For a partially ordered set $P$, a set $A\subseteq P$ is *cofinal* if for each element of $P$ there is a larger element in $A$. The *cofinality* of $P$, ${\rm cof}(P)$, is the minimal cardinality of a cofinal family in $P$.

Let $\kappa$ be an infinite cardinal number. Consider the set $\mathbb{N}^\kappa$ of all functions from $\kappa$ to $\mathbb{N}$ (equivalently, $\kappa$-sequences of natural numbers), with the pointwise partial order: $f\le g$ if $f(\alpha)\le g(\alpha)$ for all $\alpha<\kappa$.

Let $$f(\kappa):={\rm cof}(\mathbb{N}^\kappa,\le).$$ The cardinal $f(\aleph_0)$ is the well-understood dominating number $\mathfrak{d}$, which is consistently any cardinal of uncountable cofinality that is not larger than the continuum (see Blass).

Basic facts include:

- For cardinal numbers $\kappa\le\mu$, we have that $f(\kappa)\le f(\mu)$: A projection of a cofinal family in $\mathbb{N}^\mu$ on the first $\kappa$ corrdinates is cofinal in $\mathbb{N}^\kappa$.
- $\kappa<f(\kappa)$: Given $\{h_\alpha:\alpha<\kappa\}\subseteq\mathbb{N}^\kappa$, define $h(\alpha):=h_\alpha(\alpha)+1$ for all $\alpha<\kappa$.
- The value of $f(\kappa)$ with "$\le$" replaced by eventual dominance remains the same (see Comfort).

Some deeper results:

- Jech-Prikry: Assume that $2^{\aleph_0}$ is regular and smaller than $2^{\aleph_1}$. If $f(\aleph_1)=2^{\aleph_0}$, then there is an inner model with a measurable cardinal.
- If $\kappa^{\aleph_0}=\kappa$, then $f(\kappa)=2^\kappa$ (see Hathaway).

I did not conduct an extensive literature search. I would appreciate your pointing out relevant results that are not mentioned here.

**Question:**
Are there additional cardinals $\kappa$ for which $f(\kappa)$ can be evaluated in ZFC?

(The question also applies to large cadrinals. I do not request that the cardinals provably exist.)

**Update.** Intially, I also asked whether
$\kappa^{\aleph_0}=\kappa$ could be weakened to
${\rm cof}([\kappa]^{\aleph_0},\subseteq)=\kappa$.
Todd Eisworth reminds in the comments below that the latter hypothesis is true in ZFC for $\aleph_1$.

Strangely, I couldn't find an explicit reference for the following.

**Question.** Is it consistent (modulo suitable large cardinals) that $f(\aleph_1)<2^{\aleph_1}$? Assuming that (See Assaf Rinot's comment below), is there an explicit reference for that?

**Motivation:** The values $f(\kappa)$ pop up whenever I study (usually, jointly with colleagues) the character and other cardinal invariants of topological groups. They are unavoidable.

(Comment: By no means do I mean that the symbol $f(\kappa)$ should be reserved for the function thus defined. This is just a short, temporary name for brevity.)