# Ordinal-valued sheaves as internal ordinals

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 $$f whenever $$f(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 $$g on $$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 exist in 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 Logic 61 (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.

• I should maybe have noted that $\mathscr{S}_\alpha$ is not $<$-extensional (i.e., we can have $\{g:g<f\} = \{g:g<f'\}$ without having $f=f'$). Note sure about the other ones. Not sure whether this is Very Bad or simply bad. Not sure about anything. – Gro-Tsen Mar 20 at 16:27
• PS: Concerning normal semicontinuous functions and their properties, Kenneth Hardy's thesis, Rings of Normal Functions (McGill 1968) is worth looking at. – Gro-Tsen Mar 20 at 16:31
• I have been asking myself this question (or very similar things) for quite some time ! It seems reasonable (at least from your example) that $\mathscr{S}_{\alpha}$ can be described from $\mathscr{C}_{\alpha}$ as the object of order ideals. That is at least what happens for finite ordinal, I don't quitte see if it is still the case past $\omega$. Understanding the relations between these three objects might already reduce a little the problem. (PS: I think your inclusion of sheaves are in the wrong order ? ) – Simon Henry Mar 20 at 17:24
• Doesn't the natural numbers object always exist in a sheaf topos? – Andrej Bauer Mar 20 at 19:27
• @AndrejBauer It does, but I meant that an internal description of my $\mathscr{C}_\alpha$ et al. could be interpreted in a more general topos and of course the objects might then not exist. – Gro-Tsen Mar 20 at 19:29