphism $f:S\rightarrow
S^{\prime}$ making the following diagram commute:

${S}$${S^{\prime}}$${A}$$\scriptstyle{m}$$\scriptstyle{f}$$\scriptstyle{m^{\prime}}$

This poset of subobjects, $\text{Sub}(A)$, possesses additional structure,
making it a lattice.

####  11.0.2 Lattice Structure of Subobjects

A lattice is a poset in which any two elements have a greatest lower bound
(glb) or meet, and a least upper bound (lub) or join. For subobjects $S$ and
$T$ of $A$:

  * •

The meet $S\wedge T$ is given by the intersection of $S$ and $T$ in $A$.

  * •

The join $S\vee T$ is represented by the subobject generated by the union of
$S$ and $T$.

Formally, these operations can be defined through pullbacks and pushouts in
the category $\mathcal{C}$.

####  11.0.3 Heyting Algebra of Subobjects

In a Heyting category, the poset $\text{Sub}(A)$ is not just a lattice but a
Heyting algebra. This means it supports an additional operation called
implication $\rightarrow$, which is characterized by the property that for
subobjects $S$ and $T$ of $A$, there is a largest subobject $U$ such that
$S\wedge U\leq T$.

Heyting algebras generalize Boolean algebras and are essential in the internal
logic of a topos, particularly in intuitionistic logic.

####  11.0.4 Power Objects in Topos Theory

A topos can be seen as a categorical generalization of set theory,
incorporating both logical and geometrical aspects. In a topos, every
subobject poset $\text{Sub}(A)$ is a Heyting algebra, and there exists a power
object $\mathcal{P}(A)$, which internalizes the notion of the power set in set
theory.

The power object $\mathcal{P}(A)$ of an object $A$ is equipped with a
monomorphism $\in_{A}:A\times\mathcal{P}(A)\rightarrow\Omega$, where $\Omega$
is the subobject classifier of the topos. This morphism satisfies a universal
property analogous to the characteristic function of a subset in set theory.

####  11.0.5 Boolean Topos

In a Boolean topos, the internal Heyting algebra $\text{Sub}(A)$ for any
object $A$ is a Boolean algebra. This implies that every subobject has a
complement, and the internal logic of the topos aligns with classical Boolean
logic.

####  11.0.6 Internal Logic and Lawvere-Tierney Topology

The internal logic of a topos can be intuitionistic or classical, depending on
whether it forms a Heyting algebra or a Boolean algebra. The Lawvere-Tierney
topology provides a framework to study these logical structures within a
topos, defining modalities that extend the logical operations within the
category.

### References

  * •

Pettigrew, R. _An Introduction to Toposes_. [Unpublished Notes].

  * •

Awodey, S. (2010). _Category Theory_. Oxford Logic Guides 52, Oxford
University Press.

  * •

nLab: Heyting Algebra

##  Chapter 12 Kinds of Topos

###  12.1 Non-degenerate Toposes

A topos is non-degenerate if it has more than one object and more than one
morphism.

###  12.2 Well-pointed Toposes

A topos is well-pointed if the only arrow $1\rightarrow 1$ is the identity
arrow.

###  12.3 Bivalent Toposes

A topos is bivalent if it has exactly two objects and two morphisms.

###  12.4 Boolean Toposes

A topos is Boolean if its subobject classifier is a Boolean algebra.

###  12.5 Natural Number Objects

A topos has a natural number object if there is an object $N$ and arrows
$0:1\rightarrow N$ and $s:N\rightarrow N$ such that for any object $A$ and
arrows $f:1\rightarrow A$ and $g:A\rightarrow A$, there is a unique arrow
$h:N\rightarrow A$ such that $h\circ 0=f$ and $h\circ s=g\circ h$.

[1].

##  Chapter 13 Functors

###  13.1 The Definition of a Functor

A functor $F:\mathcal{C}\rightarrow\mathcal{D}$ consists of:

  * •

A function $F:\mathrm{Ob}(\mathcal{C})\rightarrow\mathrm{Ob}(\mathcal{D})$

  * •

A function $F:\mathrm{Ar}(\mathcal{C})\rightarrow\mathrm{Ar}(\mathcal{D})$

such that the following properties hold:

  * •

$F(\mathrm{id}_{A})=\mathrm{id}_{F(A)}$ for all objects $A$ in $\mathcal{C}$.

  * •

$F(g\circ f)=F(g)\circ F(f)$ for all arrows $f,g$ in $\mathcal{C}$.

###  13.2 Examples of Functors

  * •

The identity functor $\mathrm{Id}:\mathcal{C}\rightarrow\mathcal{C}$.

  * •

The constant functor $C:\mathcal{C}\rightarrow\mathcal{D}$ that sends every
object to a fixed object $D$ and every arrow to $\mathrm{id}_{D}$.

  * •

The hom-functor $\mathrm{Hom}(A,-):\mathcal{C}\rightarrow\mathbf{Set}$ that
sends every object $B$ to the set of arrows $A\rightarrow B$ and every arrow
$f:B\rightarrow C$ to the function
$\mathrm{Hom}(A,f):\mathrm{Hom}(A,B)\rightarrow\mathrm{Hom}(A,C)$ given by
composition with $f$.

###  13.3 Properties of Functors

Functors preserve the structure of categories in various ways:

  * •

They preserve isomorphisms: If $f:A\rightarrow B$ is an isomorphism in
$\mathcal{C}$, then $F(f)$ is an isomorphism in $\mathcal{D}$.

  * •

They preserve commutative diagrams: If a diagram commutes in $\mathcal{C}$,
its image under $F$ commutes in $\mathcal{D}$.

  * •

They preserve limits and colimits: If $\mathcal{C}$ has a limit (or colimit)
of a diagram $D$, then $\mathcal{D}$ has a limit (or colimit) of $F(D)$.

##  Chapter 14 Natural Transformations

###  14.1 The Definition of a Natural Transformation

A natural transformation $\eta:F\Rightarrow G$ between two functors
$F,G:\mathcal{C}\rightarrow\mathcal{D}$ consists of:

  * •

For each object $A$ in $\mathcal{C}$, an arrow $\eta_{A}:F(A)\rightarrow G(A)$
in $\mathcal{D}$

such that for every arrow $f:A\rightarrow B$ in $\mathcal{C}$, the following
diagram commutes:

${F(A)}$${F(B)}$${G(A)}$${G(B)}$$\scriptstyle{F(f)}$$\scriptstyle{\eta_{A}}$$\scriptstyle{\eta_{B}}$$\scriptstyle{G(f)}$

This commutative diagram ensures that natural transformations respect the
structure of the categories involved, providing a way to compare functors in a
coherent manner.

###  14.2 Examples of Natural Transformations

  * •

The identity natural transformation $\mathrm{Id}:F\Rightarrow F$ assigns the
identity morphism to each object, ensuring that $\eta_{A}=\mathrm{id}_{F(A)}$
for all $A$ in $\mathcal{C}$.

  * •

The composition of two natural transformations $\eta:F\Rightarrow G$ and
$\mu:G\Rightarrow H$ is a natural transformation $\mu\circ\eta:F\Rightarrow H$
defined by $(\mu\circ\eta)_{A}=\mu_{A}\circ\eta_{A}$ for each object $A$ in
$\mathcal{C}$.

  * •

For the hom-functor $\mathrm{Hom}(A,-):\mathcal{C}\rightarrow\mathbf{Set}$, a
natural transformation between $\mathrm{Hom}(A,-)$ and another functor
$\mathrm{Hom}(B,-)$ would be given by a function
$\eta:\mathrm{Hom}(A,X)\rightarrow\mathrm{Hom}(B,X)$ for each $X$ in
$\mathcal{C}$ that respects composition and identity.

###  14.3 Properties of Natural Transformations

Natural transformations have several key properties:

  * •

Naturality: The naturality condition, expressed by the commutative diagram
above, ensures that the transformation is consistent with the action of the
functors on morphisms.

  * •

Vertical and Horizontal Composition: Natural transformations can be composed
both vertically (as in the composition of $\eta$ and $\mu$) and horizontally,
which involves transforming functors applied in sequence.

[1].

##  Chapter 15 Adjoint Functors

###  15.1 Definition

Two functors $F:\mathcal{C}\rightarrow\mathcal{D}$ and
$G:\mathcal{D}\rightarrow\mathcal{C}$ are adjoint if there is a natural
isomorphism

$\mathrm{Hom}_{\mathcal{D}}(F(A),B)\cong\mathrm{Hom}_{\mathcal{C}}(A,G(B))$

for all objects $A$ in $\mathcal{C}$ and $B$ in $\mathcal{D}$.

####  15.1.1 Definition via Counit-Unit Adjunction [4]

“A counit-unit adjunction between two categories $\mathcal{C}$ and
$\mathcal{D}$ consists of two functors $F:\mathcal{D}\rightarrow\mathcal{C}$
and $G:\mathcal{C}\rightarrow\mathcal{D}$ and two natural transformations
$\varepsilon:FG\rightarrow 1_{\mathcal{C}}$ and
$\eta:1_{\mathcal{D}}\rightarrow GF$, respectively called the counit and the
unit of the adjunction, such that the compositions

$F\xrightarrow{F\eta}FGF\xrightarrow{\varepsilon F}F\quad\text{and}\quad
G\xrightarrow{\eta G}GFG\xrightarrow{G\varepsilon}G$

are the identity transformations $1_{F}$ and $1_{G}$ on $F$ and $G$
respectively. In this situation, we say that $F$ is left adjoint to $G$ and
$G$ is right adjoint to $F$, and may indicate this relationship by writing
$(\varepsilon,\eta):F\dashv G$, or simply $F\dashv G$.

In equation form, the above conditions on $(\varepsilon,\eta)$ are the counit-
unit equations:

$1_{F}=\varepsilon F\circ F\eta\quad\text{and}\quad
1_{G}=G\varepsilon\circ\eta G$

which mean that for each $X$ in $\mathcal{C}$ and each $Y$ in $\mathcal{D}$,

$1_{FY}=\varepsilon_{FY}\circ F(\eta_{Y})\quad\text{and}\quad
1_{GX}=G(\varepsilon_{X})\circ\eta_{GX}$

Note that $1_{\mathcal{C}}$ denotes the identity functor on the category
$\mathcal{C}$, $1_{F}$ denotes the identity natural transformation from the
functor $F$ to itself, and $1_{FY}$ denotes the identity morphism of the
object $FY$.”

####  15.1.2 Definition via Universal Morphisms [4]

“By definition, a functor $F:\mathcal{D}\rightarrow\mathcal{C}$ is a left
adjoint functor if for each object $X$ in $\mathcal{C}$ there exists a
universal morphism from $F$ to $X$. Spelled out, this means that for each
object $X$ in $\mathcal{C}$ there exists an object $G(X)$ in $\mathcal{D}$ and
a morphism $\varepsilon_{X}:F(G(X))\rightarrow X$ such that for every object
$Y$ in $\mathcal{D}$ and every morphism $f:F(Y)\rightarrow X$ there exists a
unique morphism $g:Y\rightarrow G(X)$ with $\varepsilon_{X}\circ F(g)=f$.

The latter equation is expressed by the following commutative diagram:

${F(Y)}$${F(G(X))}$${X}$$\scriptstyle{F(g)}$$\scriptstyle{f}$$\scriptstyle{\varepsilon_{X}}$

In this situation, one can show that $G$ can be turned into a functor
$G:\mathcal{C}\rightarrow\mathcal{D}$ in a unique way such that
$\varepsilon_{X}\circ F(G(f))=f\circ\varepsilon_{X^{\prime}}$ for all
morphisms $f:X^{\prime}\rightarrow X$ in $\mathcal{C}$; $F$ is then called a
left adjoint to $G$.

Similarly, we may define right-adjoint functors. A functor
$G:\mathcal{C}\rightarrow\mathcal{D}$ is a right adjoint functor if for each
object $Y$ in $\mathcal{D}$ there exists a universal morphism from $Y$ to $G$.
Spelled out, this means that for each object $Y$ in $\mathcal{D}$ there exists
an object $F(Y)$ in $\mathcal{C}$ and a morphism $\eta_{Y}:Y\rightarrow
G(F(Y))$ such that for every object $X$ in $\mathcal{C}$ and every morphism
$g:Y\rightarrow G(X)$ there exists a unique morphism $f:F(Y)\rightarrow X$
with $G(f)\circ\eta_{Y}=g$

The latter equation is expressed by the following commutative diagram:”

${Y}$${G(F(Y))}$${G(X)}$$\scriptstyle{\eta_{Y}}$$\scriptstyle{g}$$\scriptstyle{G(f)}$

###  15.2 Examples Concerning Set

  * •

The forgetful functor $U:\mathbf{Grp}\rightarrow\mathbf{Set}$ has a left
adjoint $F:\mathbf{Set}\rightarrow\mathbf{Grp}$ that sends each set $S$ to the
free group on $S$. This means there is a natural isomorphism
$\mathrm{Hom}_{\mathbf{Grp}}(F(S),G)\cong\mathrm{Hom}_{\mathbf{Set}}(S,U(G))$
for any set $S$ and group $G$.

  * •

The forgetful functor $U:\mathbf{Top}\rightarrow\mathbf{Set}$ has a left
adjoint $F:\mathbf{Set}\rightarrow\mathbf{Top}$ that sends each set $S$ to the
discrete topology on $S$. Here,
$\mathrm{Hom}_{\mathbf{Top}}(F(S),T)\cong\mathrm{Hom}_{\mathbf{Set}}(S,U(T))$
for any set $S$ and topological space $T$.

###  15.3 Examples Concerning Forgetful Functors

Forgetful functors between various categories often have adjoints. For
example:

  * •

The forgetful functor $U:\mathbf{Mon}\rightarrow\mathbf{Set}$ has a left
adjoint $F:\mathbf{Set}\rightarrow\mathbf{Mon}$ that sends each set $S$ to the
free monoid on $S$.

  * •

The forgetful functor $U:\mathbf{Vect}\rightarrow\mathbf{Set}$ has a left
adjoint $F:\mathbf{Set}\rightarrow\mathbf{Vect}$ that sends each set $S$ to
the free vector space on $S$.

[1].

## Bibliography

  * [1] Pettigrew, R., _An Introduction to Toposes_ , Unpublished Notes.
  * [2] Awodey, S., _Category Theory_ , Oxford Logic Guides 52, Oxford University Press, 2010.
  * [3] Jacobs, B., _Categorical Logic and Type Theory_ , Elsevier, 1999.
  * [4] Wikipedia contributors, ”Adjoint functors,” Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Adjoint_functors (accessed June 12, 2024).
  * [5] nLab contributors, _Heyting algebra_ , https://ncatlab.org/nlab/show/Heyting+algebra#to_toposes, nLab. (accessed June 12, 2024).


# Neutron scattering off spherical nuclei with global nonlocal dispersive
optical model

B. Morillon<EMAIL_ADDRESS>CEA,DAM,DIF F-91297 Arpajon, France
Université Paris-Saclay, CEA, LMCE, 91680 Bruyères-le-Châtel, France  G.
Blanchon  CEA,DAM,DIF F-91297 Arpajon, France  Université Paris-Saclay, CEA,
LMCE, 91680 Bruyères-le-Châtel, France  P. Romain  CEA,DAM,DIF F-91297
Arpajon, France  Université Paris-Saclay, CEA, LMCE, 91680 Bruyères-le-Châtel,
France  H. F. Arellano  Department of Physics-FCFM, University of Chile, Av.
Blanco Encalada 2008, RM 8370449 Santiago, Chile  CEA,DAM,DIF F-91297 Arpajon,
France

(Received: date / Revised version: date)

###### Abstract

We present a global nonlocal and dispersive optical model potential for
neutron scattering off spherical nuclei with incident energies up to 250 MeV.
This optical model is an extension of the non-dispersive Perey-Buck potential.
The imaginary components are chosen energy-dependent and the dispersive
constraints are taken into account. The surface imaginary part is nonlocal,
whereas the volume imaginary part above 10 MeV is local, allowing to reproduce
total cross sections and scattering data for high energies. We obtain a good
description of scattering observables for target-nuclei ranging from $A=16$,
up to $209$. The inclusion of nonlocal spin-orbit term enables a better
description of the analyzing power data relative to the local dispersive
model.

##  I Introduction

More than fifteen years ago, two of us introduced a global local nucleon
Optical Model Potential (OMP) Morillon and Romain (2004, 2006, 2007) including
dispersion relations Mahaux and Sartor (1991) and the local energy
approximation of Perey-Buck Perey and Buck (1962). This potential provides a
good description of integral and differential elastic cross sections for
neutron and proton scattering off spherical target-nuclei for incident energy
up to 250 MeV. Bound state energies were used as constraints for the
calibration of the potential for neutrons Morillon and Romain (2006) and
protons Morillon and Romain (2007). The calibration of the parameters was made
using the same data set of Koning-Delaroche (KD) in Ref. Koning and Delaroche
(2003), i.e. mostly stable and spherical target-nuclei. Based on the same
method, Charity et al. have studied the evolution of the OMP for target-nuclei
with large proton-neutron asymmetry Charity et al. (2006, 2007); Mueller et
al. (2011).

Since then, various attempts have been pursued to handle nonlocality
explicitly. Tian et al. have extended Perey-Buck (PB) potential Perey and Buck
(1962) to include both neutron and proton projectiles Tian et al. (2015). More
recently, Mahzoon et al. have proposed a dispersive nonlocal potential Mahzoon
et al. (2014). This potential has allowed for the first time to reproduce the
target-density and the particle number of the target-nucleus in addition to
the usual scattering observables. First applied to 40Ca for both neutron and
proton projectiles, the approach has been extended to include 48Ca Mahzoon et
al. (2017) and 208Pb Atkinson et al. (2020). In parallel, Lovell et al.
Jaghoub et al. (2018) and Jaghoub et al. Lovell et al. (2017) have studied the
energy dependence of nonlocal potentials. Those studies are all based on PB
separable potential which includes a Gaussian form factor for the nonlocality
Perey and Buck (1962). In a recent review, Hebborn et al. summarize efforts
toward improvements, within both phenomenological and microscopic optical
models approaches Hebborn et al. (2023).

In this work, we present the first global dispersive and nonlocal OMP for
neutron scattering off spherical target-nuclei with incident energy up to 250
MeV. We adopt the PB form factor as starting point. The imaginary contribution
to the OMP is taken energy-dependent whereas the real contribution is energy
dependent only through the dispersion relation. The account for nonlocality
enables us to get rid of spurious energy dependence stemming from local
approximations. The calibration of the OMP parameters is done for twenty-two
nuclei with masses ranging from $A=16$, up to $209$. Both experimental elastic
scattering observables and bound state energies are considered in the
calibration process.

We organize this work as follows. In Sec. II, we recall basics of Perey-Buck
potential model. Then we contrast the predictive power of the two most common
parametrizations: PB Perey and Buck (1962) and Tian, Pang and Ma (TPM) Tian et
al. (2015). In Sec. III, we present form factors for the real and imaginary
components of the new nonlocal dispersive OMP. The resulting parameters of
this OMP are presented and discussed in Sec. IV. Comparisons are made in Sec.
V between calculated and experimental neutron cross sections, analyzing powers
and bound single-particle neutron states. Finally, in Sec. VII we present the
main conclusions of this work. An appendix is added where we expose the method
used to solve the integro-differential Schrödinger equation is exposed in
Appendix A.

##  II Perey-Buck optical model

In the early sixties, Perey and Buck proposed a nonlocal phenomenological
potential to describe neutron scattering off nuclei with incident energies
below 28 MeV Perey and Buck (1962). The central part of PB potential reads

$\displaystyle
V(\mathbf{r},\mathbf{r}^{\prime})=U\left(\frac{\left|\mathbf{r}+\mathbf{r}^{\prime}\right|}{2}\right)H(\left|\mathbf{r}-\mathbf{r}^{\prime}\right|),$
(1)

where $H$ represents the nonlocality form factor assumed Gaussian which is
given as

$\displaystyle
H(\left|\mathbf{r}-\mathbf{r}^{\prime}\right|)=\frac{1}{\pi^{\frac{3}{2}}\beta^{3}}\exp{\left[-\frac{(\mathbf{r}-\mathbf{r}^{\prime})^{2}}{\beta^{2}}\right]}.$
(2)

Here $\beta$ is the range of the nonlocality. The radial form factor $U$ is
given the Woods-Saxon form

$\displaystyle
U(\tilde{r})=V_{V}f(\tilde{r},R,a)-iW_{S}4a\frac{df(\tilde{r},R,a)}{d\tilde{r}},$
(3)

with $f$ a two-parameter Fermi distribution given by

$\displaystyle
f(\tilde{r},R,a)=\frac{1}{1+e^{\frac{{\textstyle\tilde{r}-R}}{{\textstyle
a}}}}.$ (4)

Here $\tilde{r}=\frac{1}{2}\left|\mathbf{r}+\mathbf{r}^{\prime}\right|$, $a$
is the diffuseness parameter and $R=r_{0}A^{1/3}$ is the target-nucleus
radius, with $r_{0}$ the reduced radius and $A$ the nucleus mass. Additionally
the real volume strength $V_{V}$ and the imaginary surface strength $W_{S}$
are energy independent. No imaginary volume component was introduced in PB
between 400 keV and 28 MeV. In addition, PB adopted a local prescription for
the spin-orbit potential.

When dealing with nonlocal potentials, the scattering equations turn to be
integro-differential. One can express then scattering equations in partial
waves

$\displaystyle\frac{\hbar^{2}}{2\mu}\left[\frac{d^{2}}{dr^{2}}-\frac{l(l+1)}{r^{2}}\right]u_{jl}(r)+Eu_{jl}(r)-V_{jl}^{\mathrm{L}}(r)u_{jl}(r)-\int_{0}^{\infty}\nu_{jl}^{\mathrm{NL}}(r,r^{\prime})u_{jl}(r^{\prime})dr^{\prime}=0,$
(5)

with $u_{jl}(r)$ the radial wave function and $\mu$ the projectile-target
reduced mass Joachain (1975). For the sake of completeness, we include both
local and nonlocal contributions in Eq. (5). In the case of PB, the spin-orbit
potential is local, where

$\displaystyle V_{jl}^{\mathrm{L}}(r)=-(j(j+1)-l(l+1)-3/4)U_{so}(r),$ (6)

which makes the local contribution $(j,l)$ dependent. The form factors in Eqs.
(1) and (2) allow for analytical expressions for the potential multipoles,

$\displaystyle\nu_{l}^{\mathrm{NL}}(r,r^{\prime})=\frac{4rr^{\prime}}{\sqrt{\pi}\beta^{3}}U\left(\frac{r+r^{\prime}}{2}\right)e^{{\textstyle-\frac{(r^{2}+r^{\prime
2})}{\beta^{2}}}}i^{l}j_{l}\left(-i\frac{2rr^{\prime}}{\beta^{2}}\right).$

The method used to solve Eq. (5) is described in Appendix A.

In order to illustrate the predictive power of PB potential, in Fig. 1 we plot
experimental and calculated cross sections for ten target nuclei (Mg, Si, Ca,
Cr, Y, Zr, Nb, Sn, Pb, Bi). Solid red and dashed blue curves denote results
from PB and TPM parametrizations, respectively. The experimental total cross
section has been averaged on the energy. The effect of this averaging is
twofold. At low incident energies it filters out the compound nucleus
contribution Feshbach et al. (1954). It also allows to attenuate discrepancies
coming from merging different data sets. In Fig. 2 we plot the experimental
and calculated differential cross sections as functions of the scattering
angle in the center of mass reference frame. For the sake of conciseness, we
show only two angular distributions for each target nucleus. Here we include
results for PB and TPM parametrizations adopting the same conventions as in
Fig. 1.

While angular distributions are very well reproduced, the agreement for the
total cross section is poor in the 0.4 - 28 MeV energy range for both
parametrizations. It should be noted, however, that when Perey and Buck
developed their potential in the early sixties they did not have the current
wealth of measurements.

Figure 1: Total cross section for neutron-nucleus scattering as functions of
the beam energy. The data (symbols) are from Ref. Otuka et al. (2014). Red
solid curves correspond to PB potential Perey and Buck (1962) and blue dashed
curves to that of TPM Tian et al. (2015).

TPM potential provides a better agreement for high-energy total cross
sections. This is most likely due to the presence of a volume imaginary term
in their parametrization. Note, however, that the experimental total cross
section is systematically overestimated in the range 50-150 MeV.

Figure 2: Differential cross sections for neutron elastic scattering as
functions of the c.m. scattering angle. Red solid curves correspond to the PB
potential Perey and Buck (1962) while blue dashed curves denote TPM Tian et
al. (2015) results. Data are denoted by symbols Otuka et al. (2014).

##  III Nonlocal and dispersive optical model

Although neither microscopic nor ab-initio approaches for nucleon-nucleus
scattering are currently suited for massive and precise data evaluation, they
can provide guidance to phenomenological models. On the other side,
phenomenological approaches can become very useful mainly due to their
simplicity and flexibility. The optical model potential is connected to
microscopy through its correspondence with the self-energy Bell and Squires
(1959). The challenge here is to incorporate microscopic and ab-initio
features in the construction of phenomenological optical potentials.

One of these features is the nonlocality, emerging from Pauli exclusion
principle, in addition to dynamic polarization Fraser et al. (2008). In a
recent work it has been demonstrated that the nonlocality exhibits a bell-
shape Arellano and Blanchon (2022). These findings emerge from microscopic
descriptions of nucleon-nucleus collisions based on density-dependent g-matrix
folding model retaining nonlocalities at all stages Arellano and Blanchon
(2022, 2021). It is found that the nonlocality form factor $H$ (see Eq. (2))
is weakly energy-dependent. The leading energy dependence takes place in the
strengths of the potential. Moreover, the spin-orbit contribution results
nonlocal. In the context of PB approach Perey and Buck (1962), this
nonlocality is modeled through a Gaussian form factor. We retain a single
Gaussian nonlocality for real and imaginary components of the potential. As we
will see, this is sufficient to achieve a reasonable description of
experimental data.

Continuing the analogy with many-body physics, the optical potential can be
separated into an energy-independent Hartree-Fock and energy-dependent
polarization terms. Then, causality is ensured through a dispersion relation
Mahaux et al. (1986); Mahzoon (2015). This dispersion relation relates
scattering and bound-state properties. Hence, it allows to determine
scattering states and bound states of the target-nucleus on the same footing
using a single potential. A well-constrained phenomenological nonlocal
dispersive potential contains in practice all the correlations. Thus one can
evaluate the quality of the microscopic description not only of scattering
observables but also other physical properties such as occupation numbers,
densities and bound state energies.

###  III.1 Dispersion relation

The radial form factor $U$ in Eq. (3) is now assumed energy dependent,

$\displaystyle U(\tilde{r},E)=V(\tilde{r},E)+iW(\tilde{r},E),$ (8)

whereas the nonlocal form factor $H$ in Eq. (2) is kept energy independent.
Now, a dispersion relation applies relating the energy-dependent real and
imaginary contributions in Eq. (8),

$\displaystyle V(\tilde{r},E)$ $\displaystyle=$ $\displaystyle
V_{0}(\tilde{r})+\Delta V(\tilde{r},E),$ $\displaystyle\Delta V(\tilde{r},E)$
$\displaystyle=$ $\displaystyle\frac{\cal
P}{\pi}\int_{-\infty}^{+\infty}\frac{W(\tilde{r},E^{\prime})}{E^{\prime}-E}dE^{\prime},$
(9)

where ${\cal P}$ denotes the principal value of the integral Mahaux and Sartor
(1991). $V_{0}$ is an energy-independent potential often referred to as
Hartree-Fock, in analogy to the static self-energy in many-body physics Fetter
and Walecka (1971). When possible, the dispersive term $\Delta V$ is obtained
analytically, otherwise it is calculated by numerical integration. We assume
the energy dependence affects only the strengths of the potential. The
diffuseness, radius and nonlocality are assumed energy independent.

###  III.2 Real potentials

The real potential is assumed nonlocal with a Gaussian nonlocal form factor
$H$ as in Eq. (2) and energy independent. Its radial form factor reads

$\displaystyle V_{0}(\tilde{r})$ $\displaystyle=$ $\displaystyle
V_{V}^{\mathrm{NL}}f(\tilde{r},R,a)-V_{S}^{\mathrm{NL}}4a\frac{df(\tilde{r},R,a)}{d\tilde{r}}$
(10) $\displaystyle-$ $\displaystyle
2V_{so}^{\mathrm{NL}}\left(\frac{\hbar}{m_{\pi}c}\right)^{2}\frac{1}{\tilde{r}}\frac{df(\tilde{r},R,a)}{d\tilde{r}}\;\mathbf{l}.\mathbf{s}.$

We include a real nonlocal surface potential $V_{S}^{\mathrm{NL}}$ that
improves the agreement with the elastic angular distribution data. The spin-
orbit potential is nonlocal with the same nonlocality than the central term.
The dispersive component, $\Delta V(\tilde{r},E)$, stemming from the energy-
dependent imaginary potentials are added as described in Eq. (9).

###  III.3 Imaginary potentials

As observed in Fig. 1, the addition of a volume imaginary contribution, as
proposed by TPM Tian et al. (2015), helps to better reproduce total cross
section above 50 MeV. Following this prescription, we have tested several
values of nonlocality parameters $\beta$ for the volume imaginary potential.
We have also investigated different energy dependencies. From these studies we
conclude that a local volume imaginary potential is the best suited. The
energy-dependency retained for the volume and the surface terms are described
below.
The local imaginary volume contribution reads

$\displaystyle W^{\mathrm{L}}(r,E)$ $\displaystyle=$ $\displaystyle
W_{V}^{\mathrm{L}}(E)f(r,R,a),$ (11)

whereas the nonlocal imaginary surface contribution reads

$\displaystyle W^{\mathrm{NL}}(\tilde{r},E)$ $\displaystyle=$ $\displaystyle-
W_{S}^{\mathrm{NL}}(E)4a\frac{df(\tilde{r},R,a)}{d\tilde{r}}$ $\displaystyle-$
$\displaystyle
2W_{so}^{\mathrm{NL}}(E)\left(\frac{\hbar}{m_{\pi}c}\right)^{2}\frac{1}{\tilde{r}}\frac{df(\tilde{r},R,a)}{d\tilde{r}}\;\mathbf{l}.\mathbf{s}.$

It should be noted that the imaginary spin-orbit is chosen to be nonlocal.

####  III.3.1 Surface imaginary depth

For the depth of the nonlocal imaginary surface strength, we use a Brown-Rho
shape Brown and Rho (1981) modified by an exponential falloff,

$\displaystyle
W_{S}^{\mathrm{NL}}(E)=A_{S}^{\pm}(E-E_{F})^{n}\frac{\exp(-C_{S}\left|E-E_{F}\right|)}{(E-E_{F})^{n}+B_{S}^{n}}.$
(13)

Its energy dependence is asymmetric with respect to the neutron Fermi energy
$E_{F}=-[S_{n}(Z,N)-S_{n}(Z,N+1)]/2$. There is a single depth value
$A_{S}^{+}$ for $E>E_{F}$, and another one $A_{S}^{-}$ for $E<E_{F}$. We have
investigated different values for the parameter $n$, reaching better fits with
$n=2$. We also found that coefficients $A_{S}^{-}$, $B_{S}$ and $C_{S}$
exhibit a very weak dependence on target-nucleus mass, so that it is
reasonable to assume them constant. In contrast, $A_{S}^{+}$ decreases
linearly with respect to the mass $A$.

####  III.3.2 Volume imaginary depth

For energies around the Fermi energy, within the interval $[E_{F}-E_{V}^{-}$,
$E_{F}+E_{V}^{+}]$, the depth of the local imaginary volume component is of
Brown-Rho type Brown and Rho (1981) but with different depths $A_{V}^{\pm}$,
depending on whether the projectile energy is above ($A_{V}^{+}$) or below
($A_{V}^{-}$) the Fermi energy, namely

$W_{V}^{\mathrm{L}}(E)=\frac{{\textstyle
A_{V}^{\pm}(E-E_{F})^{2}}}{{\textstyle(E-E_{F})^{2}+B_{V}^{2}}}.$ (14)

For energies $E$ above $E_{F}+E_{V}^{+}$, the form proposed by Mahaux and
Sartor Mahaux and Sartor (1991) is applied, thus

$\displaystyle W_{V}^{\mathrm{L}}(E)$ $\displaystyle=$
$\displaystyle\frac{{\textstyle
A_{V}^{+}(E-E_{F})^{2}}}{{\textstyle(E-E_{F})^{2}+B_{V}^{2}}}$
$\displaystyle+$
$\displaystyle\alpha\left[\sqrt{E}+\frac{(E_{F}+E_{V}^{+})^{3/2}}{2E}-\frac{3}{2}\sqrt{E_{F}+E_{V}^{+}}\right].$

In the case of $E$ below $E_{F}-E_{V}^{-}$, the strength is given by

$\displaystyle W_{V}^{\mathrm{L}}(E)$ $\displaystyle=$
$\displaystyle\frac{{\textstyle
A_{V}^{-}(E-E_{F})^{2}}}{{\textstyle(E-E_{F})^{2}+B_{V}^{2}}}$ (16)
$\displaystyle\times$
$\displaystyle\left[1-\frac{\left(E-E_{F}+E_{V}^{-}\right)^{2}}{\left(E-E_{F}+E_{V}^{-}\right)^{2}+\left(E_{V}^{-}\right)^{2}}\right].$

####  III.3.3 Spin-orbit imaginary depth

For the spin orbit term, the imaginary surface strength is taken symmetrical
with respect to the Fermi energy. It reads

$\displaystyle
W_{so}^{\mathrm{NL}}(E)=\frac{A_{so}(E-E_{F})^{2}}{(E-E_{F})^{2}+C_{so}^{2}}-\frac{B_{so}(E-E_{F})^{2}}{(E-E_{F})^{2}+D_{so}^{2}},$

consisting of the difference of two Brown-Rho forms, allowing an analytical
expression for the dispersive term $\Delta V_{so}^{\mathrm{NL}}(E)$ VanderKam
et al. (2000).

##  IV Parameter calibration

Once the potential form factors are specified, we look for the parameters that
best reproduce the experimental data. For each nucleus we carry out an
iterative parameter calibration to minimize the difference between data and
the nonlocal dispersive optical model results. Then one can identify the
dependence of the parameters on the nucleus mass $A$ and the energy. The
variation of the parameters with respect to the mass and the energy is
expected to be smooth, helping the model to gain predictive power. This
parameter search is highly time-consuming, being necessary the use of parallel
computing. For each nucleus, over 2000 processors are used for twenty-four
hours to find the parameters. In addition, for low neutron energy (below about
15 MeV), compound neutron emission is described with Hauser-Feshbach formalism
Hauser and Feshbach (1952) using NLD transmission coefficients in TALYS code
Koning et al. (2008). This process is accounted for in the calibration
process.

Our search procedure makes extensive use of the comprehensive EXFOR database
Otuka et al. (2014). In the energy range from 1 keV to 250 MeV, the
experimental data of the following twenty-two elements are used to determine
our parameters: C, O, Mg, Al, Si, S, Ca, Ti, Cr, Fe, Ni, Cu, Y, Zr, Nb, Mo,
Sn, Ce, Au, Hg, Pb, Bi. About 100 energy points are needed to describe
accurately the variations of the total cross section in the energy range 1 keV
to 250 MeV. As already mentioned, for neutron energies below a few MeV and
especially for light nuclei, the experimental total cross section must be
averaged because of the presence of compound nucleus resonances. This averaged
total cross section (see Fig. 1) is retained throughout the calibration. After
each step of parameter search, we calculate bound state energies as well as
analyzing powers.

Nonlocal real depth
---
$V_{V}^{\mathrm{NL}}$ (MeV) |  $-69.71-1.140\times 10^{-2}$ A
$V_{S}^{\mathrm{NL}}$ (MeV) |  $-8.600-8.000\times 10^{-3}$ A
$V_{so}^{\mathrm{NL}}$ (MeV) |  $-9.787-1.140\times 10^{-2}$ A
Nonlocal surface imaginary depth
$W_{S}^{\mathrm{NL}}(E)=A_{S}^{\pm}(E-E_{F})^{2}\frac{{\textstyle\exp(-C_{S}\left|E-E_{F}\right|)}}{{\textstyle(E-E_{F})^{2}+B_{S}^{2}}}$
$A_{S}^{+}$ (MeV) |  $-19.62-1.500\times 10^{-2}$A, for $E_{F}<E$
$A_{S}^{-}$ (MeV) |  $-16.00$, for $E<E_{F}$
$B_{S}$ (MeV) | $11.11$
$C_{S}$ (MeV-1) | $9.200\times 10^{-3}$
Local volume imaginary depth
For $E>E_{F}+E_{V}^{+}:$
$W_{V}^{\mathrm{L}}(E)=\frac{{\textstyle
A_{V}^{+}(E-E_{F})^{2}}}{{\textstyle(E-E_{F})^{2}+B_{V}^{2}}}$
$+\alpha\left(\sqrt{E}+\frac{(E_{F}+E_{V}^{+})^{3/2}}{2E}-\frac{3}{2}\sqrt{E_{F}+E_{V}^{+}}\right).$
For $E_{F}-E_{V}^{-}<E<E_{F}+E_{V}^{+}:$
$W_{V}^{\mathrm{L}}(E)=\frac{{\textstyle
A_{V}^{\pm}(E-E_{F})^{2}}}{{\textstyle(E-E_{F})^{2}+B_{V}^{2}}}.$
For $E<E_{F}-E_{V}^{-}$:
$W_{V}^{\mathrm{L}}(E)=\frac{{\textstyle
A_{V}^{-}(E-E_{F})^{2}}}{{\textstyle(E-E_{F})^{2}+B_{V}^{2}}}$
$\times\left(1-\frac{\left(E-E_{F}+E_{V}^{-}\right)^{2}}{\left(E-E_{F}+E_{V}^{-}\right)^{2}+\left(E_{V}^{-}\right)^{2}}\right),$
with |
$A_{V}^{+}$ (MeV) |  $-32.40-2.000\times 10^{-2}$A, for $E>E_{F}$
$A_{V}^{-}$ (MeV) |  $-8.400$A, for $E<E_{F}$
$B_{V}$ (MeV) | $135.0$
$E_{V}^{+}$ (MeV) |  $40.00-9.000\times 10^{-2}$A
$E_{V}^{-}$ (MeV) | $25.50$
$\alpha$ (MeV1/2) |  $3.000\times 10^{-1}+2.000\times 10^{-3}$A
Nonlocal spin-orbit imaginary depth
$W_{so}^{\mathrm{NL}}(E)=\frac{\textstyle
A_{so}(E-E_{F})^{2}}{\textstyle(E-E_{F})^{2}+C_{so}^{2}}-\frac{\textstyle
B_{so}(E-E_{F})^{2}}{\textstyle(E-E_{F})^{2}+D_{so}^{2}}$
$A_{so}$ (MeV) | 4.893
$B_{so}$ (MeV) | 2.447
$C_{so}$ (MeV) | 50.00
$D_{so}$ (MeV) | 3.900
Table 1: Energy and mass dependency of real and imaginary potential strengths.
Geometrical parameters
---
$r_{0}$ (70 $<$ A) (fm) |  $1.1446+2.4200\times 10^{-4}$ A
$r_{0}$ (A $<$ 70) (fm) | $9.4860\times 10^{-1}+8.8000\times 10^{-3}\mathrm{A}$
| $-1.3200\times 10^{-4}\mathrm{A}^{2}+7.1000\times 10^{-7}\mathrm{A}^{3}$
$a$ (fm) |  $6.1600\times 10^{-1}-1.8200\times 10^{-4}$ A
$\beta$ (fm) | 0.915
Table 2: Nonlocality and mass dependence of depth, radii and diffuseness of
NLD potential.

The resulting parameters for strengths of the real and imaginary potentials
are summarized in Table 1. In Fig. 3, we present the strengths of the surface
[panel (a)], volume [panel (b)] and spin-orbit [panel (c)] imaginary
contributions (red curves) together with their respective dispersive
contributions (blue curves) for three target-nuclei: ${}^{40}\mathrm{Ca}$,
${}^{89}\mathrm{Y}$ and ${}^{208}\mathrm{Pb}$. In panel (a),
$W_{S}^{\mathrm{NL}}$ is non symmetrical with respect to the Fermi energy.
This asymmetry results from the fact that the phase space of particle levels
for $E\ll E_{F}$ is significantly larger than that of hole levels for $E\gg
E_{F}$. Therefore the contributions from two-particle–one-hole states for
$E\ll E_{F}$ will be larger than that for two-hole–one-particle states at
$E\gg E_{F}$ Charity et al. (2007). In panel (c), $W_{so}^{\mathrm{NL}}$ is
independent of the mass and symmetrical with respect to the Fermi energy. The
real depths of the volume surface and spin-orbit potential
($V_{V}^{\mathrm{NL}}$, $V_{S}^{\mathrm{NL}}$ and $V_{so}^{\mathrm{NL}}$) are
independent of the energy of the projectile. The variations as a function of
the mass of the nuclei are weak for the surface and volume depths, as shown in
Table 1.

Figure 3: Depths $W$ (red curve) and dispersive contribution $\Delta V$ of
surface (a), volume (b) and spin-orbit (c) imaginary potentials as functions
of the energy. Dashed, solid and dotted curves denote results for 40Ca, 89Y
and 208Pb, respectively.

The optimum nonlocality range $\beta$ is found to be 0.915 fm, larger than in
previous works Perey and Buck (1962); Tian et al. (2015). The same radius and
diffuseness are used for real and imaginary contributions. In Fig. 4 we plot
the reduced radius $r_{0}$ [panel (a)] and the diffuseness $a$ [panel (b)] as
functions of the mass $A$. The reduced radius increases with mass in contrast
to diffuseness, which is a linear function decreasing as the mass increases.
For the former, the variation is linear for masses greater than 70, while for
smaller masses the expression of the radius behaves as a polynomial of degree
3. Results are summarized in Table 2.

Figure 4: Reduced radius $r_{0}$ (a) and diffuseness $a$ (b) as functions of
the target mass $A$ for the nonlocal dispersive model.

##  V Comparison between local and nonlocal dispersive models

We examine differences in the calculated scattering observables for neutron
scattering as implied by the local dispersive (LD) model of Ref. Morillon and
Romain (2007) and the nonlocal dispersive (NLD) model presented in this work.

In Fig. 5 we plot the total cross sections for neutron-nucleus scattering as
function of the energy. We include ten targets with masses between 24 and 209.
Dashed blue curves denote results from LD model whereas solid red curves
denote NLD results (this work). We observe that both models yield comparable
results for $\sigma_{T}$ for neutron energies above $\sim\\!10$ MeV.
Differences are evidenced at the lower energies, although their agreement with
the data is comparable.

In Fig. 6 we show results for the differential cross section as function of
the scattering angle in the center-of-mass reference frame ($\theta_{c.m.}$).
The targets we include in this case are the same as in Fig. 5, with neutron
energies ranging from 6 up to 25.5 MeV. Although the angular description of
the data appears comparable from both NLD and LD models, some differences are
evidenced in their maxima and minima, with the NLD approach in closer
agreement with the data.

To compare the agreement with the data of NLD and LD approaches, we have
calculated relative differences with the data for all the cases considered in
the fit procedure. Results for these differences are shown in Fig. 7, where
panel (a) summarizes results for the total cross section, whereas panel (b)
those for the differential cross section. Red circles (blue squares) denote
results from NLD (LD) approaches. Additionally, as reference, we include in
this analysis the KD non-dispersive model (black triangles). From panel (a) we
observe that all three models yield comparable agreement in $\sigma_{T}$, of
the order of 3% for medium-mass nuclei. However, for light nuclei the NLD
model yields better agreement (by roughly $\sim\\!1$ %) relative to the LD
approach. In the case of the differential cross section, panel (b) of Fig. 7
that the NLD approach is in better agreement with the data relative to the LD
model, particularly for all targets with nuclear masses below 64.

Figure 5: Total cross section for neutron-nucleus scattering as function of
the energy. The data (symbols) are from Ref. Otuka et al. (2014). Red solid
curves correspond to NLD potential (this work) and blue dashed curves to LD
potential Morillon and Romain (2007). Figure 6: Differential cross sections
for neutron elastic scattering. Data are denoted by symbols Otuka et al.
(2014). Red solid curves correspond to NLD potential (this work) and blue
dashed curves to LD potential Morillon and Romain (2007). Figure 7: Relative
differences experimental $vs.$ calculation expressed as percentages for each
nucleus (dashed curves) for total cross section [panel (a)] and differential
cross section [panel (b)]. Results are for NLD potentials (red circles), LD
potential Morillon and Romain (2007) (blue square) and KD potential Koning and
Delaroche (2003) (black triangle).

In Fig. 8 we show the analyzing powers (red curves) obtained from our nonlocal
OMP for two incident neutron energies (10 MeV and 17 MeV) with the following
nuclei: 27Al, 40Ca, 54Fe, 58Ni, 65Cu, 89Y, 93Nb, 120Sn, 208Pb and 209Bi. These
results are compared with those calculated with LD potential (blue dashed
curves). This figure shows that the nonlocal potential better reproduces the
analyzing powers, particularly its maxima and minima.

Figure 8: Analyzing powers for neutron scattering at 10 and 17 MeV. Data are
denoted by symbols. Red solid curves are obtained with NLD potential and the
blue dashed curves with LD potential.

In Fig. 9 we show the experimental and calculated neutron single-particle
energies for 208Pb, 90Zr and 40Ca. We include results from LD and NLD global
potentials. We observe comparable agreement of these two models with the data,
leaving room for improvement.

Figure 9: Neutron single-particle energies for 208Pb, 90Zr and 40Ca determined
with LD, NLD potentials and from experiment.

##  VI Uncertainties

Figure 10: Uncertainties ($\pm 1\sigma$) in the total (red), reaction (black)
and elastic (blue) cross sections for neutron scattering of 208Pb.

To assess the effect of random variations of the parameters of the NLD model
on scattering observables, we have calculated the standard deviation for the
calculated cross sections when geometrical parameters and strengths are
varied. Specifically, in Fig. 10 we show the calculated elastic $\sigma_{E}$
(blue curves), reaction $\sigma_{R}$ (black curves) and total $\sigma_{T}$
(red curves) cross sections for n + 208Pb elastic scattering between 1 keV and
250 MeV. In this case, at each energy we allowed uniform variations for radius
(1%), diffuseness (4%), real volume and surface depth (2%) and range of
nonlocality (2%). Additionally, we allow for 20% variations of the depth of
the imaginary terms. Thus, we performed stratified samplings of parameters
over a uniform distribution, resulting in two thousand sets of parameters
$(r,a,V_{S},V_{V},\beta,W_{S},W_{V})$, each of them leading to a value for
$\sigma_{E}$, $\sigma_{R}$ and $\sigma_{T}$. In Fig. 10 we plot the mean value
(solid curves) together with its $\pm 1\sigma$ variations (shaded areas). It
is interesting to note that for energies above 0.2 MeV, the cross sections
suffer rather moderate variations under variations of parameters. Therefore,
the limitations of the model shown in Fig. 5 for $\sigma_{T}$ cannot be
attributed to the parameters but to the assumed structure in the separable
construction. This is an indication that a more complete model is needed.

##  VII Conclusions

We have presented the first global, nonlocal and dispersive optical model
potential for neutron elastic scattering off spherical nuclei. The model is
suited for beam energies up to 250 MeV and for target masses $16\leq A\leq
209$. Both central and spin-orbit terms include a nonlocal Gaussian form
factor of Perey-Buck type. Additionally, the imaginary volume potential
includes a local term. The strengths of the model are constrained by
dispersion relations, allowing a good description of integrated and angular
scattering data. The dispersion relation allows to determine scattering states
and bound states of the target-nucleus with a single potential. We are fully
aware that there remains room for improvement for the description of bound
state properties. We obtain a better description of the angular scattering
data for $A\lesssim 65$ relative to the LD approach Morillon and Romain
(2007). These improvements appear more evident in the analyzing power,
suggesting the relevance of nonlocality in the spin-orbit term. This finding
goes in line with microscopic studies of nonlocality Arellano and Blanchon
(2022), where both central and spin-orbit components are nonlocal. As already
mentioned, an important advantage of global phenomenological approaches for
nucleon scattering lies on their simplicity, making it feasible computations
which would require physical information on processes involving the whole
nuclear chart and over a broad energy range. The challenge in this approach is
to identify strengths and form factors as implied from more fundamental
approaches such as microscopic Arellano et al. (1989); Crespo et al. (1990);
Elster et al. (1990); Vorabbi et al. (2016) or ab-initio Hagen and Michel
(2012); Rotureau et al. (2017); Idini et al. (2019). Efforts along this line
are under way.

###### Acknowledgements.

This work was performed using HPC resources from CCRT. H.F.A. thanks the
hospitality of colleagues of CEA,DAM,DIF during his stay at Bruyères-le-
Châtel.

##  Appendix A Resolution of the integro-differential Schrödinger equation

The search procedure for strengths and geometrical parameters implies
comparisons between data and scattering observables resulting from the
nonlocal optical model we have discussed. This requires solving the integro-
differential Schrödinger equation in Eq. (5) for several targets, energies and
varying parameters. Thus, we need a fast and accurate method to obtain the
scattering wave functions. Along this line, Eq. (5) is solved by expanding
$u_{jl}$ on the Chebyshev polynomial basis. Since these polynomials are
defined in the interval [-1,+1] we use the new variable
$x=\frac{2}{R_{M}}r-1$. The quantity $R_{M}$ represents the maximum value of
the variable $r$ (it defines the radius where the potential becomes
negligible). Thus the radial wave function $u$ is written as a linear
combination of $N$ Chebyshev polynomials $T_{k}$ weighted by the coefficients
$C_{k}$. It reads

$\displaystyle
u\left[\frac{R_{M}}{2}(x+1)\right]\simeq\sum_{k=0}^{N-1}\overset{/}{\vphantom{a}}C_{k}T_{k}(x),\quad
x\in[-1,+1],$ (18)

where the symbol $\overset{/}{}$ indicates that the first term of the sum is
divided by a factor of 2, following the notation of Masson and Handscomb
Masson and Handscomb (2003).

To obtain the radial part of the wave function we need to find the
coefficients $C_{k}$ with $k=0,\ldots,N-1.$ That is, we need $N$ equations.
The boundary conditions at $r=0$, and $r=R_{M}$, yield two equations. Then
considering that $T_{k}(1)=1$, and $T_{k}(-1)=\left(-1\right)^{k}$, we get

$\displaystyle u(r=0)$ $\displaystyle=$
$\displaystyle\sum_{k=0}^{N-1}\overset{/}{\vphantom{a}}(-1)^{k}C_{k}=0,$ (19)
$\displaystyle u(r=R_{M})$ $\displaystyle=$
$\displaystyle\sum_{k=0}^{N-1}\overset{/}{\vphantom{a}}C_{k}=1+i,$ (20)

where $1+i$, is an arbitrary boundary value with no impact on the calculated
phaseshift Joachain (1975). It thus remains to find $N-2$ equations in order
to determine the $C_{k}$ coefficients. This is done by evaluating Eq. (5) at
each of the $N-2$ roots $x_{n}$ of the $T_{N-2}$ Chebyshev polynomial. These
roots are

$x_{n}=\cos\frac{(n-1/2)\pi}{N-2}\quad n=1,...,N-2.$ (21)

In the case of the second derivative of the radial part of the wave function,
we take advantage of the second derivative of the Chebyshev polynomials,

$\left.\frac{d^{2}u\left(\frac{R_{M}}{2}(x+1)\right)}{dx^{2}}\right|_{x=x_{n}}=\sum_{k=2}^{N-1}C_{k}\sum_{\begin{subarray}{c}m=0\\\
m-r\
\mathrm{even}\end{subarray}}^{k-2}\overset{/}{\vphantom{a}}\negthickspace(k-m)k(k+m)T_{m}(x_{n}).$
(22)

In the case of a local potential, where we denote

$\displaystyle\mathcal{V^{\mathrm{L}}}(r)=\frac{l(l+1)}{r^{2}}+\frac{2\mu}{\hbar^{2}}V(r),$
(23)

the $N-2$ equations representing Eq. (5) become

$\displaystyle\frac{4}{R_{M}^{2}}\sum_{k=2}^{N-1}C_{k}\sum_{\begin{subarray}{c}m=0\\\
k-m\
\mathrm{even}\end{subarray}}^{k-2}\overset{/}{\vphantom{a}}(k-m)k(k+m)T_{m}(x_{n})$
$\displaystyle+\sum_{k=0}^{N-1}\overset{/}{\vphantom{a}}C_{k}T_{k}(x_{n})\left(\frac{2\mu
E}{\hbar^{2}}-\mathcal{V^{\mathrm{L}}}\left(\frac{R_{M}}{2}(x_{n}+1)\right)\right)=0.$
(24)

When a nonlocal potential is considered an additional development taking
advantage of the Chebyshev basis is pursued. As for the local potential, the
nonlocal potential is also redefined as

$\displaystyle\mathcal{V}^{\mathrm{NL}}(r,r^{\prime})=\frac{2\mu}{\hbar^{2}}V(r,r^{\prime}),$
(25)

and the following term

$\displaystyle-\frac{R_{M}}{2}\negthickspace\int_{-1}^{+1}\negthickspace\negthickspace\negthickspace\mathcal{V^{\mathrm{NL}}}\left[\negthickspace\frac{R_{M}}{2}(x_{n}+1),\frac{R_{M}}{2}(y+1)\negthickspace\right]u\left[\negthickspace\frac{R_{M}}{2}(y+1)\negthickspace\right]dy,$
(26)

must be added to the Eq. (24) to obtain a Schrödinger equation with local and
nonlocal potentials. In order to perform the integration, the potential is
expressed in terms of Chebyshev polynomials. This development concerns the
variable $y$ of this potential

$\displaystyle\mathcal{V}^{\mathrm{NL}}\negthickspace\left[\negthickspace\frac{R_{M}}{2}(x+1),\frac{R_{M}}{2}(y+1)\negthickspace\right]\negthickspace\simeq\negthickspace\sum_{m=0}^{M}\overset{/}{\vphantom{a}}\negthickspace
v_{m}[\frac{R_{M}}{2}(x+1)]T_{m}(y).$

The $v_{m}$ terms are obtained by considering the $M+1$ zeros of the Chebyshev
polynomial $T_{M+1}$

$\displaystyle y_{p}=\cos\frac{(p-1/2)\pi}{M+1},\quad p=1,\ldots,M+1,$ (28)

and using the discrete orthogonality relations Masson and Handscomb (2003),
one obtains the $v_{m}$ terms,

$\displaystyle\sum_{p=1}^{M+1}\mathcal{V}^{\mathrm{NL}}\left[\frac{R_{M}}{2}(x+1),\frac{R_{M}}{2}(y_{p}+1)\right]T_{m}(y_{p})$
(29) $\displaystyle=$
$\displaystyle\sum_{p=1}^{M+1}\sum_{i=0}^{M}\overset{/}{\vphantom{a}}v_{i}\left[\frac{R_{M}}{2}(x+1)\right]T_{i}(y_{p})T_{m}(y_{p})$
$\displaystyle=$
$\displaystyle\frac{M+1}{2}v_{m}\left[\frac{R_{M}}{2}(x+1)\right].$

The calculation of the integral in Eq. (26) is thus reduced to a sum

$\displaystyle\int_{-1}^{+1}\sum_{m=0}^{M}\overset{/}{\vphantom{a}}v_{m}\left[\frac{R_{M}}{2}(x+1)\right]T_{m}(y)\sum_{k=0}^{N-1}\overset{/}{\vphantom{a}}C_{k}T_{k}(y)dy$
(30) $\displaystyle=$
$\displaystyle\sum_{m=0}^{M}\overset{/}{\vphantom{a}}v_{m}\left[\frac{R_{M}}{2}(x+1)\right]\sum_{k=0}^{N-1}\overset{/}{\vphantom{a}}C_{k}\int_{-1}^{+1}T_{m}(y)T_{k}(y)dy$
$\displaystyle=$
$\displaystyle\\!-\\!2\\!\\!\sum_{k=0}^{N-1}\overset{/}{\vphantom{a}}C_{k}\\!\\!\sum_{m=0}^{M}\overset{/}{\vphantom{a}}v_{m}\\!\left[\\!\frac{R_{M}}{2}(x+1)\right]$
$\displaystyle\times$
$\displaystyle\left.\\!\\!\frac{m^{2}+k^{2}-1}{(m^{2}+k^{2}-1)^{2}-4m^{2}k^{2}}\right|_{m+k\
\mathrm{even}}.$

The $N-2$ equations needed to determine the coefficients $C_{k}$ with local
and nonlocal potentials are therefore written with $n=1,\ldots,N-2$

$\displaystyle\frac{4}{R_{M}^{2}}\sum_{k=2}^{N-1}C_{k}\sum_{\begin{subarray}{c}m=0\\\
k-m\
\mathrm{even}\end{subarray}}^{k-2}\overset{/}{\vphantom{a}}(k-m)k(k+m)T_{m}(x_{n})$
$\displaystyle+$ $\displaystyle
R_{M}\sum_{k=0}^{N-1}\overset{/}{\vphantom{a}}C_{k}\sum_{m=0}^{M}\overset{/}{\vphantom{a}}v_{m}\left(\frac{R_{M}}{2}(x_{n}+1)\right)$
$\displaystyle\times$
$\displaystyle\left.\frac{m^{2}+k^{2}-1}{(m^{2}+k^{2}-1)^{2}-4m^{2}k^{2}}\right|_{m+k\
\mathrm{even}}$ $\displaystyle+$
$\displaystyle\sum_{k=0}^{N-1}\overset{/}{\vphantom{a}}C_{k}T_{k}(x_{n})\left(\frac{2\mu
E}{\hbar^{2}}-\mathcal{V^{\mathrm{L}}}\left(\frac{R_{M}}{2}(x_{n}+1)\right)\right)=0.$

The $C_{k}$ coefficients with $k=0,..,N-1$ are calculated by solving a system
of $N$ linear equations using the LU decomposition algorithm Press et al.
(2002, page 60). The function has been modified to allow calculation with
complex numbers. Since the energy-dependent part of the phenomenological
potential is separated from the spatial part, the decomposition of the
potential on the Chebyshev polynomials needs to be performed only once for
each nucleus. Up to 100 MeV, 30 Chebyshev polynomials are sufficient to
accurately describe the wave function. Up to 250 MeV, the function must be
developed over 50 polynomials.

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