# Cardinal Characteristics on
Bounded Generalised Baire Spaces

Tristan van der Vlugt111Fachbereich Mathematik, Universität Hamburg $-$ email:
<EMAIL_ADDRESS>$-$ The author was partially supported by the Fields
Institute, Toronto $-$ The author wishes to thank Jörg Brendle for valuable
discussions, suggestions and extensive comments. Valuable comments have been
given by Tatsuya Goto, Jing Zhang and Hiroshi Sakai.

###### Abstract

We will give an overview of four families of cardinal characteristics defined
on subspaces $\prod_{\alpha\in\kappa}b(\alpha)$ of the generalised Baire space
${}^{\kappa}\kappa$, where $\kappa$ is strongly inaccessible and
$b\in{}^{\kappa}\kappa$. The considered families are bounded versions of the
dominating, eventual difference, localisation and antilocalisation numbers,
and their dual cardinals. We investigate parameters for which these cardinals
are nontrivial and how the cardinals relate to each other and to other
cardinals of the generalised Cichoń diagram. Finally we prove that different
choices of parameters may lead to consistently distinct cardinals.

## 1\. Introduction

Cardinal characteristics of the continuum are cardinal numbers that
consistently lie strictly between $\aleph_{1}$ and the cardinality of the
reals $2^{\aleph_{0}}$. Many natural questions about the cardinality of sets
of real numbers lead to definitions of cardinal characteristics, such as the
least size of a set of reals of positive Lebesgue measure, or the least size
of an unbounded set of functions from $\omega$ to $\omega$. In studying such
cardinal characteristics, one would usually work with perfect Polish spaces,
such as the Baire space ${}^{\omega}\omega$, instead of directly working with
the set of reals.

In recent years, there have been significant developments in generalising the
theory of cardinal characteristics of the continuum to the context of the
generalised Baire space ${}^{\kappa}\kappa$, where $\kappa$ is an uncountable
cardinal. In many cases, the situation on ${}^{\kappa}\kappa$ closely
resembles the situation on ${}^{\omega}\omega$, but this is not always the
case. For instance, the meagre ideal $\mathcal{M}$, as defined on
${}^{\omega}\omega$, generalises without much trouble to a $\kappa$-meagre
ideal on ${}^{\kappa}\kappa$, but the ideal $\mathcal{N}$ of Lebesgue null
sets does not; after all, there is no adequate notion of a Lebesgue measure on
${}^{\kappa}\kappa$. Another important difference, is the rich structure of
the club filter and of stationary sets on $\kappa$, which is nonexistent on
$\omega$.

Due to the lack of a Lebesgue measure, and hence a null ideal, it is unclear
what analogue in the generalised Baire space would fit for cardinal
characteristics defined in terms of the null ideal. In the classical Baire
space ${}^{\omega}\omega$ there is a characterisation of the cardinal
characteristics $\mathrm{add}(\mathcal{N})$ and $\mathrm{cof}(\mathcal{N})$ in
terms of slaloms, that is, functions $\varphi:\omega\to\mathcal{P}(\omega)$
such that $|\varphi(n)|=2^{n}$ for each $n\in\omega$. Specifically,
$\mathrm{cof}(\mathcal{N})$ is equal to the least size of a set $W$ of slaloms
such that every $f\in{}^{\omega}\omega$ is localised by a slalom in $W$, that
is, $f(n)\in\varphi(n)$ for all but finitely many $n$. This combinatorially
defined cardinal characteristic is usually named the localisation number.

We will consider a parametrised localisation number, where we consider a set
of slaloms that localises all $f\in{}^{\omega}\omega$, with the additional
requirement that each slalom $\varphi$ in our witness set $W$ is bound by a
function $h\in{}^{\omega}\omega$, in the sense that $|\varphi(n)|\leq h(n)$
for all $n\in\omega$. The effect of this parameter $h$ is rather limited on
${}^{\omega}\omega$, since the $h$-localisation cardinal is equal to
$\mathrm{cof}(\mathcal{N})$ for any parameter $h$ that is increasing and
cofinal, and thus its cardinality does not depend on the choice of $h$. On the
generalised Baire space ${}^{\kappa}\kappa$ for $\kappa$ inaccessible, the
situation is very different, as it is consistent that there exist many
parametrised localisation numbers of different cardinalities, as shown by
Brendle, Brooke-Taylor, Friedman & Montoya in [BBTFM18] and improved by the
author in [vdV23].

On ${}^{\omega}\omega$, we gain a significant amount of complexity when we do
not consider cardinal characteristics defined on the entire space
${}^{\omega}\omega$, but on a bounded subspace $\prod_{n\in\omega}b(n)$ for
some increasing cofinal $b\in{}^{\omega}\omega$. This results in
$(b,h)$-localisation numbers, defined with $(b,h)$-slaloms $\varphi$, where
$|\varphi(n)|\leq h(n)$ and $\varphi(n)\subseteq b(n)$ for all $n\in\omega$,
localising functions $f\in\prod_{n\in\omega}b(n)$. Such cardinals were first
studied by Goldstern & Shelah [GS93]. Later, a connection between these
cardinals, Yorioka ideals and the strong measure zero ideal was given by Osuga
& Kamo [KO08, KO14] and it was shown recently by Cardona, Klausner & Mejía
[CKM21] that there can be models with $2^{\aleph_{0}}$ many
$(b,h)$-localisation numbers with mutually distinct cardinality.

The goal of this article is to study such parametrised localisation numbers in
the context of ${}^{\kappa}\kappa$ where $\kappa$ is inaccessible. Alongside,
we will also describe antilocalisation cardinals, whose (non-parametrised)
cardinality gives a combinatorial definition of the cardinal characteristics
$\mathrm{cov}(\mathcal{M})$ and $\mathrm{non}(\mathcal{M})$, and of
parametrised versions of the dominating number and the eventual difference
number (which also has non-parametrised versions that happen to be equal to
$\mathrm{cov}(\mathcal{M})$ and $\mathrm{non}(\mathcal{M})$).

After introducing the notation and giving an overview of the situation on the
(unbounded) Baire space in section 2, we will use section 3 to investigate for
which parameters $b$ and $h$ our cardinal numbers are nontrivial, in the sense
that their values lie between $\kappa^{+}$ and $2^{\kappa}$ without being
equal to either bound. Although we can describe sufficient conditions for the
cardinals to be nontrivial, there are still several open problems concerning
the necessity of these conditions.

In section 4 we will give an overview of how our cardinals relate to each
other. We do this by giving Tukey connections between relational systems. We
also introduce cardinal characteristics defined by taking the infimum or
supremum of (anti)localisation cardinals over all possible parameters.

Finally, in section 5 we will prove consistency results regarding our
cardinals. In particular we will construct a model in which there are
$\kappa^{+}$ many mutually distinct localisation cardinals (using the same
type of forcing notion as in [vdV23]), and we describe a family of $\kappa$
many parameters such that any finite set of parameters has associated
antilocalisation cardinals that can be forced to be mutually distinct (using a
generalisation of the forcing notion from [CM19]).

## 2\. Notation & Definition of Concepts

Let us establish our notation. Usually we will use the Greek letters
$\alpha,\beta,\gamma,\delta,\varepsilon,\eta,\xi$ for ordinals, while
$\kappa,\lambda,\mu,\nu$ will be used for cardinals. The class of all ordinals
is written as $\mathrm{Ord}$.

We say that a property $P$ holds for almost all $\alpha\in\kappa$ if there is
a $\beta\in\kappa$ such that $P(\alpha)$ holds for all $\alpha>\beta$, and we
abbreviate this as $\forall^{\infty}\alpha P(\alpha)$. Dually, a property is
said to hold for cofinally many $\alpha\in\kappa$ if for every
$\beta\in\kappa$ there is some $\alpha>\beta$ for which $P(\alpha)$ holds,
abbreviated as $\exists^{\infty}\alpha P(\alpha)$.

Given two functions $f,f^{\prime}$ with domain $\kappa$ and $\vartriangleleft$
a relation defined on $\mathrm{ran}(f)\times\mathrm{ran}(f^{\prime})$, we
write

  * $\bullet$

$f\mathrel{\vartriangleleft}f^{\prime}$ as a shorthand for
$\forall\alpha\in\kappa(f(\alpha)\mathrel{\vartriangleleft}f^{\prime}(\alpha))$,

  * $\bullet$

$f\mathrel{\vartriangleleft}^{*}f^{\prime}$ as a shorthand for
$\forall^{\infty}\alpha(f(\alpha)\vartriangleleft f^{\prime}(\alpha))$,

  * $\bullet$

$f\mathrel{\vartriangleleft}^{\infty}f^{\prime}$ as a shorthand for
$\exists^{\infty}\alpha(f(\alpha)\vartriangleleft f^{\prime}(\alpha))$.

The intended meaning of $\cancel{\vartriangleleft^{*}}$ and
$\cancel{\vartriangleleft^{\infty}}$ are the negations of
$\vartriangleleft^{*}$ and $\vartriangleleft^{\infty}$ respectively, as should
be clear on sight, in contrast to the ambiguously notated
$\not\vartriangleleft^{*}$ and $\not\vartriangleleft^{\infty}$. For this
reason we will henceforth use the former notation.

Unsurprisingly, we write cardinal exponentiation as $\lambda^{\mu}$ and use
the abbreviation $\lambda^{<\mu}=\bigcup_{\alpha\in\mu}\lambda^{|\alpha|}$. If
instead we want to discuss the set of functions from $X$ to $Y$, we write this
as ${}^{X}Y$. If $\alpha$ is an ordinal, then ${}^{<\alpha}Y$ denotes the set
$\bigcup_{\xi\in\alpha}{}^{\xi}Y$. We define

$\displaystyle[Y]^{\mu}$
$\displaystyle=\left\\{X\in\mathcal{P}(Y)\mid|X|=\mu\right\\}$
$\displaystyle[Y]^{<\mu}$
$\displaystyle=\left\\{X\in\mathcal{P}(Y)\mid|X|<\mu\right\\}.$

Naturally, ${}^{\leq\alpha}Y$ and $[Y]^{\leq\alpha}$ have the obvious meaning
that is implicit from the above. Given sequences $s\in{}^{\alpha}X$ and
$t\in{}^{\beta}X$, we write $s^{\frown}t\in{}^{\alpha+\beta}X$ for the
concatenation of $s$ and $t$. If $x\in X$, we write $\left\langle
x\right\rangle$ for the sequence of length $1$ containing only $x$. If
$s\in{}^{\alpha}X$ is a sequence, we write $\mathrm{ot}(s)=\alpha$ for the
length or order-type of $s$. Similarly, if $A\subseteq\mathrm{Ord}$ is a set
of ordinals, we write $\mathrm{ot}(A)$ for the order-type of $\left\langle
A,\in\right\rangle$.

An increasing function $f:\kappa\to\mathrm{Ord}$ is called continuous at
$\gamma\in\kappa$ if $f(\gamma)=\bigcup_{\alpha<\gamma}f(\alpha)$ and
discontinuous at $\gamma\in\kappa$ if
$f(\gamma)>\bigcup_{\alpha<\gamma}f(\alpha)$. If $A\subseteq\kappa$, then $f$
is (dis)continuous on $A$ if $f$ is (dis)continuous at $\gamma$ for every
limit ordinal $\gamma\in A$.

If $f,g$ are functions, we interpret arithmetical operators on the functions
elementwise, such as $f+g:\alpha\mapsto f(\alpha)+g(\alpha)$ and
$2^{f}:\alpha\mapsto 2^{f(\alpha)}$ and if $\xi$ is an ordinal
$f+\xi:\alpha\mapsto f(\alpha)+\xi$. We will often work with functions $b$
where $b(\alpha)$ is a cardinal for each $\alpha$. In such cases, we also
establish that $\mathrm{cf}(b):\alpha\mapsto\mathrm{cf}(b(\alpha))$ and
$b^{+}:\alpha\mapsto(b(\alpha))^{+}$. Finally if $\alpha$ is an ordinal, we
write $\overline{\alpha}$ for the constant function
$\kappa\to\left\\{\alpha\right\\}$.

### 2.1. Bounded Generalised Baire Spaces

As the name suggests, the generalised Baire space ${}^{\kappa}\kappa$ is a
generalisation of the classical Baire space ${}^{\omega}\omega$. Instead of
functions between the natural numbers, we consider functions between
uncountable cardinals. In our case, we are specifically interested in
generalised Baire spaces where $\kappa$ is a (strongly) inaccessible cardinal.
As such, we will fix the convention for the remainder of this article that
$\kappa$ is inaccessible. We would like to emphasise that we leave out the
mention of $\kappa$ being inaccessible from our lemmas and theorems, even if
this is a necessary assumption for the theorem to be provable.

We restrict our attention to inaccessible cardinals, because the Cichoń
diagram on ${}^{\kappa}\kappa$ behaves quite similar to its counterpart on
${}^{\omega}\omega$. We refer to [BBTFM18] for an overview of the Cichoń
diagram on the generalised Baire space and to [Bre22] for the case where
$\kappa$ is not inaccessible.

The generalised Baire space ${}^{\kappa}\kappa$ is often studied as a
topological space by endowing $\kappa$ with the discrete topology and
${}^{\kappa}\kappa$ with the ${<}\kappa$-box topology. This is the topology
generated by basic open sets of the form
$[s]=\left\\{f\in{}^{\kappa}\kappa\mid s\subseteq f\right\\}$, where
$s\in{}^{\alpha}\kappa$ for some $\alpha\in\kappa$. Note that each basic set
$[s]$ is in fact clopen.

The main focus of this article will be to study cardinal characteristics in
the context of bounded subspaces of the generalised Baire space. Given a
function $b\in{}^{\kappa}\kappa$, we consider the product space $\prod
b=\prod_{\alpha\in\kappa}b(\alpha)$ where each $b(\alpha)$ has the discrete
topology and $\prod b$ once again has the ${<}\kappa$-box topology. We will
write

$\displaystyle\textstyle\prod_{<\kappa}b=\bigcup_{\gamma\in\kappa}\left(\prod_{\alpha\in\gamma}b(\alpha)\right),$

then the topology on $\prod b$ is generated by sets $[s]=\left\\{f\in\prod
b\mid s\subseteq f\right\\}$ with $s\in\prod_{<\kappa}b$. As such, we see that
$\prod b$ is a closed subspace of ${}^{\kappa}\kappa$. We will refer to spaces
of the form $\prod b$ as bounded generalised Baire spaces. Generally we will
only consider increasing functions $b\in{}^{\kappa}\kappa$ such that
$b(\alpha)$ is an infinite cardinal for every $\alpha\in\kappa$, unless we
state otherwise.

A set $X\subseteq\prod b$ is nowhere dense if for every basic open
$[s]\subseteq\prod b$ there exists a basic open $[t]\subseteq[s]$ such that
$[t]\cap X=\varnothing$. It is easy to show that the closure of a nowhere
dense set is nowhere dense. The complement of a closed nowhere dense set is an
open dense set. We will write $\mathcal{N}_{\kappa}^{b}$ for the family of
nowhere dense subsets of $\prod b$. A set $X$ is $\kappa$-meagre if there
exists a family $\left\\{N_{\alpha}\mid\alpha\in\kappa\right\\}$ of nowhere
dense sets such that $X=\bigcup_{\alpha\in\kappa}N_{\alpha}$. Let
$\mathcal{M}_{\kappa}^{b}$ be the set of $\kappa$-meagre subsets of $\prod b$.
We will use the notation $\mathcal{M}_{\kappa}$ for the family $\kappa$-meagre
sets of ${}^{\kappa}\kappa$.

Given a ${\leq}\kappa$-complete ideal $\mathcal{I}$ on the space $X$, we
define the following four cardinal functions:

$\displaystyle\mathrm{cov}(\mathcal{I})$
$\displaystyle=\min\left\\{|\mathcal{C}|\ \middle|\
\mathcal{C}\subseteq\mathcal{I}\land\textstyle\bigcup\mathcal{C}=X\right\\}$
$\displaystyle\mathrm{non}(\mathcal{I})$ $\displaystyle=\min\left\\{|N|\
\middle|\ N\subseteq X\land N\notin\mathcal{I}\right\\}$
$\displaystyle\mathrm{add}(\mathcal{I})$
$\displaystyle=\min\left\\{|\mathcal{A}|\ \middle|\
\mathcal{A}\subseteq\mathcal{I}\land\textstyle\bigcup\mathcal{C}\notin\mathcal{I}\right\\}$
$\displaystyle\mathrm{cof}(\mathcal{I})$
$\displaystyle=\min\left\\{|\mathcal{F}|\ \middle|\
\mathcal{F}\subseteq\mathcal{I}\land\forall I\in\mathcal{I}\exists
J\in\mathcal{F}(I\subseteq J)\right\\}$

It is easy to show that families $\mathcal{M}_{\kappa}^{b}$ (and
$\mathcal{M}_{\kappa})$ are ${\leq}\kappa$-complete proper ideals. Moreover,
by the following lemma, the choice of $b\in{}^{\kappa}\kappa$ does not matter
for the value of these cardinal functions evaluated over
$\mathcal{M}_{\kappa}^{b}$, and indeed, we could work with the cardinal
functions evaluated over $\mathcal{M}_{\kappa}$ without loss of generality.

###### Lemma 2.1

There exists $M\in\mathcal{M}_{\kappa}^{b}$ such that the subspace $\prod
b\setminus M$ of $\prod b$ is homeomorphic to ${}^{\kappa}\kappa$.

  * Proof.

First, we recursively define a function
$\varphi:{}^{<\kappa}\kappa\to\prod_{<\kappa}b$. Let
$\varphi(\varnothing)=\varnothing$. Given $t\in{}^{<\kappa}\kappa$ such that
$\varphi(t)$ has been defined, let
$A=\left\\{s_{\alpha}\mid\alpha\in\kappa\right\\}\subseteq\prod_{<\kappa}b$ be
an antichain of size $\kappa$ with $\varphi(t)\subseteq s_{\alpha}$, such that
$A$ is maximal with this property, and send
$\varphi:t^{\frown}\left\langle\alpha\right\rangle\mapsto s_{\alpha}$. If
$\gamma=\mathrm{dom}(t)$ is limit and $\varphi(t\restriction\alpha)$ is
defined for each $\alpha<\gamma$, we have by construction that

$\displaystyle\textstyle\bigcup_{\alpha\in\gamma}\varphi(t\restriction\alpha)\in\prod_{<\kappa}b.$

Therefore we set
$\varphi(t)=\bigcup_{\alpha\in\gamma}\varphi(t\restriction\alpha)$.

Note that $\varphi[{}^{\alpha}\kappa]$ forms a maximal antichain in
$\prod_{<\kappa}b$: if $\alpha$ is the least ordinal such that
$\varphi[{}^{\alpha}\kappa]$ is not maximal, let
$s\notin\varphi[{}^{\alpha}\kappa]$ be such that
$\varphi[{}^{\alpha}\kappa]\cup\left\\{s\right\\}$ is an antichain, then for
any $\xi<\alpha$ there is $s_{\xi}\subseteq s$ such that
$s_{\xi}\in\varphi[{}^{\xi}\kappa]$, but then
$\bigcup_{\xi\in\alpha}s_{\xi}=s^{\prime}\subseteq s$ and
$s^{\prime}\in\varphi[{}^{\alpha}\kappa]$.

We define $\Phi:{}^{\kappa}\kappa\to\prod b$ induced by $\varphi$ as sending
$f\mapsto\bigcup_{\alpha\in\kappa}\varphi(f\restriction\alpha)$. Note that the
range $\Phi[{}^{\kappa}\kappa]$ is $\kappa$-comeagre in $\prod b$, because for
each $\alpha\in\kappa$ we can define

$\displaystyle F_{\alpha}=\left\\{f\in\textstyle\prod b\mid
f\notin\bigcup_{t\in{}^{\alpha}\kappa}[\varphi(k)]\right\\}.$

Then $F_{\alpha}$ is nowhere dense, since $\varphi[{}^{\alpha}\kappa]$ is a
maximal antichain, and $\Phi[{}^{\kappa}\kappa]=\prod
b\setminus\bigcup_{\alpha\in\kappa}F_{\alpha}$, hence
$M=\bigcup_{\alpha\in\kappa}F_{\alpha}$ is a $\kappa$-meagre set and we will
see that
$\Phi:{}^{\kappa}\kappa\mathrel{\textnormal{\guilsinglright}\mathrel{\mkern-9.0mu}\rightarrow\mathrel{\mkern-14.0mu}\rightarrow}\prod
b\setminus M$ is a homeomorphism.

It is clear that $\Phi$ is bijective. Note that $\Phi$ is open, since for any
$t\in{}^{<\kappa}\kappa$ we have $\varphi(t)\in\prod_{<\kappa}b$, thus
$\Phi[[t]]=[\varphi(t)]\setminus M$ is open in $\prod b\setminus M$. To see
that $\Phi$ is continuous, let $s\in\prod_{<\kappa}b$ with
$\mathrm{dom}(s)=\alpha$. If $t\in{}^{\alpha}\kappa$, then it follows by
construction that $\alpha\subseteq\mathrm{dom}(\varphi(t))$. Let

$\displaystyle T=\left\\{t\in{}^{\alpha}\kappa\mid
s\subseteq\varphi(t)\right\\}.$

Note that $f\in\bigcup_{t\in T}[t]$ if and only if
$s\subseteq\varphi(f\restriction\alpha)$ off $\Phi(f)\in[s]$ if and only if
$f\in\Phi^{-1}[[s]]$. ∎

###### Corollary 2.2

$\mathrm{cov}(\mathcal{M}_{\kappa})=\mathrm{cov}(\mathcal{M}_{\kappa}^{b})$,
$\mathrm{non}(\mathcal{M}_{\kappa})=\mathrm{non}(\mathcal{M}_{\kappa}^{b})$,
$\mathrm{add}(\mathcal{M}_{\kappa})=\mathrm{add}(\mathcal{M}_{\kappa}^{b})$,
$\mathrm{cof}(\mathcal{M}_{\kappa})=\mathrm{cof}(\mathcal{M}_{\kappa}^{b})$.

Apart from the $\kappa$-meagre ideal, we will also consider the ideal of
$\kappa$-strong measure zero sets. Contrary to Lebesgue measure zero sets,
strong measure zero sets can be generalised to the context of
${}^{\kappa}\kappa$. We will work in ${}^{\kappa}2$, with basic open sets
$[s]=\left\\{f\in{}^{\kappa}2\mid s\subseteq f\right\\}$ for
$s\in{}^{<\kappa}2$. A set $X\subseteq{}^{\kappa}2$ is called $\kappa$-strong
measure zero if for every $f\in{}^{\kappa}\kappa$ there exists a sequence
$\left\langle s_{\alpha}\in{}^{<\kappa}2\mid\alpha\in\kappa\right\rangle$ with
$\mathrm{ot}(s_{\alpha})=f(\alpha)$ such that
$X\subseteq\bigcup_{\alpha\in\kappa}[s_{\alpha}]$. We write
$\mathcal{S}\mathcal{N}_{\kappa}$ for the family of $\kappa$-strong measure
zero subsets of ${}^{\kappa}2$.

It is again easy to see that $\mathcal{S}\mathcal{N}_{\kappa}$ is a
${\leq}\kappa$-complete proper ideal. We conclude this section by giving an
equivalent definition of $\mathcal{S}\mathcal{N}_{\kappa}$.

###### Lemma 2.3

The following are equivalent for a set $X\subseteq{}^{\kappa}2$.

  1. (1)

$X\in\mathcal{S}\mathcal{N}_{\kappa}$

  2. (2)

For every $f\in{}^{\kappa}\kappa$ there exists a sequence
$\overline{s}=\left\langle s_{\alpha}\mid\alpha\in\kappa\right\rangle$ with
$s_{\alpha}\in{}^{f(\alpha)}2$ for each $\alpha$ such that
$X\subseteq\bigcup_{\alpha\in\kappa}[s_{\alpha}]$.

  3. (3)

For every $f\in{}^{\kappa}\kappa$ there exists a sequence
$\overline{s}=\left\langle s_{\alpha}\mid\alpha\in\kappa\right\rangle$ with
$s_{\alpha}\in{}^{f(\alpha)}2$ for each $\alpha$ such that
$X\subseteq\bigcap_{\alpha_{0}\in\kappa}\bigcup_{\beta\in[\alpha_{0},\kappa)}[s_{\beta}]$.

  * Proof.

Note that (2) is simply the definition of elements of
$\mathcal{S}\mathcal{N}_{\kappa}$. That (3) implies (2) is obvious.

To see that (2) implies (3), let
$\pi:\kappa\times\kappa\mathrel{\textnormal{\guilsinglright}\mathrel{\mkern-9.0mu}\rightarrow\mathrel{\mkern-14.0mu}\rightarrow}\kappa$
be a bijection and $f\in{}^{\kappa}\kappa$. We define
$f_{\xi}\in{}^{\kappa}\kappa$ by $f_{\xi}(\alpha)=f(\pi(\xi,\alpha))$ and use
(2) to find a sequence $\overline{s}^{\xi}=\langle
s_{\alpha}^{\xi}\mid\alpha\in\kappa\rangle$ with
$s_{\alpha}^{\xi}\in{}^{f_{\xi}(\alpha)}2$ such that
$X\subseteq\bigcup_{\alpha\in\kappa}[s^{\xi}_{\alpha}]$. Given
$\xi,\alpha\in\kappa$, we define $s_{\pi(\xi,\alpha)}=s_{\alpha}^{\xi}$ then
$s_{\beta}\in f(\beta)$ for all $\beta\in\kappa$. It follows that
$X\subseteq\bigcap_{\alpha_{0}\in\kappa}\bigcup_{\beta\in[\alpha_{0},\kappa)}[s_{\beta}]$.
∎

If $\overline{s}=\left\langle s_{\alpha}\mid\alpha\in\kappa\right\rangle$ is a
sequence such that $X\subseteq\bigcup_{\alpha\in\kappa}[s^{\xi}_{\alpha}]$, we
say $X$ is covered by $\overline{s}$. If also
$X\subseteq\bigcap_{\alpha_{0}\in\kappa}\bigcup_{\beta\in[\alpha_{0},\kappa)}[s_{\beta}]$,
then we say $X$ is cofinally covered by $\overline{s}$.

### 2.2. Cardinal Characteristics & Relational Systems

In order to study the relations between our cardinal characteristics, we will
make use of relational systems. All of the cardinal characteristics we are
interested in, can be expressed as the norm of a relational system, thus we
will use the language of relational systems to define our cardinal
characteristics. We will only give a concise description of relational systems
below, and refer to [Bla10] for a detailed description.

A relational system $\mathscr{R}=\left\langle X,Y,R\right\rangle$ is a triple
of sets $X$ and $Y$ and a relation $R\subseteq X\times Y$. We define the norm
of $\mathscr{R}$ as

$\displaystyle\left\lVert\mathscr{R}\right\rVert=\min\left\\{|W|\mid
W\subseteq Y\text{ and }\forall x\in X\exists y\in W(x\mathrel{R}y)\right\\}.$

We refer to a set $W\subseteq Y$ such that $\forall x\in X\exists y\in
W(x\mathrel{R}y)$ as a witness for
$\left\lVert\mathscr{R}\right\rVert\leq|W|$. We define a dual relation to $R$
by

$\displaystyle\cancel{R}^{-1}=\left\\{(y,x)\in Y\times X\mid(x,y)\notin
R\right\\}.$

Correspondingly we can define the dual relational system
$\mathscr{R}^{\bot}=\left\langle Y,X,\cancel{R}^{-1}\right\rangle$.

Intuitively, we can see $Y$ as a set of possible responses and $X$ as the set
of potential challenges. We want to find a set of responses
$Y^{\prime}\subseteq Y$ such that every challenge can be met with a response
from $Y^{\prime}$, and the norm expresses the least number of responses
necessary to do so. The dual relational system answers the question how many
challenges we should gather such that no single response can meet them all.

Given two relational systems $\mathscr{R}=\left\langle X,Y,R\right\rangle$ and
$\mathscr{R}^{\prime}=\left\langle
X^{\prime},Y^{\prime},R^{\prime}\right\rangle$, a Tukey connection from
$\mathscr{R}$ to $\mathscr{R}^{\prime}$ is a pair of functions $\rho_{-}:X\to
X^{\prime}$ and $\rho_{+}:Y^{\prime}\to Y$ such that for any $x\in X$ and
$y^{\prime}\in Y^{\prime}$ with $(\rho_{-}(x),y^{\prime})\in R^{\prime}$ we
also have $(x,\rho_{+}(y^{\prime}))\in R$. We let
$\mathscr{R}\preceq\mathscr{R}^{\prime}$ denote the claim that there exists a
Tukey connection from $\mathscr{R}$ to $\mathscr{R}^{\prime}$, and we let
$\mathscr{R}\equiv\mathscr{R}^{\prime}$ abbreviate
$\mathscr{R}\preceq\mathscr{R}^{\prime}\preceq\mathscr{R}$. Note that if
$\left\langle\rho_{-},\rho_{+}\right\rangle$ witnesses
$\mathscr{R}\preceq\mathscr{R}^{\prime}$, then
$\left\langle\rho_{+},\rho_{-}\right\rangle$ witnesses
$\mathscr{R}^{\prime\bot}\preceq\mathscr{R}^{\bot}$.

We will use Tukey connections to give an ordering between the norms of
relational systems, through the following lemma.

###### Lemma 2.4

If $\mathscr{R}\preceq\mathscr{R}^{\prime}$, then
$\left\lVert\mathscr{R}\right\rVert\leq\left\lVert\mathscr{R}^{\prime}\right\rVert$
and
$\left\lVert\mathscr{R}^{\prime\bot}\right\rVert\leq\left\lVert\mathscr{R}^{\bot}\right\rVert$.

We will on one occasion require a composition of several relational systems.
In particular we need a general form of the categorical product that is
defined in [Bla10]. We will state the definition and give two lemmas that are
useful to compute the norm.

Let $\mathscr{R}_{\alpha}=\left\langle
X_{\alpha},Y_{\alpha},R_{\alpha}\right\rangle$ be relational systems for each
$\alpha\in A$, where $A$ is a set of ordinals (but we will omit mention of $A$
from now on for the sake of brevity). We define the categorical product as the
system
$\bigotimes_{\alpha}\mathscr{R}_{\alpha}=\left\langle\bigcup_{\alpha}(X_{\alpha}\times\left\\{\alpha\right\\}),\prod_{\alpha}Y_{\alpha},Z\right\rangle$,
where $(x,\alpha)\mathrel{Z}\overline{y}$ off
$x\mathrel{R}_{\alpha}\overline{y}(\alpha)$. Dually we have the categorical
coproduct
$\bigoplus_{\alpha}\mathscr{R}_{\alpha}=(\bigotimes_{\alpha}\mathscr{R}_{\alpha}^{\bot})^{\bot}=\left\langle\prod_{\alpha}X_{\alpha},\bigcup_{\alpha}(Y_{\alpha}\times\left\\{\alpha\right\\}),N\right\rangle$,
where $\overline{x}\mathrel{N}(y,\alpha)$ iff
$\overline{x}(\alpha)\mathrel{R}_{\alpha}y$.

###### Lemma 2.5

$\left\lVert\bigotimes_{\alpha}\mathscr{R}_{\alpha}\right\rVert=\sup_{\alpha}\left\lVert\mathscr{R}_{\alpha}\right\rVert$.

  * Proof.

($\leq$) Let $Y^{\prime}_{\alpha}\subseteq Y_{\alpha}$ be a witness for
$|Y^{\prime}_{\alpha}|=\left\lVert\mathscr{R}_{\alpha}\right\rVert$. Let
$\lambda=\sup_{\alpha}\left\lVert\mathscr{R}_{\alpha}\right\rVert$ and let
$\sigma_{\alpha}:\lambda\rightarrow\mathrel{\mkern-14.0mu}\rightarrow
Y^{\prime}_{\alpha}$ be surjections. Define
$\overline{y}_{\xi}:\alpha\mapsto\sigma_{\alpha}(\xi)$ and let
$Y=\left\\{\overline{y}_{\xi}\mid\xi\in\lambda\right\\}\subseteq\prod_{\alpha}Y_{\alpha}$.
If $x\in X_{\alpha}$, let $y\in Y^{\prime}_{\alpha}$ such that
$x\mathrel{R}_{\alpha}y$ and find $\xi\in\lambda$ such that
$y=\sigma_{\alpha}(\xi)$, then $(x,\alpha)\mathrel{Z}\overline{y}_{\xi}$.
Hence $Y$ witnesses that
$\left\lVert\bigotimes_{\alpha}\mathscr{R}_{\alpha}\right\rVert\leq\lambda$.

($\geq$) Let $Y\subseteq\prod_{\alpha}Y_{\alpha}$ be a witness for
$|Y|=\left\lVert\bigotimes_{\alpha}\mathscr{R}_{\alpha}\right\rVert$, and let
$Y^{\prime}_{\alpha}=\left\\{\overline{y}(\alpha)\mid\overline{y}\in
Y\right\\}\subseteq Y_{\alpha}$. Then for every $x\in X_{\alpha}$ there exists
$\overline{y}\in Y$ such that $(x,\alpha)\mathrel{Z}\overline{y}$, hence
$\overline{y}(\alpha)\in Y^{\prime}_{\alpha}$ has the property that
$x\mathrel{R}_{\alpha}\overline{y}(\alpha)$, thus
$\left\lVert\mathscr{R}_{\alpha}\right\rVert\leq|Y^{\prime}|\leq|Y|=\left\lVert\bigotimes_{\alpha}\mathscr{R}_{\alpha}\right\rVert$.
∎

###### Lemma 2.6

$\left\lVert\bigoplus_{\alpha}\mathscr{R}_{\alpha}\right\rVert=\inf_{\alpha}\left\lVert\mathscr{R}_{\alpha}\right\rVert$.

  * Proof.

($\leq$) Let $Y\subseteq Y_{\alpha}$ be a witness for
$|Y|=\left\lVert\mathscr{R}_{\alpha}\right\rVert$. Let
$Y^{\prime}=\left\\{(y,\alpha)\mid y\in Y\right\\}$. If
$\overline{x}\in\prod_{\alpha}X_{\alpha}$, then there exists $y\in Y$ such
that $\overline{x}(\alpha)\mathrel{R}_{\alpha}y$ and thus
$\overline{x}\mathrel{N}(y,\alpha)$. Therefore
$\left\lVert\bigoplus_{\alpha}\mathscr{R}_{\alpha}\right\rVert\leq|Y^{\prime}|=|Y|=\left\lVert\mathscr{R}_{\alpha}\right\rVert$.

($\geq$) Let
$Y\subseteq\bigcup_{\alpha}(Y_{\alpha}\times\left\\{\alpha\right\\})$ with
$|Y|<\inf_{\alpha}\left\lVert\mathscr{R}_{\alpha}\right\rVert$. Let
$Y^{\prime}_{\alpha}=\left\\{y\in Y_{\alpha}\mid(y,\alpha)\in Y\right\\}$.
Since $|Y^{\prime}_{\alpha}|<\left\lVert\mathscr{R}_{\alpha}\right\rVert$
there is $x_{\alpha}\in X_{\alpha}$ such that
$x_{\alpha}\mathrel{\cancel{R}}_{\alpha}y$ for all $y\in Y^{\prime}_{\alpha}$.
Let $\overline{x}:\alpha\mapsto x_{\alpha}$, then
$\overline{x}\mathrel{\cancel{N}}(y,\alpha)$ for every $(y,\alpha)\in Y$. Thus
$|Y|<\left\lVert\bigoplus_{\alpha}\mathscr{R}_{\alpha}\right\rVert$. ∎

Apart from the previously mentioned cardinal functions on ideals, we will
define four other families of cardinals. In order to do so, we need to discuss
slaloms, a concept first formulated by Bartoszyński to give a combinatorial
characterisation of the additivity and cofinality of the Lebesgue null ideal
in the classical Baire space (see [Bar87, BJ95] or the next subsection).

###### Definition 2.7

Let $h\in{}^{\kappa}\kappa$ be an increasing function such that $h(\alpha)$ is
a nonzero cardinal for all $\alpha\in\kappa$. We will generally assume $h\leq
b$. Let $\mathrm{Loc}_{\kappa}^{b,h}$ be the set of functions $\varphi$ with
$\mathrm{dom}(\varphi)=\kappa$ such that $\varphi(\alpha)\subseteq b(\alpha)$
and $\left|\varphi(\alpha)\right|<h(\alpha)$ for each $\alpha\in\kappa$. If
$\varphi\in\mathrm{Loc}_{\kappa}^{b,h}$, then we call $\varphi$ a
$(b,h)$-slalom.222Note that this definition differs from the usual definition:
we define $\varphi$ such that $|\varphi(\alpha)|<h(\alpha)\leq b(\alpha)$,
instead of the traditional $|\varphi(\alpha)|\leq h(\alpha)<b(\alpha)$. Our
definition is more versatile. For example, if $h(\alpha)$ is a limit cardinal
for each $\alpha$, the resulting set of all slaloms $\varphi$ with
$|\varphi(\alpha)|<h(\alpha)$ cannot be expressed using the traditional
definition. On the other hand, the set of all traditionally defined
$(b,h)$-slaloms is the set of $(b,h^{+})$-slaloms under our definition.

Given $\varphi\in\mathrm{Loc}_{\kappa}^{b,h}$ and $f\in\prod b$, then we say
$\varphi$ localises $f$ iff $f\in^{*}\varphi$ and $\varphi$ antilocalises $f$
iff $f\mathrel{\cancel{\in^{\infty}}}\varphi$.

Similar to how we fixed the notation $\prod_{<\kappa}b$ for initial segments
of functions in $\prod b$, we will fix the notation
$\mathrm{Loc}_{<\kappa}^{b,h}$ for initial segments of $(b,h)$-slaloms, that
is, $s\in\mathrm{Loc}_{<\kappa}^{b,h}$ if and only if
$\mathrm{dom}(s)\in\kappa$ and for all $\alpha\in\mathrm{dom}(s)$ we have
$s(\alpha)\in[b(\alpha)]^{<h(\alpha)}$.

We now define the following four relational systems and associated cardinals:

$\displaystyle\mathscr{D}_{b}$ $\displaystyle=\left\langle\textstyle\prod
b,\prod b,\leq^{*}\right\rangle$
$\displaystyle\left\lVert\mathscr{D}_{b}\right\rVert$
$\displaystyle=\mathfrak{d}_{\kappa}^{b}(\leq^{*})$
$\displaystyle\left\lVert{\mathscr{D}_{b}^{\bot}}\right\rVert$
$\displaystyle=\mathfrak{b}_{\kappa}^{b}(\leq^{*})$
$\displaystyle\mathscr{E}\\!\mathscr{D}_{b}$
$\displaystyle=\left\langle\textstyle\prod b,\prod
b,\mathrel{\cancel{=^{\infty}}}\right\rangle$
$\displaystyle\left\lVert\mathscr{E}\\!\mathscr{D}_{b}\right\rVert$
$\displaystyle=\mathfrak{d}_{\kappa}^{b}(\mathrel{\cancel{=^{\infty}}})$
$\displaystyle\left\lVert{\mathscr{E}\\!\mathscr{D}_{b}^{\bot}}\right\rVert$
$\displaystyle=\mathfrak{b}_{\kappa}^{b}(\mathrel{\cancel{=^{\infty}}})$
$\displaystyle\mathscr{L}_{b,h}$ $\displaystyle=\left\langle\textstyle\prod
b,\mathrm{Loc}_{\kappa}^{b,h},\in^{*}\right\rangle$
$\displaystyle\left\lVert\mathscr{L}_{b,h}\right\rVert$
$\displaystyle=\mathfrak{d}_{\kappa}^{b,h}(\in^{*})$
$\displaystyle\left\lVert{\mathscr{L}_{b,h}^{\bot}}\right\rVert$
$\displaystyle=\mathfrak{b}_{\kappa}^{b,h}(\in^{*})$
$\displaystyle\mathscr{A}\\!\\!\mathscr{L}_{b,h}$
$\displaystyle=\left\langle\mathrm{Loc}_{\kappa}^{b,h},\textstyle\prod
b,\mathrel{\cancel{\ni^{\infty}}}\right\rangle$
$\displaystyle\left\lVert\mathscr{A}\\!\\!\mathscr{L}_{b,h}\right\rVert$
$\displaystyle=\mathfrak{d}_{\kappa}^{b,h}(\mathrel{\cancel{\ni^{\infty}}})$
$\displaystyle\left\lVert{\mathscr{A}\\!\\!\mathscr{L}_{b,h}^{\bot}}\right\rVert$
$\displaystyle=\mathfrak{b}_{\kappa}^{b,h}(\mathrel{\cancel{\ni^{\infty}}})$

We call $\mathfrak{d}_{\kappa}^{b}(\leq^{*})$ the dominating number,
$\mathfrak{b}_{\kappa}^{b}(\leq^{*})$ the unbounding number,
$\mathfrak{d}_{\kappa}^{b}(\mathrel{\cancel{=^{\infty}}})$ the eventual
difference number, $\mathfrak{b}_{\kappa}^{b}(\mathrel{\cancel{=^{\infty}}})$
the cofinal equality number333In the classical setting, the relation of being
cofinally equal is often referred to as being infinitely equal, but in our
generalised setting we require more than just an infinite amount of equality..
Both $\mathfrak{d}_{\kappa}^{b,h}(\in^{*})$ and
$\mathfrak{b}_{\kappa}^{b,h}(\in^{*})$ are sometimes called localisation
cardinals, but we believe it will be useful to distinguish between these
classes of cardinals with separate terms. We therefore opt to call
$\mathfrak{d}_{\kappa}^{b,h}(\in^{*})$ the localisation cardinal, and
$\mathfrak{b}_{\kappa}^{b,h}(\in^{*})$ the avoidance cardinal, for the reason
that a witness to $\mathfrak{b}_{\kappa}^{b,h}(\in^{*})$ is a set of functions
$F\subseteq\prod b$ that cannot be localised by a single $(b,h)$-slalom;
localisation by any specific slalom is always avoided by at least one member
of $F$. Finally this leaves the cardinals
$\mathfrak{d}_{\kappa}^{b,h}(\mathrel{\cancel{\ni^{\infty}}})$, which we will
call the anti-avoidance cardinal and
$\mathfrak{b}_{\kappa}^{b,h}(\mathrel{\cancel{\ni^{\infty}}})$, which we will
call the antilocalisation cardinal.

It should be mentioned that (anti)localisation cardinals not only suffer from
ambiguous names, they also suffer from nonuniformity of notation. In previous
literature, the notations $\mathfrak{c}_{b,h}^{\forall}$ and
$\mathfrak{v}_{b,h}^{\forall}$ have also frequently been used for
$\mathfrak{d}_{\omega}^{b,h}(\in^{*})$ and
$\mathfrak{b}_{\omega}^{b,h}(\in^{*})$ respectively, while
$\mathfrak{c}_{b,h}^{\exists}$ and $\mathfrak{v}_{b,h}^{\exists}$ have been
used for $\mathfrak{b}_{\omega}^{b,h}(\mathrel{\cancel{\ni^{\infty}}})$ and
$\mathfrak{d}_{\omega}^{b,h}(\mathrel{\cancel{\ni^{\infty}}})$ respectively.
For example, this notation was used in [Kel08, KO14, KM21, CKM21]. Here
$\mathfrak{c}$ stands for a cover of $\prod b$ with slaloms by the relations
$\in^{*}$ or $\in^{\infty}$, and $\mathfrak{v}$ stands for avoidance by or
evasion by a single slalom. We believe that avoidance forms the better antonym
to localisation, as it prevents confusion with other cardinal characteristics
known as evasion cardinals (such as described in the eponymous section of
[Bla10]). Our choice to use the notation
$\mathfrak{d}_{\kappa}^{b,h}(\in^{*})$ over $\mathfrak{c}_{b,h}^{\forall}$ has
the main benefit that the relevant relational system can be deduced from our
notation. Finally, we opted to use the relation
$\mathrel{\cancel{\ni^{\infty}}}$ instead of $\in^{\infty}$ in our relational
system for antilocalisation, since we will later see that
$\mathfrak{d}_{\kappa}^{b,h}(\mathrel{\cancel{\ni^{\infty}}})$ bears more
similarities to $\mathfrak{d}_{\kappa}^{b,h}(\in^{*})$ than to
$\mathfrak{b}_{\kappa}^{b,h}(\in^{*})$.

### 2.3. Classical and Higher Cichoń Diagram

In the next section we will investigate the provable relations between the
cardinals we have defined and what influence the parameters $b$ and $h$ have
on their cardinality. Before doing so, we would like to discuss what is known
about the case where we let $b=\overline{\kappa}$ be the constant function, in
other words, what is known about these cardinals in the (unbounded)
generalised Baire space, and how this compares to the same cardinals as
defined classically on ${}^{\omega}\omega$. When discussing the cardinals in
the context of the entire space ${}^{\kappa}\kappa$, we simply leave out the
parameter $b$ from the notation, e.g. $\mathfrak{d}_{\kappa}(\leq^{*})$
instead of $\mathfrak{d}_{\kappa}^{\overline{\kappa}}(\leq^{*})$ and
$\mathfrak{b}_{\kappa}^{h}(\mathrel{\cancel{\ni^{\infty}}})$ instead of
$\mathfrak{b}_{\kappa}^{\overline{\kappa},h}(\mathrel{\cancel{\ni^{\infty}}})$.
When discussing the cardinal in context of the classical Baire space, we
replace $\kappa$ by $\omega$, e.g.
$\mathfrak{b}_{\omega}^{b}(\mathrel{\cancel{=^{\infty}}})$.

In the context of the classical Baire space ${}^{\omega}\omega$, each of the
cardinals mentioned so far is an element of the Cichoń diagram. Let
$\mathcal{N}$ denote the ideal of sets of reals with negligible Lebesgue
measure, and let $\mathcal{M}$ denote the ($\omega$-)meagre ideal on
${}^{\omega}\omega$. The Cichoń diagram gives an overview of the relations
between our cardinals, where $\mathfrak{x}\leq\mathfrak{y}$ is provable in
$\mathsf{ZFC}$ if and only if there exists a path from $\mathfrak{x}$ to
$\mathfrak{y}$ following the arrows.

$\aleph_{1}$$\mathrm{add}(\mathcal{N})$$\mathrm{add}(\mathcal{M})$$\mathrm{cov}(\mathcal{N})$$\mathrm{non}(\mathcal{M})$$\mathfrak{b}_{\omega}(\leq^{*})$$\mathfrak{d}_{\omega}(\leq^{*})$$\mathrm{cov}(\mathcal{M})$$\mathrm{non}(\mathcal{N})$$\mathrm{cof}(\mathcal{M})$$\mathrm{cof}(\mathcal{N})$$2^{\aleph_{0}}$

Proofs for these relations and for the following facts can also be found in
[BJ95] or [Bla10].

######  Fact 2.8 ( [Tru77, Mil81])

$\mathrm{add}(\mathcal{M})=\min\left\\{\mathfrak{b}_{\omega}(\leq^{*}),\mathrm{cov}(\mathcal{M})\right\\}$
and
$\mathrm{cof}(\mathcal{M})=\max\left\\{\mathfrak{d}_{\omega}(\leq^{*}),\mathrm{non}(\mathcal{M})\right\\}$

######  Fact 2.9 ( [Bar87])

$\mathrm{cov}(\mathcal{M})=\mathfrak{d}_{\omega}(\mathrel{\cancel{=^{\infty}}})=\mathfrak{d}_{\omega}^{h}(\mathrel{\cancel{\ni^{\infty}}})$
and
$\mathrm{non}(\mathcal{M})=\mathfrak{b}_{\omega}(\mathrel{\cancel{=^{\infty}}})=\mathfrak{b}_{\omega}^{h}(\mathrel{\cancel{\ni^{\infty}}})$
and $\mathrm{cof}(\mathcal{N})=\mathfrak{d}^{h}_{\omega}(\in^{*})$ and
$\mathrm{add}(\mathcal{M})=\mathfrak{b}^{h}_{\omega}(\in^{*})$ for any
increasing cofinal $h\in{}^{\omega}\omega$.

In the context of the generalised Baire space ${}^{\kappa}\kappa$ (where we
assume, as mentioned, that $\kappa$ is inaccessible), we do not have Lebesgue
measure, thus the four cardinal invariants related to $\mathcal{N}$ cannot be
generalised. The other cardinals can be generalised, as we have defined in the
previous section, and the Cichoń diagram looks identical to the classical case
if we assume that $\kappa$ is strongly inaccessible444This assumption is
necessary for all of the cardinals of (the middle part of) the Cichoń diagram
to be nontrivial, but there are some interesting questions for $\kappa$ being
accessible, see [Bre22]. and $h$ is a cofinal increasing function.

$\kappa^{+}$$\mathfrak{b}_{\kappa}^{h}(\in^{*})$$\mathrm{add}(\mathcal{M}_{\kappa})$$\mathrm{non}(\mathcal{M}_{\kappa})$$\mathfrak{b}_{\kappa}(\leq^{*})$$\mathfrak{d}_{\kappa}(\leq^{*})$$\mathrm{cov}(\mathcal{M}_{\kappa})$$\mathrm{cof}(\mathcal{M}_{\kappa})$$\mathfrak{d}_{\kappa}^{h}(\in^{*})$$2^{\kappa}$

Proofs for these relations can be found in [BBTFM18].

######  Fact 2.10 ( [BBTFM18])

$\mathrm{add}(\mathcal{M}_{\kappa})=\min\left\\{\mathfrak{b}_{\kappa}(\leq^{*}),\mathrm{cov}(\mathcal{M}_{\kappa})\right\\}$
and
$\mathrm{cof}(\mathcal{M}_{\kappa})=\max\left\\{\mathfrak{d}_{\kappa}(\leq^{*}),\mathrm{non}(\mathcal{M}_{\kappa})\right\\}$

######  Fact 2.11 ( [BHZ05], [Lan92])

$\mathrm{non}(\mathcal{M}_{\kappa})=\mathfrak{b}_{\kappa}(\mathrel{\cancel{=^{\infty}}})$
and
$\mathrm{cov}(\mathcal{M}_{\kappa})=\mathfrak{d}_{\kappa}(\mathrel{\cancel{=^{\infty}}})$.

Contrary to the classical case, the parameter $h$ of
$\mathfrak{d}_{\kappa}^{h}(\in^{*})$ matters in the generalised case. If
$\mathsf{id}:\alpha\mapsto|\alpha|$ and $\mathsf{pow}:\alpha\mapsto
2^{|\alpha|}$, then the corresponding cardinals are consistently different:

######  Fact 2.12 ( [BBTFM18])

It is consistent that
$\mathfrak{d}_{\kappa}^{\mathsf{pow}}(\in^{*})<\mathfrak{d}_{\kappa}^{\mathsf{id}}(\in^{*})$.

This consistency result can be improved considerably.

######  Fact 2.13 ( [vdV23])

(Consistently) there exists a family $H\subseteq{}^{\kappa}\kappa$ with
$|H|=\kappa^{+}$ such that for any enumeration $\left\langle
h_{\xi}\mid\xi\in\kappa^{+}\right\rangle$ of $H$ there exists a forcing
extension in which
$\mathfrak{d}_{\kappa}^{h_{\xi}}(\in^{*})<\mathfrak{d}_{\kappa}^{h_{\xi^{\prime}}}(\in^{*})$
for all $\xi<\xi^{\prime}$.

A dual result to the above would be the consistency of
$\mathfrak{b}_{\kappa}^{\mathsf{id}}(\in^{*})<\mathfrak{b}_{\kappa}^{\mathsf{pow}}(\in^{*})$,
but the forcing techniques used to give the above-mentioned consistency
results cannot be dualised, and other forcing methods are hard to use due to a
lack of preservation theorems in the context of ${}^{\kappa}\kappa$.

Meanwhile, the parameter $h$ does not influence the cardinality of
$\mathfrak{d}_{\kappa}^{h}(\mathrel{\cancel{\ni^{\infty}}})$ or
$\mathfrak{b}_{\kappa}^{h}(\mathrel{\cancel{\ni^{\infty}}})$, since we have
the following result, which we will prove with Corollary 4.4.

###### Fact 2.14

$\mathfrak{d}_{\kappa}^{h}(\mathrel{\cancel{\ni^{\infty}}})=\mathrm{cov}(\mathcal{M}_{\kappa})$
and
$\mathfrak{b}_{\kappa}^{h}(\mathrel{\cancel{\ni^{\infty}}})=\mathrm{non}(\mathcal{M}_{\kappa})$
for any choice of $h\in{}^{\kappa}\kappa$.

We will see that the parameter $b$ does make a difference, and that there can
be consistently different cardinalities of the form
$\mathfrak{d}_{\kappa}^{b,h}(\mathrel{\cancel{\ni^{\infty}}})$ for different
values of $b,h\in{}^{\kappa}\kappa$.

## 3\. Trivial Choices of Parameters

It is perhaps not very surprising that some choices of parameters $b$ and $h$
will result in the cardinal characteristics having trivial values. What we
mean precisely with a cardinal characteristic $\mathfrak{x}$ having a
‘trivial’ value, is that $\mathfrak{x}$ is undefined, or that it is provable
in $\text{``}\,\mathsf{ZFC}+\kappa\text{ is inaccessible}\,\text{''}$ that
$\mathfrak{x}\leq\kappa^{+}$ or $\mathfrak{x}=2^{\kappa}$.

It is perhaps not evident that the determination of $b$ and $h$ such that our
cardinals are nontrivial, is itself not a trivial task. Indeed, this section
contains several nontrivial open questions regarding the triviality of
cardinal characteristics.

We will establish a general pattern that it is possible to give a complete
characterisation of the cases in which the cardinals
$\mathfrak{b}_{\kappa}^{b}(\leq^{*})$, $\mathfrak{b}_{\kappa}^{b,h}(\in^{*})$,
$\mathfrak{b}_{\kappa}^{b,h}(\mathrel{\cancel{\ni^{\infty}}})$ and
$\mathfrak{b}_{\kappa}^{b}(\mathrel{\cancel{=^{\infty}}})$ are trivial. For
each of these families of cardinals we are able to formulate a trichotomy of
the case where the cardinal is ${<}\kappa$, the case where it is exactly
$\kappa$ and the case where the cardinal is ${>}\kappa$. In the last case we
can show that the cardinal is nontrivial and we give an independence proof.

For the cardinals $\mathfrak{d}_{\kappa}^{b}(\leq^{*})$,
$\mathfrak{d}_{\kappa}^{b,h}(\in^{*})$,
$\mathfrak{d}_{\kappa}^{b,h}(\mathrel{\cancel{\ni^{\infty}}})$ and
$\mathfrak{d}_{\kappa}^{b}(\mathrel{\cancel{=^{\infty}}})$, the natural
conjecture is that these are trivial exactly when their $\mathfrak{b}$-duals
are trivial, but this turns out to be hard to prove in each case. We will give
partial results and some related problems. On the other hand, we can show that
these $\mathfrak{d}$-side cardinals are nontrivial whenever the
$\mathfrak{b}$-side cardinals are nontrivial.

### 3.1. Domination & Unboundedness

Let us start with $\mathfrak{d}_{\kappa}^{b}(\leq^{*})$ and
$\mathfrak{b}_{\kappa}^{b}(\leq^{*})$. It is clear that value of these
cardinals only depends on the cofinality of $b(\alpha)$ by the following
lemma.

###### Lemma 3.1

$\mathscr{D}_{b}\equiv\mathscr{D}_{\mathrm{cf}(b)}$.

  * Proof.

For each $\alpha\in\kappa$ let
$\langle\beta^{\alpha}_{\xi}\mid\xi<\mathrm{cf}(b(\alpha))\rangle$ be a
strictly increasing sequence of ordinals that is cofinal in $b(\alpha)$ and
for any $\eta\in b(\alpha)$ let
$\xi^{\alpha}_{\eta}=\min\\{\xi\in\mathrm{cf}(b(\alpha))\mid\eta\leq\beta^{\alpha}_{\xi}\\}$

For $\mathscr{D}_{b}\preceq\mathscr{D}_{\mathrm{cf}(b)}$ let
$\rho_{-}(f):\alpha\mapsto\xi^{\alpha}_{f(\alpha)}$ and
$\rho_{+}(g^{\prime}):\alpha\mapsto\beta^{\alpha}_{g^{\prime}(\alpha)}$.

For $\mathscr{D}_{\mathrm{cf}(b)}\preceq\mathscr{D}_{b}$ let
$\rho_{-}(f^{\prime}):\alpha\mapsto\beta^{\alpha}_{f^{\prime}(\alpha)}$ and
$\rho_{+}(g):\alpha\mapsto\xi^{\alpha}_{g(\alpha)}$. ∎

It also becomes apparent quite immediately why we require
$b\in{}^{\kappa}\kappa$ to be such that $b(\alpha)$ is an infinite cardinal
for (almost) all $\alpha\in\kappa$:

###### Lemma 3.2

If $b(\alpha)$ is a successor ordinal for almost all $\alpha$, then
$\mathfrak{d}_{\kappa}^{b}(\leq^{*})=1$ and
$\mathfrak{b}_{\kappa}^{b}(\leq^{*})$ is undefined.

  * Proof.

Let $b(\alpha)=f(\alpha)+1$ for almost all $\alpha\in\kappa$, then clearly $f$
dominates all functions in $\prod b$, hence
$\mathfrak{d}_{\kappa}^{b}(\leq^{*})=1$ and
$\mathfrak{b}_{\kappa}^{b}(\leq^{*})$ is undefined. ∎

The following lemma gives a complete characterisation of the functions $b$ for
which $\mathfrak{b}_{\kappa}^{b}(\leq^{*})$ is trivial. Note that the cases
(i), (ii) and (iii) form a trichotomy.

###### Lemma 3.3

For each regular cardinal $\lambda<\kappa$ let
$D_{\lambda}=\left\\{\alpha\in\kappa\mid\mathrm{cf}(b(\alpha))=\lambda\right\\}$.

  1. (i)

If there exists a least regular cardinal $\lambda<\kappa$ such that
$D_{\lambda}$ is cofinal in $\kappa$, then
$\mathfrak{b}_{\kappa}^{b}(\leq^{*})=\lambda$.

  2. (ii)

If $D_{\lambda}$ is bounded for all regular $\lambda<\kappa$ and there exists
a stationary set $S$ such that for each $\xi\in S$ there exists
$\alpha_{\xi}\geq\xi$ with $\mathrm{cf}(b(\alpha_{\xi}))\leq\xi$, then
$\mathfrak{b}_{\kappa}^{b}(\leq^{*})=\kappa$.

  3. (iii)

If $D_{\lambda}$ is bounded for all regular $\lambda<\kappa$ and there exists
a club set $C$ such that for each $\xi\in C$ we have
$\mathrm{cf}(b(\alpha))>\xi$ for all $\alpha\geq\xi$, then
$\mathfrak{b}_{\kappa}^{b}(\leq^{*})\geq\kappa^{+}$.

  * Proof.

(i) For each $\gamma\in D_{\lambda}$ let
$\left\langle\delta_{\gamma}^{\alpha}\mid\alpha\in\lambda\right\rangle$ be an
increasing cofinal sequence in $\gamma$. Let $f_{\alpha}\in\prod b$ be any
function such that $f_{\alpha}(\gamma)=\delta_{\gamma}^{\alpha}$ for each
$\gamma\in D_{\lambda}$, then we claim that
$B=\left\\{f_{\alpha}\mid\alpha\in\lambda\right\\}$ is unbounded. Let
$g\in\prod b$. By the pigeonhole principle there exists $\alpha\in\lambda$
such that $g(\gamma)<\delta_{\gamma}^{\alpha}$ for cofinally many $\gamma\in
D_{\lambda}$, hence we see that $f_{\alpha}\mathrel{\cancel{\leq^{*}}}g$.

On the other hand, if $|B|<\lambda$, then let $\alpha_{0}$ be large enough
such that $\mathrm{cf}(b(\alpha))\geq\lambda$ for all $\alpha\geq\alpha_{0}$,
then $|\left\\{f(\alpha)\mid f\in B\right\\}|<\mathrm{cf}(b(\alpha))$ for all
$\alpha\geq\alpha_{0}$, thus we can pick $g\in\prod b$ such that
$g(\alpha)=\sup\left\\{f(\alpha)\mid f\in B\right\\}<b(\alpha)$ for each
$\alpha\geq\alpha_{0}$ to see that $f\leq^{*}g$ for all $f\in B$.

(ii) Since each $D_{\lambda}$ is bounded, we may assume that
$\alpha_{\xi}\neq\alpha_{\xi^{\prime}}$ for each $\xi\in S$. For each $\xi\in
S$, let $\\{\beta_{\xi}^{\eta}\mid\eta\in\xi\\}$ be a cofinal subset of
$b(\alpha_{\xi})$ (not necessarily increasing). Given $\alpha\in\kappa$ we
define $f_{\alpha}$ such that $f_{\alpha}(\alpha_{\xi})=\beta_{\xi}^{\alpha}$
for all $\xi\in S$ with $\alpha<\xi$ and arbitrary otherwise. We claim that
$B=\left\\{f_{\alpha}\mid\alpha\in\kappa\right\\}$ is unbounded.

Let $g\in\prod b$, then for each $\xi\in S$ there is a minimal
$\eta_{\xi}<\xi$ such that
$g(\alpha_{\xi})<\beta_{\xi}^{\eta_{\xi}}=f_{\eta_{\xi}}(\alpha_{\xi})$, hence
$g^{\prime}:\xi\mapsto\eta_{\xi}$ is a regressive function on $S$. Therefore
there is stationary $S^{\prime}\subseteq S$ such that $g^{\prime}$ is constant
on $S^{\prime}$, say with value $\eta$, then we see that
$f_{\eta}\mathrel{\cancel{\leq^{*}}}g$.

On the other hand, suppose that $B\subseteq\prod B$ with $\mu=|B|<\kappa$ and
let $\lambda>\mu$ be regular, then $D_{\lambda}$ is bounded, thus there exists
$\alpha_{0}$ such that $\mathrm{cf}(b(\alpha))>\mu$ for all
$\alpha\geq\alpha_{0}$. Let $g\in\prod b$ be such that
$g(\alpha)=\sup\left\\{f(\alpha)\mid f\in B\right\\}$ for each
$\alpha\geq\alpha_{0}$ to see that $f\leq^{*}g$ for all $f\in B$.

(iii) Let $B\subseteq\prod b$ with $|B|=\kappa$ and enumerate $B$ as
$\left\\{f_{\eta}\mid\eta\in\kappa\right\\}$. Let
$\left\langle\alpha_{\xi}\mid\xi\in\kappa\right\rangle$ be an increasing
enumeration of $C$. We define $g(\alpha)=\bigcup_{\eta\in\xi}f_{\eta}(\alpha)$
for all $\alpha\in[\alpha_{\xi},\alpha_{\xi+1})$ and $\xi\in\kappa$, then
$g(\alpha)$ is the supremum of a sequence of length
$\xi\leq\alpha_{\xi}<\mathrm{cf}(b(\alpha))$, hence $g\in\prod b$. Clearly
$f_{\xi}\leq^{*}g$ for each $\xi\in\kappa$, so $B$ is bounded. ∎

Hence, we see that $\mathfrak{b}_{\kappa}^{b}(\leq^{*})$ is trivial in cases
(i) and (ii). We will later prove that
$\kappa^{+}<\mathfrak{b}_{\kappa}^{b}(\leq^{*})$ is consistent in case (iii),
but let us take a look at the dual $\mathfrak{d}_{\kappa}^{b}(\leq^{*})$
first.

If $\mathfrak{d}_{\kappa}^{b}(\leq^{*})$ behaves dually to
$\mathfrak{b}_{\kappa}^{b}(\leq^{*})$, then we expect
$\mathfrak{d}_{\kappa}^{b}(\leq^{*})<2^{\kappa}$ to be inconsistent in cases
(i) and (ii). Remember that we assume that $b$ is increasing, hence by Lemma
3.1, we can reduce case (i) to the situation where $b$ is a constant function
with a regular cardinal as value. In other words, we have to study the
dominating number as defined in the space ${}^{\kappa}\lambda$ where
$\lambda<\kappa$ is a regular cardinal.

Note that even if we drop the assumption that $b$ is increasing, the following
Tukey connection shows that the behaviour on the space ${}^{\kappa}\lambda$ is
essentially the relevant part of the (in)consistency of
$\mathfrak{d}_{\kappa}^{b}(\leq^{*})<2^{\kappa}$.

###### Lemma 3.4

If $\left\langle\alpha_{\xi}\mid\xi\in\kappa\right\rangle\in{}^{\kappa}\kappa$
is a strictly increasing sequence and $b^{\prime}:\xi\mapsto b(\alpha_{\xi})$,
then $\mathscr{D}_{b^{\prime}}\preceq\mathscr{D}_{b}$.

  * Proof.

Easy. ∎

Dominating numbers in the space ${}^{\kappa}\lambda$ have been studied by many
in the past. Brendle showed in the last section of [Bre22] (using different
notation where the roles of $\kappa$ and $\lambda$ are reversed) that if
$\lambda<\kappa$ and $\lambda$ is regular uncountable, then
$\mathfrak{d}_{\kappa}^{\overline{\lambda}}(\leq^{*})<2^{\kappa}$ is actually
consistent. An example for a model where this holds, is the model resulting
from adding $\kappa^{++}$ many $\mu$-Cohen reals over $\mathsf{GCH}$, where
$\mu<\lambda$. Because this also destroys the inaccessibility of $\kappa$, we
cannot use this in our context.

The question whether
$\mathfrak{d}_{\kappa}^{\overline{\lambda}}(\leq^{*})<2^{\kappa}$ is
consistent with $\kappa\geq 2^{<\lambda}$ is mentioned as Question 16 in
[Bre22]. Moreover, the special case where $\kappa=\omega_{1}$ and
$\lambda=\omega$ is a famous open problem of Jech and Prikry [JP79] that is
still unsolved almost half a century later. We will give a partial answer and
prove that $\mathfrak{d}_{\kappa}^{\overline{\lambda}}(\leq^{*})=2^{\kappa}$
when $\kappa$ is inaccessible.

###### Theorem 3.5

If $\lambda<\kappa$ is regular and $\mathrm{cf}(b(\alpha))=\lambda$ for
cofinally many $\alpha\in\kappa$, then
$\mathfrak{d}_{\kappa}^{b}(\leq^{*})=2^{\kappa}$.

We will delay the essential part of the proof of this theorem to the next
subsection, since it will be a corollary of Lemma 3.18, which states that
$\mathfrak{d}_{\kappa}^{b,h}(\in^{*})=2^{\kappa}$ if $h(\alpha)=\lambda$ for
cofinally many $\alpha\in\kappa$. This is related to
$\mathfrak{d}_{\kappa}^{b}(\leq^{*})$ through the following lemma.

###### Lemma 3.6

If $b=^{*}h$, then $\mathscr{L}_{b,h}\preceq\mathscr{D}_{b}$.

  * Proof.

We define $\rho_{-}:\prod b\to\prod b$ and $\rho_{+}:\prod
b\to\mathrm{Loc}_{\kappa}^{b,h}$ as a Tukey connection for
$\mathscr{L}_{b,h}\preceq\mathscr{D}_{b}$.

We will have $\rho_{+}$ sending $g\in\prod b$ to
$\varphi\in\mathrm{Loc}_{\kappa}^{b,h}$ where $\varphi(\alpha)=g(\alpha)$
whenever $h(\alpha)=b(\alpha)$ and arbitrary otherwise. Since $h=^{*}b$ and
$g(\alpha)<b(\alpha)$ for all $\alpha\in\kappa$, this is well-defined. Let
$\rho_{-}:f\mapsto f+1$.

If $f,g\in\prod b$, let $\varphi=\rho_{+}(g)$. Then $f+1\leq^{*}g$ implies
$f<^{*}g$, hence $f\in^{*}\varphi$. ∎

  * Proof of Theorem 3.5.

Let $\lambda<\kappa$ be regular and $\mathrm{cf}(b(\alpha))=\lambda$ for
cofinally many $\alpha\in\kappa$. By Lemma 3.1 and Lemma 3.4 we may assume
that $b(\alpha)=\lambda$ for all $\alpha\in\kappa$. Let $h=b$, then the
conditions of Lemmas 3.18 and 3.6 are satisfied, thus
$2^{\kappa}=\mathfrak{d}_{\kappa}^{b,h}(\in^{*})\leq\mathfrak{d}_{\kappa}^{b}(\leq^{*})\leq
2^{\kappa}$. ∎

We currently do not know whether $\mathfrak{d}_{\kappa}^{b}(\leq^{*})$ also is
trivial in case (ii) of Lemma 3.3.

###### Question 3.7

Let
$D_{\lambda}=\left\\{\alpha\in\kappa\mid\mathrm{cf}(b(\alpha))=\lambda\right\\}$
be bounded for all regular $\lambda<\kappa$ and $S$ be stationary such that
for each $\xi\in S$ there exists $\alpha_{\xi}\geq\xi$ with
$\mathrm{cf}(b(\alpha_{\xi}))\leq\xi$. Is
$\mathfrak{d}_{\kappa}^{b}(\leq^{*})<2^{\kappa}$ consistent?

To conclude this subsection, we will give an independence result to show that
both $\mathfrak{b}_{\kappa}^{b}(\leq^{*})$ and
$\mathfrak{d}_{\kappa}^{b}(\leq^{*})$ are nontrivial in case (iii). We prove
this with a generalisation of Hechler forcing.

###### Definition 3.8

Let $C\subseteq\kappa$. We define $(b,C)$-Hechler forcing
$\mathbb{D}_{\kappa}^{b}(C)$ as the forcing notion with conditions $p=(s,f)$
where $s\in\prod_{<\kappa}b$ is such that $\mathrm{dom}(s)\in C$ and
$f:\kappa\setminus\mathrm{dom}(s)\to\kappa$ is such that $f(\alpha)\in
b(\alpha)$ for all $\alpha\in\mathrm{dom}(f)$, where the ordering is defined
as $(t,g)\leq(s,f)$ iff $s\subseteq t$ and $f(\alpha)\leq g(\alpha)$ for all
$\alpha\in\mathrm{dom}(g)$ and $f(\alpha)\leq t(\alpha)$ for all
$\alpha\in\mathrm{dom}(t)\setminus\mathrm{dom}(s)$.

It is clear from the definition, under the assumptions of case (iii), that
$\mathbb{D}_{\kappa}^{b}(C)$ adds a generic element of $\prod b$ that
dominates all functions in $\prod b$ from the ground model. We will show that
$\mathbb{D}_{\kappa}^{b}(C)$ preserves cardinals and cofinalities. Since
$\mathbb{D}_{\kappa}^{b}(C)$ is not ${<}\kappa$-closed in general, we need a
slightly weaker variant of closure.

###### Definition 3.9

A forcing notion $\mathbb{P}$ is strategically ${<}\kappa$-closed if White has
a winning strategy for $\mathcal{G}(\mathbb{P},p)$ for all $p\in\mathbb{P}$.
Here $\mathcal{G}(\mathbb{P},p)$ is the game of length $\kappa$ as follows: at
stage $\alpha\in\kappa$, White chooses a condition $p_{\alpha}$ such that
$p_{\alpha}\leq p^{\prime}_{\xi}$ for all $\xi<\alpha$ (and $p_{0}\leq p$) and
Black subsequently chooses a condition $p^{\prime}_{\alpha}\leq p_{\alpha}$.
White wins $\mathcal{G}(\mathbb{P},p)$ if White can make a move at every stage
$\alpha\in\kappa$ of the game.

###### Lemma 3.10

Let $C$ be a club set such that for each $\xi\in C$ we have
$\mathrm{cf}(b(\alpha))>\xi$ for all $\alpha\geq\xi$, then
$\mathbb{D}_{\kappa}^{b}(C)$ is strategically ${<}\kappa$-closed and
${<}\kappa^{+}$-c.c.

  * Proof.

For strategic ${<}\kappa$-closure, at stage $\alpha$ of the game
$\mathcal{G}(\mathbb{D}_{\kappa}^{b}(C),p)$, let $\langle
p_{\xi}\mid\xi<\alpha\rangle$ and $\langle
p^{\prime}_{\xi}\mid\xi<\alpha\rangle$ be the sequences of previous moves by
White and Black respectively. The winning strategy for White will be to choose
$(s_{\alpha},f_{\alpha})$ such that $\mathrm{dom}(s_{\alpha})\geq\alpha$ in
successor stages. Under this strategy, if $\alpha$ is a limit ordinal, then we
have $\xi\leq\mathrm{dom}(s_{\xi})\in C$ for each $\xi\in\alpha$. It follows
that $\alpha\leq\mathrm{dom}(s_{\alpha})$ and hence
$\mathrm{cf}(b(\beta))>\alpha$ for all $\beta\geq\mathrm{dom}(s_{\alpha})$ by
the properties of $C$. Therefore $\left\langle
f_{\xi}(\beta)\mid\xi\in\alpha\right\rangle$ is not cofinal in $b(\beta)$,
thus we can define
$f_{\alpha}:\beta\mapsto\bigcup_{\xi\in\alpha}f_{\xi}(\beta)$ for each
$\beta\in\kappa\setminus\mathrm{dom}(s_{\alpha})$.

For ${<}\kappa^{+}$-c.c., note that for any
$(s,f),(s,g)\in\mathbb{D}_{\kappa}^{b}(C)$ we can choose
$h(\alpha)=\max\left\\{f(\alpha),g(\alpha)\right\\}$ for all
$\alpha\in\kappa\setminus\mathrm{dom}(s)$ to see that have $(s,h)\leq(s,f)$
and $(s,h)\leq(s,g)$. Thus, if
$\mathcal{A}\subseteq\mathbb{D}_{\kappa}^{b}(C)$ is an antichain and
$(s,f),(t,g)\in\mathcal{A}$ are distinct, then we must have $s\neq t$, hence
$|\mathcal{A}|\leq\kappa$. ∎

###### Corollary 3.11

$\mathbb{D}^{b}_{\kappa}(C)$ preserves cardinals and cofinality and does not
add any elements to ${}^{<\kappa}\kappa$.

###### Fact 3.12

Iterations of arbitrary length with ${<}\kappa$-support preserve strategic
${<}\kappa$-closure and ${<}\kappa^{+}$-c.c., and hence preserve cardinals,
cofinality and do not add any elements to ${}^{<\kappa}\kappa$.

This implies especially that iteration will not destroy the inaccessibility of
$\kappa$.

###### Theorem 3.13

Let $C$ be a club set such that for each $\xi\in C$ we have
$\mathrm{cf}(b(\alpha))>\xi$ for all $\alpha\geq\xi$, then it is consistent
that $\mathfrak{b}_{\kappa}^{b}(\leq^{*})>\kappa^{+}$ and that
$\mathfrak{d}_{\kappa}^{b}(\leq^{*})<2^{\kappa}$.

  * Proof.

Let
$\mathbb{P}=\langle\mathbb{P}_{\alpha},\dot{\mathbb{Q}}_{\alpha}\mid\alpha\in\kappa^{++}+\kappa^{+}\rangle$
be a ${<}\kappa$-support iteration with
$\mathbb{P}_{\alpha}\Vdash\text{``}\,\dot{\mathbb{Q}}_{\alpha}=\dot{\mathbb{D}}_{\kappa}^{b}(C)\,\text{''}$
for each $\alpha$ and let
$\mathbf{V}\vDash\text{``}\,\mathsf{GCH}\,\text{''}$. If $B\subseteq(\prod
b)^{\mathbf{V}^{\mathbb{P}_{\kappa^{++}}}}$ is such that $|B|\leq\kappa^{+}$,
then $B\in\mathbf{V}^{\mathbb{P}_{\alpha}}$ for some $\alpha<\kappa^{++}$,
since the $\kappa^{++}$-th stage of the iteration does not add any
$\kappa$-reals. The generic $\mathbb{D}_{\kappa}^{b}(C)$-real that is added in
the $\alpha+1$-th stage then dominates all elements of $B$, hence $B$ is not
unbounded. This shows that
$\mathbf{V}^{\mathbb{P}_{\kappa^{++}}}\vDash\text{``}\,\mathfrak{b}_{\kappa}^{b}(\leq^{*})>\kappa^{+}\,\text{''}$.

Next, let $D$ consist of the $\kappa$-reals that are added in the final
$\kappa^{+}$ stages of the iteration, then $D$ clearly forms a dominating
family of size $\kappa^{+}$ in $\mathbf{V}^{\mathbb{P}}$, and it is easy to
see that
$\mathbf{V}^{\mathbb{P}}\vDash\text{``}\,2^{\kappa}=\kappa^{++}\,\text{''}$,
showing that
$\mathbf{V}^{\mathbb{P}}\vDash\text{``}\,\mathfrak{d}_{\kappa}^{b}(\leq^{*})<2^{\kappa}\,\text{''}$.
∎

### 3.2. Localisation & Avoidance

With localisation and avoidance cardinals, we have not only the parameter $b$,
but also the parameter $h$ giving the width of our slaloms. As with the
dominating and unbounding numbers, there are certain choices of these
parameters for which we have trivial cardinal characteristics.

We will motivate our assumption that $h\leq b$ with the following two lemmas:

###### Lemma 3.14

$b<^{*}h$ if and only if $\mathfrak{d}_{\kappa}^{b,h}(\in^{*})=1$.
If $b<^{*}h$, then $\mathfrak{b}_{\kappa}^{b,h}(\in^{*})$ is undefined.

  * Proof.

If $b<^{*}h$, let $B=\left\\{\alpha\in\kappa\mid
b(\alpha)<h(\alpha)\right\\}$, and choose some
$\varphi\in\mathrm{Loc}_{\kappa}^{b,h}$ such that $\varphi(\alpha)=b(\alpha)$
for all $\alpha\in B$. Since almost all $\alpha\in\kappa$ are in $B$, we see
that $f\in\prod b$ implies $f\in^{*}\varphi$. Hence $\varphi$ localises the
entirety of $\prod b$, making $\mathfrak{d}_{\kappa}^{b,h}(\in^{*})=1$ and
$\mathfrak{b}_{\kappa}^{b,h}(\in^{*})$ undefined.

Reversely, if there is a strictly increasing sequence
$\left\langle\alpha_{\xi}\mid\xi\in\kappa\right\rangle$ with
$h(\alpha_{\xi})\leq b(\alpha_{\xi})$ for all $\xi$, and
$\varphi\in\mathrm{Loc}_{\kappa}^{b,h}$, then there is $\gamma_{\xi}\in
b(\alpha_{\xi})\setminus\varphi(\alpha_{\xi})$ for each $\xi$ since
$|\varphi(\alpha_{\xi})|<h(\alpha_{\xi})\leq b(\alpha_{\xi})$. Therefore, if
$f\in\prod b$ is such that $f(\alpha_{\xi})=\gamma_{\xi}$ for all
$\xi\in\kappa$, then $f\mathrel{\cancel{\in^{*}}}\varphi$. ∎

We will state two essential properties relating two relational systems of the
form $\mathscr{L}_{b,h}$ with different parameters to each other. The first
deals with cofinal subsets of indices (cf. Lemma 3.4) and the second shows
monotonicity with regards to $\leq^{*}$. The proofs are elementary.

###### Lemma 3.15

If $\left\langle\alpha_{\xi}\mid\xi\in\kappa\right\rangle\in{}^{\kappa}\kappa$
is a strictly increasing sequence and $b^{\prime}:\xi\mapsto b(\alpha_{\xi})$
and $h^{\prime}:\xi\mapsto h(\alpha_{\xi})$, then
$\mathscr{L}_{b^{\prime},h^{\prime}}\preceq\mathscr{L}_{b,h}$.

###### Lemma 3.16

Let $h\leq^{*}h^{\prime}$ and $b\geq^{*}b^{\prime}$, then there exist Tukey
connections ${\mathscr{L}_{b^{\prime},h^{\prime}}}\preceq{\mathscr{L}_{b,h}}$.

Mirroring the situation of Lemma 3.3, we will give a complete characterisation
of the cases in which $\mathfrak{b}_{\kappa}^{b,h}(\in^{*})$ is trivial.

###### Lemma 3.17

For each $\lambda<\kappa$ define $D_{\lambda}=\left\\{\alpha\in\kappa\mid
h(\alpha)=\lambda\right\\}$.

  1. (i)

If there exists a least cardinal $\lambda<\kappa$ such that $D_{\lambda}$ is
cofinal in $\kappa$, then $\mathfrak{b}_{\kappa}^{b,h}(\in^{*})=\lambda$.

  2. (ii)

If $D_{\lambda}$ is bounded for all $\lambda<\kappa$ and $h$ is continuous on
a stationary set, then $\mathfrak{b}_{\kappa}^{b,h}(\in^{*})=\kappa$,

  3. (iii)

If $D_{\lambda}$ is bounded for all $\lambda<\kappa$ and $h$ is discontinuous
on a club set, then $\kappa^{+}\leq\mathfrak{b}_{\kappa}^{b,h}(\in^{*})$.

  * Proof.

(i) Assume that $\lambda$ is minimal such that $D_{\lambda}$ is cofinal. For
each $\eta<\lambda$, define $f_{\eta}\in\prod b$ elementwise by
$f_{\eta}:\alpha\mapsto\eta$ for all $\alpha\in D_{\lambda}$ and
$f_{\eta}:\alpha\mapsto 0$ otherwise. If
$\varphi\in\mathrm{Loc}_{\kappa}^{b,h}$ and $\alpha\in D_{\lambda}$, then
$|\varphi(\alpha)|<h(\alpha)=\lambda$, so there exists $\eta<\lambda$ for
which $\eta\notin\varphi(\alpha)$, hence
$f_{\eta}\mathrel{\cancel{\in^{*}}}\varphi$. Therefore
$\mathcal{F}=\left\\{f_{\eta}\mid\eta<\lambda\right\\}$ witnesses that
$\mathfrak{b}_{\kappa}^{b,h}(\in^{*})\leq|\mathcal{F}|=\lambda$.

On the other hand, if $\mathcal{F}\subseteq\prod b$ and
$|\mathcal{F}|<\lambda$, then by minimality of $\lambda$ we see that
$|\mathcal{F}|<h(\alpha)$ for almost all $\alpha\in\kappa$, thus we can choose
$\varphi\in\mathrm{Loc}_{\kappa}^{b,h}$ such that
$\varphi:\alpha\mapsto\left\\{f(\alpha)\mid f\in\mathcal{F}\right\\}$ whenever
$|\mathcal{F}|<h(\alpha)$, then we see that $f\in^{*}\varphi$ for all
$f\in\mathcal{F}$, proving that
$|\mathcal{F}|<\mathfrak{b}_{\kappa}^{b,h}(\in^{*})$.

(ii) Since each $D_{\lambda}$ is bounded, it follows that
$\left\\{\alpha\in\kappa\mid h(\alpha)\leq\lambda\right\\}$ is bounded for
every $\lambda<\kappa$. Let $\mathcal{F}\subseteq\prod b$ with
$|\mathcal{F}|<\kappa$, and let $\beta\in\kappa$ be such that
$h(\alpha)\leq|\mathcal{F}|$ implies $\alpha<\beta$. If
$\varphi\in\mathrm{Loc}_{\kappa}^{b,h}$ and $\xi\geq\beta$, then
$|\mathcal{F}|<h(\xi)$, hence we can define $\varphi(\xi)=\left\\{f(\xi)\mid
f\in\mathcal{F}\right\\}$ for all $\xi\geq\beta$. It is clear that
$f\in^{*}\varphi$ for all $f\in\mathcal{F}$, thus
$|\mathcal{F}|<\mathfrak{b}_{\kappa}^{b,h}(\in^{*})$. Since $\mathcal{F}$ was
arbitrary such that $|\mathcal{F}|<\kappa$, we see that
$\kappa\leq\mathfrak{b}_{\kappa}^{b,h}(\in^{*})$.

To see $\kappa=\mathfrak{b}_{\kappa}^{b,h}(\in^{*})$, consider
$\mathcal{F}=\left\\{f_{\eta}\mid\eta\in\kappa\right\\}$ where
$f_{\eta}:\alpha\mapsto\eta$ when $\eta\in b(\alpha)$, and
$f_{\eta}:\alpha\mapsto 0$ otherwise. Since we assume $h\leq^{*}b$ and $h$ is
cofinal, for every $\eta\in\kappa$ there is $\alpha_{0}$ such that
$b(\alpha)>\eta$ for all $\alpha\geq\alpha_{0}$, thus $f_{\eta}(\alpha)=\eta$
for almost all $\alpha\in\kappa$. Since $h$ is continuous on a stationary set
$S_{0}$, the set of fixed-points $S_{1}=\left\\{\alpha\in S_{0}\mid
h(\alpha)=\alpha\right\\}$ is stationary, implying that
$\alpha\mapsto|\varphi(\alpha)|$ is regressive on the stationary $S_{1}$.
Consequently Fodor’s lemma tells us that there exists a stationary set
$S_{2}\subseteq S_{1}$ and $\lambda\in\kappa$ such that
$|\varphi(\alpha)|<\lambda$ for all $\alpha\in S_{2}$. If
$f_{\eta}\in^{*}\varphi$ for all $\eta\in\lambda$, then
$N_{\eta}=\left\\{\alpha\in\kappa\mid\eta\notin\varphi(\alpha)\right\\}$ is
nonstationary for each $\eta\in\lambda$, so $\bigcup_{\eta\in\lambda}N_{\eta}$
is nonstationary, and thus $S_{2}\setminus\bigcup_{\eta\in\lambda}N_{\eta}$ is
stationary. However, for any $\alpha\in
S_{2}\setminus\bigcup_{\eta\in\lambda}N_{\eta}$ we have
$\lambda\subseteq\varphi(\alpha)$, which contradicts that
$|\varphi(\alpha)|<\lambda$ for $\alpha\in S_{2}$. Therefore $\mathcal{F}$
forms a witness for $\mathfrak{b}_{\kappa}^{b,h}(\in^{*})\leq\kappa$.

(iii) Let $\mathcal{F}\subseteq\prod b$ with $|\mathcal{F}|=\kappa$ and
enumerate $\mathcal{F}$ as $\left\langle
f_{\eta}\mid\eta\in\kappa\right\rangle$. Let $C$ be a club set containing no
successor ordinals such that $h$ is discontinuous on $C$, and let
$\left\langle\alpha_{\xi}\mid\xi\in\kappa\right\rangle$ be the increasing
enumeration of $C\cup\left\\{0\right\\}$. For each $\xi\in\kappa$ let
$\lambda_{\xi}=\bigcup_{\alpha\in\alpha_{\xi}}h(\alpha)$, where we have the
convention that $\bigcup\varnothing=0$, then $\lambda_{\xi}<h(\alpha_{\xi})$
for all $\xi\in\kappa$ by discontinuity. Given $\alpha\in\kappa$, let $\xi$ be
such that $\alpha\in[\alpha_{\xi},\alpha_{\xi+1})$, which exists by $C$ being
club, and let
$\varphi(\alpha)=\left\\{f_{\eta}(\alpha)\mid\eta\in\lambda_{\xi}\right\\}$.
Since $h$ is increasing,
$|\varphi(\alpha)|\leq\lambda_{\xi}<h(\alpha_{\xi})\leq h(\alpha)$, thus
$\varphi\in\mathrm{Loc}_{\kappa}^{b,h}$. Finally $h$ is cofinal, thus
$\left\langle\lambda_{\xi}\mid\xi\in\kappa\right\rangle$ is cofinal, hence for
every $\eta\in\kappa$ we have $f_{\eta}\in^{*}\varphi$, showing that
$\kappa<\mathfrak{b}_{\kappa}^{b,h}(\in^{*})$. ∎

We will once again see that $\mathfrak{b}_{\kappa}^{b,h}(\in^{*})$ is
nontrivial in case (iii), but first let us consider
$\mathfrak{d}_{\kappa}^{b,h}(\in^{*})$.

In case (i) we have $\mathfrak{d}_{\kappa}^{b,h}(\in^{*})=2^{\kappa}$. The
proof is a generalisation based on Lemmas 1.8, 1.10 and 1.11 from [GS93],
where the analogous theorem is proved for ${}^{\omega}\omega$.

###### Lemma 3.18

Suppose that there exists some $\lambda<\kappa$ such that
$D_{\lambda}=\left\\{\alpha\in\kappa\mid h(\alpha)=\lambda\right\\}$ is
cofinal in $\kappa$. Then $\mathfrak{d}_{\kappa}^{b,h}(\in^{*})=2^{\kappa}$.

  * Proof.

We are only interested in finding a lower bound of
$\mathfrak{d}_{\kappa}^{b,h}(\in^{*})$, thus by Lemma 3.15 we could restrict
our attention to a cofinal subset of $\kappa$. We therefore assume without
loss of generality that $D_{\lambda}=\kappa$, that is, $h(\alpha)=\lambda$ for
all $\alpha\in\kappa$. To prove this lemma we assume furthermore without loss
of generality that $b=h$. This suffices to prove the lemma, by Lemma 3.16.

Let $b^{\prime}$ be defined by $b^{\prime}(\alpha)=2^{|\alpha|}$ for all
$\alpha\in\kappa$. We start with proving that
$\mathfrak{d}_{\kappa}^{b^{\prime},h}(\in^{*})=2^{\kappa}$.

Let
$\pi_{\alpha}:{}^{\alpha}2\mathrel{\textnormal{\guilsinglright}\mathrel{\mkern-9.0mu}\rightarrow}b^{\prime}(\alpha)$
be an injection for every $\alpha\in\kappa$. For some arbitrary
$g\in{}^{\kappa}2$, define $f_{g}\in\prod b^{\prime}$ to be such that
$f_{g}(\alpha)=\pi_{\alpha}(g\restriction\alpha)$ for all $\alpha\in\kappa$.
If $g,g^{\prime}\in{}^{\kappa}2$ are distinct, then $g\restriction\alpha\neq
g^{\prime}\restriction\alpha$ for almost all $\alpha\in\kappa$, hence
$f_{g}(\alpha)\neq f_{g^{\prime}}(\alpha)$ for almost all $\alpha\in\kappa$.
Therefore, if $\varphi\in\mathrm{Loc}_{\kappa}^{b^{\prime},h}$, then there are
at most $\lambda$ many functions $g\in{}^{\kappa}2$ such that
$f_{g}\in^{*}\varphi$. Since ${}^{\kappa}2$ cannot be the union of less than
$2^{\kappa}$ sets of size $\lambda$, we see that
$\mathfrak{d}_{\kappa}^{b^{\prime},h}(\in^{*})=2^{\kappa}$.

We will construct a Tukey connection
$\mathscr{L}_{b^{\prime},h}\preceq\mathscr{L}_{b,h}$. Let $\left\langle
W_{\alpha}\mid\alpha\in\kappa\right\rangle$ be a partition of $\kappa$ such
that $|W_{\alpha}|=\lambda^{2^{|\alpha|}}$ and let
$\left\langle\Phi_{\xi}^{\alpha}\mid\xi\in W_{\alpha}\right\rangle$ be an
enumeration of all functions $2^{|\alpha|}\to\lambda$. We let $\rho_{-}:\prod
b^{\prime}\to\prod b$ send a function $f^{\prime}$ to the function $f$ defined
as follows: for $\xi\in\kappa$, let $\alpha$ be such that $\xi\in W_{\alpha}$,
then we let $f(\xi)=\Phi_{\xi}^{\alpha}(f^{\prime}(\alpha))$. We let
$\rho_{+}:\mathrm{Loc}_{\kappa}^{b,h}\to\mathrm{Loc}_{\kappa}^{b^{\prime},h}$
send a slalom $\varphi$ to the slalom $\varphi^{\prime}$, where
$\varphi^{\prime}(\alpha)=\left\\{\eta\in b^{\prime}(\alpha)\mid\forall\xi\in
W_{\alpha}(\Phi_{\xi}^{\alpha}(\eta)\in\varphi(\xi))\right\\}$. We show that
$\varphi^{\prime}$ is indeed a $(b^{\prime},h)$-slalom by proving that
$|\varphi^{\prime}(\alpha)|<h(\alpha)=\lambda$.

Assume towards contradiction that there exists a sequence
$\left\langle\eta_{\beta}\mid\beta\in\lambda\right\rangle$ of distinct
elements of $\varphi^{\prime}(\alpha)$. We define a function
$\Phi:b^{\prime}(\alpha)\to\lambda$ by sending $\eta_{\beta}\mapsto\beta$ and
$\eta\mapsto 0$ if $\eta\neq\eta_{\beta}$ for all $\beta\in\lambda$. Note that
$\mathrm{dom}(\Phi)=b^{\prime}(\alpha)=2^{|\alpha|}$, hence there is $\xi\in
W_{\alpha}$ such that $\Phi=\Phi_{\xi}^{\alpha}$. Since
$|\varphi(\xi)|<h(\xi)=\lambda$ there is some
$\beta\in\lambda\setminus\varphi(\xi)$, but then
$\Phi(\eta_{\beta})=\Phi_{\xi}^{\alpha}(\eta_{\beta})\notin\varphi(\xi)$,
which implies the contradictory $\eta_{\beta}\notin\varphi^{\prime}(\alpha)$.

Finally we have to prove that $\rho_{-}$, $\rho_{+}$ form a Tukey connection.
If we assume that $f\in^{*}\varphi$, then
$\Phi_{\xi}^{\alpha}(f^{\prime}(\alpha))\in\varphi(\xi)$ for almost all
$\xi\in\kappa$, where $\alpha$ is such that $\xi\in W_{\alpha}$. This means
that for almost all $\alpha\in\kappa$ we have
$\Phi_{\xi}^{\alpha}(f^{\prime}(\alpha))\in\varphi(\xi)$ for all $\xi\in
W_{\alpha}$. Therefore, for almost all $\alpha\in\kappa$ we have
$f^{\prime}(\alpha)\in\varphi^{\prime}(\alpha)$, showing that
$f^{\prime}\in^{*}\varphi^{\prime}$. ∎

The situation in case (ii) appears to be more complicated, and currently our
best lower bound is given by the maximal size of almost disjoint families of
functions. A family $\mathcal{A}\subseteq\prod b$ is called almost disjoint if
$f=^{\infty}g$ implies that $f=g$ for all $f,g\in\mathcal{A}$.

###### Lemma 3.19

If $D_{\lambda}=\left\\{\alpha\in\kappa\mid h(\alpha)=\lambda\right\\}$ is
bounded for all $\lambda\in\kappa$ and $h$ is increasing and continuous on a
stationary set and $\mathcal{A}\subseteq\prod b$ is an almost disjoint family,
then $|\mathcal{A}|\leq\mathfrak{d}_{\kappa}^{b,h}(\in^{*})$.

  * Proof.

Since we assume $h\leq b$ and $h$ is cofinal, $b$ is also cofinal. Note that
there exists an almost disjoint family $\mathcal{A}\subseteq\prod b$ with
$|\mathcal{A}|=\kappa$: let $f_{\eta}\in\prod b$ send $\alpha\mapsto\eta$ if
$\eta\in b(\alpha)$ and $\alpha\mapsto 0$ otherwise, then
$\mathcal{A}=\left\\{f_{\eta}\mid\eta\in\kappa\right\\}$ suffices. We will
therefore assume without loss of generality that $\mathcal{A}$ is an almost
disjoint family with $|\mathcal{A}|\geq\kappa$.

Given $\varphi\in\mathrm{Loc}_{\kappa}^{b,h}$, let $\lambda_{\varphi}$ be
minimal such that there is stationary $S_{\varphi}$ with
$|\varphi(\alpha)|=\lambda_{\varphi}$ for all $\alpha\in S_{\varphi}$. Fix
some arbitrary $A\subseteq\mathcal{A}$ with $|A|=\lambda_{\varphi}^{+}$,
enumerate $A$ as $\left\langle
f_{\alpha}\mid\alpha\in\lambda_{\varphi}^{+}\right\rangle$ and define
$\xi_{\alpha,\beta}=\min\left\\{\xi\in\kappa\mid\forall\eta\in[\xi,\kappa)(f_{\alpha}(\eta)\neq
f_{\beta}(\eta))\right\\}$. Since $\xi_{\alpha,\beta}<\kappa$ for all distinct
$\alpha,\beta\in\lambda_{\varphi}^{+}$, and $\lambda_{\varphi}^{+}<\kappa$ and
$\kappa$ is inaccessible, we see that
$\xi=\bigcup_{\alpha\in\lambda_{\varphi}^{+}}\bigcup_{\beta\in\alpha}\xi_{\alpha,\beta}\in\kappa$.
If $\eta\geq\xi$, then $f_{\alpha}(\eta)$ is distinct for each
$\alpha\in\lambda_{\varphi}^{+}$, thus
$\left|\left\\{f_{\alpha}(\eta)\mid\alpha\in\lambda_{\varphi}^{+}\right\\}\right|=\lambda_{\varphi}^{+}$.
For every $\eta\in S_{\varphi}\setminus\xi$ there is
$\alpha\in\lambda_{\varphi}^{+}$ such that
$f_{\alpha}(\eta)\notin\varphi(\eta)$, hence by the pigeonhole principle there
exists $\alpha\in\lambda_{\varphi}^{+}$ such that
$f_{\alpha}\mathrel{\cancel{\in^{*}}}\varphi$.

Thus, we see that $\mathcal{A}_{\varphi}=\left\\{f\in\mathcal{A}\mid
f\in^{*}\varphi\right\\}$ has $|\mathcal{A}_{\varphi}|\leq\lambda_{\varphi}$.
If $\Phi\subseteq\mathrm{Loc}_{\kappa}^{b,h}$ is of minimal cardinality to
witnesses $\mathfrak{d}_{\kappa}^{b,h}(\in^{*})$, then
$\bigcup_{\varphi\in\Phi}\mathcal{A}_{\varphi}=\mathcal{A}$, and thus
$\kappa\leq|\mathcal{A}|\leq|\Phi|\cdot\sup_{\varphi\in\Phi}\lambda_{\varphi}=|\Phi|$.
For the last equality, note that
$\sup_{\varphi\in\Phi}\lambda_{\varphi}\leq\kappa$ and that this inequality
would be strict if $|\Phi|<\kappa$. ∎

Note that an almost disjoint family $\mathcal{A}\subseteq\prod b$ with
$|\mathcal{A}|=2^{\kappa}$ exists when there exists a continuous strictly
increasing sequence $\left\langle\alpha_{\xi}\mid\xi\in\kappa\right\rangle$
such that $2^{|\xi|}\leq|b(\alpha_{\xi})|$ for all $\xi\in\kappa$. The
construction for such a family is done by fixing bijections
$\pi_{\xi}:{}^{\xi}2\mathrel{\textnormal{\guilsinglright}\mathrel{\mkern-9.0mu}\rightarrow\mathrel{\mkern-14.0mu}\rightarrow}b(\alpha_{\xi})$
and for any $f\in{}^{\kappa}2$ considering the function $f^{\prime}\in\prod b$
given by $f^{\prime}:\alpha\mapsto\pi_{\xi}(f\restriction\xi)$ for each
$\alpha\in[\alpha_{\xi},\alpha_{\xi+1})$, then $\left\\{f^{\prime}\mid
f\in{}^{\kappa}2\right\\}$ forms an almost disjoint family.

Even if $b$ does not grow fast enough such that the above construction can be
done, it is still possible to add an almost disjoint family of size
$2^{\kappa}$ with forcing. Let us focus on the case where
$b=\mathsf{id}:\alpha\mapsto\alpha$ is the identity function, and look at a
forcing notion $\mathbb{AD}$ that adds a $\lambda$-sized almost disjoint
family of regressive functions, that is, elements of $\prod\mathsf{id}$.

###### Definition 3.20

The forcing notion $\mathbb{AD}$ has the conditions
$p:X_{p}\times\beta_{p}\to\kappa$ such that $X_{p}\in[\lambda]^{<\kappa}$,
$|X_{p}|\leq\beta_{p}\in\kappa$, and $p(\xi,\alpha)<\alpha$ for all $\xi\in
X_{p}$ and $0<\alpha\in\beta_{p}$. Given $\xi\in X_{p}$, we write
$p_{\xi}:\beta_{p}\to\kappa$ for the (regressive) function
$p_{\xi}:\alpha\mapsto p(\xi,\alpha)$. The ordering on $\mathbb{AD}$ is given
by $q\leq p$ iff $p\subseteq q$ (implicitly $X_{p}\subseteq X_{q}$ and
$\beta_{p}\leq\beta_{q}$), and $q_{\xi}(\alpha)\neq q_{\xi^{\prime}}(\alpha)$
for any $\alpha\in\beta_{q}\setminus\beta_{p}$ and distinct
$\xi,\xi^{\prime}\in X_{p}$.

###### Lemma 3.21

The set of all $p\in\mathbb{AD}$ such that $(\xi,\alpha)\in\mathrm{dom}(p)$,
is dense for any $\xi\in\lambda$ and $\alpha\in\kappa$.

  * Proof.

Let $p\in\mathbb{AD}$ and $\beta_{p}\leq\alpha\in\kappa$, then we will first
find $q\leq p$ with $X_{q}=X_{p}$ and $\beta_{q}=\alpha+1$. Fix an enumeration
$\left\langle\xi_{\eta}\mid\eta<|X_{p}|\right\rangle$ of $X_{p}$, then for any
$\gamma\in[\beta_{p},\alpha]$ note that $|X_{p}|\leq\beta_{p}\leq\gamma$,
hence we can define $q_{\xi_{\eta}}(\gamma)=\eta<\gamma$, then $q\leq p$.

Next we show how to increase $X_{p}$. Let $p\in\mathbb{AD}$ and $\mu<\kappa$,
then by the above there exists $q\leq p$ with $X_{q}=X_{p}$ and
$|X_{p}|+\mu\leq\beta_{q}$. If $X\in[\lambda\setminus X_{p}]^{\mu}$, we can
find $r\leq q$ with $X_{r}=X_{p}\cup X$ and $\beta_{r}=\beta_{q}$ simply by
letting $r_{\xi}(\alpha)=0$ for all $\alpha\in\beta_{q}$ and $\xi\in X$. ∎

###### Lemma 3.22

If $G$ is an $\mathbb{AD}$-generic filter over $\mathbf{V}$, and we define
$f_{\xi}=\bigcup\left\\{p_{\xi}\mid p\in G\land\xi\in X_{p})\right\\}$, then
$f_{\xi}\in{}^{\kappa}\kappa$ is a regressive function and
$\left\\{f_{\xi}\mid\xi\in(\lambda)^{\mathbf{V}}\right\\}$ is almost disjoint.

  * Proof.

It is clear from the definition of $\mathbb{AD}$ and the above lemma that
$f_{\xi}\in{}^{\kappa}\kappa$ and that $f_{\xi}$ is regressive. If
$\xi,\xi^{\prime}\in(\lambda)^{\mathbf{V}}$ are distinct, then let $p\in G$ be
such that $\xi,\xi^{\prime}\in X_{p}$ and let $\alpha\geq\beta_{p}$. For any
$q\leq p$ with $q\in G$ and $\alpha\in\beta_{q}$ we have $q_{\xi}(\alpha)\neq
q_{\xi^{\prime}}(\alpha)$. By the above lemma there exist such $q\leq p$ with
$q\in G$, hence $f_{\xi}(\alpha)\neq f_{\xi^{\prime}}(\alpha)$ for any
$\alpha>\beta_{p}$. Therefore $f_{\xi}$ and $f_{\xi^{\prime}}$ are almost
disjoint. ∎

###### Lemma 3.23

$\mathbb{AD}$ is ${<}\kappa$-closed and has the ${<}\kappa^{+}$-c.c..

  * Proof.

First, let us prove ${<}\kappa$-closure. Let $\gamma\in\kappa$ and let
$\left\langle p^{\eta}\in\mathbb{AD}\mid\eta<\gamma\right\rangle$ be a
descending chain of conditions. It is clear that
$X=\bigcup_{\eta\in\gamma}X_{p^{\eta}}\in[\lambda]^{<\kappa}$ and
$|X|\leq\beta=\bigcup_{\eta\in\gamma}\beta_{p^{\eta}}\in\kappa$, and that
$p=\bigcup_{\eta\in\gamma}p^{\eta}:X\times\beta\to\kappa$ and
$p(\xi,\eta)<\eta$ for all $(\xi,\eta)\in X\times\beta$. If
$\xi,\xi^{\prime}\in X_{p^{\eta}}$ are distinct and
$\alpha\in\beta\setminus\beta_{p^{\eta}}$, then there is $\zeta>\eta$ such
that $\alpha\in\beta_{p^{\zeta}}$. Since the chain of conditions is descending
we have $p^{\zeta}\leq p^{\eta}$, which implies that
$p_{\xi}(\alpha)=p^{\zeta}_{\xi}(\alpha)\neq
p^{\zeta}_{\xi^{\prime}}(\alpha)=p_{\xi^{\prime}}(\alpha)$, so $p\leq
p_{\eta}$.

As for ${<}\kappa^{+}$-c.c., suppose $\mathcal{B}\subseteq\mathbb{AD}$ is a
subset with $|\mathcal{B}|=\kappa^{+}$. We let
$\mathcal{A}=\left\\{\mathrm{dom}(p)\mid p\in\mathcal{B}\right\\}$, then by
$|X_{p}|<\kappa$ and $\beta_{p}<\kappa$ for each $p\in\mathcal{B}$ we see that
$\mathcal{A}$ is a family of sets of cardinality ${<}\kappa$. Applying the
$\Delta$-system lemma on $\mathcal{A}$, we know that there exists
$\mathcal{B}_{0}\subseteq\mathcal{B}$ such that $|\mathcal{B}_{0}|=\kappa^{+}$
and some sets $X\in[\lambda]^{<\kappa}$ and $\beta\in\kappa$ such that
$\mathrm{dom}(p)\cap\mathrm{dom}(p^{\prime})=X\times\beta$ for all distinct
$p,p^{\prime}\in\mathcal{B}_{0}$.

If $X=\varnothing$, then any $p,p^{\prime}\in\mathcal{B}_{0}$ have some $q\leq
p$ and $q^{\prime}\leq p^{\prime}$ such that
$|X_{p}|+|X_{p^{\prime}}|\leq\beta_{q}=\beta_{q^{\prime}}$ using Lemma 3.21.
Then $q\cup q^{\prime}\leq p$ and $q\cup q^{\prime}\leq p^{\prime}$, thus
$\mathcal{B}$ is not an antichain.

On the other hand, if $X\neq\varnothing$, then
$\beta=\beta_{p}=\beta_{p^{\prime}}$ for all $p,p^{\prime}\in\mathcal{B}_{0}$.
Since $|X\times\beta|<\kappa$ and $\kappa^{<\kappa}=\kappa$, there exists
$\mathcal{B}_{1}\subseteq\mathcal{B}_{0}$ with $|\mathcal{B}_{1}|=\kappa^{+}$
such that $p\restriction(X\times\beta)=p^{\prime}\restriction(X\times\beta)$
for all $p,p^{\prime}\in\mathcal{B}_{1}$. Once again, using Lemma 3.21 we can
see that $p,p^{\prime}$ are compatible, and thus $\mathcal{B}$ is not an
antichain. ∎

###### Corollary 3.24

$\mathbb{AD}$ preserves cardinals and cofinality and does not add any elements
to ${}^{<\kappa}\kappa$.

###### Corollary 3.25

It is consistent that there exists an almost disjoint family
$\mathcal{A}\subseteq\prod\mathsf{id}$ of cardinality $2^{\kappa}=\lambda$ for
any $\lambda$ with $\mathrm{cf}(\lambda)>\kappa$.

  * Proof.

Starting with a model where
$\mathbf{V}\vDash\text{``}\,2^{\kappa}=\lambda\,\text{''}$, let $G$ be
$\mathbb{AD}$-generic over $\mathbf{V}$, and let
$\mathcal{F}=\left\\{f_{\xi}\mid\xi\in(2^{\kappa})^{\mathbf{V}}\right\\}$ be
the almost disjoint family described in Lemma 3.22. An argument by counting
names shows that $(2^{\kappa})^{\mathbf{V}}=(2^{\kappa})^{\mathbf{V}[G]}$. ∎

Although the ${<}\kappa$-closure implies that $\mathbb{AD}$ preserves the
inaccessibility of $\kappa$, the same cannot be said for stronger large
cardinal assumptions on $\kappa$. Indeed, it is inconsistent that an almost
disjoint family $\mathcal{A}\subseteq\prod\mathsf{id}$ of size larger than
$\kappa$ exists for measurable cardinals, as was pointed out to me by Jing
Zhang.

###### Theorem 3.26

If $\kappa$ is measurable, $b\in{}^{\kappa}\kappa$ is continuous and
$\mathcal{A}\subseteq\prod b$ is almost disjoint, then
$|\mathcal{A}|\leq\kappa$.

  * Proof.

Let $\mathcal{U}\subseteq\mathcal{P}(\kappa)$ be a ${<}\kappa$-complete
nonprincipal normal ultrafilter on $\kappa$. Assume towards contradiction that
$\mathcal{A}\subseteq\prod b$ is an almost disjoint family with
$|\mathcal{A}|=\kappa^{+}$. Since $b$ is continuous, the set of fixed points
of $b$ contains a club set, thus every $f\in\mathcal{A}$ is regressive on a
club set. Therefore there exists $X_{f}\in\mathcal{U}$ such that $f$ is
regressive on $X_{f}$, and since $\mathcal{U}$ is normal there exists
$Y_{f}\in\mathcal{U}\restriction X_{f}$ and $\gamma_{f}\in\kappa$ such that
$\mathrm{ran}(f\restriction Y_{f})=\left\\{\gamma_{f}\right\\}$. By the
pigeonhole principle there exists $\mathcal{A}^{\prime}\subseteq\mathcal{A}$
and $\gamma\in\kappa$ with $|\mathcal{A}^{\prime}|=|\mathcal{A}|$ such that
$\gamma_{f}=\gamma$ for all $f\in\mathcal{A}^{\prime}$. Then for any distinct
$f,f^{\prime}\in\mathcal{A}^{\prime}$ we have $Y_{f}\cap
Y_{f^{\prime}}\in\mathcal{U}$, contradicting that $f$ and $f^{\prime}$ are
almost disjoint. ∎

In the end, we do not know whether the conditions of (ii) imply that
$\mathfrak{d}_{\kappa}^{b,h}(\in^{*})$ is trivial, that is, the following
question remains open:

###### Question 3.27

Is $\mathfrak{d}_{\kappa}^{b,h}(\in^{*})<2^{\kappa}$ consistent with $b$
continuous on a club set?

Finally, in case (iii) we can prove that each of
$\kappa^{+}<\mathfrak{b}_{\kappa}^{b,h}(\in^{*})$ and
$\mathfrak{d}_{\kappa}^{b,h}(\in^{*})<2^{\kappa}$ are consistent using a
generalisation of localisation forcing. This mirrors what we saw for
domination and unboundedness in Theorem 3.13.

###### Definition 3.28

Let $C\subseteq\kappa$. We define $(b,h,C)$-Localisation forcing
$\mathbb{L}\mathbbm{oc}^{b,h}_{\kappa}(C)$ as the forcing notion with
conditions $p=(s,F)$, where $s\in\mathrm{Loc}_{<\kappa}^{b,h}$ is a function
with domain $\gamma\in C$ and $F\subseteq\prod b$ with $|F|<h(\gamma)$. The
ordering is given by $(t,G)\leq(s,F)$ iff $s\subseteq t$ and $F\subseteq G$
and for all $\alpha\in\mathrm{dom}(t)\setminus\mathrm{dom}(s)$ and $f\in F$ we
have $f(\alpha)\in t(\alpha)$.

It is clear from the definition, under the assumptions of case (iii), that
$\mathbb{L}\mathbbm{oc}^{b,h}_{\kappa}(C)$ adds a generic element of
$\mathrm{Loc}_{\kappa}^{b,h}$ that localises all elements of $\prod b$ from
the ground model.

###### Lemma 3.29

Let $C$ be a club set such that $h$ is discontinuous on $C$, then
$\mathbb{L}\mathbbm{oc}_{\kappa}^{b,h}(C)$ is strategically ${<}\kappa$-closed
and ${<}\kappa^{+}$-c.c..

  * Proof.

For strategic ${<}\kappa$-closure, at stage $\alpha$ of
$\mathcal{G}(\mathbb{L}\mathbbm{oc}_{\kappa}^{b,h}(C),p)$, let $\langle
p_{\xi}=(s_{\xi},F_{\xi})\mid\xi\in\alpha\rangle$ and $\langle
p^{\prime}_{\xi}=(s_{\xi}^{\prime},F_{\xi}^{\prime})\mid\xi\in\alpha\rangle$
be the sequences of previous moves by White and Black respectively. We will
describe the winning strategy for White at stage $\alpha$.

If $\alpha=\beta+1$ is successor, White will decide some $\gamma\in C$ such
that $h(\mathrm{dom}(s^{\prime}_{\beta}))<\gamma$ and $\alpha<\gamma$. Since
$|F_{\beta}^{\prime}|<h(\mathrm{dom}(s_{\beta}^{\prime}))$ and $h$ is
increasing, we can define $s_{\alpha}\supseteq s_{\beta}^{\prime}$ by letting
$s_{\alpha}(\eta)=\\{f(\eta)\mid f\in F_{\beta}^{\prime}\\}$ for each
$\eta\in[\mathrm{dom}(s_{\beta}^{\prime}),\gamma)$ and let
$F_{\alpha}=F_{\beta}^{\prime}$.

We have to show that White can make a move at limit $\alpha$ as well. We claim
that $s_{\alpha}=\bigcup_{\xi\in\alpha}s_{\xi}$ and
$F_{\alpha}=\bigcup_{\xi\in\alpha}F_{\xi}$ works. Clearly
$\mathrm{dom}(s_{\alpha})\in C$ by $C$ being club, and

$\displaystyle|F_{\alpha}|=|\alpha|\cdot\textstyle\sup_{\xi\in\alpha}|F_{\xi}|\leq\bigcup_{\xi\in\alpha}h(\mathrm{dom}(s_{\xi}))<h(\bigcup_{\xi\in\alpha}\mathrm{dom}(s_{\xi}))=h(\mathrm{dom}(s_{\alpha}))$

Here the strict inequality follows from $h$ being discontinuous on $C$.
Therefore, $(s_{\alpha},F_{\alpha})$ is indeed a valid move for White,
implying that White has a winning strategy.

For ${<}\kappa^{+}$-c.c., note that for any
$(s,F),(s,G)\in\mathbb{L}\mathbbm{oc}_{\kappa}^{b,h}(C)$ with
$\mathrm{dom}(s)=\gamma$ we have $|F|<h(\gamma)$ and $|G|<h(\gamma)$, hence
since $h(\gamma)$ is infinite also $|F\cup G|<h(\gamma)$. Therefore $(s,F\cup
G)$ is a condition below both $(s,F)$ and $(s,G)$. Thus if
$\mathcal{A}\subseteq\mathbb{L}\mathbbm{oc}_{\kappa}^{b,h}(C)$ is an antichain
and $(s,F),(t,G)\in\mathcal{A}$ are distinct, then we must have $s\neq t$,
hence $|\mathcal{A}|\leq\kappa$. ∎

###### Corollary 3.30

$\mathbb{L}\mathbbm{oc}^{b,h}_{\kappa}(C)$ preserves cardinals and cofinality
and does not any elements to ${}^{<\kappa}\kappa$.

###### Theorem 3.31

Let $C$ be a club set such that $h$ is discontinuous on $C$, then it is
consistent that $\mathfrak{b}_{\kappa}^{b,h}(\in^{*})>\kappa^{+}$ and that
$\mathfrak{d}_{\kappa}^{b,h}(\in^{*})<2^{\kappa}$.

  * Proof.

Analogous to Theorem 3.13, with $\mathbb{L}\mathbbm{oc}_{\kappa}^{b,h}(C)$
taking the role of $\mathbb{D}_{\kappa}^{b}(C)$. ∎

### 3.3. Antilocalisation & Anti-avoidance

We can give similar results for
$\mathfrak{d}_{\kappa}^{b,h}(\mathrel{\cancel{\ni^{\infty}}})$ and
$\mathfrak{b}_{\kappa}^{b,h}(\mathrel{\cancel{\ni^{\infty}}})$ to what we
discussed in the previous sections. Firstly, we make three observations,
without proof, analogous to Lemmas 3.14, 3.15 and 3.16. As a consequence we
will once again assume that $h\leq b$ is always the case.

###### Lemma 3.32

$b<^{\infty}h$ if and only if
$\mathfrak{d}_{\kappa}^{b,h}(\mathrel{\cancel{\ni^{\infty}}})=1$.
If $b<^{\infty}h$, then
$\mathfrak{b}_{\kappa}^{b,h}(\mathrel{\cancel{\ni^{\infty}}})$ is undefined.

###### Lemma 3.33

If $\left\langle\alpha_{\xi}\mid\xi\in\kappa\right\rangle\in{}^{\kappa}\kappa$
is a strictly increasing sequence and $b^{\prime}:\xi\mapsto b(\alpha_{\xi})$
and $h^{\prime}:\xi\mapsto h(\alpha_{\xi})$, then
$\mathscr{A}\\!\\!\mathscr{L}_{b^{\prime},h^{\prime}}\preceq\mathscr{A}\\!\\!\mathscr{L}_{b,h}$.

###### Lemma 3.34

Let $h\geq^{*}h^{\prime}$ and $b\leq^{*}b^{\prime}$, then there exist Tukey
connections
${\mathscr{A}\\!\\!\mathscr{L}_{b^{\prime},h^{\prime}}}\preceq{\mathscr{A}\\!\\!\mathscr{L}_{b,h}}$.

A comprehensive overview of the trivial values for these cardinals in the
classical context has been given by Cardona and Mejía in section 3 of [CM19].
However, the classical characterisation uses a substantial amount of finite
arithmetic and thus appears quite different from the characterisation of the
trivial values of
$\mathfrak{b}_{\kappa}^{b,h}(\mathrel{\cancel{\ni^{\infty}}})$, given below.
We once again have three cases, making this similar to Lemmas 3.3 and 3.17.

###### Lemma 3.35

For each $\lambda<\kappa$ define

$\displaystyle D_{\lambda}=\left\\{\alpha\in\kappa\mid
b(\alpha)\leq\lambda\right\\}\ \cup\ \left\\{\alpha\in\kappa\mid
h(\alpha)=b(\alpha)\text{ and }\mathrm{cf}(b(\alpha))\leq\lambda\right\\}.$

  1. (i)

If there exists a least cardinal $\lambda<\kappa$ such that $D_{\lambda}$ is
cofinal in $\kappa$, then
$\mathfrak{b}_{\kappa}^{b,h}(\mathrel{\cancel{\ni^{\infty}}})=\lambda$.

  2. (ii)

If $D_{\lambda}$ is bounded for all $\lambda<\kappa$ and there is a stationary
set $S$ such that

    1. (a)

$b$ is continuous on $S$, or

    2. (b)

for each $\xi\in S$ there exists $\alpha_{\xi}\geq\xi$ with
$h(\alpha_{\xi})=b(\alpha_{\xi})$ and $\mathrm{cf}(b(\alpha_{\xi}))\leq\xi$,

then $\mathfrak{b}_{\kappa}^{b,h}(\mathrel{\cancel{\ni^{\infty}}})=\kappa$.

  3. (iii)

If $D_{\lambda}$ is bounded for all $\lambda<\kappa$ and there is a club set
$C$ such that

    1. (a)

$b$ is discontinuous on C, and

    2. (b)

for each $\xi\in C$ and $\alpha\geq\xi$, if $h(\alpha)=b(\alpha)$, then
$\mathrm{cf}(b(\alpha))>\xi$,

then
$\kappa^{+}\leq\mathfrak{b}_{\kappa}^{b,h}(\mathrel{\cancel{\ni^{\infty}}})$.

  * Proof.

(i) Assume that $\lambda$ is minimal such that $D_{\lambda}$ is cofinal. Let
$D_{\lambda}^{\prime}=\left\\{\alpha\in\kappa\mid b(\alpha)=\lambda\right\\}$
and $D_{\lambda}^{\prime\prime}=\left\\{\alpha\in\kappa\mid
h(\alpha)=b(\alpha)\text{ and }\mathrm{cf}(b(\alpha))=\lambda\right\\}$. Note
that at least one of $D_{\lambda}^{\prime}$ or $D_{\lambda}^{\prime\prime}$ is
cofinal. We will assume without loss of generality that
$D_{\lambda}=D^{\prime}_{\lambda}\cup D^{\prime\prime}_{\lambda}$.

First, we will show that
$\mathfrak{b}_{\kappa}^{b,h}(\mathrel{\cancel{\ni^{\infty}}})\leq\lambda$.

If $\alpha\in D_{\lambda}^{\prime\prime}$, then
$\mathrm{cf}(b(\alpha))=\lambda$, thus we can find a continuous sequence
$\langle\beta_{\xi}^{\alpha}\mid\xi<\lambda\rangle$ that is cofinal in
$b(\alpha)$. In the other case where $\alpha\in D_{\lambda}^{\prime}$, let
$\beta_{\xi}^{\alpha}=\xi$ for all $\xi<\lambda=b(\alpha)$.

Since $|[\beta_{\xi}^{\alpha},\beta_{\xi+1}^{\alpha})|<h(\alpha)\leq
b(\alpha)$ for all $\alpha\in D_{\lambda}$, we can pick some
$\varphi_{\xi}\in\mathrm{Loc}_{\kappa}^{b,h}$ for each $\xi<\lambda$ such that
$\varphi_{\xi}(\alpha)=[\beta_{\xi}^{\alpha},\beta_{\xi+1}^{\alpha})$ for all
$\alpha\in D_{\lambda}$. For $f\in\prod b$, let $F_{f}:D_{\lambda}\to\lambda$
map $\alpha\in D_{\lambda}$ to the $\xi\in\lambda$ such that
$f(\alpha)\in\varphi_{\xi}(\alpha)$, then by the pigeonhole principle
$F_{f}^{-1}(\xi)$ is cofinal for some $\xi\in\lambda$, and thus for this $\xi$
we have $f\in^{\infty}\varphi_{\xi}$. Therefore
$\left\\{\varphi_{\xi}\mid\xi<\lambda\right\\}$ witnesses that
$\mathfrak{b}_{\kappa}^{b,h}(\mathrel{\cancel{\ni^{\infty}}})\leq\lambda$.

Next we show that
$\lambda\leq\mathfrak{b}_{\kappa}^{b,h}(\mathrel{\cancel{\ni^{\infty}}})$. Let
$\left\\{\varphi_{\xi}\mid\xi\in\mu\right\\}\subseteq\mathrm{Loc}_{\kappa}^{b,h}$
for some $\mu<\lambda$.

If $D_{\lambda}^{\prime}$ is cofinal, then we must have $b(\alpha)\geq\lambda$
for almost all $\alpha\in\kappa$ by minimality of $\lambda$. Suppose that
$\lambda\leq b(\alpha)$ and
$\bigcup_{\xi<\mu}\varphi_{\xi}(\alpha)=b(\alpha)$, then
$\mathrm{cf}(b(\alpha))\leq\mu$, and furthermore for every $\nu<b(\alpha)$
there is some $\xi\in\mu$ such that $\nu\leq|\varphi_{\xi}(\alpha)|$, implying
that $h(\alpha)=b(\alpha)$. But then $\alpha\in D^{\prime\prime}_{\mu}$, which
is bounded by minimality of $\lambda$. Therefore
$\bigcup_{\xi<\mu}\varphi(\alpha)\neq b(\alpha)$ for almost all
$\alpha\in\kappa$.

If $D_{\lambda}^{\prime}$ is bounded and $D_{\lambda}^{\prime\prime}$ is
cofinal, then for almost all $\alpha\notin D_{\lambda}^{\prime\prime}$ we must
have $h(\alpha)\neq b(\alpha)$ or $\mathrm{cf}(b(\alpha))>\lambda$. Hence we
have $\max\left\\{h(\alpha),\lambda\right\\}<b(\alpha)$ or
$\mathrm{cf}(b(\alpha))\geq\lambda$ for almost all $\alpha\in\kappa$. If
$\max\left\\{h(\alpha),\lambda\right\\}<b(\alpha)$, then also $\mu\cdot
h(\alpha)<b(\alpha)$, thus $\bigcup_{\xi<\mu}\varphi_{\xi}(\alpha)\neq
b(\alpha)$. On the other hand if $\mathrm{cf}(b(\alpha))\geq\lambda$, then
clearly $\bigcup_{\xi<\mu}\varphi_{\xi}(\alpha)\neq b(\alpha)$ as well.

In either case we find $f\in\prod b$ such that
$f\mathrel{\cancel{\in^{\infty}}}\varphi_{\xi}$ for all $\xi<\mu$, and
consequently
$\mu<\mathfrak{b}_{\kappa}^{b,h}(\mathrel{\cancel{\ni^{\infty}}})$.

(ii) We first prove that
$\mathfrak{b}_{\kappa}^{b,h}(\mathrel{\cancel{\ni^{\infty}}})\leq\kappa$ in
case (a), then in case (b), and then we show the reverse direction, that
$\kappa\leq\mathfrak{b}_{\kappa}^{b,h}(\mathrel{\cancel{\ni^{\infty}}})$.

(a) Assume $b$ is increasing and continuous on stationary $S\subseteq\kappa$.
Then the set of fixed points $S^{\prime}=\left\\{\alpha\in S\mid
b(\alpha)=\alpha\right\\}$ is also stationary.

For each $\eta\in\kappa$ let $\varphi_{\eta}\in\mathrm{Loc}_{\kappa}^{b,h}$ be
defined as $\varphi_{\eta}(\xi)=\left\\{\eta\right\\}$ if $\eta<b(\xi)$ and
arbitrary otherwise. If $f\in\prod b$, then $f(\xi)<b(\xi)$ for every
$\xi\in\kappa$, therefore $f$ is regressive on $S^{\prime}$. By Fodor’s lemma
then there exists stationary $S^{\prime\prime}\subseteq S^{\prime}$ such that
$f\restriction S^{\prime\prime}$ is constant. Let $\eta$ be such that
$f(\xi)=\eta$ for all $\eta\in S^{\prime\prime}$, then clearly
$f\in^{\infty}\varphi_{\eta}$. Hence
$\left\\{\varphi_{\eta}\mid\eta<\kappa\right\\}$ witnesses
$\mathfrak{b}_{\kappa}^{b,h}(\mathrel{\cancel{\ni^{\infty}}})\leq\kappa$.

(b) Assume $S$ is stationary and for each $\xi\in S$ let $\alpha_{\xi}\geq\xi$
be such that $h(\alpha_{\xi})=b(\alpha_{\xi})$ and
$\mathrm{cf}(b(\alpha_{\xi}))\leq\xi$. We define for each $\eta\in\kappa$ a
slalom $\varphi_{\eta}\in\mathrm{Loc}_{\kappa}^{b,h}$ in such a way that for
each $\xi\in S$ we have
$\bigcup_{\eta<\xi}\varphi_{\eta}(\alpha_{\xi})=b(\alpha_{\xi})$, which is
possible since $\mathrm{cf}(b(\alpha_{\xi}))\leq\xi$ and
$h(\alpha_{\xi})=b(\alpha_{\xi})$.

If $f\in\prod b$, then for each $\xi\in S$ we define $f^{\prime}(\xi)$ to be
the least $\eta<\xi$ such that
$f(\alpha_{\xi})\in\varphi_{\eta}(\alpha_{\xi})$. Now $f^{\prime}$ is
regressive on $S$, so by Fodor’s lemma there exists stationary
$S^{\prime}\subseteq S$ such that $f^{\prime}\restriction S^{\prime}$ is
constant, say with value $\eta$. Then
$f(\alpha_{\xi})\in\varphi_{\eta}(\alpha_{\xi})$ for each $\xi\in S^{\prime}$,
thus $f\in^{\infty}\varphi_{\eta}$. Hence
$\left\\{\varphi_{\eta}\mid\eta<\kappa\right\\}$ witnesses
$\mathfrak{b}_{\kappa}^{b,h}(\mathrel{\cancel{\ni^{\infty}}})\leq\kappa$.

Finally for the reverse direction, let $\lambda<\kappa$ and
$\left\\{\varphi_{\xi}\mid\xi<\lambda\right\\}\subseteq\mathrm{Loc}_{\kappa}^{b,h}$.
Since $D_{\lambda}$ is bounded, we can find $\alpha_{0}\in\kappa$ such that
for every $\alpha>\alpha_{0}$ we have $\lambda<b(\alpha)$ and either
$h(\alpha)<b(\alpha)$ or $\lambda<\mathrm{cf}(b(\alpha))$. If
$h(\alpha)<b(\alpha)$, then $\lambda\cdot h(\alpha)<b(\alpha)$, meaning
$\bigcup_{\xi<\lambda}\varphi_{\xi}(\alpha)\neq b(\alpha)$. On the other hand,
if $\lambda<\mathrm{cf}(b(\alpha))$, then by
$|\varphi_{\xi}(\alpha)|<h(\alpha)\leq b(\alpha)$ we see that once again
$\bigcup_{\xi<\lambda}\varphi_{\xi}(\alpha)\neq b(\alpha)$. Hence we can
construct $f\in\prod b$ such that
$f\mathrel{\cancel{\in^{\infty}}}\varphi_{\xi}$ for all $\xi<\lambda$, which
proves that
$\kappa\leq\mathfrak{b}_{\kappa}^{b,h}(\mathrel{\cancel{\ni^{\infty}}})$.

(iii) Let $C$ be a club set with the properties mentioned in (iii) and let
$\left\\{\varphi_{\eta}\mid\eta\in\kappa\right\\}\subseteq\mathrm{Loc}_{\kappa}^{b,h}$.
We can enumerate $C$ increasingly as
$\left\langle\alpha_{\xi}\mid\xi\in\kappa\right\rangle$, then every
$\alpha\in\kappa$ has some $\xi\in\kappa$ such that
$\alpha\in[\alpha_{\xi},\alpha_{\xi+1})$. Note that
$|\bigcup_{\eta\in\xi}\varphi_{\eta}(\alpha)|\leq|\xi|\cdot\sup_{\eta\in\xi}|\varphi_{\eta}(\alpha)|$.
Since $b$ is increasing and discontinuous on $C$ and
$\xi\leq\alpha_{\xi}\leq\alpha$, we see that $\xi<b(\alpha_{\xi})\leq
b(\alpha)$. If $h(\alpha)<b(\alpha)$, then it is clear that
$\sup_{\eta\in\xi}|\varphi_{\eta}(\alpha)|<b(\alpha)$ as well. Else, if
$h(\alpha)=b(\alpha)$, then by the properties of $C$ we see that
$\mathrm{cf}(b(\alpha))>\alpha_{\xi}\geq\xi$, thus it also follows that
$\sup_{\eta\in\xi}|\varphi_{\eta}(\alpha)|<b(\alpha)$. We can therefore
conclude that $|\bigcup_{\eta\in\xi}\varphi_{\eta}(\alpha)|<b(\alpha)$ for all
$\alpha\in\kappa$, and thus we can define $f(\alpha)$ to be some value in
$b(\alpha)$ that lies outside of $\varphi_{\eta}(\alpha)$ for each $\eta<\xi$.
Then clearly $f\mathrel{\cancel{\in^{\infty}}}\varphi_{\eta}$ for all
$\eta\in\kappa$. ∎

As with domination and localisation, it is not known whether cases (i) and
(ii) of Lemma 3.35 imply that
$\mathfrak{d}_{\kappa}^{b,h}(\mathrel{\cancel{\ni^{\infty}}})=2^{\kappa}$. If
we consider the constant function $\overline{2}$, then we see that
$\mathfrak{d}_{\kappa}^{b,\overline{2}}(\mathrel{\cancel{\ni^{\infty}}})\leq\mathfrak{d}_{\kappa}^{b,h}(\mathrel{\cancel{\ni^{\infty}}})$
by Lemma 3.34. Since
$\mathfrak{d}_{\kappa}^{b,\overline{2}}(\mathrel{\cancel{\ni^{\infty}}})=\mathfrak{d}_{\kappa}^{b}(\mathrel{\cancel{=^{\infty}}})$
(see Lemma 3.41), we will discuss the cases (i) and (ii) in the next section.
Let us mention here, that if, by exception, we allow that $b(\alpha)$ is
finite for cofinally many $\alpha$, then we may use a connection with
localisation cardinals to prove that
$\mathfrak{d}_{\kappa}^{b,h}(\mathrel{\cancel{\ni^{\infty}}})=2^{\kappa}$ (see
Lemma 3.44). We will also give a partial answer to case (i) in the form of the
following lemma:

###### Lemma 3.36

If $\lambda$ is regular and $D=\left\\{\alpha\in\kappa\mid
b(\alpha)=h(\alpha)=\lambda\right\\}$ is cofinal, then
$\mathscr{D}_{\overline{\lambda}}\preceq\mathscr{A}\\!\\!\mathscr{L}_{b,h}$.

  * Proof.

Enumerate $D$ as $\left\\{\alpha_{\xi}\mid\xi\in\kappa\right\\}$. We need
$\rho_{-}:{}^{\kappa}\lambda\to\mathrm{Loc}_{\kappa}^{b,h}$ and
$\rho_{+}:\prod b\to{}^{\kappa}\lambda$.

Given $g\in{}^{\kappa}\lambda$, let
$\rho_{-}(g)\in\mathrm{Loc}_{\kappa}^{b,h}$ be such that
$\rho_{-}(g):\alpha_{\xi}\mapsto g(\xi)$ for all $\xi\in\kappa$. Given
$f\in\prod b$, let $\rho_{+}(f):\xi\mapsto f(\alpha_{\xi})$ for each
$\xi\in\kappa$. Since $\alpha_{\xi}\in D$ for each $\xi\in\kappa$, we have
$h(\alpha_{\xi})=b(\alpha_{\xi})=\lambda$, thus $\rho_{-}$ and $\rho_{+}$ are
well-defined. If $\varphi=\rho_{-}(g)$ and $f^{\prime}=\rho_{+}(f)$, and
$f\mathrel{\cancel{\in^{\infty}}}\varphi$, then it is easy to see that
$g\leq^{*}f^{\prime}$, so this is a Tukey connection. ∎

Since we already know that
$\mathfrak{d}_{\kappa}^{\overline{\lambda}}(\leq^{*})=2^{\kappa}$ from Theorem
3.5, the assumptions of the above lemma imply that
$\mathfrak{d}_{\kappa}^{b,h}(\mathrel{\cancel{\ni^{\infty}}})=2^{\kappa}$.

###### Definition 3.37

Let $C\subseteq\kappa$. We define $(b,h,C)$-antilocalisation forcing
$\mathbb{AL}\mathbbm{oc}_{\kappa}^{b,h}(C)$ as the forcing with conditions
$p=(s,\Phi)$ where $s\in\prod_{<\kappa}b$ is a function with domain $\gamma\in
C$ and $\Phi\subseteq\mathrm{Loc}_{\kappa}^{b,h}$ with $|\Phi|\leq\gamma$. The
ordering is given by $(t,\Psi)\leq(s,\Phi)$ if $s\subseteq t$ and
$\Phi\subseteq\Psi$ and for each
$\alpha\in\mathrm{dom}(t)\setminus\mathrm{dom}(s)$ and $\varphi\in\Phi$ we
have $t(\alpha)\notin\varphi(\alpha)$.

It is clear from the definition, under the assumptions of case (iii), that
$\mathbb{AL}\mathbbm{oc}_{\kappa}^{b,h}(C)$ adds a generic function in $\prod
b$ that is antilocalised by every ground model $(b,h)$-slalom.

###### Lemma 3.38

Let $C$ be a club set with such that the assumptions of (iii) from Lemma 3.35
are satisfied. Then $\mathbb{AL}\mathbbm{oc}_{\kappa}^{b,h}(C)$ is
strategically ${<}\kappa$-closed and ${<}\kappa^{+}$-c.c..

  * Proof.

For strategic ${<}\kappa$-closure, at stage $\alpha$ of
$\mathcal{G}(\mathbb{AL}\mathbbm{oc}_{\kappa}^{b,h}(C),p)$, let $\langle
p_{\xi}=(s_{\xi},\Phi_{\xi})\mid\xi\in\alpha\rangle$ and $\langle
p^{\prime}_{\xi}=(s_{\xi}^{\prime},\Phi_{\xi}^{\prime})\mid\xi\in\alpha\rangle$
be the sequences of previous moves by White and Black respectively. We will
describe the winning strategy for White at stage $\alpha$.

If $\alpha=\beta+1$ is successor, White will decide some $\gamma\in C$ such
that $\alpha\leq\gamma$. Let
$\mathrm{dom}(s_{\beta}^{\prime})=\gamma^{\prime}\in C$, then if
$\xi\in[\gamma^{\prime},\gamma)$ we have
$|\bigcup_{\varphi\in\Phi_{\beta}^{\prime}}\varphi(\xi)|\leq|\gamma^{\prime}|\cdot\sup_{\varphi\in\Phi_{\beta}^{\prime}}|\varphi(\xi)|<b(\xi)$
through the same reasoning as in the proof of Lemma 3.35 (iii). Therefore, we
can find some value $s_{\alpha}(\xi)\in
b(\xi)\setminus\bigcup_{\varphi\in\Phi_{\beta}^{\prime}}\varphi(\xi)$ to
define $s_{\alpha}$ with $\mathrm{dom}(s_{\alpha})=\gamma$, and let
$\Phi_{\alpha}=\Phi_{\beta}^{\prime}$.

We have to show that White can make a move at limit $\alpha$ as well. We claim
that $s_{\alpha}=\bigcup_{\xi\in\alpha}s_{\xi}$ and
$\Phi_{\alpha}=\bigcup_{\xi\in\alpha}\Phi_{\xi}$ works. Clearly
$\mathrm{dom}(s_{\alpha})\in C$ by $C$ being club, and

$\displaystyle|\Phi_{\alpha}|\leq|\alpha|\cdot\textstyle\sup_{\xi\in\alpha}|\Phi_{\xi}|=|\alpha|\cdot\bigcup_{\xi\in\alpha}\mathrm{dom}(s_{\xi})\leq|\alpha|\leq\mathrm{dom}(s_{\alpha})$

Therefore, $(s_{\alpha},F_{\alpha})$ is indeed a valid move for White, so
White has a winning strategy.

For ${<}\kappa^{+}$-c.c., note that for any
$(s,\Phi),(s,\Psi)\in\mathbb{AL}\mathbbm{oc}_{\kappa}^{b,h}(C)$ with
$\mathrm{dom}(s)=\gamma$ we have $|\Phi|\leq\gamma$ and $|\Psi|\leq\gamma$. We
may assume without loss of generality that $\gamma$ is infinite, then also
$|\Phi\cup\Psi|\leq\gamma$, hence $(s,\Phi\cup\Psi)$ is a condition below both
$(s,\Phi)$ and $(s,\Psi)$. Thus if
$\mathcal{A}\subseteq\mathbb{AL}\mathbbm{oc}_{\kappa}^{b,h}(C)$ is an
antichain and $(s,\Phi),(t,\Psi)\in\mathcal{A}$ are distinct, then we must
have $s\neq t$, hence $|\mathcal{A}|\leq\kappa$. ∎

###### Corollary 3.39

$\mathbb{AL}\mathbbm{oc}_{\kappa}^{b,h}(C)$ preserves cardinals and cofinality
and does not add any elements to ${}^{<\kappa}\kappa$.

###### Theorem 3.40

Let $C$ be a club set and $b,h\in{}^{\kappa}\kappa$ be such that the
assumptions of (iii) from Lemma 3.35 are satisfied, then it is consistent that
$\mathfrak{b}_{\kappa}^{b,h}(\mathrel{\cancel{\ni^{\infty}}})>\kappa^{+}$ and
that
$\mathfrak{d}_{\kappa}^{b,h}(\mathrel{\cancel{\ni^{\infty}}})<2^{\kappa}$.

  * Proof.

Analogous to Theorem 3.13, with $\mathbb{AL}\mathbbm{oc}_{\kappa}^{b,h}(C)$
taking the role of $\mathbb{D}_{\kappa}^{b}(C)$. ∎

### 3.4. Eventual Difference & Cofinal Equality

Eventual difference is a special case of antilocalisation, where the function
$h$ that determines the size of the sets in the slaloms is as small as
possible:

###### Lemma 3.41

$\mathscr{E}\\!\mathscr{D}_{b}\equiv\mathscr{A}\\!\\!\mathscr{L}_{b,\overline{2}}$.

  * Proof.

Any slalom $\varphi\in\mathrm{Loc}_{\kappa}^{b,\overline{2}}$ is a sequence of
singletons, and thus we can define $f_{\varphi}\in\prod b$ such that
$\left\\{f_{\varphi}(\alpha)\right\\}=\varphi(\alpha)$ for all
$\alpha\in\kappa$. It follows that $g\in^{\infty}\varphi$ if and only if
$g=^{\infty}f_{\varphi}$. ∎

It follows that Lemma 3.35 immediately gives us the following characterisation
of the trivial values of
$\mathfrak{b}_{\kappa}^{b}(\mathrel{\cancel{=^{\infty}}})$, since we can
ignore the case where $h(\alpha)=b(\alpha)$ if we let $h=\overline{2}$.

######  Corollary 3.42 (cf. Lemma 3.35)

For each $\lambda<\kappa$ define $D_{\lambda}=\left\\{\alpha\in\kappa\mid
b(\alpha)\leq\lambda\right\\}$.

  1. (i)

If there exists a least cardinal $\lambda<\kappa$ such that $D_{\lambda}$ is
cofinal in $\kappa$, then
$\mathfrak{b}_{\kappa}^{b}(\mathrel{\cancel{=^{\infty}}})=\lambda$.

  2. (ii)

If $D_{\lambda}$ is bounded for all $\lambda<\kappa$ and $b$ is continuous on
a stationary set, then
$\mathfrak{b}_{\kappa}^{b}(\mathrel{\cancel{=^{\infty}}})=\kappa$.

  3. (iii)

If $D_{\lambda}$ is bounded for all $\lambda<\kappa$ and $b$ is discontinuous
on a club set, then
$\kappa^{+}\leq\mathfrak{b}_{\kappa}^{b}(\mathrel{\cancel{=^{\infty}}})$.

It is equally immediate that the forcing construction from Theorem 3.40 gives
us a consistency result in case (iii) of Corollary 3.42.

######  Corollary 3.43 (Cf. Theorem 3.40)

Let $b\in{}^{\kappa}\kappa$ be such that the assumptions of (iii) from
Corollary 3.42 are satisfied, then it is consistent that
$\mathfrak{b}_{\kappa}^{b}(\mathrel{\cancel{=^{\infty}}})>\kappa^{+}$ and that
$\mathfrak{d}_{\kappa}^{b}(\mathrel{\cancel{=^{\infty}}})<2^{\kappa}$.

We can use Lemma 3.18 and a special connection between localisation and
antilocalisation for functions with finite values to prove the following
lemma, giving a condition for which
$\mathfrak{d}_{\kappa}^{b}(\mathrel{\cancel{=^{\infty}}})$ and
$\mathfrak{d}_{\kappa}^{b,h}(\mathrel{\cancel{\ni^{\infty}}})$ are trivial.
Note that this is significantly weaker than the assumption in case (i), since
$b$ is not only bounded on a cofinal set, but even finite cofinally often.

###### Lemma 3.44

If $D=\left\\{\alpha\in\kappa\mid b(\alpha)<\omega\right\\}$ is cofinal, then
$\mathfrak{d}_{\kappa}^{b}(\mathrel{\cancel{=^{\infty}}})=\mathfrak{d}_{\kappa}^{b,h}(\mathrel{\cancel{\ni^{\infty}}})=2^{\kappa}$.

  * Proof.

We want to give a lower bound to
$\mathfrak{d}_{\kappa}^{b,h}(\mathrel{\cancel{\ni^{\infty}}})$, thus by Lemma
3.34 we may assume that $h=\overline{2}$, and moreover by Lemma 3.33 and $D$
being cofinal, we may assume that $b=\overline{n}$ for some $n\in\omega$.

Note that
$\mathscr{L}_{\overline{n},\overline{n}}\equiv\mathscr{A}\\!\\!\mathscr{L}_{\overline{n},\overline{2}}$.
Namely, if $f\in\prod b$, let $\varphi_{f}:\alpha\mapsto
n\setminus\left\\{f(\alpha)\right\\}$, then
$\varphi_{f}\in\mathrm{Loc}_{\kappa}^{\overline{n},\overline{n}}$, and on the
other hand, if $\varphi\in\mathrm{Loc}_{\kappa}^{\overline{n},\overline{2}}$,
then let $f_{\varphi}$ be the unique function such that
$\varphi(\alpha)=\left\\{f_{\varphi}(\alpha)\right\\}$ for all
$\alpha\in\kappa$. It is clear that $f\mathrel{\cancel{\in^{\infty}}}\varphi$
if and only if $f_{\varphi}\in^{*}\varphi_{f}$.

It follows that
$2^{\kappa}=\mathfrak{d}_{\kappa}^{\overline{n},\overline{n}}(\in^{*})\leq\mathfrak{d}_{\kappa}^{\overline{n}}(\mathrel{\cancel{=^{\infty}}})\leq\mathfrak{d}_{\kappa}^{b}(\mathrel{\cancel{=^{\infty}}})=\mathfrak{d}_{\kappa}^{b,\overline{2}}(\mathrel{\cancel{\ni^{\infty}}})\leq\mathfrak{d}_{\kappa}^{b,h}(\mathrel{\cancel{\ni^{\infty}}})$
if $D$ is cofinal. Here the first equality is given by Lemma 3.18. ∎

The above argument is able to relate eventual difference to localisation by
removing a single element from a finite set, which changes its cardinality.
Naturally, removing a single element from an infinite set does not change the
cardinality of the infinite set, thus the technique from this proof does not
give us insight on how to prove
$\mathfrak{d}_{\kappa}^{b}(\mathrel{\cancel{=^{\infty}}})=2^{\kappa}$ if $b$
is only bounded on a cofinal set by an infinite value.

It is equally unclear whether
$\mathfrak{d}_{\kappa}^{b}(\mathrel{\cancel{=^{\infty}}})=2^{\kappa}$ in case
(ii) of Corollary 3.42.

###### Question 3.45

Is $\mathfrak{d}_{\kappa}^{b}(\mathrel{\cancel{=^{\infty}}})<2^{\kappa}$
consistent in case (i) or (ii) of Corollary 3.42?

## 4\. More Relations Between Cardinals

In this section we will give an overview of some additional relations between
the cardinals we have mentioned so far.

### 4.1. Relations for Fixed Parameters

We do this by defining Tukey connections. The following theorem gives an
overview of relations between the cardinals we have discussed so far, for a
fixed pair of parameters $b,h$ such that the relevant cardinals are
nontrivial.

###### Theorem 4.1

Let $b,h\in{}^{\kappa}\kappa$ be such that the conditions of cases (iii) of
Lemmas 3.35, 3.3 and 3.17 are satisfied, then the following Tukey connections
exist, where the relations marked by † require the additional assumption that
$h\leq^{*}\mathrm{cf}(b)$.

$\mathscr{A}\\!\\!\mathscr{L}_{b,h}$$\mathscr{D}_{b}$$\mathscr{L}_{b,h}$$\mathscr{E}\\!\mathscr{D}_{b}$$\mathscr{D}^{\bot}_{b}$$\mathscr{A}\\!\\!\mathscr{L}^{\bot}_{b,h}$$\preceq^{\dagger}$$\preceq^{\dagger}$${\preceq}^{\dagger}$$\preceq$$\preceq$$\preceq$$\preceq$

  * Proof.

The relation between eventual difference and antilocalisation was already
established with Lemma 3.41, and combined with Lemma 3.34 we get
$\mathscr{E}\\!\mathscr{D}_{b}\preceq\mathscr{A}\\!\\!\mathscr{L}_{b,h}$.

It is also easy to see for $f,g\in\prod b$, that if $g\leq^{*}f$, then
$f+1\mathbin{\cancel{\leq^{*}}}g$ and $g\mathrel{\cancel{=^{\infty}}}f+1$.
This implies that there are Tukey connections
$\mathscr{D}^{\bot}_{b}\preceq\mathscr{D}_{b}$ and
$\mathscr{E}\\!\mathscr{D}_{b}\preceq\mathscr{D}_{b}$.

Next, for $f\in\prod b$ and $\varphi\in\mathrm{Loc}_{\kappa}^{b,h}$, we have
that $f\in^{*}\varphi$ implies $f\in^{\infty}\varphi$, and thus we get a Tukey
connection
$\mathscr{A}\\!\\!\mathscr{L}^{\bot}_{b,h}\preceq\mathscr{L}_{b,h}$.

Finally we will show that
$\mathscr{A}\\!\\!\mathscr{L}_{b,h}\preceq\mathscr{D}_{b}\preceq\mathscr{L}_{b,h}$
as long as we assume that $h\leq^{*}\mathrm{cf}(b)$. Note that this also
implies
$\mathscr{A}\\!\\!\mathscr{L}^{\bot}_{b,h}\succeq\mathscr{D}_{b}^{\bot}$ by
duality.

We define $\rho_{-}^{0}:\mathrm{Loc}_{\kappa}^{b,h}\to\prod b$ and
$\rho_{+}^{0}:\prod b\to\prod b$ as a Tukey connection for
$\mathscr{A}\\!\\!\mathscr{L}_{b,h}\preceq\mathscr{D}_{b}$, and
$\rho_{-}^{1}:\prod b\to\prod b$ and
$\rho_{+}^{1}:\mathrm{Loc}_{\kappa}^{b,h}\to\prod b$ as a Tukey connection for
$\mathscr{D}_{b}\preceq\mathscr{L}_{b,h}$.

We will have $\rho_{-}^{0}=\rho_{+}^{1}$ sending
$\varphi\in\mathrm{Loc}_{\kappa}^{b,h}$ to $g\in\prod b$ where
$g(\alpha)=\sup(\varphi(\alpha))$ if $\sup(\varphi(\alpha))<b(\alpha)$ and
arbitrary otherwise. Since $h\leq^{*}\mathrm{cf}(b)$, we see that
$\varphi(\alpha)$ is not cofinal in $b(\alpha)$ for almost all $\alpha$, so
$g(\alpha)=\sup(\varphi(\alpha))$ for almost all $\alpha\in\kappa$. We let
$\rho_{-}^{1}$ be the identity function, and $\rho_{+}^{0}:f\mapsto f+1$.

If $f\in\prod b$ and $\varphi\in\mathrm{Loc}_{\kappa}^{b,h}$, let
$g=\rho_{-}^{0}(\varphi)=\rho_{+}^{1}(\varphi)$ and $f+1=\rho_{+}^{0}(f)$.
Then $f\in^{*}\varphi$ implies $f(\alpha)\leq\sup(\varphi(\alpha))=g(\alpha)$
for almost all $\alpha\in\kappa$, hence $f\leq^{*}g$. On the other hand, if
$g\leq^{*}f$, then $g<^{*}f+1$, hence $\sup(\varphi(\alpha))<f(\alpha)+1$ for
almost all $\alpha\in\kappa$, implying
$f+1\mathrel{\cancel{\in^{\infty}}}\varphi$. ∎

In summary, we can draw the cardinal characteristics related to these
relational systems in the diagram below, where the dashed lines require that
$h\leq^{*}\mathrm{cf}(b)$. Note that
$\mathfrak{b}_{\kappa}^{b}(\mathrel{\cancel{=^{\infty}}})\leq\mathrm{non}(\mathcal{M}_{\kappa})$
and
$\mathrm{cov}(\mathcal{M}_{\kappa})\leq\mathfrak{d}_{\kappa}^{b}(\mathrel{\cancel{=^{\infty}}})$
follow from Lemmas 3.41 and 3.34 and Fact 2.11.

$\kappa^{+}$$\mathfrak{b}_{\kappa}^{b}(\leq^{*})$$\mathfrak{b}_{\kappa}^{b}(\mathrel{\cancel{=^{\infty}}})$$\mathfrak{b}_{\kappa}^{b,h}(\mathrel{\cancel{\ni^{\infty}}})$$\mathfrak{b}_{\kappa}^{b,h}(\in^{*})$$\mathfrak{d}_{\kappa}^{b}(\leq^{*})$$\mathfrak{d}_{\kappa}^{b}(\mathrel{\cancel{=^{\infty}}})$$\mathfrak{d}_{\kappa}^{b,h}(\mathrel{\cancel{\ni^{\infty}}})$$\mathfrak{d}_{\kappa}^{b,h}(\in^{*})$$\mathrm{cov}(\mathcal{M}_{\kappa})$$\mathrm{non}(\mathcal{M}_{\kappa})$$2^{\kappa}$
Figure 1: Diagram of the relations between cardinals on bounded spaces

We saw that
$\mathscr{A}\\!\\!\mathscr{L}_{b,h}\preceq\mathscr{D}_{b}\preceq\mathscr{L}_{b,h}$
if $h\leq^{*}\mathrm{cf}(b)$, and in Lemma 3.6 we showed that reversely
$\mathscr{L}_{b,h}\preceq\mathscr{D}_{b}$ if $h=^{*}b$. Under the same
assumption, we can also prove that
$\mathscr{D}_{b}\preceq\mathscr{A}\\!\\!\mathscr{L}_{b,h}$. Therefore, the
dashed arrows in the above diagram collapse to become equalities in the case
where $b=^{*}\mathrm{cf}(b)=^{*}h$.

###### Theorem 4.2

If $h=^{*}b$, then $\mathscr{D}_{b}\preceq\mathscr{A}\\!\\!\mathscr{L}_{b,h}$.

  * Proof.

We define $\rho_{-}:\prod b\to\mathrm{Loc}_{\kappa}^{b,h}$ and $\rho_{+}:\prod
b\to\prod b$ as a Tukey connection for
$\mathscr{D}_{b}\preceq\mathscr{A}\\!\\!\mathscr{L}_{b,h}$.

We will have $\rho_{-}$ sending $g\in\prod b$ to
$\varphi\in\mathrm{Loc}_{\kappa}^{b,h}$ where $\varphi(\alpha)=g(\alpha)$
whenever $h(\alpha)=b(\alpha)$ and arbitrary otherwise. This is well-defined,
since $|g(\alpha)|<b(\alpha)=h(\alpha)$ for almost all $\alpha\in\kappa$. Let
$\rho_{+}$ be the identity function.

If $f,g\in\prod b$, let $\varphi=\rho_{-}(g)$. Then
$f\mathrel{\cancel{\in^{\infty}}}\varphi$ implies
$f(\alpha)\notin\varphi(\alpha)=g(\alpha)$ for almost all $\alpha\in\kappa$,
hence $g\leq^{*}f$. ∎

The relation
$\mathscr{E}\\!\mathscr{D}_{b}\preceq\mathscr{A}\\!\\!\mathscr{L}_{b,h}$ can
also be reversed for certain choices of $h$ and $b$, by the following theorem.

###### Theorem 4.3

Let $b$ be increasing (not strictly) and $\left\langle
I_{\alpha}\mid\alpha<\kappa\right\rangle$ be an interval partition of $\kappa$
with $|I_{\alpha}|=h(\alpha)$ for each $\alpha\in\kappa$ such that
$b(\alpha)=b(\xi)=b(\alpha)^{h(\alpha)}$ for all $\xi\in I_{\alpha}$ and
$\alpha\in\kappa$, then
$\mathscr{E}\\!\mathscr{D}_{b}\equiv\mathscr{A}\\!\\!\mathscr{L}_{b,h}$.

  * Proof.

We already know
$\mathscr{E}\\!\mathscr{D}_{b}\preceq\mathscr{A}\\!\\!\mathscr{L}_{b,h}$ from
Theorem 4.1, thus we show that
$\mathscr{A}\\!\\!\mathscr{L}_{b,h}\preceq\mathscr{E}\\!\mathscr{D}_{b}$. We
do this by giving a Tukey connection
$\rho_{-}:\mathrm{Loc}_{\kappa}^{b,h}\to\prod b$ and $\rho_{+}:\prod b\to\prod
b$. For each $\alpha\in\kappa$ let
$\pi_{\alpha}:b(\alpha)\to{}^{I_{\alpha}}b(\alpha)$ be a bijection, which
exist by $b(\alpha)^{h(\alpha)}=b(\alpha)$.

Given $\varphi\in\mathrm{Loc}_{\kappa}^{b,h}$, let
$\lambda_{\alpha}=|\varphi(\alpha)|$ and enumerate each
$\varphi(\alpha)=\\{x^{\alpha}_{\xi}\in
b(\alpha)\mid\xi\in\lambda_{\alpha}\\}$. We define
$g_{\xi}^{\alpha}=\pi_{\alpha}(x^{\alpha}_{\xi})\in{}^{I_{\alpha}}b(\alpha)$
for all $\alpha\in\kappa$ and $\xi\in\lambda_{\alpha}$. Fix
$\iota_{\alpha}=\min(I_{\alpha})$, then for every $\xi\in\lambda_{\alpha}$ we
see that $\iota_{\alpha}+\xi\in I_{\alpha}$, since
$\lambda_{\alpha}<h(\alpha)=|I_{\alpha}|$. Let $g\in\prod b$ be a function
such that $g(\iota_{\alpha}+\xi)=g^{\alpha}_{\xi}(\iota_{\alpha}+\xi)$ for
each $\alpha\in\kappa$ and $\xi\in\lambda_{\alpha}$. This is well defined,
because $b(\alpha)=b(\xi)$ for all $\xi\in I_{\alpha}$. We let
$\rho_{-}(\varphi)=g$.

We define $\rho_{+}(f)\in\prod b$ sending
$\alpha\mapsto\pi_{\alpha}^{-1}(f\restriction I_{\alpha})$.

Now suppose that $\varphi\in\mathrm{Loc}_{\kappa}^{b,h}$ and $f\in\prod b$ and
let $g=\rho_{-}(\varphi)$ and $f^{\prime}=\rho_{+}(f)$. Suppose that
$f^{\prime}\in^{\infty}\varphi$ and $\alpha_{0}\in\kappa$, then there is
$\alpha>\alpha_{0}$ such that $f^{\prime}(\alpha)\in\varphi(\alpha)$, so pick
$\xi\in\lambda_{\alpha}$ such that $x^{\alpha}_{\xi}=f^{\prime}(\alpha)$. Then
we see

$\displaystyle
g^{\alpha}_{\xi}=\pi_{\alpha}(x_{\xi}^{\alpha})=\pi_{\alpha}(f^{\prime}(\alpha))=\pi_{\alpha}(\pi^{-1}_{\alpha}(f\restriction
I_{\alpha}))=f\restriction I_{\alpha}.$

In particular
$f(\iota_{\alpha}+\xi)=g_{\xi}^{\alpha}(\iota_{\alpha}+\xi)=g(\iota_{\alpha}+\xi)$,
and since $\iota_{\alpha}+\xi>\alpha_{0}$ and $\alpha_{0}$ was arbitrary, it
follows that $f=^{\infty}g$. ∎

Note that the proof does not require the assumption that $b(\alpha)<\kappa$
for $\alpha\in\kappa$. Indeed, the above theorem holds even if we let
$b:\kappa\to\left\\{\kappa\right\\}$, since $\kappa$ is inaccessible. It
follows that the cardinal characteristics
$\mathfrak{d}_{\kappa}^{h}(\mathrel{\cancel{\ni^{\infty}}})$ and
$\mathfrak{b}_{\kappa}^{h}(\mathrel{\cancel{\ni^{\infty}}})$ from the
(unbounded) generalised Baire space do not depend on the choice of
$h\in{}^{\kappa}\kappa$, and can be related to the $\kappa$-meagre ideal as a
consequence of Fact 2.11.

###### Corollary 4.4

$\mathfrak{d}_{\kappa}^{h}(\mathrel{\cancel{\ni^{\infty}}})=\mathrm{cov}(\mathcal{M}_{\kappa})$
and
$\mathfrak{b}_{\kappa}^{h}(\mathrel{\cancel{\ni^{\infty}}})=\mathrm{non}(\mathcal{M}_{\kappa})$
for any choice of $h\in{}^{\kappa}\kappa$.

We will conclude this subsection with a relationship between antilocalisation
and localisation for different parameters.

###### Theorem 4.5

Let $h\cdot h^{\prime}<^{*}b$ and $b^{g}\leq^{*}b^{\prime}$ for all $g<h$,
then there exists a Tukey connection
${\mathscr{A}\\!\\!\mathscr{L}_{b,h}}\preceq{\mathscr{L}_{b^{\prime},h^{\prime}}}$.

  * Proof.

Let $\alpha_{0}\in\kappa$ be large enough such that $h(\alpha)\cdot
h^{\prime}(\alpha)<b(\alpha)$ and $b(\alpha)^{<h(\alpha)}\leq
b^{\prime}(\alpha)$ for all $\alpha\geq\alpha_{0}$. For each
$\alpha\geq\alpha_{0}$, we fix an injection
$\iota_{\alpha}:[b(\alpha)]^{<h(\alpha)}\mathrel{\textnormal{\guilsinglright}\mathrel{\mkern-9.0mu}\rightarrow}b^{\prime}(\alpha)$.

Given $\varphi\in\mathrm{Loc}_{\kappa}^{b,h}$, let
$\rho_{-}(\varphi)=f^{\prime}\in\prod b^{\prime}$ map
$\alpha\mapsto\iota_{\alpha}(\varphi(\alpha))$ for any $\alpha\geq\alpha_{0}$,
and arbitrary otherwise. Given
$\varphi^{\prime}\in\mathrm{Loc}_{\kappa}^{b^{\prime},h^{\prime}}$, let
$\rho_{+}(\varphi)=f\in\prod b$ have $f(\alpha)\in
b(\alpha)\setminus\bigcup_{\xi\in\varphi^{\prime}(\alpha)\cap\mathrm{ran}(\iota_{\alpha})}\iota_{\alpha}^{-1}(\xi)$
for any $\alpha\geq\alpha_{0}$, and arbitrary otherwise. Note that this is
well-defined, since $|\varphi^{\prime}(\alpha)|<h^{\prime}(\alpha)$ and
$|\iota_{\alpha}^{-1}(\xi)|<h(\alpha)$, therefore
$|\bigcup_{\xi\in\varphi^{\prime}(\alpha)\cap\mathrm{ran}(\iota_{\alpha})}\iota_{\alpha}^{-1}(\xi)|\leq
h(\alpha)\cdot h^{\prime}(\alpha)<b(\alpha)$.

If $\alpha\geq\alpha_{0}$ and $f^{\prime}(\alpha)\in\varphi^{\prime}(\alpha)$,
then
$\varphi(\alpha)=\iota_{\alpha}^{-1}(f^{\prime}(\alpha))\subseteq\bigcup_{\xi\in\varphi^{\prime}(\alpha)\cap\mathrm{ran}(\iota_{\alpha})}\iota_{\alpha}^{-1}(\xi)$,
so $f(\alpha)\notin\varphi(\alpha)$. Therefore
$f^{\prime}\in^{*}\varphi^{\prime}$ implies
$f\mathrel{\cancel{\in^{\infty}}}\varphi$. ∎

### 4.2. Infima and Suprema

Remember that we have the following monotonicity results from Lemmas 3.16 and
3.34. Let $b\leq b^{\prime}$. Then:

$\displaystyle\mathfrak{b}_{\kappa}^{b^{\prime},h}(\in^{*})$
$\displaystyle\leq\mathfrak{b}_{\kappa}^{b,h}(\in^{*})$
$\displaystyle\mathfrak{d}_{\kappa}^{b,h}(\in^{*})$
$\displaystyle\leq\mathfrak{d}_{\kappa}^{b^{\prime},h}(\in^{*})$
$\displaystyle\mathfrak{b}_{\kappa}^{b,h}(\mathrel{\cancel{\ni^{\infty}}})$
$\displaystyle\leq\mathfrak{b}_{\kappa}^{b^{\prime},h}(\mathrel{\cancel{\ni^{\infty}}})$
$\displaystyle\mathfrak{d}_{\kappa}^{b^{\prime},h}(\mathrel{\cancel{\ni^{\infty}}})$
$\displaystyle\leq\mathfrak{d}_{\kappa}^{b,h}(\mathrel{\cancel{\ni^{\infty}}})$

This motivates the definition of the following cardinal characteristics.

###### Definition 4.6

Given $h\in{}^{\kappa}\kappa$, let:

$\displaystyle\mathfrak{inf}_{\kappa}^{h}(\in^{*})$
$\displaystyle=\inf\\{\mathfrak{b}_{\kappa}^{b,h}(\in^{*})\mid
b\in{}^{\kappa}\kappa\\}$ $\displaystyle\mathfrak{sup}_{\kappa}^{h}(\in^{*})$
$\displaystyle=\sup\\{\mathfrak{d}_{\kappa}^{b,h}(\in^{*})\mid
b\in{}^{\kappa}\kappa\\}$
$\displaystyle\mathfrak{inf}_{\kappa}^{h}(\mathrel{\cancel{\ni^{\infty}}})$
$\displaystyle=\inf\\{\mathfrak{d}_{\kappa}^{b,h}(\mathrel{\cancel{\ni^{\infty}}})\mid
b\in{}^{\kappa}\kappa\\}$
$\displaystyle\mathfrak{sup}_{\kappa}^{h}(\mathrel{\cancel{\ni^{\infty}}})$
$\displaystyle=\sup\\{\mathfrak{b}_{\kappa}^{b,h}(\mathrel{\cancel{\ni^{\infty}}})\mid
b\in{}^{\kappa}\kappa\\}$
$\displaystyle\mathfrak{inf}_{\kappa}(\mathrel{\cancel{=^{\infty}}})$
$\displaystyle=\inf\\{\mathfrak{d}_{\kappa}^{b}(\mathrel{\cancel{=^{\infty}}})\mid
b\in{}^{\kappa}\kappa\\}$
$\displaystyle\mathfrak{sup}_{\kappa}(\mathrel{\cancel{=^{\infty}}})$
$\displaystyle=\sup\\{\mathfrak{b}_{\kappa}^{b}(\mathrel{\cancel{=^{\infty}}})\mid
b\in{}^{\kappa}\kappa\\}$

Note that for $\in^{*}$ we take infima over the $\mathfrak{b}$-side cardinals,
while for $\mathrel{\cancel{\ni^{\infty}}}$ and
$\mathrel{\cancel{=^{\infty}}}$ we take infima over the $\mathfrak{d}$-side
cardinals, and vice versa for the suprema. Note also that
$\mathfrak{inf}_{\kappa}(\mathrel{\cancel{=^{\infty}}})=\mathfrak{inf}_{\kappa}^{\overline{2}}(\mathrel{\cancel{\ni^{\infty}}})$
and
$\mathfrak{sup}_{\kappa}(\mathrel{\cancel{=^{\infty}}})=\mathfrak{sup}_{\kappa}^{\overline{2}}(\mathrel{\cancel{\ni^{\infty}}})$
by Lemma 3.41.

In [CM19] it is proved that the parameter $h$ does not matter if we work in
${}^{\omega}\omega$. This is based on two claims which we repeat below.

###### Fact 4.7

For any $b,h,h^{\prime}\in{}^{\omega}\omega$ such that $h\leq h^{\prime}$
there exists $b^{\prime}\in{}^{\omega}\omega$ such that
$\mathfrak{d}_{\omega}^{b,h}(\in^{*})\leq\mathfrak{d}_{\omega}^{b^{\prime},h^{\prime}}(\in^{*})$
and
$\mathfrak{b}_{\omega}^{b^{\prime},h^{\prime}}(\in^{*})\leq\mathfrak{b}_{\omega}^{b,h}(\in^{*})$.

###### Fact 4.8

For any $b^{\prime},h\in{}^{\omega}\omega$ such that $\overline{2}\leq h$
there exists $b\in{}^{\omega}\omega$ such that
$\mathfrak{d}_{\omega}^{b,h}(\mathrel{\cancel{\ni^{\infty}}})\leq\mathfrak{d}_{\omega}^{b^{\prime}}(\mathrel{\cancel{=^{\infty}}})$
and
$\mathfrak{b}_{\omega}^{b^{\prime}}(\mathrel{\cancel{=^{\infty}}})\leq\mathfrak{b}_{\omega}^{b,h}(\mathrel{\cancel{\ni^{\infty}}})$.

In the generalised context, Fact 4.7 can only be partially generalised, and
indeed we will see that differences in the parameter $h$ can lead to
consistently different cardinals. Meanwhile Fact 4.8 is completely
generalisable and we will look at this first. One could compare the next lemma
to Theorem 4.3.

###### Lemma 4.9

For any $b^{\prime},h\in{}^{\kappa}\kappa$ such that $\overline{2}\leq h$
there exists $b\in{}^{\kappa}\kappa$ such that
$\mathscr{A}\\!\\!\mathscr{L}_{b,h}\preceq\mathscr{E}\\!\mathscr{D}_{b^{\prime}}$.

  * Proof.

Let $\left\langle I_{\alpha}\mid\alpha\in\kappa\right\rangle$ be the (unique)
interval partition of $\kappa$ such that $I_{\alpha}\subseteq\min(I_{\beta})$
for $\alpha<\beta$ and $|I_{\alpha}|=h(\alpha)$ for all $\alpha\in\kappa$. We
define $b(\alpha)=\left|\prod_{\xi\in I_{\alpha}}b^{\prime}(\xi)\right|$ and
let
$\pi_{\alpha}:b(\alpha)\mathrel{\textnormal{\guilsinglright}\mathrel{\mkern-9.0mu}\rightarrow\mathrel{\mkern-14.0mu}\rightarrow}\prod_{\xi\in
I_{\alpha}}b^{\prime}(\xi)$ be a bijection for each $\alpha\in\kappa$.

For each $\varphi\in\mathrm{Loc}_{\kappa}^{b,h}$, let
$\rho_{-}(\varphi)=g_{\varphi}\in\prod b^{\prime}$ be defined as follows. For
each $\alpha\in\kappa$ and we take some surjection
$\iota_{\alpha}^{\varphi}:I_{\alpha}\rightarrow\mathrel{\mkern-14.0mu}\rightarrow\varphi(\alpha)$.
Given $\xi\in I_{\alpha}$, let
$\beta=\iota_{\alpha}^{\varphi}(\xi)\in\varphi(\alpha)$, then we define
$g_{\varphi}(\xi)=\pi_{\alpha}(\beta)(\xi)$.

For $f^{\prime}\in\prod b^{\prime}$, we let $\rho_{+}(f^{\prime})=f\in\prod b$
be given by $f(\alpha)=\pi_{\alpha}^{-1}(f^{\prime}\restriction I_{\alpha})$.

Now $(\rho_{-},\rho_{+})$ forms a Tukey connection. If
$\rho_{+}(f^{\prime})=f\in^{\infty}\varphi$, let $\alpha\in\kappa$ be
arbitrarily large such that $f(\alpha)\in\varphi(\alpha)$. Take $\xi\in
I_{\alpha}$ such that $f(\alpha)=\iota_{\alpha}^{\varphi}(\xi)$, then
$g_{\varphi}(\xi)=\pi_{\alpha}(f(\alpha))(\xi)=f^{\prime}(\xi)$, thus since
$\alpha$ is arbitrarily large, we see that $f^{\prime}=^{\infty}g_{\varphi}$.
∎

###### Corollary 4.10

$\mathfrak{inf}_{\kappa}^{h}(\mathrel{\cancel{\ni^{\infty}}})=\mathfrak{inf}_{\kappa}(\mathrel{\cancel{=^{\infty}}})$
and
$\mathfrak{sup}_{\kappa}^{h}(\mathrel{\cancel{\ni^{\infty}}})=\mathfrak{sup}_{\kappa}(\mathrel{\cancel{=^{\infty}}})$
for each $h\in{}^{\kappa}\kappa$.

  * Proof.

By Lemma 3.34 we see that
$\mathfrak{inf}_{\kappa}(\mathrel{\cancel{=^{\infty}}})\leq\mathfrak{inf}_{\kappa}^{h}(\mathrel{\cancel{\ni^{\infty}}})$
and
$\mathfrak{sup}_{\kappa}(\mathrel{\cancel{=^{\infty}}})\geq\mathfrak{sup}_{\kappa}^{h}(\mathrel{\cancel{\ni^{\infty}}})$.

By Lemma 4.9, for any $b^{\prime}\in{}^{\kappa}\kappa$ we can find some
$b\in{}^{\kappa}\kappa$ such that
$\mathfrak{d}_{\kappa}^{b,h}(\mathrel{\cancel{\ni^{\infty}}})\leq\mathfrak{d}_{\kappa}^{b^{\prime},\overline{2}}(\mathrel{\cancel{\ni^{\infty}}})$,
thus
$\mathfrak{inf}_{\kappa}(\mathrel{\cancel{=^{\infty}}})\geq\mathfrak{inf}_{\kappa}^{h}(\mathrel{\cancel{\ni^{\infty}}})$.
Similar for
$\mathfrak{sup}_{\kappa}(\mathrel{\cancel{=^{\infty}}})\leq\mathfrak{sup}_{\kappa}^{h}(\mathrel{\cancel{\ni^{\infty}}})$.
∎

An equality between $\mathfrak{inf}_{\omega}(\mathrel{\cancel{=^{\infty}}})$
and $\mathrm{non}(\mathcal{S}\mathcal{N}_{\kappa})$ was first found by
Rothberger [Rot41]. Miller [Mil81] used this result to remark that
$\mathrm{add}(\mathcal{M})=\min\left\\{\mathfrak{b}_{\omega}(\leq^{*}),\mathfrak{inf}_{\omega}(\mathrel{\cancel{=^{\infty}}})\right\\}$.
One could also show that
$\mathrm{cof}(\mathcal{M})=\max\left\\{\mathfrak{d}_{\omega}(\leq^{*}),\mathfrak{sup}_{\omega}(\mathrel{\cancel{=^{\infty}}})\right\\}$.
We show that each of these three claims generalises.

Instead of proving
$\mathrm{non}(\mathcal{S}\mathcal{N}_{\kappa})=\mathfrak{inf}_{\kappa}(\mathrel{\cancel{=^{\infty}}})$
directly, we show one direction by use of a Tukey connection. Let
$\mathcal{X}_{0}=\left\\{(g,b)\in{}^{\kappa}\kappa\times{}^{\kappa}\kappa\mid
g<b\right\\}$ and
$\mathcal{X}_{1}=\left\\{\tau:{}^{\kappa}\kappa\to{}^{\kappa}\kappa\mid\forall
b\in{}^{\kappa}\kappa(\tau(b)<b)\right\\}$, and define a relation
$J_{\infty}\subseteq\mathcal{X}_{0}\times\mathcal{X}_{1}$ by
$(f,b)\mathrel{J}_{\infty}\tau$ iff $f=^{\infty}\tau(b)$. We use this to
define the relational system $\mathscr{J}_{\infty}$, which is equivalent to
the categorical product
$\bigotimes_{b\in{}^{\kappa}\kappa}\mathscr{E}\\!\mathscr{D}_{b}^{\bot}$. We
also define a relational system whose norms are the covering and uniformity
numbers of the $\kappa$-strong measure zero ideal.

$\displaystyle\mathscr{J}_{\infty}$
$\displaystyle=\left\langle\mathcal{X}_{0},\mathcal{X}_{1},J_{\infty}\right\rangle$
$\displaystyle\left\lVert\mathscr{J}_{\infty}\right\rVert$
$\displaystyle=\mathfrak{sup}_{\kappa}(\mathrel{\cancel{=^{\infty}}})$
$\displaystyle\left\lVert\mathscr{J}_{\infty}^{\bot}\right\rVert$
$\displaystyle=\mathfrak{inf}_{\kappa}(\mathrel{\cancel{=^{\infty}}})$
$\displaystyle\mathscr{S}\\!\\!\mathscr{N}$
$\displaystyle=\left\langle{}^{\kappa}2,\mathcal{S}\mathcal{N}_{\kappa},\in\right\rangle$
$\displaystyle\left\lVert\mathscr{S}\\!\\!\mathscr{N}\right\rVert$
$\displaystyle=\mathrm{cov}(\mathcal{S}\mathcal{N}_{\kappa})$
$\displaystyle\left\lVert\mathscr{S}\\!\\!\mathscr{N}^{\bot}\right\rVert$
$\displaystyle=\mathrm{non}(\mathcal{S}\mathcal{N}_{\kappa})$

The following Tukey connection proves
$\mathfrak{inf}_{\kappa}(\mathrel{\cancel{=^{\infty}}})\geq\mathrm{non}(\mathcal{S}\mathcal{N}_{\kappa})$
and