Let $X$ be a topological space (feel free to add some separation axioms like “completely regular” if they help in answering the questions). Let $\alpha$ be an ordinal, identified as usual with $\{\beta \in \mathit{Ord} : \beta<\alpha\}$. Consider the following sheaves of $\alpha$-valued functions on $X$:

$\mathscr{C}_\alpha$ takes an open set $U\subseteq X$ to the set of continuous functions $U\to\alpha$ where $\alpha$ is given the usual (=order) topology.

$\mathscr{S}_\alpha$ takes an open set $U\subseteq X$ to the set of upper semicontinuous functions $U\to\alpha$, where “upper semicontinuous” means that $f(x) = \operatorname{lim\,sup}_{y\to x} f(y)$ for all $x\in U$ and of course $\operatorname{lim\,sup}_{y\to x} f(y) := \inf_{V\ni x} \sup_{y\in V} f(y)$ where the $\inf$ ranges over all (open, say) neighborhoods $V$ of $x$ in $U$.

$\mathscr{N}_\alpha$ takes an open set $U\subseteq X$ to the set of normal upper semicontinuous functions $U\to\alpha$, where “normal upper semicontinuous” means that $f(x) = \operatorname{lim\,sup}_{y\to x} \operatorname{lim\,inf}_{z\to y} f(z)$ for all $x\in U$ (of course, $\operatorname{lim\,inf}$ is defined dually to $\operatorname{lim\,sup}$).

The restriction maps are pointwise restriction in each case. Since each of these is defined by a local property on functions, they are, indeed, sheaves of sets, i.e., objects of the topos $\operatorname{Sh}(X)$ of sheaves of sets on $X$.

We have $\mathscr{C}_\alpha \subseteq \mathscr{N}_\alpha \subseteq \mathscr{S}_\alpha$ where $\subseteq$ means “is a subsheaf of (for the obvious inclusion maps)”.

**Examples:**

$\mathscr{C}_\omega$, the set of locally constant $\mathbb{N}$-valued functions on $X$, is the natural numbers object of the topos $\operatorname{Sh}(X)$. And $\mathscr{C}_n$, for finite $n$, is the $n$-fold coproduct $1+\cdots+1$ of the terminal sheaf.

$\mathscr{S}_2$ takes an open set $U \subseteq X$ to the set of open sets $V \subseteq U$ (identifying an open set $V$ with the upper semicontinuous function $\mathbf{1}_{U\setminus V}$) with restriction defined by intersection, i.e., it is the subobject classifier $\Omega$ of $\operatorname{Sh}(X)$. More generally, $\mathscr{S}_n(U)$ is the set of all chains $U \supseteq V_0\supseteq \cdots\supseteq V_{n-1} = \varnothing$ of open sets.

$\mathscr{N}_2$ (as a subsheaf of $\mathscr{S}_2$) consists of

*regular*open sets $V \subseteq U$, and can be identified with the internal subset $\{p\in\Omega : \neg\neg p = p\}$ of $\Omega$.

Each of these sheaves comes equipped with a strict ordering $<$ defined by $f<g$ whenever $f(x)<g(x)$ for *all* $x\in U$. (There is also a lax ordering $\leq$ defined analogously, and the two satisfy axioms (1)–(6) from this question, and in the case of $\mathscr{C}_\alpha$, even (7). But I believe $<$ is more interesting than $\leq$.)

Furthermore, note that $<$ is well-founded in the sense that if $\mathscr{P}$ is a subsheaf of $\mathscr{F}_\alpha$ (any one of $\mathscr{C}_\alpha, \mathscr{S}_\alpha, \mathscr{N}_\alpha$) such that “any $f \in \mathscr{F}_\alpha(U)$ for which ‘every $g\in\mathscr{F}_\alpha(V)$ with $V\subseteq U$ such that $g<f$ on $V$ belongs to $\mathscr{P}(V)$’ itself belongs to $\mathscr{P}(U)$” then in fact $\mathscr{P}=\mathscr{F}_\alpha$. (This is proved by induction on $\alpha$, and, of course, the case of $\mathscr{S}_\alpha$ suffices. Note that if we had used *lower* semicontinuous functions, I think this would not work, e.g., among the lower semicontinuous $\mathbb{N}$-valued functions, the locally bounded ones form a subsheaf which appears to satisfy the induction hypothesis but not generally its conclusion.)

Questions:

Can we provide an internal description of $\mathscr{C}_\alpha, \mathscr{S}_\alpha, \mathscr{N}_\alpha$ (i.e., couched in the language of intuitionistic set theory) that makes sense in an arbitrary elementary topos? (Not demanding, of course, that these objects

existin an arbitrary topos: e.g., $\mathscr{C}_\omega$ should be the natural numbers object, which doesn't always exist. Also, not demanding that the description be “uniform” in $\alpha$: it can iterate “externally” over $\alpha$.)How do these objects relate to the various kinds of intuitionistic ordinals considered in Paul Taylor's paper “Intuitionistic Sets and Ordinals” (

J. Symbolic Logic61(1996) 705–744)?

**Motivation / Meta:** I am mostly trying to get an intuitive image of how intuitionistic ordinals work, and $\mathscr{C}_\alpha, \mathscr{S}_\alpha, \mathscr{N}_\alpha$ are not only objects I can (sort of) visualize but they also seem fairly natural, hence the question. I am a little confused by the fact that upper semicontinuous functions seem to be necessary to make $<$ well-founded (see above), while lower semicontinuous functions would have made the connection with $\Omega$ more transparent — I may have missed something stupid. If a closely related object is better behaved, I am interested in answers concerning it.

**Edit (2019-03-21) / Meta:** In a previous edit to this question (see revision history) I thought I had defined a sheaf $\mathscr{T}_\alpha$ of *lower* semicontinuous $\alpha$-valued functions which was better behaved (notably, both well-founded and $<$-extensional; with $\mathscr{T}_2$ being $\Omega$ in the more obvious way $V \mapsto \mathbf{1}_V$ and $\mathscr{T}_\omega$ being *locally bounded* lower semicontinuous $\mathbb{N}$-valued functions). It turns out my definition was nonsensical, so I'm deleting it. If I find a satisfactory definition, I'll edit again. But maybe such an object cannot be defined and that's part of the problem. Sorry about the noise.

Rings of Normal Functions(McGill 1968) is worth looking at. $\endgroup$ – Gro-Tsen Mar 20 at 16:31