# Showing that a certain simplicial set has levelwise small cardinality

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!

$$\mathcal{U}_\beta(\Delta[n])$$ 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 $$\beta<\alpha$$.