Showing that a certain simplicial set has levelwise small cardinality

Question

My question concerns Section 2 of the article "The Simplicial Model of Univalent Foundations (after Voevodsky)" (https://arxiv.org/pdf/1211.2851.pdf).

Let $\alpha$ be a strongly inaccessible cardinal.

Let $f : X \to Y$ be a map of simplicial sets.

1. We say that $f$ is well-ordered if it is equipped with a well-ordering of $Y_x := f_n^{-1}(x)$ for each simplex $x\in X_n$.

2. We say that $f$ is $\alpha$-small if $\left\lvert{Y_x}\right\rvert < \alpha$ for every simplex $x$.

Let $f: X \to Y$ and $g : Z \to Y$ be well-ordered simplicial maps. A morphism $f \to g$ is a fiber-preserving simplicial map $h : X \to Z$ such that $h_n : f_n^{-1}(y) \to g_n^{-1}(y)$ is order-preserving for every natural number $n$ and every $y\in Y_n$.

Define the functor $\mathcal{U}_{\alpha} : \mathbf{sSet}^{\text{op}} \to \mathbf{Set}$ so that $\mathcal{U}_{\alpha}(X)$ consists of all isomorphism classes of $\alpha$-small well-ordered Kan fibrations $Y\to X$. Also, let $$ \mathrm{U}_{\alpha} = \mathcal{U}_{\alpha}\circ \mathcal{Y}^{\text{op}} : \varDelta^{\text{op}} \to \mathbf{Set} $$ where $\mathcal{Y} :\varDelta \to \mathbf{sSet}$ denote the Yoneda embedding.

The authors say that if $\beta <\alpha$ is also inaccessible, then the unique map $\mathrm{U}_{\beta} \to 1$ is $\alpha$-small (p. 23, near the bottom). This amounts to saying that the set

$$ \left(\mathrm{U}_{\beta}\right)_n = \mathcal{U}_{\beta}(\Delta[n]) $$ has cardinality $<\alpha$ for each $n$.

I am unable, however, to see a set-theoretic justification for this. I'd be grateful if someone could provide one!

Answer 1

$\mathcal{U}_\beta(\Delta[n])$ is the set of isomorphism classes of $\beta$-small well-ordered fibrations over $\Delta[n]$. Such a thing is uniquely determined by an isomorphism class of $\beta$-small well-orderings over each element of $\Delta[n]$, together with the face and degeneracy maps between them. There are $\beta$ isomorphism classes of $\beta$-small well-orderings, and there are countably many elements of $\Delta[n]$. Since $\beta \cdot \aleph_0 = \beta$, there are $\beta$ choices of these fibers. Now for every face and degeneracy map, the domain and codomain are $\beta$-small, hence (since $\beta$ is inaccessible) there are $<\beta$ possible maps. Since there are countably many face and degeneracy maps, there are no more than $\beta \cdot \aleph_0 = \beta$ choices of the face and degeneracy maps. Thus there are overall $\beta$ choices, and $\beta<\alpha$.